# Sample PhD Projects in Singularity Theory

## General Interests

Geometry and topology of singular spaces, differential and complex hyperbolic geometry, categories.

### Professor Victor Goryunov

**1. Local first order invariants of smooth mappings**

Finite type invariants of smooth maps were introduced by Victor Vassiliev in his seminal works on knot invariants. So far, classifications of first order local invariants have been carried out for maps of surfaces to R^2 (Ohmoto-Aicardi) and into R^3 (myself), and for maps between 3-manifolds (myself, and Oset Sinha & Romero-Fuster). In the latter case, there are still quite a few questions on geometrical sense of the basic invariants. The next on the agenda are Lagrangian maps to R^3 and R^3.

**2. Local first order invariants of wavefronts in the 3-space**

This is similar to the 1st theme, but this time for Legendrian maps from R^2 to R^3, and in a variety of (co)orientation settings.

**3. Hyperbolic reflection groups as monodromy groups of functions with symmetry**

In his recent PhD dissertation, Joel Haddley considered cyclic symmetries of the 14 exceptional unimodular function singularities. (The strange duality observed by Arnold for the 14 singularities provided the first example of mirror symmetry.) The main result of J.Haddley’s thesis was a classification of all hyperbolic reflection groups arising as monodromy groups acting on the relevant subspaces in the vanishing homology. The next task is identification of the groups within the set of those known, and proving the analog of Looijenga’s theorem that an appropriate period map establishes an isomorphism between the base of a symmetric versal deformation of a function singularity and the orbit space of the reflection group.

### Dr Oleg Karpenkov

**1. Toric singularities and lattice geometry**

It is well-known that congruence classes of integer polytopes are in one-to-one correspondence with projective toric varieties. Recent results in geometry of lattice polytopes provided global relations on toric singularities of complex toric surfaces. In particular, for surfaces of the Euler characteristic 3, the complete description of possible triplets of singularities was obtained. There are several directions for further research. First, one should study the cases of greater Euler characteristic. Secondly, to investigate the situation with three-dimensional projective complex toric varieties. The investigation will involve tools of lattice geometry, some basic knowledge of algebraic geometry is desirable.**2. Algebraic properties of qubic extensions with similar tori decompositions**

Recent progress in the study of periodic Klein-Voronoi continued fractions resulted in detection of special regular subfamilies in the space of all qubic extensions of the field of rational numbers. The goal of this project is to provide the systematic study of such regular subfamilies and to study the common properties of cubic extensions in the families. This subject touches, geometry of numbers, convex polytopes, generalized Euclidean algorithms.**3. Integer angles in integer simplices**

The classification of integer angles in integer triangles provides the classification of toric projective complex surfaces whose Euler characteristic equals 3. In the higher dimensional case the situation is reacher. For instance in three-dimensional case one should classify both the angles at vertices and the angles between faces. We consider this question as a starting point in the research in this subject. The challenges of this area are the four-dimensional White's theorem on empty lattice polytopes and several versions of lonely runner problem.**4. Maximal commutative group approximation**

How to find "best rational approximations" of maximal commutative subgroups of GL(n,R)? The first steps in the study of this problem were made in our recent paper with Prof. A.Vershik.

It contains both classical problems of Diophantine and simultaneous approximations as particular subcases but in generally is much wider. The main topic here is to develop effective methods to construct best approximations of maximal commutative subgroups. The most interesting case correspond to integer groups of algebraic extensions of rational numbers.

### Dr Anna Pratoussevitch

**1. Complex hyperbolic geometry**

The geometry of the complex hyperbolic spaces is very interesting as they exhibit some of the flexibility properties of the real hyperbolic spaces as well as some of the rigidity properties of the quaternionic hyperbolic geometry. In the 1970s work of Mostow led to the celebrated discovery of non-arithmetic complex hyperbolic lattices. In recent years the interest in the area has been revived through the work of W. Goldman, R. Schwartz, J. Parker and others. There are many open questions about discreteness of groups of isometries of the complex hyperbolic spaces, for example about complex hyperbolic triangle groups, i.e. groups generated by 3 complex reflections. The aim of the project is to study geometric and arithmetic properties of such groups via fundamental domain constructions, trace formulas and trace rings.

**2. Real forms of Higher spin bundles on Riemann surfaces**

Complex functions that appear in physics usually have real origins. Objects are usually characterized by real parameters (impulse, energy etc.) and relations are described by real functions of the parameters. Invariant mathematical structures that describe these properties are real forms of complex functions. Properties of real forms reflect hidden properties of the complex functions. The systematic study of properties of real forms of complex functions and complex manifolds only started in the last decades of the 20th century. A real form of a complex algebraic curve is a pair that consists of the complex algebraic curve and an anti-holomorphic involution on it. For example, if a complex algebraic curve is given by the equation F(x,y)=0, the complex conjugation of both components x and y induces an anti-holomorphic involution. The involution acts on all structures connected with the Riemann surface, such as complex vector bundles. The aim of the research project is to develop a theory of real forms of line bundles on Riemann surfaces with cone points and to apply this theory to classify real forms of Gorenstein quasi-homogeneous surface singularities.

### Professor Aleksandr Pukhlikov

See above under Algebraic Geometry.

### Dr Jon Woolf

**1. Studying the homotopy theory of stratified spaces**

In order to detect properties of a stratification of a space one needs to modify traditional homotopy theory. There are two approaches, either considering paths which 'wind outwards from deeper strata' or paths which are transversal to all strata. Both theories are quite new and there are many interesting projects, both theoretical and example-based. For instance a recent PhD student studied the transversal homotopy theory of spheres (stratified by a marked point and its complement), which turns out to be very closely related to the study of tangles and to Baez-Dolan's Tangle Hypothesis.

**2. Studying spaces of Bridgeland stability conditions on a triangulated category**

To any triangulated category (for instance a derived category of modules for an algebra or of sheaves) one can associate a complex manifold of stability conditions. The geometry of this reflects many algebraic properties of the triangulated category. For example there is a stratification of the space whose combinatorial structure is closely related to that of the poset of t-structures in the category. A possible project would be to investigate this relationship in specific examples (coming from quiver algebras and/or constructible sheaves on simple stratified spaces).

**3. Describing the structure of Witt groups of perverse sheaves with applications to signature formulae for singular spaces**

The category of perverse sheaves on a stratified space (or a space with a suitable class of stratifications) has a very rich structure relating to, amongst other things, stratified Morse theory and intersection cohomology. A natural invariant to consider is its Witt group, which is generated by non-degenerate symmetric bilinear forms (in the category of perverse sheaves). From this one can extract information about signatures and L-classes of the underlying spaces.