# Past Departmental Colloquium

## 2014/2015

Talk #19

Time and location: May 15th 2015, 4:30, Room G16, Department of Mathematical Sciences

Title: Automated reasoning for unknot detection

Abstract: We explore the application of automated reasoning techniques to unknot detection, a classical problem of computational topology. We adopt a two-pronged experimental approach, using a theorem prover to try to establish a positive result (i.e. that a knot is the unknot), whilst simultaneously using a model finder to try to establish a negative result (i.e. that the knot is not the unknot). The theorem proving approach utilises equational reasoning, whilst the model finder searches for a minimal size counter-model. We present and compare experimental data using the involutary quandle of the knot, as well as comparing with alternative approaches. The talk is based on the paper [1] and some more recent results will be mentioned as well. [1] Andrew Fish, Alexei Lisitsa: Detecting Unknots via Equational Reasoning, I: Exploration. CICM 2014: 76-91; arxiv version: http://arxiv.org/pdf/1405.4211v1.pdf

Speaker: Dr. A.Lisitsa (University of Liverpool, Department of Computer Sciences)

Talk #18

Title: 2-generator groups acting on the complex hyperbolic space

Abstract: We consider the differential equation $w'' + Aw = 0$ and study the question when there are two linearly independent solutions with ”few” zeros. After reviewing the results for a polynomial coefficient $A$ we discuss various results due to Bank, Laine and others dealing with the case of a transcendental entire coefficient $A$. We then discuss the disproof of a conjecture of Bank and Laine on the topic. This is joint work with Alexandre Eremenko.

Speaker: Prof. W.Bergweiler (Kiel University)

Talk #17

Time and location: May 1st 2015, 4:00, Room G16, Department of Mathematical Sciences

Title: An excursion with beta-expansions

Abstract: In this talk we give an overview of the various properties of expansions in non-integer base. One important property is the non-uniqueness of the expansion. We give various deterministic and random algorithms that generate such expansions, and we highlight their ergodic properties. We end the talk with an amusing application in $A/D$ conversion and fractal geometry

Speaker: Prof. K.Dajani (University of Utrecht)

Talk #16

Time and location: March 27th 2015, 4:00, Room G16, Department of Mathematical Sciences

Title: 2-generator groups acting on the complex hyperbolic space

Abstract: In 1933 Lehmer formulates the following problem: does there exist a constant $c>0$ such that the Mahler measure M$(\alpha)$ of any nonzero algebraic integer $\alpha$, not being a root of unity, satisfies M$(\alpha)\geq 1+c$. The Conjecture of Lehmer claims that it is the case. To test it, many families of algebraic numbers tending to 1 have been considered. It is a limit problem and a problem of minoration of M (or of the height for elliptic curves or abelian varieties). Another open limit problem is the characterization of the first derived set of the set T of Salem numbers. A first conjecture of Boyd claims that the union of S ~$\cup$~ T of the sets of Pisot numbers and Salem numbers is closed. A second conjecture of Boyd claims that the first derived set of the set of Salem numbers is the set of Pisot numbers. To each real algebraic number $\beta>1$ is associated the Artin-Mazur zeta function of the beta-transformation $\zeta_{\beta}(z)$, which arises from the R'enyi-Parry numeration system. Recent properties of $\zeta_{\beta}(z)$, deeply linked to heights and studies of asymptotic lacunarity in $\beta$-expansions of 1 by Diophantine Approximation techniques, will be recalled. If $(\beta_i)$ is a convergent sequence of algebraic numbers, a fundamental question is to know how to give a sense to ``$\lim_i \zeta_{\beta_i}$" and whether such limit dynamical zeta function may bring solutions or a new light to these questions, to the arithmetics of the limit point $\beta = \lim_i \beta_i$. In the context of a recent theorem of Flammang which solves a Conjecture of C. Smyth on height one trinomials, we will take the example of a family F of Perron numbers, which tends to 1, dominant roots of trinomials of height 1, non reciprocal, of small Mahler measure. We will show that the asymptotic expansions (of Poincar'e) of the poles of the dynamical zeta functions $\zeta_{\beta_i}(z)$ allow to obtain the asymptotic expansion of the Mahler measure and to give a direct proof that the conjecture of Lehmer is true for the family F.

Speaker: Prof. J.-L.Verger-Gaugry (University of Savoie Mont Blanc and Institut Fourier)

Talk #15

Time and location: March 13th 2015, 4:00, Room G16, Department of Mathematical Sciences

Title: 2-generator groups acting on the complex hyperbolic space

Abstract: In this talk, I will expose a classification of 2-generator groups in SU(2;1) that relies on the use of algebraic invariants such as traces. These groups appear as a natural generalisation of sub-groups of PSL(2;R), which act by isometries on the Poincare disc. The talk will be elementary, and I will review the classical case of SL(2;R) before moving to complex hyperbolic space.

Speaker: Dr. P.Will (Institute Fourier, Grenoble)

Talk #14

Time and location: March 6th 2015, 4:00, Room G16, Department of Mathematical Sciences

Title: The true cost of over-the-counter derivatives

Abstract: This talk will attempt to assess the true cost of over-the-counter derivatives for the global economy. It is based on 7 years of practical expert witness work for clients of toxic derivatives suing the issuing global banks in several EU jurisdictions. I will explain what these structured financial products are and sketch the mathematical and computational techniques used in their valuation. Using Bank of International Settlements Data, the per annum rent extracted from the global real economy by derivatives dealers is estimated to be about 3 trillion dollars, or about 5% of global GDP!

Speaker: Prof. M.A.H.Dempster (University of Cambridge and Cambridge Systems Associates Limited)

Talk #13

Time and location: February 27th 2015, 4:00, Room G16, Department of Mathematical Sciences

Title: Case studies in the mathematical modelling of collective cell behaviour in biology

Abstract: This talk will review three biological problems in which cell movement plays an important role: (i) acid-mediated cancer cell invasion, (ii) cranial neural crest cell invasion, (iii) intestinal crypt dynamics. Each example will illustrate a particular modelling technique, ranging from (i) coupled systems of partial differential equations, to (ii) a simple hybrid agent-based model, to (iii) an individual discrete-based model. It will be shown that, at least for simple cases, all these models have the same mathematical form for the cell density, namely a nonlinear partial differential equation where the specific form of nonlinearity in the diffusion term is determined by the hypothesised behaviour of cells at the discrete level.

Speaker: Prof. P. Maini (University of Oxford)

Talk #12

Time and location: February 20th 2015, 4:00, Room G16, Department of Mathematical Sciences

Title: Challenges in stochastics and finance: the case of dependence dynamics in credit modelling

Abstract: We introduce derivatives markets, discussing their current size and the history of derivatives mathematical models. We briefly illustrate the Nobel awarded option pricing paradigm of Black, Scholes and Merton. We explain which assumptions of the framework have been superseded by market developments over the years. We present as a fundamental example the case of credit derivatives, and Collateralized Debt Obligations (CDOs) in particular, whose valuation poses challenging modeling problems related to systemic risk scenarios and extreme losses. We analyze the industry models for such products pre- and in-crisis. We show that poor mathematical treatment of clustering and modes in the tail of the loss distribution may lead to inaccurate valuation, both pre-and especially in crisis. As a solution, we propose a stochastic loss model originally introduced by us in 2006, whose arbitrage-free dynamics leads naturally to clustering in the distribution tail. We illustrate the calibration of such Generalized Poisson Loss model in 2006-2010. We draw conclusions n the way research and crises may be poorly described and investigated by mainstream press and media. We finally enlarge he picture and comment on current challenges for Mathematical Finance, including the recent developments in nonlinear valuation, requiring semi-linear PDE or FBSDEs theory.

Speaker: Prof. D. Brigo (Imperial College London)

Talk #11

Time and location: February 13th 2015, 4:00, Room G16, Department of Mathematical Sciences

Title: Robust Stability and Boundedness of Nonlinear Hybrid Stochastic Differential Delay Equations

Abstract: One of the important issues in the study of hybrid stochastic differential delay equations (SDDEs) is the automatic control, with consequent emphasis being placed on the asymptotic analysis of stability and boundedness. In the study of asymptotic properties, the robust stability has received a great deal of attention. The theory of robust stability shows how much perturbation a given stable hybrid SDDE can tolerate so that its perturbed system remains stable. Almost all results so far on the robust stability require that the underlying SDDEs be either linear or nonlinear with linear growth condition. However, little is known on the robust stability of nonlinear hybrid SDDEs without the linear growth condition, which is one of the key topics in this paper. The other key topic is the robust boundedness. The aim here is to answer the question: how much perturbation can a given asymptotically bounded hybrid SDDE tolerate so that its perturbed system remains asymptotically bounded?

Speaker: Prof. L.Hu (Donghua University)

Talk #10

Time and location: February 6th 2015, 4:00, Room G16, Department of Mathematical Sciences

Title: Partial Differential Equations of Mixed Type: From Mechanics to Geometry

Abstract: Many nonlinear partial differential equations arising in science and engineering naturally are of mixed hyperbolic-elliptic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear equations. Important examples include shock reflection-diffraction problems in fluid mechanics (the Euler equations), isometric embedding problems in differential geometry (the Gauss-Codazzi-Ricci equations), among many others. In this talk we will present natural connections of nonlinear partial differential equations of mixed type with these longstanding problems from mechanics to geometry, and will discuss some recent developments in the analysis of these nonlinear equations through the examples with emphasis on identifying/developing mathematical approaches, ideas, and techniques to deal with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed.

Speaker: Prof. G.-Q.G.Chen (University of Oxford)

Talk #9

Time and location: December 5th 2014, 4:00, Room G16, Department of Mathematical Sciences

Title: Coxeter groups, quiver mutations and hyperbolic manifolds.

Abstract: Mutations of quivers were introduced by Fomin and Zelevinsky at the beginning of 2000's in the context of cluster algebras. Since then, mutations appear (sometimes completely unexpectedly) in various domains of mathematics and physics. Using mutations of quivers, Barot and Marsh recently constructed a series of presentations of Coxeter groups. I will discuss a geometric interpretation of this result: it occurs that these presentations give rise to a construction of geometric manifolds with large symmetry groups, in particular to some hyperbolic manifolds of small volume with proper actions of Coxeter groups.

Speaker: Dr. P.Tumarkin (Durham University)

Talk #8

Time and location: November 21st 2014, 4:00, Room MATH-105, Department of Mathematical Sciences

Title: Geometry of hyperbolic manifolds and the discreteness problem for Schottky groups

Abstract: Here is an old problem: given two matrices in SL(2,C), when is the group they generate free and discrete? If it is, the group is called a Schottky group. We shall describe how this problem can be solved in a special symmetrical case using Keen-Series pleating rays, based on ideas about the geometry of hyperbolic 3-manifolds from the school of Thurston. Along the way we will explain some of the many remarkable advances in this field which have been made over the last 20 years.

Speaker: Prof. C. Series (University of Warwick)

Talk #7

Time and location: November 12th 2014, 4:00, Room MATH-106, Department of Mathematical Sciences

Title: Seeing the wood for the trees with mathematical modelling

Abstract: The eye is a complex organ and, as such, represents a rich source of fascinating problems for applied mathematicians, interested in understanding its anatomy and physiology and how these change during ageing and in disease. In this talk attention will focus on photoreceptors, light-sensing retinal cells whose length fluctuates on a daily basis. I will start by presenting a simple mathematical model, formulated as a free boundary problem, which can be used to determine whether the observed fluctuations in healthy photoreceptors may be attributed to changes in oxygen demand during periods of light and dark. I will then focus on retinitis pigmentosa, a degenerative disease that targets the photoreceptors and causes progressive loss of visual function. I will present a second mathematical model developed in order to determine whether hyperoxia, exposure to elevated oxygen levels, may be responsible for the patterns of photoreceptor degeneration associated with retinitis pigmentosa. If time permits, I will also explain how we are using mathematical modelling to investigate how visual cues may regulate the growth of the eyeball during childhood.

Speaker: Prof. H. Byrne (University of Oxford)

Talk #6

Time and location: November 7th 2014, 4:00, Room G16, Department of Mathematical Sciences

Title: Emergent geometry from strongly coupled quantum dynamics

Abstract: I will describe to what extent we understand the dynamical origin of higher dimensional space-time geometry from quantum field theory computations in a particular example of the gauge/gravity duality, namely the large N limit of maximally supersymmetric Yang Mills theory. I will go in particular through the description of extended geometric objects from field theory setups. These extended object are interpreted as a special collection of D-branes and the shortest strings stretching between them in the dual geometry.

Speaker: Prof. D. Berenstein (UC Santa Barbara)

Talk #5

Time and location: October 31st 2014, 4:00, Room G16, Department of Mathematical Sciences

Title: Particle Physics in the Multiverse

Abstract: Is our universe a single entity or part of a multiverse? Can the Standard Model of Particle Physics be derived uniquely from a fundamental theory, or is it just one of many possibilities? Insights from inflation and string theory in the past three decades suggests that it is not unique, and that the number of alternatives may be enormous. This would have important implications for our attempts at understanding the Standard Model and the expectations for “new physics”. It also implies that we cannot ignore an important bias in observations: the existence of observers. This leads inevitably to a controversial issue: the anthropic principle.

Speaker: Prof. B. Schellekens (NIKEF)

Talk #4

Time and location: October 24th 2014, 4:00, Room G16, Department of Mathematical Sciences

Title: Long Wave Theories for Thin and Periodic Structures

Abstract: The connection between long-wave asymptotic procedures for thin and periodic elastic structures is discussed. In particular, it is claimed that the well-established low-frequency limit in plate and shell theories is similar in a sense to the conventional homogenisation routine. It is also shown that the high-frequency long-wave limit earlier developed for thin walled structures can be applied to periodic continua and discrete lattices. The latter limit is oriented to analysis of micro-scale dynamic phenomena and finds important applications in modelling of meta-materials. As an illustration, common features of Rayleigh-Lamb and Floquet-Bloch spectra are revealed. Similarity of localised phenomena in non-homogeneous thin and periodic structures is also demonstrated.

Speaker: Prof. Ju. Kaplunov (Keele University)

Talk #3

Time and location: October 17th 2014, 4:00, Room G16, Department of Mathematical Sciences

Title: Models of biological pattern formation: from elementary steps to complex structures

Abstract: Development of higher organisms starts with a single cell and leads to an overwhelming complexity of differentiated cells and tissues. It will be shown that mathematically-based modeling provides an important tool to understand the underlying principles. Using classical experiments we have proposed molecular realistic interactions that account for elementary steps and the observed regulatory properties. Models will be outlined and illustrated by computer simulations that describe pattern formation in initially homogeneous tissue, the formation the orthogonal body axes, regeneration of removed parts, stable cell differentiation, segmentation and the initiation of substructures such as legs and wings. Crucial are non-linear self-enhancing reactions that are coupled with antagonistic reactions of longer range. The resulting patterns are very robust. Complex patterns result from linking many pattern-forming reactions; each pattern, once formed, generates the precondition and asymmetry for the subsequent patterning step. In many cases normal patterns can be restored even after severe perturbation, in agreement with many observations. Meanwhile many of these models found direct support by molecular-genetic observations. The principles we found also have counterparts in many other processes, from the formation of sand dunes to social interactions.

Speaker: Prof. H. Meinhardt (Max Plank Institute for Developmental Biology)

Talk #2

Time and location: October 10th 2014, 4:00, Room G16, Department of Mathematical Sciences

Title: Density-based cluster analysis: an interface between Statistics, Differential Topology and Dynamical Systems

Abstract: Despite its popularity, the investigation of some theoretical aspects of clustering has been relatively sparse. One of the main reasons for this lack of theoretical results is surely the fact that, unlike the situation with other statistical problems such as regression or classification, for some of the clustering methodologies it is difficult to specify the population goal to which the data-based clustering algorithms try to get close. In this talk we investigate the theoretical foundations of clustering by focusing on two main objectives: first, to provide an explicit formulation for the ideal population goal of density-based clustering, which understands clusters as regions of high density (here, Morse theory plays a crucial role); and second, to present two new risk functions, applicable to any clustering methodology, to evaluate the performance of a data-based clustering algorithm with respect to the ideal population goal. In particular, it is shown that only mild conditions on a sequence of density estimators are needed to ensure that the sequence of clusterings that they induce is consistent.

Speaker: Prof. J.E. Chacon (University of Extremadura)

Talk #1

Time and location: October 3rd 2014, 4:00, Room G16, Department of Mathematical Sciences

Title: Expectiles as risk measures

Abstract: In this talk we consider expectiles as risk measures. Expectiles are a well-known concept in statistics, but seem to be hardly known in risk analysis. They are generalizations of the expectation in quite a similar fashion as quantiles (alias value at risk) are generalizations of the median. Thus it is quite natural to investigate their properties as risk measures. In this talk we show interesting properties of these risk measures, including coherence and their dual representation as worst case expectation. We also compare their properties to properties of other popular coherent risk measures like conditional value at risk.

Speaker: Prof. A.Mueller (University of Siegen)

## 2013/2014

Talk #25

Time and location: 9th May 2014, Room 105, Department of Mathematical Sciences

Title: Stochastic approximations in life annuity portfolios

Abstract: In portfolios of life annuity contracts, the payments made by an annuity provider (an insurance company or a pension fund) are driven by the random number of survivors.

This talk discusses approximations for the present value of the payments made by the annuity provider. These approximations account not only for systematic longevity risk but also for the diversifiable fluctuations around the unknown life table. They provide the practitioner with a useful tool avoiding the problem of simulations within simulations in, for instance, Solvency 2 calculations, valid whatever the size of the portfolio. This talk is based on research with Professors J. Dhaene, E. Frostig, S. Haberman and A. Renshaw, as well as on work in progress with S. Gbari.

Speaker: Prof.M. Denuit (Universit´e Catholique de Louvain, Belgium)

Talk #24 (Special talk)

Time and location: 2nd May 2014, Room 105, Department of Mathematical Sciences

Title: Exponential Stability of Compactly Coupled Wave Equations with Delay terms in the Boundary Feedbacks

Abstract: We consider a linear system of compactly coupled wave equations with Neumann feedback controllers that contain delay terms. First, we prove under some assumptions that the closed-loop system generates a C_0 semigroup of contractions on an appropriate Hilbert space. Then, under further assumptions, we show that the closed-loop system is exponentially stable. This result is obtained by introducing a suitable energy function and by using an observability estimate.

Speaker: Prof. Salah-Eddine Rebiai (University of Batna, Algeria)

Talk #23

Time and location: 30th April 2014, Room 105, Department of Mathematical Sciences

Title: Modelling plant cell and tissue growth

Abstract: Plant growth typically occurs through the coordinated anisotropic expansion of plant cells. Growth is regulated by hormones and is driven by high intracellular pressures generated by osmosis. This machinery allows a plant primary root, for example, to penetrate soil in a direction guided by gravity, while seeking out nutrients and avoiding obstacles. Multiscale computational models provide a powerful framework for understanding such processes. I will describe the biomechanical aspects of a model for root gravitropism that incorporates descriptions of cell walls as fibre-reinforced viscoelastic polymer networks and adopts upscaling approaches to efficiently describe growth of multicellular tissues.

Speaker: Prof. O. Jensen (University of Manchester)

Talk #22

Time and location: 4th April 2014, Room 105, Department of Mathematical Sciences

Title: The Cramér-Lundberg and the dual risk models: Ruin, dividend problems and duality feature

Abstract: In the present paper we study some existing duality features between two very well known models in Risk Theory. The classical Cramér–Lundberg risk model with application to insurance, and the dual risk model with (some) financial application. For simplicity the former will be referred as the primal model. It has been of extensive treatment in the literature, it assumes that a given surplus process has constant deterministic gains (premiums) and random loses (claims) that come at random times. On the other hand, the latter, called as dual model, works in opposite direction, losses (costs) are constant and deterministic, and the gains (earnings) are random and come at random times. Sometimes this one is called the negative model. Similar quantities, with similar mathematical properties, work in opposite direction and have different meanings. There is however an important feature that makes the two models quite distinct, either in their application or in their nature: the loading condition, positive or negative, respectively. The primal model has been worked extensively and focuses essentially in ruin problems (in many different aspects) whereas the dual model has developed more recently and focuses on dividend payments. I most cases, they have been worked apart, however they have many connection points that allow us to use methods and results from one to another. Basically form the first to the second. Identifying the right connection, or duality, is crucial so that we transport methods and results. In the work by Afonso et al. (2013) this connection is first addressed in the case when the times between claims/gains follow an exponential distribution. We can easily understand that the ruin time in the primal has a correspondence to the dividend time in the latter. On the opposite side the time to hit an upper barrier in the primal model has a correspondence to the time to ruin in the dual model. Another interesting feature is the severity of ruin in the former and the size of the dividend payment in the latter.

Speaker: Prof. Alfredo D: Egídio dos Reis (ISEG Lisbon)

Talk #21

Time and location: 28th March 2014, Room 105, Department of Mathematical Sciences

Title: Homology of Curves and Surfaces in hyperbolic 3-manifolds

Abstract: By definition, every homologically trivial closed curve contained in a given manifold bounds a singular surface. In recent joint work with Liu, we have shown that if such a curve lives in a closed hyperbolic 3-manidold then it bounds an essential surface (a surface is essential if its fundamental group injects in the fundamental group of the 3-manifold). I will explain the background of this result and also present its applications. Also, I will talk about our other result that says that every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by a connected closed quasi-Fuchsian subsurface.

Speaker: Prof. Vladimir Markovic (University of Cambridge)

Talk #20

Time and location: 21st March 2014, Room 105, Department of Mathematical Sciences

Title: On McKean-Vlasov stochastic equations

Abstract: New existence and uniqueness theorems for McKean-Vlasov equations will be established, both weak and strong. One of the main tools is Girsanov's transformation of measures. One more basis is strong solutions of ordinary Ito's equations

Speaker: Prof. Alexandre Veretennikov (University of Leeds)

Talk #19

Time and location: 14th March 2014, 4:00, Room 105, Department of Mathematical Sciences

Title: A functional analytic approach to singular perturbation problems: a quasi-linear heat transmission problem in a periodic dilute two-phase composite

Abstract: This talk is dedicated to the analysis of boundary value problems on singularly perturbed domains by an approach which is alternative to those of asymptotic analysis and of homogenization theory. In particular, we consider a temperature transmission problem for a composite material which fills the Euclidean space. The composite has a periodic structure and consists of two materials. In each periodicity cell one material occupies an inclusion of size $\epsilon$, and the second material fills the remaining part of the cell. We assume that the thermal conductivities of the materials depend nonlinearly upon the temperature. We show that for $\epsilon$ small enough the problem has a solution, \textit{i.e.}, a pair of functions which determine the temperature distribution in the two materials. Then we analyze the behaviour of such a solution as $\epsilon$ approaches $0$.

Speaker: Prof. Massimo Lanza de Cristoforis (University of Padova)

Talk #18

Time and location: 7th March 2014, 4:00, Room 105, Department of Mathematical Sciences

Title: How the fish got its spots

Abstract: In 1952 Alan Turing proposed a mathematical framework for understanding certain chemical systems. Importantly, he demonstrated that through a counter-intuitive mechanism the chemical concentrations could be made to form patterns. He coined the term ‘morphogens’ for these chemicals, suggesting that they instructed animal cells to adopt different fates, depending on the concentration of morphogen to which they were exposed. Thus, a new field of research was born, leading to novel mathematical developments and new biological experiments that continue to hunt for a physical example of Turing's mechanism, where these morphogens can be identified. Here, we briefly review sixty years of research inspired by his seminal paper.

Speaker: Dr. T. Woolley (University of Oxford)

Talk #17

Time and location: 4th March 2014, 2:00, Room 104, Department of Mathematical Sciences

Title: The quasiconformal universe of Sierpinski carpets

Abstract: Sierpinski carpets are self-similar fractals that appear in many areas of mathematics. For example, these fractals arise as Julia sets of rational functions or as limit sets of Kleinian groups. While the topology of Sierpi'nski carpets has been well understood for a long time, a deeper insight into their quasiconformal geometry has been gained only recently. This is particularly relevant and interesting for many questions in dynamics and geometry, but many problems remain open. In my talk I will give an introduction to this subject and will report on some recent developments.

Speaker: Prof. Mario Bonk from the University of California, Los Angeles

Talk #16

Time and location: 28th February 2014, 4:00, Room 105, Department of Mathematical Sciences

Title: The large-scale geometry of random groups

Abstract: If we consider a group presentation with (say) two generators, and one relation of a certain length, what properties do we expect the group to have? As the length and/or number of relations grows, what properties become more and more likely? The study of "random groups" is concerned with answering such questions. In this talk we will survey some of the results in this area, focussing on Gromov's density model for random groups. We'll be particularly interested in the large-scale geometry of these groups.

Talk #15

Time and location: 21st February 2014, 4:00, Room 105, Department of Mathematical Sciences

Title: Save for the bad times or consume as long as you have ? Worst-case consumption

Abstract: Worst-case portfolio optimization has been introduced in Korn and Wilmott (2002) and is based on distinguishing between random stock price fluctuations and market crashes which are subject to Knightian uncertainty. In normal times, the stock price dynamics follow a classical model such as the Black-Scholes type one or any other desired model. At the crash time, the stock prices fall by an individual unknown percentage of which we only know upper bounds, but make no distributional assumption. Due to the absence of full probabilistic information, a worst-case portfolio problem is considered that will be solved completely. The corresponding optimal strategy is of a multi-part type and makes an investor indifferent between the occurrence of the worst possible crash and no crash at all. We will consider various generalizations of this setting and - as a very recent result - will in particular answer the question ”Is it good to save for bad times or should one consume more as long as one is still rich?” Surprisingly, this question has two answers that are both very reasonable from an economic point of view. The mathematical methods used are classical stochastic control methods combined with abstract controller-stopper games and indifference arguments.

Talk #14

Time and location: 14th February 2014, 4:00, Room 105, Department of Mathematical Sciences

Title: Learning what’s best

Abstract: Game theory is widely used in disciplines ranging from Economics to Behavioural Biology to model interactions between competing players. Although it is easy to define the standard ‘Nash equilibrium’ solution for such games, it is less clear how players might learn to play their best policy. Here we look at the theory and application of a variety of simple reinforcement rules for learning optimal behaviour, and explore their convergence through the convergence of the corresponding dynamical systems. The rules we consider vary in the amount of differentiation between participants and the amount of information available to players – some require the players to have full knowledge of all aspects of the process, while in others the strategies learned will converge to a Nash distribution without players knowing the pay-off structure of the game or even knowing that they are playing a game.

Speaker: Dr. E. J. Collins (University of Bristol)

Talk #13

Time and location: 7th February 2014, 4:00, Room 105, Department of Mathematical Sciences

Title: The superconformal bootstrap for structure constants

Speaker: Prof.Luis Fernando Alday (University of Oxford)

Talk #12

Time and location: 31st January 2014, 4:00, Room 105, Department of Mathematical Sciences

Title: Computation of R-estimators

Abstract: In this talk, we discuss a simple iterative algorithm for computing R-estimates of the parameters of the linear regression models. The algorithm can be applied routinely to compute R-estimates based on any score function (both bounded and unbounded but square-integrable). We apply this to some well-known datasets and can identify outliers which would not have been detected using least squares.

Speaker: Dr. Kanchan Mukherjee (University of Lancaster)

Talk #11

Time and location: 13th December 2013, 4:00, Room 105, Department of Mathematical Sciences

Title: Rigidity of moduli of pointed curves in arbitrary characteristic and applications

Abstract: Let M_{g,n} be the quasiprojective variety of nonsingular genus g curves with n distinct marked points; it has a natural compactification \bar M_{g,n} parametrizing nodal curves, due to Deligne and Mumford. Both are defined over the integers, hence over an arbitrary field k. Hacking showed that \bar M{g,n} is rigid over an algebraically closed field of char. $0$. In joint work with A. Massarenti (IMPA), we show that \bar M_{0,n} is rigid in any characteristic. As an application, we show that (over an arbitrary field) for $n\ge 5$ the only automorphisms of $\bar M_{0,n}$ are the permutations of the points, extending results by Bruno and Mella in char. $0$. If time allows, we will discuss how the rigidity of any $\bar M_{g,n}$ implies the rigidity of all the (smaller) alternative compactifications of $\bar M_{g,n}$ defined by Hassett.

Speaker: Prof. B. Fantechi (SISSA)

Talk #10

Time and location: 11th December 2013, 4:00, Room 105, Department of Mathematical Sciences

Title: Active matters

Abstract: Active materials, such as cells and microorganisms, create their own energy. These systems naturally operate out of thermodynamic equilibrium and hence provide a testing ground for theories of non-equilibrium statistical physics. They also have potential as micro-machines, creating mechanical work from chemical energy. Because of their size bacteria and other tiny swimmers move at low Reynolds number in a world dominated by viscosity. As a result they need special strategies for swimming and for stirring the surrounding fluid. We are interested in asking questions such as: Are there particular swimming strokes which help a bacterium to move in a viscoelastic environment such as the gut? Have microswimmers evolved to be good stirrers, agitating the surrounding fluid to increase their nutrient supply, or to be stealth swimmers, producing only a local disturbance as they swim?

A dense suspension of bacteria shows turbulent-like behaviour, with a velocity field that is continuously changing, with swirls and jets forming and decaying. Very similar flow fields are seen in other active systems, on widely varying length and time scales, from suspensions of microtubules and molecular motors, to agitated granular matter, schools of fish, and flocks of birds. Normally turbulence is a consequence of inertia, absent at low Reynolds numbers, and ‘active turbulence’ needs a different explanation. Currently there are indications that the formation and decay of topological defects might underlie the turbulent patterns in some active materials.

Speaker: Prof. J. Yeomans (University of Oxford)

Talk #9

Time and location: 29th November 2013, 4:00, Room 105, Department of Mathematical Sciences

Title: Set a thief to catch a thief: can we make use of parasites to control vector-borne diseases?

Abstract: Experiments and field trials have shown that the intracellular bacterium Wolbachia may be introduced into populations of the mosquito Aedes aegypti, the primary vector for dengue fever. In the absence of Wolbachia, a mosquito acquiring the dengue virus from an infected human enters an exposed (infected but not infectious) period before becoming infectious itself. A Wolbachia-infected mosquito that acquires dengue (i) may have a reduced lifespan, so that it is less likely to survive the exposed period and become infectious, and (ii) may have a reduced ability to transmit dengue, even if it has survived the exposed period. Wolbachia introduction has therefore been suggested as a potential dengue control measure. We set up a mathematical model for the system to investigate this suggestion and to evaluate the desirable properties of the Wolbachia strain to be introduced. We show that Wolbachia has excellent potential for dengue control in areas where R_0 is not too large. However, if R_0 is large, Wolbachia strains that reduce but do not eliminate dengue transmission have little effect on endemic steady states or epidemic sizes. Unless control measures to reduce R_0 by reducing mosquito populations are also put in place, it may be worth the extra effort in such cases to introduce Wolbachia strains that eliminate dengue transmission completely.

Speaker: Prof. N. Britton (University of Bath)

Talk #8

Time and location: 22nd November 2013, 4:00, Room 105, Department of Mathematical Sciences

Title: The role of non-explosiveness in Markov processes and parametrised Markov processes

Abstract: Many modern applications of Markov processes require unbounded jump intensities as a function of state. This allows for possible explosiveness of the process. Non-explosiveness guarantees stochasticity of the transition function of a conservative Markov process with values in a countable space. As a result, it can be used to classify functions for which an integral version of the Kolmogorov forward equation, the Dynkin formula, holds. The Dynkin formula plays a role in e.g. showing that solutions to the continuous time Poisson equation exist, as well as the uniqueness of these solutions in the Banach space of functions with a finite weighted supremum norm. A well-known sufficient condition for non-explosiveness is a the existence of a moment drift function, V say. This means that the function is unbounded off finite sets, and it satisfies QV<= cV, for some constant c, where Q denotes the generator of the Markov process. Interestingly enough, it can be shown that this condition is also necessary.

Also in the context of parametrised Markov processes, non-explosiveness plays an important role. It seems to be necessary to guarantee that continuity properties of the generator as a function of the parameter carry over to continuity of the associated transition function. This is e.g. required for showing convergence of performance measures of perturbed Markov processes as the perturbation parameter converges to 0. We will prove the various properties mentioned, as well as applications to optimal control problems.

Speaker: Dr. F. Spieksma (University of Leiden)

Talk #7

Time and location: 8th November 2013, 4:00, Room 105, Department of Mathematical Sciences

Title: Exponential growth rate for a singular linear stochastic delay differential equation

Abstract: First we briefly review sufficient criteria for the uniqueness of an invariant measure of a stochastic delay differential equation obtained in joint work with Martin Hairer and Jonathan Mattingly. Then, we study the very simple one-dimensional equation $dX(t)=X(t-1)dW(t)$ in more detail and establish the existence of a deterministic exponential growth rate of a suitable norm of the solution via a Furstenberg-Hasminskii-type formula.

Speaker: Prof. M.Scheutzow (Technische Universität Berlin)

Talk #6

Time and location: 1st November 2013, 4:00, Room 105, Department of Mathematical Sciences

Title: Models for the formation of a critical layer in water-wave propagation

Abstract: Two models are proposed for the formation of critical layers where initially none existed, described within the classical theory of water-wave propagation over constant vorticity. In one model, the pressure is controlled (mimicking the passage of a storm); in the other, the pressure decreases with height above the surface (although the physics is sound, this does not correspond to the typical air-over-water scenario). Formal asymptotic expansions are constructed, based on the fundamental parameters in the problem, producing a model that describes the appearance of stagnation points (on the top or bottom), critical levels and associated closed streamlines.

Speaker: Prof. Robin Johnson (University of Newcastle)

Talk #5

Time and location: 25th October 2013, 4:00, Room 105, Department of Mathematical Sciences

Title: The Coupon Collector Problem

Abstract: This classical problem in probability theory looks at how many (independent and identically distributed, equally likely) coupons need to be collected by a collector in order to get a complete set of n coupons. I will briefly review the classical problem and some related questions such as how many coupons does the collector only have one copy of? I will discuss new results for the classical problem and extending results to the case where the coupons are not all equally likely.

Speaker: Dr. Peter Neal (University of Lancaster)

Talk #4

Time and location: 18th October 2013, 4:00, Room 105, Department of Mathematical Sciences

Title: Dispersion in the presence of interfacial discontinuities

Abstract: This talk will focus on probability questions arising in the geophysical and biological sciences concerning dispersion in highly heterogeneous environments, as characterized by abrupt changes (discontinuities) in the diffusion coefficient. Some specific phenomena observed in laboratory and field experiments involving breakthrough curves (first passage times), occupation times, and local times will be addressed within a probabilistic framework largely founded on the Ito-McKean-Feller classic skew Brownian motion and Stroock-Varadhan martingale theory. This is based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, and Brian Wood at Oregon State University.

Speaker: Prof. E. Waymire (Oregon State University)

Talk #3

Time and location: 11th October 2013, 4:00, Room 105, Department of Mathematical Sciences

Title: Scheduling in systems with time-varying service rate

Abstract: We address the problem of developing a well-performing and implementable scheduler of heterogeneous users with wireless connection to the base station. The main feature of such real-life systems is that the quality conditions of the user channels are time-varying, which turn into the time-varying service rate. We assume that this phenomenon follows a Markovian law. We use constrained Markov decision processes to model it, solve its Langrangian relaxation to provide a mathematically grounded solution and propose its two practical approximations. One of them is a simple "index rule", which generalizes the well-known Smith's rule, which is optimal if service rates are constant. We illustrate the performance of the proposed schedulers and existing alternatives in a variety of simulation scenarios. We further discuss open problems in studying the performance of solutions to such non-work-conserving systems analytically.

Speaker: Dr. Peter Jacko (University of Lancaster and BCAM)

Talk #2

Time and location: 4th October 2013, 4:00, Room 105, Department of Mathematical Sciences

Title: The paradoxical world of the black box group theory

Abstract: The talk will describe a peculiar and perhaps unique phenomenon in mathematics: the so-called "black box" probabilistic algorithms of computational finite group theory. A typical object of black box methods is a group generated by several matrices of large size, say, 100 by 100, over a finite field. Individual elements of such a group can be easily manipulated by a computer; however, the size of the whole group is astronomical, and arguments leading to identification of the structure of the group are being de facto carried out in an infinite object. In my talk, I will focus on those aspects of the theory where finer structure of black box groups mimics that of connected compact Lie groups, bridging the gap between finite and infinite.

Speaker: Prof. A. Borovik (University of Manchester)

Talk #1

Time and location: 27th September 2013, 4:00, Room 105, Department of Mathematical Sciences

Title: Pathwise Stability of Degenerate Stochastic Evolutions

Abstract: For linear stochastic evolution equations with linear multiplicative noise, a new method is presented for estimating the pathwise Lyapunov exponent. The method consists of finding a suitable (quadratic) Lyapunov function by means of solving an operator inequality. One of the appealing features of this approach is the possibility to show stabilizing effects of degenerate noise. The results are illustrated by applying them to the examples of a stochastic partial differential equation and a stochastic differential equation with delay. In the case of a stochastic delay differential equation our results improve upon earlier results.

Speaker: Prof. J. Bierkens (Radboud Universiteit Nijmegen)