 # Pure Mathematics Seminars

Thursday 14th October 2021 - Collatz orbits of sparser numbers

Joint seminar between the Dynamical Systems and the Stochastic groups

Speaker: Professor Mary Rees (University of Liverpool)

13:00 via Zoom

Abstract

The Collatz map sends an integer n to n/2 if n is even and 3n+1 if n is odd. So the orbit of 1 is 1,4,2,1,4,2,1..

The orbit of 3 is 3,10,5,16,8,4,2,1.. The famously intractable Collatz conjecture, which is thought to have been first circulated by word of mouth in 1950, states that the orbit of every strictly positive integer ends in the cycle  1,4,2..

It is not even known if this problem is decidable. John Conway proved in the 1950's that a more general problem is not decidable. What positive  information we have about the Collatz map is largely, although not entirely, of a probabilistic nature. It was shown in the 1970's that, for most positive integers n, in the sense of positive density, the Collatz orbit of n passes below n. Terence Tao's 2019 paper Almost all Collatz orbits attain almost bounded values'' is a powerful strengthening of this basic result.

Most work on the Collatz conjecture is focussed on typical'' integers, which essentially means those n such that the orbit of  n starts by passing below n in a reasonable amount of time. Tao's method shows that the orbits  such numbers typically continue to decrease for some time.

This talk will concentrate on numbers which are not necessarily typical -- by looking not at the standard measures of uniform density on intervals of integers, but at other measures on integers: the so called geometric measures, and integers which are typical for such a measure. These are numbers whose Collatz orbits  start by increasing, depending on the mean of the geometric measure. A chain of conjectures will be stated which, if true, show that the Collatz orbits of these sparser numbers do later decrease for some time. A first weaker version of the final conjecture in the chain has been proved: not enough to be the basis of an induction, but a step in the right direction.

Thursday 5th November 2020 - A Sierpinski Mandelbrot Spiral

Dynamical Systems

Speaker: Eric Chang, Northwestern University

15:00 via MS Teams

Abstract: The Sierpinski carpet fractal and Mandelbrot set are fascinating images with diverse applications.  For iterated functions, fixing a parameter determines the dynamics.  However, we will use the dynamical plane to talk about the parameter plane.  In particular, for the family of singularly perturbed rational maps $z^n + \lambda / z^d$ for z and $\lambda$ complex, n at least 4 is even, and d at least 3 is odd, we identify patterns of Sierpinski holes and Mandelbrot sets via dynamical constructions, based on the work of Bob Devaney.

Thursday 29th October 2020 - The Collatz conjecture

Dynamical Systems

Speaker: Mary Rees, University of Liverpool

13:00 via MS Teams

Abstract: The Collatz map on the integers sends an even integer to n/2 and an odd integer to 3n+1.
Thus 1 is sent to 4, 2 to 1, 3 to 10, 4 to 2, and so on. There is therefore a periodic cycle of  1 to 2 to 4 to 1. The most basic form of the conjecture is:

Does the Collatz orbit of every strictly positive integer end in the periodic orbit of 1?

This problem emerged, initially by word of mouth, in the 1950's. Lothar Collatz had considered maps of this type since the 1930's.  Erdos considered it "Hopeless. Absolutely hopeless''. John Conway showed that, in general, such problems could not be solved algorithmically. There are, however,  positive results of a probabilistic nature. Terence Tao showed that almost every positive integer attains almost bounded values. He comments in his introduction that removing the second "almost'' would probably be as hard as proving the full conjecture.

Thursday 22nd October 2020 - The Permutable Mystery Tour for transcendental meromorphic functions

Dynamical Systems

Speaker: Gustavo Rodrigues Ferreira, The Open University

13:00 via MS Teams

Abstract: Which pairs of analytic functions commute? This question has been studied for a century now, and we have several reasonable answers for polynomials and rational functions. For transcendental meromorphic functions (entire and non-entire), however, little is known. In this talk, I will focus on a dynamical approach to this question: do permutable transcendental meromorphic functions have the same Julia set? As we will see, in many cases the answer is "yes", but the most general case remains open. If time allows, we will also see how symmetries of Julia sets relate to this problem.

Thursday 15th October 2020 - A generalised Blanchard's program

Dynamical Systems

Speaker: Rafael Alcaraz Barrera, IF-UASLP

13:00 via MS Teams

Abstract: In 1989, F. Blanchard sumarised results regarding a classification theorem for the symbolic dynamics of the class of beta-shifts. In particular, beta shifts of finite type, sofic, beta-shifts with the specification property and synchronised beta-shifts were characterised using the combinatorial properties of the quasi-greedy expansion of 1 in a non-integer base q. During the talk, we describe some recent progress in extending such classification to the class of intermediate beta-shifts on two symbols. If time permits, we will also mention some results and problems regarding such classification result for an arbitrary set of symbols.

Thursday 1st October 2020 - Endpoints of Julia sets

Dynamical Systems

Speaker: David Lipham, Auburn University at Montgomery

13:00 via MS Teams

Abstract: In this talk I will present several new results about the topologies of endpoint sets which are generated by functions in the exponential class. I will:

• compute their topological dimensions,
• determine whether they are homeomorphic to Erdős spaces, and
• discuss their relevance to some open problems in topology.
Thursday 2nd July 2020 - On the dimension of sets of points escaping to infinity at given rate under exponential iteration

Dynamical Systems

Speaker: Krzysztof Barański (Warsaw University)

13:00 via MS Teams

Abstract: We determine the Hausdorff and packing dimension of sets of points which escape to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we answer a question of Sixsmith from 2016. This is a joint work with Bogusława Karpińska.

Thursday 18th June 2020 - Constructing examples of oscillating wandering domains

Dynamical Systems

Speaker: Vasiliki Evdoridou (The Open University)

13:00 via MS Teams

Abstract: Let U be a Fatou component of a transcendental entire function. If U is not eventually periodic then it is called a wandering domain. Although Sullivan's celebrated result showed that rational maps have no wandering domains, transcendental entire functions can have wandering domains. The first wandering domain of oscillating type was constructed by Eremenko and Lyubich in 1987. Motivated by their construction and the recent classification of simply connected wandering domains obtained by Benini, E., Fagella, Rippon and Stallard, we give a general technique, based on Approximation Theory, for the construction of bounded oscillating wandering domains. We show that this technique can be used to produce examples of oscillating wandering domains of all six different types that arise by the classification. This is joint work with P. Rippon and G. Stallard.

Thursday 11th June 2020 - Cantor sets and Julia sets

Dynamical Systems

Speaker: Alastair Fletcher (Northern Illinois University)

16:00 via MS Teams

Abstract: Cantor sets embedded in R^n can arise dynamically in various ways, for example as Julia sets of uniformly quasiregular (uqr) mappings, or as attractor sets of iterated function systems. The main theme of this talk will be to try and recognize when an embedded Cantor set is a Julia set of a uqr map, or look for geometric properties of the Cantor set that preclude this. This is based on joint work with Vyron Vellis (UTK).

Thursday 28th May 2020 - Moonshine beyond the Monster

Speaker: Dr Michael Mertens (University of Liverpool)

16:00 via Microsoft Teams

Abstract: Ever since its discovery in the 1978, Moonshine and in particular Monstrous Moonshine has been a fascinating subject, connecting several seemingly unrelated areas of Mathematics, most obvious among which are Representation Theory of finite groups and Number Theory. In this talk, we will give an overview about some aspects of the subject, both classical as well as more recent. In particular we shall discuss some novel connections between finite (simple) groups and arithmetic of elliptic curves.

Thursday 7th May 2020 - Docile entire functions

Dynamical Systems

Speaker: Lasse Rempe, University of LIverpool

13:00 via MS Teams

Abstract: For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree $d\geq 2$ is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle $d$-tupling on the circle.

For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of “docile” functions. We also show docility for a large class of transcendental entire functions with bounded postsingular sets (those that are “strongly geometrically finite”). This is joint work with Mashael Alhamed and Dave Sixsmith.

In this talk, I will focus on explaining the definition of docile functions, and its motivation.

Thursday 30th April 2020 - Iteration in tracts

Dynamical Systems

Speaker: James Waterman (University of Liverpool)

13:00 via Microsoft Teams

Abstract: By a result of Rippon and Stallard, for a transcendental entire function there exist points in the escaping set whose orbit escapes to infinity arbitrarily slowly. We will discuss extending this result to prove the existence of points which escape to infinity arbitrarily slowly within a particular type of domain, called a tract.

Friday 6th March 2020 - Symplectic geometry and algebraic K-theory

Algebraic Geometry

Speaker: Dr Yanki Lekili (Imperial College)

16:00 MATH-106

Abstract: Waldhausen's delooping of algebraic K-theory is a packaging of representations of A_n-quiver into a simplicial category. I will first explain this by drawing some pictures of arcs on a disk.  Once we interpret these pictures as an appropriate Fukaya category, we can draw more complicated pictures which give rise to higher deloopings of K-theory. This is joint work with T. Dyckerhoff and G. Jasso and based on our recent arXiv preprint.

Thursday 27th February 2020 - Invariant sets from Kerekjarto/ Birkhoff

Dynamical Systems

Speaker: Professor Mary Rees

13:00 MATH-117

Abstract: Bela Kerekjarto was a Hungarian topologist active in the 1930's. He is well-known in some quarters, but I first came across him while researching the work of Yael Naim Dowker, who taught at Imperial from the mid 1950's until her retirement in the mid 1980's. Some of her work, in particular with her Ph D student George Lederer, impinged  on topological dynamics. Their joint work used a remarkable result which she attributed to Kerekjarto, which in essence shows the existence of nontrivial stable/unstable sets for any compact invariant set, some 25 years before the publication of the now-classical corresponding results for the differentiable category. I shall discuss the implications for the structure of the set of invariant sets of a homeomorphism of a compact metric space. There will be more questions than answers.

Thursday 13th February 2020 - Natural extensions of unimodal maps are virtually sphere homeomorphisms

Dynamical Systems

Speaker: Dr Toby Hall (University of Liverpool)

13:00 MATH-117

Abstract: Let f be a unimodal map of the interval. The inverse limit construction replaces f with a self-homeomorphism - the natural extension of f - of its inverse limit space. These inverse limit spaces have complicated and intricate topology: for example, they typically contain indecomposable continua.

I'll discuss recent work with Philip Boyland (University of Florida) and André de Carvalho (University of Sao Paulo), in which we construct mild semi-conjugacies from these natural extensions to sphere homeomorphisms. The family of sphere homeomorphisms constructed in this way includes examples of Thurston's pseudo-Anosov maps; of generalized pseudo-Anosovs; and of (further generalized) measurable pseudo-Anosovs.

Thursday 6th February 2020 - Tangent curves, degenerations, and blowups

Algebraic Geometry

Speaker: Dr Nabijou (University of Glasgow)

16:00 MATH-106

Abstract: It is well-known that every smooth plane cubic E supports precisely 9 flex lines. By analogy, we may ask: "How many degree d curves intersect E in a single point?" The problem of calculating such numbers of tangent curves has fascinated enumerative geometers for decades. Despite being an extremely classical and concrete problem, it was not until the advent of Gromov-Witten invariants in the 1990s that a general method was discovered. The resulting theory is incredibly rich, and the curve counts satisfy a suite of remarkable properties, some proven and some still conjectural.

In this talk, I will discuss two distinct projects which take inspiration from this geometry. In the first, joint with Lawrence Barrott, we study the behaviour of tangent curves as the cubic E degenerates to a cycle of lines. Using the machinery of logarithmic Gromov-Witten theory, we obtain detailed information concerning how the tangent curves degenerate along with E. The resulting theorems are purely classical, with no reference to Gromov-Witten theory, but they do not appear to admit a classical proof. In a separate project, joint with Dhruv Ranganathan, we perform iterated blowups of moduli spaces to prove the so-called local-logarithmic conjecture for hyperplane sections; this gives access to a large number of previously unknown enumerative theories.

No prior knowledge of Gromov-Witten theory will be assumed.

Thursday 30th January 2020 - Rationally connected rational double covers of primitive Fano varieties

Algebraic Geometry

Speaker: Professor Aleksandr Pukhlikov (University of Liverpool)

16:00 MATH-106

Abstract: We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no rational maps $X\dashrightarrow V$ of degree 2, where $X$ is a rationally connected variety. This fact is true for many other families of primitive Fano varieties either. It generalizes easily for rationally connected Galois rational covers with an abelian Galois group and motivates a conjecture on absolute rigidity of primitive Fano varieties.

Thursday 30th January 2020 - Julia continua of transcendental entire functions

Dynamical Systems

Speaker: Tania Gricel Benitez Lopez (University of Liverpool)

13:00 MATH-117

Abstract: Recently Rempe-Gillen gave an almost complete description of the possible topology of the Julia continua of disjoint-type functions combining well-studied concepts from continuum theory and new techniques of transcendental dynamics. In particular, he constructed a function where all Julia continua are pseudo-arcs, however this arises as a special case of a more general construction. In this presentation, we discuss how to construct a disjoint-type function such that all Julia continua are pseudo-arcs using a different technique which is more explicit, and as a result we obtain better control over the lower order of growth of the function.

Thursday 21st November 2019 - Entire functions whose maximum modulus set has prescribed discontinuities

Dynamical Systems

Speaker: Leticia Pardo Simon (Polish Academy of Sciences)

13:00 MATH-117

Abstract: In 1909, Hardy gave an example of a transcendental entire function f so that the set of points where f achieves its maximum modulus, M(f), has infinitely many discontinuities. This is one of only two known examples of such a function. In this talk, we will significantly generalise these examples. In particular, we will show that, given an increasing sequence of positive real numbers, tending to infinity, there is a transcendental entire function, f, such that M(f) has discontinuities with moduli at all these values. This is joint work with Dave Sixsmith.

Thursday 14th November 2019 - Bounds on singularities from symplectic topology

Algebraic Geometry

Speaker: Dr Jonny Evans (University of Lancaster)

16:00 MATH-106

Abstract: Joint work with Ivan Smith. The moduli space of surfaces of general type (with fixed topological invariants) M_{K^2,\chi} can be compactified by adding in stable surfaces. These stable surfaces can have singularities, and it is a nontrivial problem to understand what kinds of singularities can appear for surfaces at the boundary of M_{K^2,\chi}. We focus on the Wahl singularities (a particular class of isolated singularities with Milnor number zero) and show that the length of the Wahl singularities for surfaces at the boundary of M_{K^2,\chi} is bounded above by 4K^2+7. This is an improvement on the previously-known bound of 400(K^2)^2 (due to Y. Lee) and comes close to what is now known to be the optimal bound of 4K^2+1 (due to Rana--Urzua). The novelty of our approach is that we translate the problem into the language of symplectic geometry and establish the bound using pseudoholomorphic curve techniques and Seiberg-Witten theory.

Thursday 14th November 2019 - Unique ergodicity and zero entropy of irregular almost automorphic symbolic extensions of irrational rotations and applications to quasicrystals

Dynamical Systems

Speaker: Gabriel Fuhrmann (Imperial College)

13:00 MATH-117

Abstract: A classical result by Markley and Paul states that irregularalmost automorphic symbolic systems over irrational rotations are typically not uniquely ergodic and have positive entropy. By constructing a particular Cantor set, we prove that for each irrational rotation there still are almost automorphic extensions which are mean-equicontinuous (and hence have zero entropy and are uniquely ergodic).

The talk aims to be accessible by a broad audience. To that end, we will first motivate the study of almost automorphic systems by taking a look at well-studied representatives: dynamical systems associated to quasicrystals (i.e. Delone dynamical systems). After a short review of the results of Markley and Paul, we will eventually discuss the construction of the above mentioned Cantor set.

This is a joint work with Eli Glasner, Tobias Jäger and Christian Oertel.

Wednesday 13th November 2019 - Cylinders in del Pezzo surfaces with Du Val singularities

Algebraic Geometry

Speaker: Dr Grigory Belousov (Higher Economics School, Moscow)

16:00 MATH-106

Abstract: In this talk, I will show how to prove that a del Pezzo surface S with du Val singularities has a $-K_S$-polar cylinder if and only if there exist a tiger such that the support of this tiger does not contain anti-canonical divisor. Also I will describe all del Pezzo surfaces with Du Val singularities that do not contain any cylinders.

Friday 8th November 2019 - Algebra vs Geometry (2019 Terry Wall Lecture)

Speaker: Prof Caucher Birkar (University of Cambridge)

16:30 Rotblat Lecture Theatre

Abstract: The fields of algebra and geometry interact in numerous ways. This has a long history going back to thousands of years ago. In this talk I will discuss some of those interactions with a particular emphasis on algebraic geometry.

Caucher Birkar is a professor at the University of Cambridge. He grew up on a subsistence farm in the Kurdistan province of Iran during the Iran-Iraq war, and relocated to the UK as a refugee in 2000. In 2004 he received his PhD from Nottingham University. He has made fundamental contributions to birational geometry and the minimal model program. Among many honors, he was awarded the 2010 Leverhulme Prize in mathematics and statistics for "his outstanding contributions to fundamental research in algebraic geometry", the 2010 Prize of the Fondation Sciences Mathématiques de Paris, the 2016 AMS Moore Prize, and became a fellow of the Royal Society in 2019. In 2018 he received the Fields Medal, the highest honor in mathematics, "for the proof of the boundedness of Fano varieties and for contributions to the minimal model program".

Tea and coffee will be served outside the Lecture Theatre from 4pm.

Thursday 7th November - Wiman-Valiron discs and the Hausdorff dimension of Julia sets of meromorphic functions

Speaker: Dr James Waterman (University of Liverpool)

13:00 MATH-117

Abstract: The Hausdorff dimension of the Julia set of transcendental entire and meromorphic functions has been widely studied. We review results concerning the Hausdorff dimension of these sets starting with those of Baker in 1975 and continuing to recent work of Bishop. In particular, Baranski, Karpinska, and Zdunik proved that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic function with a particular type of domain called a logarithmic tract is greater than one. We discuss generalizing this result to meromorphic maps with a simply connected direct tract and certain restrictions on the singular values of these maps. In order to accomplish this, we develop tools from Wiman-Valiron theory, showing that some tracts contain a dramatically larger disk about maximum modulus points than previously known.

Thursday 31st October 2019 - Stability Index and Weierstrass Functions

Speaker: Charles Walkden, University of Manchester

13:00 MATH-117

Abstract: A dynamical system may have multiple attractors. The basin of attraction for a given attractor is the set of points which, under interation, converge to that attractor.  In the case where there are multiple attractors, the basins may have very complicated local structure, for example the boundary between them could be highly irregular.  The stability index (introduced numerically by Alexander-Yorke-You-Kan, Sommerer-Ott and others in physical systems in the early 1990s and studied more theoretically by Podvigina-Ashwin (2011) and Keller (2015, 2017)) is a number, behaving like a local dimension, that captures the complexity of the local structure of the basins of attraction.  This talk will discuss how one can explicitly calculate the stability index in the case of skew-product dynamical systems, using classical tools and techniques from hyperbolic dynamics and thermodynamic formalism.

Thursday 24th October 2019 - Complex differential equations and their singularities

Dynamical Systems

Speaker: Thomas Kecker (University of Portmouth)

13:00 MATH-117

Abstract: We will consider certain classes of 2nd-order non-linear, non-autonomous, ordinary differential equations in the complex domain and discuss what types of singularities the solutions of these can develop when analytically continued. In particular, we will study the local and global behaviour of the solutions in the complex plane. Some of the classes of equations discussed include the famous Painlevé equations, which have found many applications in mathematical physics recently.

Monday 14th October 2019 - Dynamics of transcendental entire functions with escaping singular orbits

Dynamical Systems

Speaker: Leticia Pardo Simon (University of Liverpool)

14:00 MATH-G16

Thursday 10th October 2019 - Building surfaces from equilateral triangles

Dynamical Systems

Speaker: Prof Lasse Rempe-Gillen (University of Liverpool)

13:00 MATH-117

Abstract: In this talk, we consider the following natural question. Suppose that we glue a (finite or infinite) collection of equilateral triangles together in such a way that each edge is identified with precisely one other edge, each vertex is identified with only finitely many other vertices. If the resulting surface is connected, it naturally has the structure of a Riemann surface, i.e., a one-dimensional complex manifold. We ask which surfaces can arise in this fashion.

The answer in the compact case is given by a famous classical theorem of Belyi, which states that a compact surface can arise from this construction if and only if it is defined over a number field. These Belyi surfaces and their associated “Belyi functions” have found applications across many fields of mathematics.

In joint work with Chris Bishop, we give a complete answer of the same question for the case of infinitely many triangles (i.e., for non-compact Riemann surfaces).

Thursday 3rd October 2019 - Uniformization of quasitrees

Speaker: Dr Daniel Meyer (University of Liverpool)

13:00-14:00 MATH117

Abstract: A quasisymmetry maps balls in a controlled manner. These maps are generalizations of conformal maps and may be viewed as a global versions of quasiconformal maps. Originally, they were introduced in the context of geometric function theory, but appear now in geometric group theory and analysis on metric spaces among others. The quasisymmetric uniformization problem asks when a given metric space is quasisymmetric to some model space. Here we consider "quasitrees''. We show that any such tree is quasisymmetrically equivalent to a geodesic tree. Under additional assumptions it is quasisymmetric to the "continuum self-similar tree''. This is joint work with Mario Bonk.

Friday 9th August 2019 - An Extended Fatou-Shishikura inequality

Speaker: Anna Benini (University of Barcelona)

14:00 MATH-117

Abstract: It is classically known that the existence of non-repelling periodic points is connected to the presence of a singular value 'near by'- for example, attracting and parabolic basins need to contain at least one singular value. For repelling periodic points, singular values come into play when the periodic point in question is not the landing point of any periodic ray. In this talk we will show how to associate a specific singular orbit to every non-repelling cycle, as well as to every repelling cycle whose points are not landing points of periodic rays. This gives a version of the Fatou-Shishikura inequality which takes into account such repelling cycles. This is joint work with N. Fagella.

Friday 9th August 2019 - Decompositions of log-correlated fields with applications

Speaker: Eero Saksman (University of Helsinki)

13:00 MATH-117

Abstract: We consider a simple idea to decompose of log-correlated Gaussian fields into two-parts, both of which behave well in suitable sense. Applications include Onsager type inequalities in all dimensions, analytic dependence and existence of critical chaos measures for a large class of log-correlated fields. Talk is based on joint work with Janne Junnila (EPFL) and Christian Webb (Aalto University).

Monday 8th July 2019 - Holomorphic dynamics 'Scheme 3' meeting

Venue: MATH106

Speakers:

1:00 - 2:00 - Alastair Fletcher "Julia sets, Cantor sets, and Spiders' Webs of Doughnuts"

2:00 - 3:00 - François Berteloot - "Lyapunov exponents and bifurcations near Lattès maps"

3:30 - 4:00 - Alexandre De Zotti - "Hausdorff dimension for irrationally indifferent attractors"

4:00 - 4:30 - Yannis Dourekas - "A new family of entire functions with no wandering domains"

4:30 - 5:30 - Gwyneth Stallard - "The components of the escaping set in transcendental dynamics"

Wednesday 5th June 2019 - Relations between Discrete Differential Geometry and Architectural Geometry

Speaker: Christian Mueller (TU Wien, Austria)

16:00 MATH-MAGIC Room

Abstract: In modern architectural freeform surfaces gained a lot of attraction. Many tasks that arise in the generation and manufacturing of such surfaces, like surface segmentation, paneling, support structure construction, etc., are surprisingly closely related to discrete differential geometry. We will discuss such relations exemplified on some special cases, like curved support structures and surfaces with constant ratio of principal curvatures, cladding facades with surfaces isometric to surfaces of revolution, and by subdividing meshes in a conformal way.

Wednesday 15th May 2019 - Transpennine Topology Triangle meeting

MATH-103

11:30-12:00 Tea/coffee

12:00-13:00 Ian Leary (University of Southampton)

Abstract: in the 1990's, Bestvina-Brady solved a long-standing problem by constructing the first examples of groups that are almost finitely presented' but not finitely presented.  I'll describe their construction, my recent generalization of it, and an application related to the Higman Embedding Theorem.

13:00-14:00 Lunch

14:00-15:00 Toby Hall (University of Liverpool)

Inverse limits of tent maps and sphere homeomorphisms

Abstract: The inverse limit construction allows the study of a dynamical system - a self-map of a topological space - to be converted into the study of a self-homeomorphism of the inverse limit. The inverse limit is usually much more complicated than the original space: for example, inverse limits of tent maps - simple piecewise affine self-maps of an interval - typically contain indecomposable continua.

In this talk I'll describe how mild identifications on the inverse limits of tent maps turn them into 2-spheres, and make it possible to construct a parameterized family of sphere homeomorphisms corresponding to the parameterized family of tent maps. I'll discuss the dynamics of these homeomorphisms, which include examples of Thurston's pseudo-Anosov maps, so-called generalized pseudo-Anosovs, and the (further generalized) measurable pseudo-Anosovs.

The talk will start with an extended introduction to relevant ideas in dynamical systems theory, aimed at non-specialists. This is joint work with Philip Boyland (University of Florida) and Andre' de Carvalho (University of Sao Paulo).

15:00-15:30 Tea/coffee

15:30-16:30 Ieke Moerdijk (University of Sheffield/Utrecht)

Dendroidal spaces and mapping spaces between little cubes operads

Thursday 9th May 2019 - Algebraic and Tropical Prym Varieties

Algebraic Geometry

Speaker: Yoav Len (Georgia Tech University, USA)

MATH-103 17:00

Abstract: My talk will revolve around combinatorial aspects of Abelian varieties. I will focus on Pryms, a class of Abelian varieties that occurs in the presence of double covers, and have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. I will explain how problems concerning Pryms may be reduced, via tropical geometry, to problems on metric graphs. As a consequence, we obtain new results concerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill{Noether loci. This is joint work with Martin Ulirsch.

Thursday 28th March 2019 - Classification of degenerations and Picard lattices of Kahlerian K3 surfaces with small finite symplectic automorphism groups

Algebraic Geometry

Speaker: Prof Viacheslav V. Nikulin (University of Liverpool and Steklov Mathematical Institute, Moscow)

17:00 MATH-103

Abstract: Such classification for finite symplectic automorphism groups of order > 6 was done in my papers (see arXiv: 1403.6061, 1504.00326, 1608.04373, 1707.05677). For remaining small groups: D_6, C_4, (C_2)^2, C_3, C_2 such classification is very difficult and requires a lot of calculations and considerations of  many cases. In my talk, I shall consider such classification for groups D_6 and C_4. See my preprint arXiv: 1804.00991 for some details.

Wednesday 27th March 2019 - Grassmann convexity and multiplicative Sturm theory

Speaker: Professor Michael Shapiro (Michigan State University)

14:00 MATH-210

Abstract: In this paper we settle a special case of the Grassmann convexity conjecture formulated in 2000 by B. Shapiro and myself. We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. We show that this formula gives the lower bound for the required total number of real zeros for equations of an arbitrary order and, using our results on the Grassmann convexity, we prove that the aforementioned formula is correct for equations of orders 4 and 5. (joint with N. Saldanha and B. Shapiro)

Thursday 21st March 2019 - A component of the Hilbert scheme of hyperelliptic Jacobians

Algebraic Geometry

Speaker: Dr Andrea Ricolfi (SISSA/ISAS, Trieste)

MATH-LT027 16:00

Abstract: Let C be a smooth curve of genus g > 2, embedded in its Jacobian J via an Abel-Jacobi map. The component H(C) of the Hilbert scheme of J containing the Abel-Jacobi embedding has the same closed points of J itself. If C is non-hyperelliptic, H(C) = J as schemes, but this no longer holds in the hyperelliptic case, due to ramification of the Torelli morphism along the hyperelliptic locus. In this talk we describe the scheme structure on H(C); as an application, we determine the scheme structure on the moduli space of Picard sheaves on a hyperelliptic Jacobian studied by Mukai.

Thursday 14th March 2019 - Computing classes of admissible covers

Algebraic Geometry

Speaker: Dr Jason Van Zelm (Humboldt University Berlin)

17:00 MATH-103

Abstract: Let Adm(g,h,G) be the space of degree admissible G covers C → D of a genus h curve D by genus g curves C. There is a natural map f : Adm(g,h,G) →  Mgnbar into the moduli space of stable curves taking the source curve of an admissible cover and forgetting everything else. When the class [f(Adm(g,h,G))] is tautological we can try to express this class in terms of a known basis for the tautological ring of Mgnbar. We will discuss several strategies for making these computations and give a number of examples.

Wednesday 6th March 2019 - Fibrations in Segre fourfolds and families of del Pezzo surface of degree 6

Algebraic Geometry

Speaker: Marcello Bernardara (University of Toulouse)

16:00 MATH-105

Abstract: Fibrations X \to M in del Pezzo surfaces of degree 6 are an interesting example of Mori fibre space. For example, Addington-Hassett-Tschinkel-Varilly Alvarado have constructed such a (rational) fibration on cubic foufolds containing an elliptic scroll, thus proving the rationality of some of them. I will describe how to construct such fibrations over a general base M starting from fibrations Y \to M in Segre fourfolds, and how to study their derived categories and the associated Brauer classes on ramified covers of M. This is a joint work in progress with Addington, Auel, and Faenzi.

Thursday 21st February 2019 - Primes, knots, and exodromy

Algebraic Geometry

Speaker: Clark Barwick (University of Edinburgh)

17:00 MATH-103

Abstract: Half a century ago, Barry Mazur and David Mumford suggested a remarkable dictionary between prime numbers and knots. I will explain how the story of exodromy permits one to make this dictionary precise, and I will describe some applications.

Thursday 21st February 2019 - Curves on K3 surfaces

Algebraic Geometry

Speaker: Dr Soheyla Feizbakhsh (Imperial College, London)

16:00 MATH-103

Abstract: I will briefly explain the notion of Bridgeland stability conditions on K3 surfaces. Then I will show some of its recent applications in classical algebraic geometry, including a new upper bound for the number of global sections of sheaves on K3 surfaces and computing higher rank Clifford indices of curves on K3 surfaces.

Friday 1st February 2019 - Algebraic links in the Poincaré sphere and the Alexander polynomials

Speaker: Sabir Gusein-Zade (Moscow State University)

13:30-14:30 Room MATH-211

Abstract: The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the E_8-surface singularity (S,0) in (C^3,0) with the 5-dimensional sphere \S_{\epsilon}^5 of radius \epsilon in C^3. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in (S,0) with the sphere \S_{\epsilon}^5 of radius \epsilon small enough. We discuss to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It is possible to show that, if the strict transform of a curve in (S,0) does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding E_8-diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere is determined by the Poincaré series of the filtration defined by the corresponding curve valuations. (They coincides with each other for a reducible curve singularity and differ by the factor (1-t) for an irreducible one.) It was shown that, under conditions similar for those for curves, the Poincaré series of a collection of divisorial valuations determines the combinatorial type of the minimal resolution of the collection.

The talk is based on a joint work with A.Campillo and F.Delgado.

Wednesday 16th January 2019 - Pseudolattices, del Pezzo surfaces, and fibrations over discs

Algebraic Geometry

Speaker: Dr Alan Thompson (University of Loughborough)

16:00-17:00 Room MATH-211

Abstract: The theory of pseudolattices was initiated by Kuznetsov, building on work of many other authors. The canonical example is the numerical Grothendieck group associated to the bounded derived category of coherent sheaves on a smooth variety. In this talk I will discuss a special class of pseudolattices, which may be thought of as numerical Grothendieck groups associated to bounded derived categories of coherent sheaves on smooth rational surfaces which admit a smooth anticanonical divisor. Such pseudolattices may be classified and, perhaps unsurprisingly, their classification parallels the well-known classification of del Pezzo surfaces.

However, this special class of pseudolattices also arises naturally in a second context, associated to a certain class of elliptic Lefschetz fibrations over complex discs. I will show that the classification result above allows one to classify such elliptic Lefschetz fibrations up to diffeomorphism. This setting should be thought of as mirror, in a homological sense, to the original context of smooth rational surfaces admitting smooth anticanonical divisors. This work is joint with Andrew Harder.

Wednesday 12th December 2018 - Positivity of higher codimension cycles

Algebraic Geometry

Speaker: Artie Prendergast-Smith (University of Loughborough)

16:00-17:00 Room MATH-104

Abstract: Recent years have seen substantial development in the theory of positivity for algebraic cycles of codimension greater than 1. My talk will survey some of this progress and explain some new examples in joint work with Norbert Pintye.

Wednesday 5th December 2018 - t-structures and twisted complexes on derived injectives

Algebraic Geometry

Speaker: Francesco Genovese (University of Hasselt)

16:00-17:00 Room MATH-105

Abstract: I will report on a joint work with Wendy Lowen and Michel Van den Bergh. An abelian category with enough injectives can be reconstructed as the category of finitely presented modules over the category of its injective objects. This is a key result of the deformation theory of abelian categories developed by Wendy Lowen and Michel Van den Bergh: in fact, deformations of abelian categories with enough injectives can be understood as deformations of their category of injectives. We show a generalisation of this reconstruction result in the derived setting, which involves t-structures and a suitable derived enhancement of injective objects. Namely, given a pretriangulated dg-category with a non-degenerate left bounded t-structure with enough derived injectives, it can be reconstructed as the dg-category of left bounded twisted complexes of derived injectives. This is the foundational step toward developing the deformation theory of triangulated categories with a t-structure.

Wednesday 28th November 2018 - Derived categories of Hilbert squares and their deformations

Algebraic Geometry

Speaker: Theo Raedschelders (University of Glasgow)

16:00-17:00 Room MATH-104

Abstract:Hitchin showed that for a smooth projective surface S with an exceptional structure sheaf, there is an intimate connection between the noncommutative deformation theory of S, and the usual (commutative) deformation theory of the Hilbert scheme of 2 points on S. In this talk, I will explain this result from the point of view of derived categories, and use this as an ansatz to discuss what happens in higher dimensions. As is often the case, it turns out that some things generalise, and some don't. This is joint work with Pieter Belmans and Lie Fu.

Monday 26th November 2018 - Rigidity of linearly constrained frameworks

Speaker: Tony Nixon (University of Lancaster)

17:00-18:00 Room MATH-104

Abstract: A bar-joint framework is a realisation of a graph in \R^d with edges represented by stiff bars and vertices by universal joints. The framework is rigid if every bar-length-preserving continuous deformation of the framework joints arises from an isometry of \R^d. I will describe the general theory of rigid bar-joint frameworks in \R^d. I will also consider the problem of characterising the generic rigidity of bar-joint frameworks in \R^d in which each vertex is constrained to lie in a given affine subspace. The special case when d=2 was previously solved by Ileana Streinu and Louis Theran in 2010. We will extend their characterisation to the case when d\geq 3 and each vertex is constrained to lie in an affine subspace of dimension t, when t=1,2 and also when t\geq 3 and d\geq t(t-1). This is joint work with James Cruickshank, Hakan Guler and Bill Jackson.

Wednesday 21st November 2018 - Local structures of algebraic stacks and destackification

Algebraic Geometry

Speaker: Dr David Rydh (KTH Stockholm)

16:00-17:00 Room MATH-104

Abstract: Algebraic stacks generalize equivariant algebraic geometry: the action of an algebraic group on a scheme. In this talk I will give a survey of two types of results for algebraic stacks: (1) local structure theorems and (2) destackification theorems.

The local structure theorems state that certain algebraic stacks locally look like the quotient of a scheme by a group action. This is related to Luna's slice theorem in equivariant geometry.

Destackification is about eliminating or reducing the stabilizer groups of a stack via some well-controlled operations such as blow-ups in smooth centers and root stacks along smooth divisors. This is closely related to resolution of singularities for simplicial and non-simplicial toric varieties and, for infinite stabilizers, Kirwan's partial desingularization of good quotients in Geometric Invariant Theory.

Time permitting, I will also mention some applications to weak factorization, derived categories, Grothendieck rings of varieties, and Donaldson--Thomas invariants.

The results in (1) are joint work with Jarod Alper and Jack Hall. The results in (2) are joint work with Daniel Bergh and Dan Edidin.

All are welcome!

Wednesday 14th November 2018 - Towards a universal tropical Picard variety

Algebraic Geometry

Speaker: Martin Ulrisch (Goethe University Frankfurt)

16:00-17:00 Room MATH-104

Abstract: The main topic of this talk is the analogy between classical geometry of algebraic curves and and the very recent and still expanding geometry of tropical curves, focussing in particular on the theory of divisors. In particular, there is a notion of a Picard group on a tropical curve that has drawn significant attention recently, since it both encodes the combinatorics of compactified algebraic Picard varieties as well as the combinatorics of chip firing on metric graphs. After a concise introduction to the basic theory, I will report on work-in-progress with Margardia Melo, Sam Molcho, Filippo Viviani, and Jonathan Wise where we show how natural modular subdivisions of the tropical Picard group (associated to certain stability conditions) give rise to a universal tropical Picard variety with good geometric properties as well as to a new logarithmic construction of compactified algebraic Picard varieties, expanding e.g. on the work of Kass-Pagani.

Wednesday 31st October 2018 - Clifford type inequality for curves on K3 and Fano surfaces

Speaker: Chunyi Li (University of Warwick)

16:00 Room MATH-029

Abstract: The higher rank Clifford’s Theorem states that for a semistable bundle E on a curve C with rank r, degree d and slope \mu\in[0,2g-2], the dimension of its global section h^0(E)\leq r+d/2. In the recent thirty years, much better/sharp bounds for h^0(E) for different type of curves have been established by Newstead, Lange, Mercat, Brambila-Paz, Farkas and several other people.

Motivated by the project on constructing Bridgeland stability conditions on Calabi-Yau threefolds, we deduce the main technical part to the study of these Clifford type inequalities for curves on K3 and some Fano surfaces. I’ll introduce our new methods and results for the Clifford type inequalities on these types of curves. If time permits, I’ll also explain a recent result on the existence of stability conditions on Calabi-Yau threefolds and why the Clifford type inequality is the core difficulty in this program. This is a joint work with Soheyla Feyzbakhsh.

Thursday 25th October 2018 - Mating quadratic maps with the modular group

Algebraic Geometry

Speaker: Luna Lomonaco (University of Sao Paulo)

10:00-11:00 Room CTH-LTD

Abstract: In 1994 S. Bullett and C. Penrose introduced a one complex parameter family of holomorphic correspondences, which we denote F_a, and proved that for every real parameter in the connectedness locus such correspondence is a mating between a quadratic polynomial and the modular group. They conjectured that this is the case for every parameter in the connectedness locus.

We show here that matings between the modular group and rational maps in the parabolic quadratic family Per_1(1) provide a better model: we prove that every member of the family F_a which has the parameter in the connectedness locus is such a mating. Moreover, we develop a dynamical theory for such a family which parallels the Douady-Hubbard theory of quadratic polynomials. This is a joint work with S. Bullett.

Friday 19th October 2018 - Taming Infinities

Speaker: Professor Martin Hairer (Imperial College, London) - Terry Wall Annual Lecture

16:00 Room Rotblat Lecture Theatre, Chadwick Building

Abstract: Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! What's worse, this doesn't just happen for some exotic theories, but in the standard theories describing some of the most fundamental aspects of nature. Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the conceptual and mathematical aspects of these techniques and we will see how they have recently been used to study equations whose meaning was not even clear until now.

Monday 17th September 2018 - Extending the Abel-Jacobi map

Speaker: Dr David Holmes

16:00 Room MATH-029

Abstract: It is well-known that the Abel-Jacobi map does not extend over the boundary of the moduli space of stable marked curves. We approach this problem by making (logarithmic) blowups of the moduli space, constructing a universal’ resolution of the indeterminacies of the Abel-Jacobi map. After considering some elementary examples, we will show how to apply this construction to the problem of compactifying the double ramification cycle, and show that the resulting compactification is compatible with various conjectures and formulae in the literature.

Tuesday 17th July 2018 - Transpennine Topology Triangle meeting

Speaker: Anna Pratoussevitch (University of Liverpool) - Spaces of Higher Spin Klein Surfaces

12:00 Room MATH-029

Abstract: We will discuss the space of higher spin bundles on hyperbolic Klein surfaces. A Klein surface is a generalisation of a Riemann surface to the case of non-orientable surfaces or surfaces with boundary. The category of Klein surfaces is isomorphic to the category of real algebraic curves. An m-spin bundle on a Klein surface is a complex line bundle whose m-th tensor power is the cotangent bundle. We describe all connected components of the space of higher spin bundles on hyperbolic Klein surfaces in terms of their topological invariants and prove that any connected component is homeomorphic to a quotient of R^d by a discrete group. The spaces of higher spin bundles on Klein surfaces have connections with singularity theory and real algebraic geometry. We will discuss an application to real forms of Gorenstein quasi-homogeneous surface singularities, in particular to Brieskorn-Pham singularities. This is joint work with S. Natanzon.

Speaker: Anwar Alameddin (University of Liverpool) - Motivic Measures through Motivic Homotopy Theory

14:00 Room MATH-029

Abstract: I will recall an extension of the stable homotopy category, given by its motivic counterpart (over the complex numbers). Then, I will explain how some Euler-Poincaré characteristics factorise through a non-connected K-theory of the motivic stable homotopy category. Time permits, I will discuss how this can be applied to address some questions in algebraic geometry.

Speaker: Julian Holstein (University of Lancaster) - A tour of infinity local systems

15:30 Room MATH-029

Abstract: The category of local systems on a topological space has several well-known descriptions, as locally constant sheaves, representations of the fundamental group or vector bundles with flat connection. There is a natural derived analogue, the infinity category of infinity local systems, which has a multitude of equivalent definitions. In this talk I will describe infinity local systems in several different ways and talk about relations between the different characterisations.

In particular I will talk about recent work with Joe Chuang and Andrey Lazarev describing infinity local systems as certain categories of modules over dg algebras.

Thursday 5th July 2018 - Multipoint Okounkov Bodies

Algebraic Geometry

Speaker: Antonio Trusiani (Chalmers University)

16:00 Room MATH-029

Abstract: Starting from the data of a big line bundle L on a projective manifold X with a choice of N different points on X, I’ll give a new construction of N Okounkov bodies that encodes important geometric features of (L → X, p1, . . . , pN ) such as the volume of L, the (moving) multipoint Seshadri constant of L at p1, . . . , pN , and the possibility to construct a K¨ahler packings centered at p1, . . . , pN . In the toric case the multipoint Okounkov bodies at N torus-fixed points can be obtained subdividing the polytope, while in the surface case it is possible to show their polyhedrality, and in the case X = P2 their shape is strongly connected to the renowned Nagata’s Conjecture.

Thursday 7th June 2018 - PGL(2)-invariants of collections of torsion points of elliptic curves

Algebraic Geometry

Speaker: Professor Fedor Bogomolov - Courant Institute (New York) and Higher School of Economics (Moscow)

16:00 Room MATH-029

Abstract: The main object of the talk is a (complex) elliptic curve E with a standard degree 2 projection on P^1. Assuming that we fix one of the ramification points as a zero we obtain a subset PE_{tors} of the images of torsion points on E inside P^1.

This sets are different and we have shown jointly with Yuri Tschinkel that these sets are very different for different elliptic curves - they have finite intersection for any two nonisomorphic elliptic curves. However some subsets of the above PE_{tors} are PGL(2) equivalent. This holds for the images of points of order 3 and order 4.

In this talk I am going to discuss a general problem of the behavior of PGL(2)-invariants of the k-tuples of the images of torsion points of different order.

Thursday 7th June 2018 - Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational

Algebraic Geometry

Speaker: Ciaran Meachan (University of Glasgow)

15:00 Room MATH-029

Abstract: Starting with two non-birational derived equivalent K3 surfaces, one can ask whether their Hilbert schemes of points are birational. In this talk, we will show that in some cases they are but in most cases they are not. This is joint work with Giovanni Mongardi and Kota Yoshioka.

Thursday 24th May 2018 - Bridgeland stability for semiorthogonal decompositions, hyperkaehler manifolds and cubic fourfolds

Algebraic Geometry

Speaker: Paolo Stellari (University of Milan)

16:00 Room MATH-104

Abstract: We illustrate a new method to induce stability conditions on semiorthogonal decompositions and apply it to the Kuznetsov component of the derived category of cubic fourfolds. We use this to generalize results of Addington-Thomas about cubic fourfolds and to study the rich hyperkaehler geometry associated to these hypersurfaces. This is the content of joint works with Arend Bayer, Howard Nuer, Marti’ Lahoz, Emanuele Macri’ and Alex Perry.

Tuesday 15th May 2018 - A moduli perspective on Grassmannians and matroids in idempotent algebra

Speaker: Jeffrey Giansiracusa (University of Swansea)

16:00 Room MATH-104

Abstract: Grassmannians are an important example of moduli spaces in algebraic geometry (over rings). In this talk I will explore and contrast several distinct notions of Grassmannians that appear when passing to the setting of idempotent semirings. There is: (1) a moduli space of locally free quotients, which has an interesting topology but might not be representable; (2) the Dressian, which is described elegantly in terms of an idempotent version of exterior algebra; and (3) various tropicalizations of Grassmannians. I will discuss these objects from a scheme-theoretic and moduli-theoretic point of view.

Thursday 10th May 2018 - On Shimura subvarieties of A_g contained in the Prym locus

Algebraic Geometry

Speaker: Dr. Paola Frediani (University of Pavia)

17:00 Room MATH-104

Abstract: I will present some results obtained in collaboration with E. Colombo, A. Ghigi and M. Penegini on Shimura subvarieties of Ag generically contained in the Prym locus. I will explain the construction of 1 dimensional families of double covers compatible with a fixed group action on the base curve C such that the quotient of C by the group is the projective line. I will give a simple criterion for the image of these families under the Prym map to be a Shimura curve. I will show that this criterion allows us to construct several examples of Shimura curves generically contained in the Prym locus in A g for g < 13.

Thursday 10th May 2018 - Differential-geometric aspects of the Torelli map

Algebraic Geometry

Speaker: Dr. Alessandro Ghigi (University of Pavia)

16:00 Room MATH-104

Abstract: In the first part of the talk I will briefly describe some results obtained by various people in the last years about the second fundamental form of the Torelli map and about totally geodesic submanifolds of A_g contained the Jacobian locus. Next I will discuss a recent result relating divisors in the Jacobian locus to totally geodesic submanifolds.

Thursday 26th April 2018 - Cantor Bouquets and escaping singular orbits in Class B

Dynamical Systems

Speaker: Leticia Pardo Simon (University of Liverpool)

13:00 Room MATH-514

Abstract: As a partial answer to Eremenko's conjecture, it is known for functions in Class B of finite order that every point in their escaping set can be connected to infinity by an escaping curve. Even if those curves, called "hairs" or "rays" not always land, this has been positively proved for some functions with bounded postsingular set by showing that their Julia set is structured as a Cantor Bouquet, that is, an embedding in the plane of a straight brush. In this talk I will consider certain functions with unbounded postsingular set whose singular orbits escape at some minimum speed. In this setting, some hairs will split when they hit critical points. Still, I will conclude that the existence of a map on their parameter space whose Julia set is a Cantor Bouquet guarantees that such hairs, if maybe now with split ends, still land.

Thursday 26th April 2018 - Singularities and global geometry of moduli of curves with spin and principal bundles

Algebraic Geometry

Speaker: Mattia Galeotti (Jussieu, Paris)

16:00 Room MATH-104

Abstract: In a series of recent papers, Chiodo, Farkas and Ludwig carried out a deep analysis of the singular locus of the moduli space of stable (twisted) curves with an $\ell$-torsion line bundle. They showed that for $\ell\leq 6$ and $\ell\neq 5$ pluricanonical forms extend over any desingularization. This opens the way to a computation of the Kodaira dimension without desingularizing, as done by Farkas and Ludwig for $\ell=2$, and by Chiodo, Eisenbud, Farkas and Schreyer for $\ell=3$.

We generalize this works in two directions.At first we treat roots of line bundles on the universal curve systematically: we consider the moduli space of curves C with a line bundle L such that $L^{\ell}\cong\omega_C^{\xx k}$. New loci of canonical and non-canonical singularities appear for any k not a multiple of $\ell$ and $\ell>2$, we provide a set of combinatorial tools allowing us to completely describe the singular locus in terms of dual graphs.

Furthermore, we treat moduli spaces of curves with a G-cover where G is any finite group. In particular for G=S_3 we approach the evaluation of the Kodaira dimension of the moduli space, and list the remaining obstacles to calculate it.

Thursday 19th April 2018 - Entropy conjecture in uniformly quasiregular dynamics

Dynamical Systems

Speaker: Tuomas Sahlsten (University of Manchester)

13:00 Room MATH-514

Abstract: We establish Shub's entropy conjecture for all uniformly quasiregular mappings on compact connected oriented Riemannian manifolds without boundary such that their rational cohomology is not isomorphic to that of the sphere. The proof builds on the recent absolute continuity theorem of Kangasniemi for the equilibrium measure Okuyama and Pankka constructed for uniformly quasiregular maps and cohomology results by Kangasniemi and Pankka for manifolds supporting uniformly quasiregular dynamics.

This is a joint work with Ilmari Kangasniemi (Helsinki), Yusuke Okuyama (Kyoto) and Pekka Pankka (Helsinki)

Thursday 12th April 2018 - The geometry of De Jonquières divisors on projective curves

Algebraic Geometry

Speaker: Mara Ungureanu (Humboldt University, Berlin)

16:00 Room MATH-104

Abstract: De Jonquières divisors are divisors with prescribed multiplicity contained in a linear series on a smooth projective curve. Viewing the curve as embedded in projective space by the linear series, the points in the support of these divisors determine hyperplanes that intersect the curve with prescribed multiplicities at the points of intersection. In this talk we give a broad overview of many interesting results pertaining to this setup, starting from de Jonquières' classical enumerative formula from 1866, and up to contemporary results involving the moduli space of curves.

Thursday 12th April 2018 - Arclike and Indecomposable Dynamic Rays

Dynamical Systems

Speaker: Stephen Worsley (University of Liverpool)

13:00 Room MATH-514

Abstract: The dynamic rays of a function give structure its dynamics. The topology of these rays may not, however, be straightforward even for simple maps. I have studied an example of the transcendental entire function E(z)=2\pi i e^z, for which it is known that the closures of such rays may be an indecomposable continuum, a particularly pathological kind of continuum. I have shown that, furthermore, that the closure (in an appropriate compactification of the complex plane) of any such ray of E must be Arclike and I have given conditions on precisely which rays have indecomposable closure.

Friday 2nd March 2018 - Newton-Okounkov bodies and toric degenerations of Mori dream spaces

Algebraic Geometry

Speaker: Dr Elisa Postinghel (University of Loughborough)

16:00 Room MATH-104

Abstract: Building on work of Okounkov from the 1990s, in 2008 Kaveh and Khovanskii, Lazarsfeld and Mustata showed how to associate to an n-dimensional algebraic variety X and a line bundle a convex body in n-dimensional Euclidean space, the Newton-Okounkov body. In the first part of this talk we will revise construction and main properties of these bodies. In the second part of the talk we will see that for Mori dream spaces, Newton-Okounkov bodies are particularly nice and give rise to toric degenerations. This is joint work with Stefano Urbinati.

Thursday 1st March 2018 - Homogeneous dynamics and S-arithmetic quantitative Oppenheim conjecture

Dynamical Systems

Speaker: Keivan Mallahi-Karai (Jacobs University)

13:00 Room MATH-514

Abstract: Let $q$ be a non-degenerate indefinite quadratic form in $n>2$ variables over $\mathbb{R}$, which is not a multiple of a rational form. Answering a longstanding question of Oppenheim, Margulis proved in 1986 that the set of values $q( \mathbb{Z}^n)$ is a dense subset of $\mathbb{R}$. Quantifying this result, Eskin, Margulis, and Mozes obtained the asymptotic behavior of the number of integral vectors $v$ of norm at most $T$ satisfying $q(v) \in (a,b)$. Both works are dynamical in nature, and rely heavily on features of the unipotent flow dynamics on homogenous spaces.

In this talk, I will elaborate on this history and then discuss a recent generalisation of the theorem of Eskin, Margulis, and Mozes, in which instead of one real quadratic form, a finite number of quadratic forms over different completions of the set of rational numbers (real and $p$-adic) is considered. This talk is based on a joint work with Seonhee Lim and Jiyoung Han.

Friday 23rd February 2018 - Affine quadrics and the Picard group of the motivic category

Algebraic Geometry

Speaker: Dr Alexander Vishik (University of Nottingham)

17:00 Room MATH-104

Abstract: Affine quadrics can be considered as a non-split algebro-geometric spheres. Over algebraic closure these consist of just two cells. And, as was shown by Bachmann, their reduced motives are invertible. It appears that these motives behave better than that of projective quadrics. In particular, the motive of affine quadrics {q=1} determines the respective quadratic form. This gives an embedding of the Grothendieck-Witt group GW(k) of quadratic forms over k (or, which is the same, the (0)-stable homotopy group of spheres) into the Picard group Pic(DM(k,Z/2)) of the Voevodsky's motivic category. In topology, the respective map is an isomorphism. In algebraic geometry, it is no longer the case. In my talk I will describe the subgroup of the Pic generated by the image. Quite unexpectedly, all the relations are described in terms of indecomposable direct summands in the motives of projective quadrics (so, in terms of some Chow-motivic information).

Friday 16th February 2018 - Cubic surfaces over finite fields

Algebraic Geometry

Speaker: Dr Daniel Loughran (University of Manchester)

17:00 Room MATH-104

Abstract: Serre has asked what are the possibilities for the number of rational points on cubic surfaces over finite fields. In this talk we give a complete solution to this problem, building on special cases treated by Swinnerton-Dyer. This is joint work with Barinder Banwait and Francesc Fité.

Thursday 15th February 2018 - Cutting sequences on Veech surfaces

Dynamical Systems

Speaker: Irene Pasquinelli (University of Durham)

13:00 Room MATH-514

Abstract: Consider the dynamical system given by the geodesic flow on a flat surface. Given a polygonal representation for a surface, one can code the trajectory using the sides of the polygons and thus obtain a cutting sequence. A natural question to ask then, is whether any sequence one picks can come from a certain trajectory. In other words, can we characterise the set of cutting sequences in the set of all sequences in the alphabet? And when the answer is yes, can we recover the direction of the trajectory? In this talk we will give an overview of the cases where these questions have been answered.

Wednesday 14th February 2018 - Deligne-Mostow lattices and cone metrics on the sphere

Speaker: Irene Pasquinelli (University of Durham)

16:00 Room MATH-104

Abstract: Finding lattices in PU(n,1) has been one of the major challenges of the last decades. One way of constructing a lattice is to give a fundamental domain for its action on the complex hyperbolic space. One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle. In this talk we will see how this construction can be used to build fundamental polyhedra for all Deligne-Mostow lattices in PU(2,1).

Friday 9th February 2018 - Geometrically finite Kleinian groups and dimension

Dynamical Systems

Speaker: Jonathan Fraser (University of St. Andrews)

11:00 Room MATH-104

Abstract: Kleinian groups act discretely on hyperbolic space and give rise to beautiful and intricate mathematical objects, such as tilings and fractal limit sets. The dimension theory of these limit sets has a particularly interesting history, the first calculation of the Hausdorff dimension going back to seminal work of Patterson from the 1970s.  In the geometrically finite case, the Hausdorff, box-counting, and packing dimensions are all given by the Poincare exponent.  I will discuss recent work on the Assouad dimension, which is not necessarily given by the Poincare exponent in the presence of parabolic points.

Thursday 8th February 2018 - Automorphisms of pointless surfaces

Algebraic Geometry

Speaker: Costya Shramov (Higher School of Economics and Steklov Institute, Moscow)

17:00 Room MATH-103

Abstract: I will speak about finite groups acting by birational automorphisms of surfaces over algebraically non-closed fields, mostly function fields. One of important observations here is that  a smooth geometrically rational surface S is either birational to a product of a projective line and a conic (in particular, S is rational provided that it has a point), or finite subgroups of its birational automorphism group are bounded. We will also discuss some particular types of surfaces with interesting automorphism groups, including Severi-Brauer surfaces.

Monday 5th February 2018 - Tilting relative generators for birational morphisms

Algebraic Geometry

Speaker: Dr Agnieszka Bodzenta  (Warsaw)

16:00 Room MATH-104

Abstract: For a birational morphism of smooth varieties f: X \to Y with the dimension of fibers bounded by one, the derived category of X admits a relative tilting object over Y. It is a direct sum of copies of the canonical line bundle restricted to relative canonical divisors of partial contractions g:X \to Z. It endows the derived category of X with a t-structure related to the map f. I will show that Y is the fine moduli space of simple quotients of O_X in the heart of this t-structure. I will also prove that the t-structures for f and any partial contraction g are related by two tilts in torsion pairs. This is a joint work with A. Bondal.

Thursday 1st February 2018 - On compactifications of strata of abelian differentials

Algebraic Geometry

Speaker: Quentin Gendron (MPI- Bonn)

16:00 Room MATH-104

Abstract: Gendron_Abstract Besides Algebraic Geometry, the work that Quentin is going to discuss is motivated by Teichmuller dynamics on flat surfaces.

Monday 29th January 2018 - Geometric realizations of quiver mutations

Singularity Therory

14:00 Venue: TBC

Speaker: Dr. Anna Felikson (University of Durham)

Abstract: A quiver is a weighted oriented graph, a mutation of a quiver is a simple combinatorial transformation arising in the theory of cluster algebras. In this talk we connect mutations of quivers to reflection groups acting on linear spaces and to groups generated by point symmetries in the hyperbolic plane. We show that any mutation class of rank 3 quivers admits a geometric presentation via such a group and that the properties of this presentation are controlled by the Markov number p^2+q^2+r^2-pqr, where p,q,r are the weights of the arrows in the quiver. This is a joint work with Pavel Tumarkin.

Thursday 14th December 2017 - Slow escaping points for quasimeromorphic mappings

Dynamical Systems

Speaker: Luke Warren (University of Nottingham)

11:00 Room TP-117

Abstract: For a transcendental meromorphic function f, the escaping set is given by I(f) = {x in C : f^n(x) is defined for all n,  |f^n(x)| tends to infinity as n tends to infinity}.

It has been shown by Rippon and Stallard that for such f, there exists a point in J(f) that escapes arbitrarily slowly. More recently, a result by Nicks states that the slow escaping result also holds for quasiregular mappings of transcendental type, which are higher dimensional analogues of analytic mappings with an essential singularity at infinity.

Following a similar method of Nicks, combined with some ideas on the escaping set of  quasimeromorphic mappings with an infinite number of poles, we shall extend this result and show that there exists a point that escapes arbitrarily slowly for quasimeromorphic mappings with an essential singularity at infinity. This will include the proof of a new growth result for quasiregular mappings near an essential singularity.

Thursday 7th December 2017 - Hausdorff dimension of the boundary of Siegel disks

Dynamical Systems

Speaker: Alexandre de Zotti (Imperial College, London)

11:00 Room TP-117

Abstract: In this work I will present my work in progress with Davoud Cheraghi towards a proof of the existence of Siegel disks of quadratic polynomials with a boundary of Hausdorff dimension two.

Thursday 30th November 2017 - Cohomological Rigidity and the Anosov-Katok construction

Dynamical Systems

Speaker: Nikos Karaliolios (Imperial College, London)

11:00 Room TP-117

Abstract: Let f be a smooth volume preserving diffeomorphism of a compact manifold and \phi a known smooth function of zero integral with respect to the volume. The linear cohomological equation over f is \psi \circ f - \psi = \phi where the solution \psi is required to be smooth. Diffeomorphisms f for which a smooth solution \psi exists for every such smooth function \phi are called Cohomologically Rigid. Herman and Katok have conjectured that the only such examples up to conjugation are Diophantine rotations in tori.

We study the relation between the solvability of this equation and the fast approximation method of Anosov-Katok and prove that fast approximation cannot construct counter-examples to the conjecture

Monday 29th November 2017 - LMS Singularity Day (dedicated to the memory of Ragnar-Olaf Buchweitz)

Singularity Therory

Venue: MATH-106

13.30 - 14.30  Eleonore Faber (University of Leeds)
Noncommutative desingularizations of discriminants of reflection groups

Abstract: This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls. Let $G$ be a finite subgroup of $GL(n,K)$ for a field $K$ whose characteristic does not divide the order of $G$. The group $G$ acts linearly on the polynomial ring $S$ in $n$ variables over $K$. When $G$ is generated by reflections, then the discriminant $D$ of the group action of $G$ on $S$ is a hypersurface with a singular locus of codimension 1, in particular, $D$ is a so-called free divisor. In this talk we give a natural construction of a noncommutative resolution of singularities of the coordinate ring of $D$ as a quotient of the skew group ring $A=S*G$. We will explain this construction, which gives a new view on Knörrer's periodicity theorem for matrix factorizations and allows to extend Auslander's theorem about the algebraic version of the McKay correspondence to reflection groups.

14.30 - 15.30  Alexandr Buryak (University of Leeds)
Extended r-spin theory and the deformed superpotential of the A-singularity

Abstract: It is well-known that the parameter space of the miniversal deformation of the A-singularity carries a Frobenius manifold structure. The potential of this Frobenius manifold can be described as the generating series of certain integrals over the moduli space of r-spin curves. This is the simplest case of the Landau-Ginzburg mirror symmetry. I will show that the deformed superpotential of the A-singularity also has a geometric interpretation in terms of a certain extension of the r-spin theory. The talk is partially based on a joint work with E. Clader and R. J. Tessler.

15.30 - 16.00 Tea/coffee/biscuits, room 304

16.00 - 17.00
Firuza Mamedova (Hanover)
Equivariant indices of 1-forms on varieties

Abstract: For a $G$-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic $G$-variety with an isolated singular point ($G$ is a finite group) one has notions of the equivariant homological index and of the (reduced) equivariant radial index as elements of the ring of complex representations of the group. During my talk I will show that on a germ of a smooth complex-analytic $G$-variety these indices coincide. This permits to consider the difference between them as a version of the equivariant Milnor number of a germ of a $G$-variety with an isolated singular point. The talk is based on a joint work (arXiv:1701.01827) with Sabir M. Gusein-Zade.

Monday 27th November 2017 - Uniformization of metric surfaces

Dynamical Systems

Speaker: Kai Rajala (University of Jyväskylä)

11:00 Room TP-117

Abstract: We discuss extensions of the classical uniformization theorem to metric spaces that are topological surfaces and have locally finite two-dimensional Hausdorff measure.

Thursday 23rd November 2017 - Some properties of Falconer’s formula for the Hausdorff dimension of self-affine fractals

Algebraic Geometry

Speaker: Ian Morris (University of Surrey)

11:00 Room TP-117

Abstract: The dimension theory of self-similar sets (under suitable separation conditions) has been well-understood since the 1980s, when J.E. Hutchinson gave a simple formula for the Hausdorff and box dimensions in terms of the contraction ratios of the similarity transformations. The dimension theory of self-affine sets is much less well-understood and remains an active topic of research. In 1988 Falconer devised a more subtle pressure formula which provides the value of the Hausdorff dimension of a self-affine set in generic cases. I will describe some recent research on Falconer's dimension formula and its implications for the dimension theory of self-affine sets.

Friday 17th November 2017 - Classification of Picard lattices of K3 surfaces

Speaker: Viacheslav V. Nikulin (Steklov Mathematical Institute, Moscow and University of Liverpool)

17:00-18:00 Room MATH-104

Abstract: Using our recent results about classification of degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups, we classify Picard lattices of Kahlerian K3 surfaces. By classification, we understand classification depending on their possible finite symplectic automorphism groups and their non-singular rational curves if a Picard lattice is negative definite.

Thursday 16th November 2017 - A landing theorem for entire functions with bounded postsingular sets

Speaker: Professor Lasse Rempe-Gillen (University of Liverpool)

11:00 Room MATH-117

Abstract: Let f be a polynomial in one complex variable, of degree 2. Assume that all critical points of f have bounded orbits under iteration of f. Then the Julia set of f is connected, and its unbounded connected component (the basin of infinity) is stratified by so-called “external rays” to infinity, which are the gradient lines for the Green’s function. These rays have played a crucial role in the study of polynomial dynamics for more than three decades, including in celebrated results of Yoccoz, McMullen and Lyubich.

A theorem of Douady states that every repelling periodic point of f is the landing point of a periodic external ray. This theorem has been the cornerstone of the above-mentioned breakthroughs in polynomial dynamics. We establish an analogous result for transcendental entire functions.

Of course, here the Julia set is unbounded, and there is no longer a basin of infinity. It was proposed already thirty years ago that certain curves, called “hairs”, on which the iterates tend to infinity can play the same roles as external rays in this setting. However, we now know that there are entire functions for which there are no such hairs at all. Instead, we use a notion of “dreadlocks” – certain unbounded connected sets – to prove the following result:

Suppose that f is a transcendental entire function, and assume that all singular values of f have bounded orbits under the iteration of f. Then every repelling periodic point of f is the landing point of a periodic dreadlock.

My goal in this talk  is to give only a short introduction to make sense of the results, and then explain a crucial element of the proof of the theorem. (Joint work with Anna Benini).

Friday 10th November 2017 - Specialization of (stable) rationality

Speaker: Evgeny Shinder (Sheffield)

17:00-18:00 Room MATH-104

Abstract: The specialization question for rationality is the following one: assume that very general fibers of a flat proper morphism are rational, does it imply that all fibers are rational? I will talk about recent solution of this question in characteristic zero due to myself and Nicaise, and Kontsevich-Tschinkel. The method relies on a construction of various specialization morphisms for the Grothendieck ring of varieties (stable rationality) and the Burnside ring of varieties (rationality).

Thursday 2nd November 2017 - Counting the number of trigonal curves of genus five over finite fields

Speaker: Thomas Wennink (University of Liverpool)

16:00-17:00 Room MATH-104

Abstract: The trigonal curves of genus five form a closed subscheme of M_5, the moduli space of smooth curves of genus five. The Hodge Euler characteristic of these spaces can be found by counting the number of points they have over finite fields. To count trigonal curves of genus 5 we use the fact that they correspond to projective plane quintics that have precisely one singularity, which is of delta-invariant one. In my master thesis https://arxiv.org/abs/1701.00375 I counted these projective plane quintics over finite fields using a partial sieve method.

Tuesday 31st October 2017 - Moduli space of curves, tautological relations and integrable systems

Speaker: Paolo Rossi

17:00-18:00 Room MATH-106

Abstract: In the study of the topology of moduli space of stable curves and its tautological ring, a surprising feature is the appearence of integrable systems of PDEs (typically in terms of generating functions of intersection numbers of various types of cohomology classes). Beside being a remarkable bridge towards mathematical physics, this fact brings new powerful techniques to the field. In a recent series of papers with A. Buryak, B. Dubrovin and J. Guéré, we construct an integrable system from any given cohomological field theory using various tautological classes (including the double ramification cycle) and we compare it with the more classical Dubrovin-Zhang integrable hierarchy. This comparison suggests a new, large family of conjectural tautological relations in all genera and number of marked points. I will report on our progress in proving them and on their applications.

Thursday 26th October 2017 - There are no Diophantine Quintuples

Speaker: Prof. Volker Ziegler (Salzburg University)

15:00-16:00 Room MATH-104

Abstract: An m-tuple of distinct positive integers (a_1,...,a_m) is called a Diophantine m-tuple if a_ia_j+1 is a perfect square for all i not equal to j. It was a long outstanding question whether a Diophantine quintuple exists. In a recent paper joint with Bo He and Alain Togbè we recently proved that none exists. After a short introduction to the problem we present the new ideas that led to the proof of the so-called Diophantine quintuple conjecture.

Thursday 26th October 2017 - On recurrence statistics and Poisson laws for dynamical systems

Speaker: Mark Holland (Exeter University)

11:00-12:00 Room MATH-117

Abstract: For a time series of observations (X_n) generated by a measure preserving dynamical system, we consider the maxima process M_n=Max(X_1,...X_n), and examine the probabilistic limit laws that can arise for M_n under suitable time normalization. We consider the implication these laws have on the recurrence statistics for the dynamical system, such as: extreme and record statistics, Poisson laws, and Borel Cantelli results.

Thursday 19th October 2017 - Waldhausen Ring Spectra of Varieties

Speaker: Dr Anwar Alameddin (University of Edinburgh)

16:00-17:00 Room MATH-104

Abstract: The Grothendieck ring of varieties encodes essential information about the varieties, including their stable birational classes, ℓ-adic characteristic, Hodge characteristic, etc. In this talk, I will explain how some Euler-Poincaré characteristics arise through Waldhausen K-theories. More specifically, I will show that a cosheaf with respect to the Grothendieck topology generated by abstract blow-ups, with values in a Waldhausen category, admits a compactly supported extension. Then, I will explain how the (modified) Grothendieck ring of varieties extends to the ring spectra of some monoidal Waldhausen categories, generated by the homotopy category of varieties with respect to universal homeomorphisms.

Thursday 12th October 2017 - The derived period map

Speaker: Julien Holstein (University of Lancaster)

16:00-17:00 Room MATH-104

Abstract: Griffiths’ period map expresses how the Hodge filtration on cohomology varies in a family of smooth projective varieties. Thus one can linearize moduli problems of varieties and study them using the moduli of Hodge structure. In this talk I will describe how to extend the period map to the realm of derived geometry and construct a  period map for a smooth projective map of derived stacks. This generalises both Griffiths classical period map and the infinitesimal derived period map that was introduced by Fiorenza and Manetti. This is joint work with Carmelo Di Natale.
No knowledge of derived algebraic geometry will be assumed.

Wednesday 4th October 2017 - An approach to the motivic version of the BPQ theorem

Speaker: Dr Joe Palacios Baldeon (IMCA, Lima, Peru)

17:00-18:00 Room MATH-103

Abstract: We follow Schlichtkrull's approach to deal with the motivic version of the BPQ theorem. One difficulty that arises in this context is the motivic version of Freudenthal suspension theorem, as it does not follow straightforwardly from the topological set-up.

Tuesday 3rd October 2017 - An introduction to A1-homotopy theory of schemes

Speaker: Dr Joe Palacios Baldeon (IMCA, Lima, Peru)

17:00-18:00 Room MATH-103

Abstract: This lecture will be a brief overview on the basics of the A1-homotopy theory of schemes, needed to understand a possible approach to the motivic version of BPQ, to be delivered in the third lecture. We will highlight what is the homotopy theory of schemes, how to contract the affine line to a point, and what exactly the word "motivic" means.

Reference: F. Morel and V. Voevodsky.  A1-homotopy theory of schemes. Inst. Hautes Etudes Sci. Publ. Math. (1999).

Monday 2nd October 2017 - The Barratt-Priddy-Quillen (BPQ) theorem in topology

Speaker: Dr Joe Palacios Baldeon (IMCA, Lima, Peru)

17:00-18:00 Room MATH-103

Abstract: The BPQ theorem says that the stable homotopy groups of spheres can be expressed in terms of the classifying space of the infinite symmetric group. We will explain the Schlichtkrull's approach to the BPQ-theorem which involves the group completion of homotopy infinite symmetric powers of pointed CW-complexes.

Reference:  C. Schlichtkrull. The homotopy infinite symmetric product represents stable homotopy. Algebr. Geom. Topol. (2007).

Friday 29th September 2017 - Automorphisms of certain affine complements in the projective space

Speaker: Professor Aleksandr Pukhlikov

16:00-17:00 Room MATH-104

Abstract: We prove that every biregular automorphism of the complement in the $M$-dimensional projective space ($M\geqslant 3$) of a hypersurface $S$ of degree $m\geqslant M+1$ with a unique singular point of multiplicity $(m-1)$, resolved by one blow up, is a restriction of some automorphism of the projective space, preserving the hypersurface $S$; in particular, for a general such hypersurface $S$ the group of automorphisms of its complement is trivial.

Friday 30th June 2017 - Transpennine Topology Triangle

11:30 - 16:30 MATH-104

12:00 - Jon Woolf (Liverpool)

Title: Stratified Homotopy Theory

Abstract: Stratified spaces arise in many contexts within topology, geometry, and algebra. By fixing suitable stratifications of geometric simplices one can construct stratified versions of geometric realisation and of the total singular complex functor, giving an adjunction between simplicial sets and stratified spaces. The Joyal model structure on simplicial sets can be cofibrantly transferred acrss this to obtain a model structure on the category of stratified spaces. The cofibrant-fibrant spaces and weak equivalences in this model structure are closely related to classical' notions in the theory of stratified spaces, however one also gains new insights, in particular a new notion of stratified fibration which has better properties than previous ones. This is joint work with Stephen Nand-Lal.

14:00 - Stephen Nand-Lal (Liverpool)

Title: Notions of Basepoint' for a Stratified Space

Abstract: There are a number of potential approaches to basing a stratified space. In this talk I will explain why some plausible approaches are unsatisfactory, and introduce the notion of basing for a stratified space which we believe to be correct. As evidence for this, there is an adjunction between stratified suspension and loop space functors, which allows us to construct, for suitably nice stratified spaces, an $\mathbb{N}$-indexed family of categories that behave analogously to the homotopy groups of a connected space. This is joint work with Jon Woolf.

14:30 - Alessio Cipriani (Liverpool)

Title: Perverse Sheaves as Modules

Abstract: I will introduce the abelian category of perverse sheaves on a topologically stratified space together with some important properties and examples. I will then explain the construction of the projective cover of a simple perverse sheaf under the assumption that all strata have finite fundamental group. This is the key ingredient needed to show that the cateory of perverse sheaves is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if, and only if, all strata have finite fundamental group.

15:30 - Thomas Eckl (Liverpool)

Title: Topological classification of isolated holomorphic foliation singularities

Abstract: Isolated singularities of holomorphic foliations can be topologically described by intersecting the leaves of the foliation with (small) spheres centered in the singularities. This works particularly well for holomorphic foliation singularities of Poincare type, but is also useful for other types. After discussing results along these lines we give a complete classification of isolated singularities of plane holomorphic foliations in the Poincare domain and present partial results and conjectures in higher dimensions.

Thursday 15th June 2017 - Computer Assisted and Experimental Mathematics: Case for Automated Reasoning

Speaker: Dr Alexei Lisitsa (Computer Sciences)

2:00PM MATH-105

Abstract: Computers  have been used in Mathematics for decades. Recent years have witnessed  rise of mathematical applications of non-numerical symbolic computations and automated and interactive reasoning. While interactive reasoning using proof assistants appears as a  main vehicle in formalization and validation of existing mathematical knowledge (cf. recent efforts on formalization of proofs of Kepler conjecture and 4-Colorability Theorem), automated reasoning finds new applications in proving new results.

After short overview of recent trends in computer assisted mathematics, I will focus on two applications of automated reasoning to tackle  open problems.

First application  is a major progress in the Erdos Discrepancy Problem (EDP), obtained by encoding the problem into Boolean satisfiability and applying the state of art SAT (satisfiability)  solver (joint work with Boris Konev, 2013-2015). The work attracted wide publicity, in particular due to enormous size of the generated proof of the C=2 case of the EDP, and caused discussions of the very nature of computer assisted mathematical proofs.

Second application is ongoing investigation in the Andrews-Curtis conjecture, a problem in combinatorial group theory open since 1965, where automated theorem proving appears as the most powerful method for the experimental exploration of the conjecture.

Tuesday 30th May 2017 - Igusa quartic and Wiman-Edge sextics

Speaker: Ivan Cheltsov (University of Edinburgh)

5:00PM MATH-G16

Abstract: The automorphism group of Igusa quartic is the symmetric group of degree 6. There are other quartic threefolds that admit a faithful action of this group. One of them is the famous Burkhardt quartic threefold. Together they form a pencil that contains all $\mathfrak{S}_6$-symmetric quartic threefolds.

Arnaud Beauville proved that all but four of them are irrational, while Burkhardt and Igusa quartic are known to be rational.  Cheltsov and Shramov proved that the remaining two threefolds in this pencil are also rational.  In this talk, I will give an alternative proof of both these (irrationality and rationality) results.

To do this, I will describe Q-factorizations of the double cover of the four-dimensional projective space branched over the Igusa quartic, which is known as Coble fourfold.  Using this, I will show that $\mathfrak{S}_6$-symmetric quartic threefolds are birational to conic bundles over quintic del Pezzo surfaces whose degeneration curves are contained in the pencil studied by Wiman and Edge.  This is a joint work with Sasha Kuznetsov and Costya Shramov from Moscow..

Friday 19th May 2017 - E-polynomials of character varieties and applications

Speaker: Marina Logares (University of Plymouth)

4:00PM MATH-103

Abstract: Character varieties have been studied largely by means of their correspondence to the moduli space of Higgs bundles. In this talk we will report on a method to study their Hodge structure, in particular to compute their E- polynomials. Moreover, we will explain an application of the given method such as, the study of the topology of the moduli space of doubly periodic instantons. This is based on joint work with V. Muñoz and P. Newstead.

Friday 12th May 2017 - Lyapunov spectrum of Markov tree

Speaker: Prof. Alexander Veselov (Loughborough University)

1:00PM MATH-G16

Abstract: Markov triples are the integer solutions of the Markov equation $$x^2+y^2+z^2=3xyz.$$

They surprisingly appeared in many areas of mathematics initially in classical number theory, but more recently in hyperbolic and algebraic geometry, the theory of Teichmueller spaces, Frobenius manifolds and Painleve equations.

Markov numbers can be naturally represented using planar binary tree, so their growth depends on the paths on such tree, which can be labelled by the points of real projective line.

I will discuss some recent results about the corresponding Lyapunov exponents found jointly with K. Spalding.

Thursday 11th May 2017 - Kähler packings and Seshadri constants II

Speaker: Thomas Eckl (University of Liverpool)

4:00PM MATH-103

Abstract: I will discuss packing problems of complex manifolds with a given finite number of balls with equal radius relying on Riemannian metrics, symplectic forms and Kähler metrics in the most basic case.

Thursday 4th May 2017 - Kähler packings and Seshadri constants I

Speaker: Thomas Eckl (University of Liverpool)

4:00PM MATH-103

Abstract: I will discuss packing problems of complex manifolds with a given finite number of balls with equal radius relying on Riemannian metrics, symplectic forms and Kähler metrics in the most basic case.

Tuesday 25th April 2017 - Divisor theory on tropical and log smooth curves

Speaker: Mattia Talpo (Simon Fraser University - Vancouver)

4:00PM MATH-104

Abstract: Tropical geometry is a relatively new branch of algebraic geometry, that aims to prove facts about algebraic varieties by studying their "tropicalizations", which are piecewise linear objects, amenable to combinatorial study. A prominent topic in recent research in the area, that is leading to new insights about "classical" open questions, is a theory of divisors on tropical curves. In this talk I will survey some of the related ideas, and explain how they are connected to line bundles on log smooth curves (joint work with Foster, Ranganathan and Ulirsch).

Thursday 23rd March 2017 - Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac--Moody algebras

Speaker: Viacheslav V. Nikulin (University of Liverpool and Steklov Mathematical Institute, Moscow)

Venue: 5:00PM MATH-103

Abstract: Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give examples of lattice-polarized K3 surfaces with automorphic discriminant. See arXiv:1702.07551 for some details. These are joint results with Valery Gritsenko.

Thursday 16th March 2017 - The $4n^2$-inequality for complete intersection singularities

Speaker: Aleksandr Pukhlikov (University of Liverpool)

5:00PM MATH-103

Abstract: The famous $4n^2$-inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is greater than $4n^2\mu$, where $\mu$ is the multiplicity of the singular point.

Thursday 16th February 2017 - The space of rational curves and Manin’s conjecture

Speaker: Sho Tanimoto (University of Copenhagen)

5:00PM MATH-103

Abstract: Recently there are many progresses regarding exceptional sets of rational points in Manin’s conjecture by Hacon, Jiang, Lehmann, Tanimoto, Tschinkel. We found that the study of rational points has applications in the study of rational curves, e.g., the dimension of the space of rational curves, and the behavior of the space of rational curves can be summarized as a version of Manin’s conjecture for rational curves. In this talk, I will talk about these developments. This is joint work with Brian Lehmann.

Thursday 3rd February 2017 - GLEN meeting 2017

Venue: Hele-Shaw lecture theatre, Walker Building, School of Engineering

1:00PM Roberto Fringuelli, University of Edinburgh - The Picard group of the universal moduli space of vector bundles on stable curves.

In this talk, we present the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its Picard group. As a consequence, we obtain an explicit description of the Picard groups of the universal moduli stack of vector bundles on smooth curves and of the Schmitt's compactification over the stack of stable curves. We show some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group. In particular, we give necessary and sufficient conditions for the existence of a universal family of an open substack of the rigidification. In the remaining time, we discuss some consequences for the associated moduli varieties.

2:15PM Liana Heuberger, Imperial College London - Fano varieties, general elephants and classification.

We start by giving a short overview on the classification of Fano varieties. The smooth case is known up to dimension three, and we discuss the role that the general elephant, i.e. a general divisor D in |-K_X|, plays in this classification. We state the problem and current advances in dimension four. We then investigate what happens if we allow the Fano variety to have mild singularities and why this makes sense in the bigger picture of classification of algebraic varieties.

4:00PM Jeffrey Giansiracusa, University of Swansea - Tropical geometry and algebra over idempotent semirings.

Tropical geometry is a tool that can reduce problems in algebraic geometry to piecewise polyhedral geometry and combinatorics, but it is also a new world of geometry in its own right.  In this talk I will introduce the emerging picture of this kind of geometry as parallel to Grothendieck's vision of algebraic geometry.  Here rings are replaced by idempotent semirings, and the role of linear algebra in classical commutative algebra is replaced by the combinatorics of matroids.

Thursday January 19th 2016 - Expanding Thurston Maps

Speaker: Dr Daniel Meyer

Venue: Room MATH-211 1:00PM

Abstract: In this talk I will give an overview of my (recent and not so recent) work, done jointly with Mario Bonk. An expanding Thurston map'' is a postcritically finite branched covering map of the 2-sphere that is expanding in a suitable sense. We study these maps with methods from geometric group theory. A major motivation comes from Cannon's conjecture''. This well-known conjecture stipulates that a group that behaves topologically'' as a Kleinian group is'' a Kleinian group. This conjecture corresponds to Thurston's characterization of rational maps among Thurston maps. Closely related is the problem when a metric sphere is quasisymmetrically equivalent to the standard unit sphere, i.e., a quasisphere''. We define visual metrics'' for an expanding Thurston map f. The geometry of the sphere S^2 equipped with such a metric d encodes properties of f. For example, (S^2,d) is a quasisphere if and only if f is topologically conjugate to a rational map.

Thursday December 15th 2016 - Singularity Day 2016

1:30PM - 5:00PM Room G16 (MAGIC)

### Nicola Pagani (University of Liverpool) - Wall-crossing on compactified universal Jacobians

The last 20 years have seen huge develpoments in the enumerative geometry of the moduli spaces of stable curves. In this talk, we will discuss the beginning of a similar programme for the universal Jacobian parameterizing line bundles on stable curves. The universal Jacobian admits many natural compactifications, each of which should play an important role in the enumerative geometry, thus giving rise to interesting wall-crossing phenomena.  We will discuss our first results in this research programme and their application. We have an explicit picture of the combinatorics of the stability space and of the walls that govern all different compactifications, and we understand how the wall-crossing works for codimension-1 cycles.  This is research in progress with Jesse Kass (South Carolina).

### James Montaldi (University of Manchester) - An application of the versal deformation of the '3 lines singularity' to classical mechanics

The famous problem of the rotating free rigid body allows for a reduction to Euler's equations, a dynamical system in R^3. The set of equilibria of this equation form the three coordinate axes in R^3, which is a determinantal singularity (and not a complete intersection).  There are interesting perturbations of this system (corresponding to the addition of gyroscopic terms) and the resulting deformation is also determinantal. We describe this (well-known) deformation and its consequence for the dynamics, and then proceed to discuss the stability of these perturbed equilibria, which becomes a problem in deforming the critical points of a function on a singular variety.

Reference: Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry.  (2014) http://dx.doi.org/10.3934/jgm.2014.6.237

### Oleg Karpenkov (University of Liverpool) - Configuration spaces of tensegrities

In this talk we consider a natural stratification of the configuration space of tensegrities. We discuss several results about the structure of the configuration space of (mostly two-dimensional) tensegrities with a small number of points. In particular we briefly describe the technique of surgeries that is used to find geometric conditions for tensegrities. We conclude the talk with several open problems related to the stratification of the space of tensegrities.

Tuesday 6th December 2016 - Transversality

Speaker: Adam Epstein (University of Warwick)

3:00PM MAGIC Room G-16

Abstract: Thurston's infinitesimal rigidity argument, and extensions due to the speaker and various other researchers, yields transversality principles via the Inverse Function Theorem. It has been suggested that our own account constitutes folklore more appropriate to a conference proceedings than to a research journal. We compare our own results to those subsequently claimed by others, and highlight differences in applicability and reliability.

Tuesday 29th November 2016 - Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections

Speaker: Viacheslav V. Nikulin (University of Liverpool and Steklov Mathematical Institute, Moscow)

4:00PM Room MATH-103

Abstract: This is a continuation of our papers with V.A. Gritsenko in 1997 - 2002 where we mainly considered the case of hyperbolic lattices of the rank 3. Here, for all ranks, we classify 2-reflective hyperbolic lattices S of elliptic type with a lattice Weyl vector. They define the corresponding hyperbolic Kac--Moody algebras of arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac--Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identity is given by automorphic forms with 2-reflective divisors. We give exact construction of these automorphic forms. All these considerations are related to interesting classes of algebraic K3 surfaces and their mirror symmetry. See our recent preprint with V.A. Gritsenko: arXiv: 1602.08359 for some details.

Tuesday 29th November 2016 - Several Constructions in the Eremenko-Lyubich Class

Speaker: Kirill Lazebnik (Stony Brook University)

3:00PM MAGIC Room G-16

Abstract: In 1985, Sullivan proved the no-wandering domain theorem for rational maps of the Riemann Sphere. Using the same ideas, (Lyubich and Eremenko) and (Goldberg and Keen), proved a no-wandering domain theorem for entire functions with finite singular set. Whereas wandering domains were known to exist for more general entire functions, it was unknown whether wandering domains occur for entire functions with bounded (but possibly infinite) singular set. Such a function was constructed by Bishop in 2012 using the "folding theorem" (proven in the same paper) for constructing entire functions. We will first discuss the folding theorem and the wandering domain example. Next we will discuss the topology of the wandering domain. Finally we will show how these methods can be used to construct several functions in the Eremenko-Lyubich class with other dynamical properties we will prescribe.

Tuesday 22nd November 2016 - Quasiperiodic sums and products

Speaker: Paul Verschueren (Open University)

3:00PM MAGIC Room G-16

Abstract: Quasiperiodic Sums (Birkhoff Sums over a rotation) and Products arise in many areas of mathematics including the study of Strange Non-Chaotic Attractors, Critical KAM Theory, Quantum Chaos, q-series, Partition Theory, and Diophantine Approximation.

Graphs of these functions can form intriguing geometrically strange and self-similar structures. They are easy and rewarding to investigate numerically, and suggest many avenues for investigation. However they prove resistant to rigorous analysis.

In this talk we will survey some of the most important examples, and focus on the most heavily studied  example, Sudler's product of sines. We will also report on new approaches which allow us to settle negatively an open question of Erdős & Szekeres  from 1959, and to prove a number of experimental results reported recently by Knill & Tangerman (2011).

Tuesday 15th November 2016 - Holomorphic motions of Julia sets: a lambda-lemma in several complex variables

Speaker: Fabrizio Bianchi (Imperial College, London)

3:00PM in Room G16 (MAGIC)

Abstract: For a family of rational maps, results by Lyubich, Mané-Sad-Sullivan and DeMarco provide a fairly complete understanding of dynamical stability. A central tool in this description is the celebrated lambda-lemma, which allows one to promote a holomorphic motion of a set to one of its closure.

Starting from the basics, and paying particular attention to the differences with respect to the transcendental case, I will review the one-dimensional theory and depict a general panorama about dynamical stability in several variables. I will focus on the arguments that do not readily generalise to this setting, and introduce the tools and ideas that allow one to overcome these problems.

Tuesday 8th November 2016 - On class A and class B

Speaker:  David Sixsmith (University of Liverpool)

3:00PM in Room G16 (MAGIC)

Abstract: In 1970 MacLane asked if it is possible for a locally univalent function in the class A to have an arc tract. This question remains open, but several results about it have been given. We significantly strengthen these results, in particular replacing the condition of local univalence by the more general condition that the set of critical values is bounded. Also, we adapt a recent powerful technique of Bishop in order to show that there is a function in the Eremenko-Lyubich class for the disc that is not in the class A.

Friday 4th November 2016 - Motivic stable homotopy groups of spheres

Speakers:  Professor Oliver Roendings (University of Onsabruck)

4:00PM in Room MATH-105

Abstract: Fabien Morel determined the zeroth motivic stable homotopy group over an arbitrary field in terms of units of the field (Milnor K-theory) and quadratic forms (Grothendieck-Witt groups). In joint work with Markus Spitzweck and Paul Arne Ostvar, we determine the first motivic stable homotopy group in terms of Milnor K-theory and higher Grothendieck-Witt groups. The computation is based on the spectral sequence coming from Voevodsky's slice filtration, and work of Levine and Voevodsky on the slices of the motivic sphere spectrum. One potential application, as Asok and Fasel show, is a solution of Murthy's conjecture of splitting vector bundles over smooth affine varieties in the unstable range.

Tuesday 1st November 2016 - Birational geometry of del Pezzo fibrations of degree 2

Speakers:  Igor Krylov (University of Edinburgh)

4:00PM in Room MATH-103

Abstract: I will discuss rationality of rationally connected varieties. Their minimal models are Mori fiber spaces. In dimension three these are Fano varieties, del Pezzo fibrations, and conic bundles. The rationality question in dimension three is almost solved for smooth varieties. I discuss possible generalizations: including singularities or considering stable rationality and discuss recent results for del Pezzo fibrations.

Tuesday 1st November 2016 - The escaping set of transcendental self-maps of the punctured plane

Speaker:  David Marti Pete (Open University)

3:00PM in Room G16 (MAGIC)

Abstract: We study the iteration of holomorphic self-maps of C*, the complex plane with the origin removed, for which both zero and infinity are essential singularities. The escaping set of such maps consists of the points whose orbit accumulates to zero and/or infinity following what we call essential itineraries. We show that the Julia set always contains escaping points with every essential itinerary. The concept of essential itinerary leads to a partition of the escaping set into uncountably many disjoint sets, the boundary of each of which is the Julia set. Under certain hypotheses, each of these sets contains uncountably many curves to zero and infinity. We also use approximation theory to provide examples of functions with escaping Fatou components.

Tuesday 25th October 2016 - Speiser class Julia sets with dimension near 1

Speaker: Simon Albrecht (University of Liverpool)

3:00PM in Room G-16 (MAGIC)

Abstract: We prove that for any d>0, there exists a transcendental entire function f with only finitely many singular values such the Hausdorff dimension of the Julia set of f is less than 1+d. This is joint work with Christopher J. Bishop from Stony Brook.

Tuesday 18th October 2016 - Inverse limits and measurable pseudo-Anosov maps

Speaker:  André Salles de Carvalho (University of São Paulo, Brazil)

3:00PM in Room G16 (MAGIC)

Abstract: I'll present a construction which starts with a graph endomorphism and, using a quotient of its inverse limit, yields:

1. a pseudo-Anosov map if the graph map is a train track map;
2. a generalized pseudo-Anosov map if the graph map is post-critically finite and has an irreducible aperiodic transition matrix;
3. an interesting type of surface homeomorphisms which generalizes both the previous classes otherwise.

In particular, this produces a unified construction of surface homeomorphisms whose dynamics mimics that of the tent family of interval endomorphisms, completing an earlier construction of unimodal generalized pseudo-Anosov maps in the post-critically finite case. This is joint work with Phil Boyland and Toby Hall.

Tuesday 11th October 2016 - Birationally rigid complete intersections of codimension two

Speakers:  Daniel Evans and Aleksandr Pukhlikov (University of Liverpool)

5:00PM in Room MATH-104

Abstract: We prove that in the parameter space of $M$-dimensional Fano complete intersections of index one and codimension two the locus of varieties that are not birationally superrigid has codimension at least $(M-10)(M-11)/2-1$.

Monday 12th September 2016 - What is a formula?

Speaker:  Igor Pak (UCLA)

3:00PM in Room G16 (MAGIC)

Abstract: Integer sequences arise in a large variety of combinatorial problems as a way to count combinatorial objects. Some of them have nice formulas, some have elegant recurrences, and some have nothing interesting about them at all. Can we characterize when? Can we even formalize what is a "formula"? I will give a mini-survey aiming to answer these questions. At the end, I will present some recent results counting certain walks in graphs and certain permutation classes, and finish with open problems.

Tuesday 6th September 2016 - Quasiregular maps and Zorich maps with one attracting fixed point

Speaker:  Patrick Comdühr (Kiel University)

10:45AM in Room MATH-117.

Abstract: The theory of holomorphic functions in one complex variable is well known by students and is of rich structure. We want to focus on functions in R^d, with d > 2, which behave in a similar way as their holomorphic counterparts. More precisely we look at a quasiregular analogue of the exponential map and use iteration theory to understand its behaviour.

Wednesday 29th June 2016 - How to count zeroes arithmetically

Speaker:  Jesse Kass (South Carolina)

2:00PM in Room MATH-104.

Abstract: A celebrated result of Eisenbud—Khimshaishvili—Levine computes the local degree of a smooth function f : Rn —> Rn as the signature of the residue pairing, an explicit symmetric bilinear form.  We prove a parallel result computing the local A1-degree of a polynomial function, answering a question posed by David Eisenbud in 1978.  This talk will present this result and then discuss applications to singularity theory, including a definition of an arithmetic analogue of the Milnor number, if time permits.  This is joint work with Kirsten Wickelgren.

Tuesday 24th May 2016 - Birationally rigid fibrations into Fano double spaces

Speaker: Aleksandr Pukhlikov (University of Liverpool)

5:00PM in Room MATH-104

Abstract: We develop the quadratic technique of proving birational rigidity of Fano-Mori fibre spaces over a higher-dimensional base. As an application, we prove birational rigidity of generic fibrations into Fano double spaces of dimension $M\geqslant 4$ and index one over a rationally connected base of dimension at most $\frac12 (M-2)(M-1)$. An estimate for the codimension of the subset of hypersurfaces of a given degree in the projective space with a positive-dimensional singular set is obtained, which is close to the optimal one.

Wednesday 18th May 2016 - Self-affine sets: topology and arithmetic

Speaker:  Dr Nikita Sidorov (University of Manchester)

3:00pm in Room 106

Abstract: Let M be an n x n real matrix with eigenvalues less than 1 in modulus. Consider the iterated function system (IFS) {Mx-v, Mx+v} with some vector v such that it is non-degenerate. In my talk I will address the questions related to the topology of the attractor of this IFS (connectedness, non-empty interior) as well as connections to beta-expansions and similar number-theoretic objects for n=2. This talk is based on my three recent papers with Kevin Hare (Waterloo).

Wednesday 11th May 2016 - Dimension gaps in self-affine sponges

Speaker:  David Simmons (University of York)

3:00PM in Room MATH-106.

Abstract: In this talk, I will discuss a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension, as well as my recent result showing that the answer is negative. The counterexample is a self-affine sponge in $\mathbb R^3$ coming from anaffine iterated function system whose coordinate subspace projections satisfy the strong separation condition. Its dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, which implies that sponges with a dimension gap represent a nonempty open subset of the parameter space. This work is joint with Tushar Das (Wisconsin - La Crosse).

Tuesday 19th April 2016 - Phase transitions in random networks

Speaker:  Olga Valba (National Research University, Moscow)

4:00PM in Room MATH-106

Abstract: We consider an equilibrium ensemble of large Erdős-Renyi topological random networks with fixed vertex degree and two types of vertices, black and white, prepared randomly with the bond connection probability p. The network energy is a sum of all unicolor triples (either black or white), weighted with chemical potential of triples μ. Minimizing the system energy, we see for some positive μ the formation of two predominantly unicolor clusters, linked by a string of Nbw black-white bonds. We have demonstrated that the system exhibits critical behavior manifested in the emergence of a wide plateau on the Nbw(μ) curve, which is relevant to a spinodal decomposition in first-order phase transitions.

Wednesday 13th April 2016 - Sturmian measures in ergodic optimization

Speaker: Vasso Anagnostopoulou (Imperial College, London)

3:00PM in Room MATH-106

Tuesday 12th April 2016 - A first view on proof assistants

Speaker:  Thomas Eckl (University of Liverpool)

4:00PM in Room G16 (MAGIC)

Abstract: In the last decade proof assistants like COQ and HOL Light became useful instruments in more and more areas of mathematics. In particular, they were used to (re)prove the 4-Colour-Theorem and Kepler's Conjecture (on the packing of balls), with interesting consequences for the original proof strategies.

In this talk I will demonstrate how the COQ proof assistant works, by proving the irrationality of $\sqrt{2}$. On the way, I will discuss type theory as the mathematical foundation of proof assistants, and why the conclusions of COQ are safe. Finally, I will speculate on how machine learning might improve the efficiency of COQ as an instrument of mathematical research.

Wednesday 9th March 2016 - Fatou’s web and non-escaping endpoints

Speaker:  Vassiliki Evdoridou (Open University)

3:00PM in Room MATH-106

Abstract: Let f be Fatou’s function, that is, f(z)= z+1+ exp(-z). We show that the escaping set of f, which consists of all points that tend to infinity under iteration, has a structure known as a spider’s web. We discuss a consequence of this result concerning the non-escaping endpoints of the Julia set of f. More specifically, we prove that the set of non-escaping endpoints together with infinity form a totally disconnected set. Finally, we show how these techniques can be adapted in order to show that a similar result holds for some functions in the exponential family.

Tuesday 1st March 2016 - Birationally rigid pfaffian Fano 3-folds; what next?

5:00PM in Room MATH-104

Abstract: I will give an overview of the geometry of Fano 3-folds after Mori theory. After discussing past approaches to the classification, I will highlight why such attempts seem hopeless. Building on recent advances in the geometry of Fanos, I introduce a new viewpoint on the classification problem. A main emphasis will be given to the unpredicted behaviour of the first examples of non-complete intersection Fanos, discovered in a joint work with Takuzo Okada.

24th February 2016 - Straightening the square

Speaker:  Arnaud Chéritat (Toulouse)

3:00PM in Room MATH-106

17th February 2016 - Landing of rays and dreadlocks for transcendental entire functions

Speaker:  Lasse Rempe-Gillen (University of Liverpool)

3:00PM in Room MATH-106

Abstract: There is a famous theorem, which we shall call the "Douady-Hubbard landing theorem", which states that every repelling or parabolic periodic point of a polynomial  with bounded postcritical set can be accessed by a certain periodic curve (an  “external  ray”); conversely every such ray lands at a repelling or parabolic point. (I will give a  short introduction to these concepts and the result in my talk.) The theorem has been a cornerstone of polynomial dynamics, being central to the famous “Yoccoz puzzle” which  has  been used to study the local connectivity of the Mandelbrot set, among other things.

We prove a version of this result for transcendental entire functions with bounded postsingular sets. In the case where the function has finite order, our result implies again  that every repelling (or parabolic) periodic point is the landing point of a certain periodic “hair”. However, in the right formulation  our result applies even to functions where the Julia set contains no nontrivial curves, where the role of the hairs is taken on by objects we call “dreadlocks”. Furthermore, we more  generally  establish landing of hairs or dreadlocks at points in hyperbolic subsets of the Julia set. (This is joint work with Anna Miriam Benini.)

10th February 2016 - Fractal dimensions of an overlapping generalization of Barański Carpets

Speaker:  Leticia Pardo Simón (University of Liverpool)

Abstract: We will study the Hausdorff, packing and box dimension of a family of self-affine sets generalizing Barański carpets. More specifically we fix a Barański system and allow both vertical and horizontal random translations, while preserving the rows and columns structure. The alignment kept in the construction lets us give formulas for the fractal dimensions outside of a small set of exceptional translations. These formulae will coincide with those for the non-overlapping case, and thus provide us with examples where the box-counting and Hausdorff dimension do not necessarily agree. (This is joint work with Thomas Jordan).

Tuesday 12th January 2016 - Feynman periods: numbers and geometry

Speaker:  Dr Dmitry Doryn (Center for Geometry and Physics, Institute of Basic Science, Uni. Postech, Pohang)

4:00PM in Room MATH-104.

Abstract: I will speak on the Feynman periods, the values of Feynman integrals in (massless, scalar) phi^4 theory, from the number-theoretical perspective. Then I define a closely related geometrical object, the graph hypersurface. One can try to study the geometry of these hypersurfaces (cohomology, Grothendieck ring, number of rational points over finite fields) and to relate it to the periods. The most interesting results come out from the study of the c_2 invariant (on the arithmetical side).

Wednesday 16th December 2015 - Singularity Day Conference (sponsored by the LMS)

11:00- 17:30 Room G16 (MAGIC)

### Oleg Karpenkov (Liverpool) - Finite and infinitesimal flexibility of semidiscrete surfaces

In this talk we will discuss infinitesimal and finite flexibility for generic semidiscrete surfaces. These surfaces are combined of smooth ribbons, they are in a sense limit shapes of quadrilateral graphs. It turns out that a generic 2-ribbon semidiscrete surface has one degree of infinitesimal and finite flexibility, which leads to construction of curious flexible mechanisms. Generic n-ribbon surfaces with n>2 do not possess any flexibility at all. Addressing this, we show a necessary condition for infinitesimal flexibility of a 3-ribbon surface.

### Juan José Nuño Ballesteros (Valencia) - Equisingularity of map germs from a surface to the plane

Let (X,0) be an ICIS of dimension 2 and let f: (X,0) --> (C^2,0) be a map germ with an isolated instability. We look at the invariants that appear when X_s is a smoothing of (X,0) and f_s: X_s --> C^2 is a stabilization of f. We find relations between these invariants and also give necessary and sufficient conditions for a 1-parameter family F to be Whitney equisingular. As an application, we show that a family (X_t,0) is Zariski equisingular if and only if it is Whitney equisingular and the numbers of cusps and double folds of a generic linear projection do not depend on t.

### David Mond (Warwick) - Invariants of the disentanglement of a map-germ (C^3,0)--> (C^4,0)

This is a very concrete talk focusing on a single example. We calculate the ranks of homology groups of spaces associated with the disentanglement of a map germ (C^3,0) --> (C^4,0). The talk will therefore give an introduction to the geometry of such map-germs. Some of the spaces involved have non-isolated singularities, and a number of classical and not-so-classical techniques must be brought to bear. After using every technique we can think of, some key questions remain, and it seems that some new ideas are called for. We end by describing what we think is needed. (Work in progress by Isaac Bird and David Mond).

### Anna Pratoussevitch (Liverpool) - Traces and discreteness for certain subgroups of PU(2,1)

While discrete subgroups of the group PSL(2,R)=PU(1,1) of isometries of the real hyperbolic plane are classified, the discreteness of subgroups of the group PU(2,1) of isometries of the complex hyperbolic plane is not well understood. We discuss a class of subgroups of PU(2,1) generated by complex reflections and show some non-discreteness results by considering the traces of elements in the group.

### Victor Goryunov (Liverpool) - On planar caustics

We study local invariants of planar caustics, that is, invariants of Lagrangian maps from surfaces to R^2 whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of the caustics. Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space L of the Lagrangian maps.

We obtain a description of the spaces of the discriminantal cycles (possibly non-trivial) for the Lagrangian maps of an arbitrary surface, both for the integer and mod2 coefficients. For the majority of these cycles we find homotopy-independent interpretations which guarantee the triviality required.

As an application, we use the discriminantal cycles to establish non-contractibility of certain loops in L. (This is a joint work with Katy Gallagher).

Thursday 12th November 2015 - On Functions in the Speiser Class with One Tract

Speaker: Simon Albrecht (Stony Brook University)

Abstract: The Speiser class consists of all transcendental entire functions with finite singular set. A tract of an entire function f is a connected component of {z : |f(z)|>R} for some R>0. We construct functions in the Speiser class with a prescribed tract by using quasiconformal folding, a method introduced by C. Bishop in 2011.