# Seminars & Events for Department Colloquium

##### Contact structures in dimension 3 and the Seiberg-Witten equations

I hope to give some indication of how the Seiberg-Witten equations are used to study the dynamics of vector fields on 3-dimensional manifolds. One result of this research is a proof of Alan Weinstein's conjecture about the existence of closed integral curves of the Reeb vector field for a contact 1-form.

##### Projective geometry on manifolds

Rich classes of geometric structures on manifolds are defined by coordinate atlases taking values in a fixed homogeneous space. The existence and classification of such structures leads to a moduli space, which itself is modelled on the algebraic variety of representations of the fundamental group in the automorphism group of the geometry. Topological symmetries lead to actions of mapping class groups on the moduli spaces, whose dynamics reflects the topology and the geometry. This talk will present various examples of the general classification problem, in dimensions 2 and 3.

##### Volume of polytopes, operator analogues, and Arthur's trace formula

There are two ways (among many others) to compute the volume of a (convex) polytope. One using a formula of Brion and another using an argument of P. McMullen and R. Schneider. The ensuing identity suggests a non-commutative generalization which we can currently prove for Coxeter zonotopes (e.g. a permutahedron). This algebraic equality plays a role in Arthur's trace formula. This has applications to spectral asymptotics of locally symmetric spaces. No prior knowledge of these subjects is assumed. Joint work with Tobias Finis and Werner Muller.

##### Finite and infinite representations of surface groups and cross ratios

In this talk, I explain how cross ratios—special functions of four arguments—on the boundary at infinity of surfaces groups describe both finite (in SL(nR)) and infinite (in a group related to diffeomorphisms of the circle) dimensional representations of surface groups. If time permits, I will explain what are the generalisation of McShane's identity in that context as well as some conjectural picture relating these representations to complex analysis.

##### Integral Apollonian circle packings

Apollonian circle packings are infinite packings of circles, constructed recursively from an initial configuration of four mutually touching circles by adding circles externally tangent to triples of such circles. Configurations of four mutually touching circles were studied by Descartes in 1643. If the initial four circles have integer curvatures, so do all the circles in the packing. If in addition the circles have rational centers so do all the circles in the packing. Why? This talk describes results in geometry, group theory and number theory arising from such packings. (This is joint work with Ron Graham, Colin Mallows, Allan Wilks, and Catherine Yau.)

##### The harmonic mean curvature flow of a 2-dimensional hypersurface

The harmonic mean curvature flow is the flow that moves a hypersurface embedded in $R^3$ by the speed given by a ratio of the Gauss and the mean curvature of the given surface in the direction of its normal. It is a fully nonlinear, weakly parabolic equation, degenerate at the points at which our hypersurface changes its convexity and fast diffusion when the mean curvature tends to zero. We prove a short time existence of such a flow in a nonconvex case. We also prove that if the mean curvature does not go to zero, the flow becomes strictly convex at some time and shrinks to a round point.

##### Chaoticity of the Teichmüller flow

A non-zero Abelian differential on a compact Riemann surface determines an atlas, outside the singularities, whose coordinate changes are translations. The vertical flow with respect to this translation structure generalizes the genus one notion of rational and irrational flows on tori. A fundamental tool in the understanding of the dynamics of vertical flows is the Teichmüller flow (acting on the moduli space of Abelian differentials), regarded as a renormalization operator. The chaotic nature of the dynamics of the Teichmüller flow has been a much researched topic, and currently it is known that it displays exponential decay of correlations.

##### Entropy and the localization of eigenfunctions

We study the behaviour of the eigenfunctions of the laplacian, on a compact negatively curved manifold, and for large eigenvalues. The Quantum Unique Ergodicity conjecture predicts that the probability measures defined by these eigenfunctions should converge weakly to the Riemannian volume. We prove an entropy lower bound on these probability measures, which shows for instance that it is difficult for them to concentrate on closed geodesics.

##### Algebraic cobordism: applications and perspectives

We will survey our theory, with F. Morel, of algebraic cobordism. This is the algebraic analog of complex cobordism, and may be viewed as a refinement of the Chow ring, replacing algebraic cycles with algebraic manifolds. We will discuss its relation with the Chow ring and the Grothendieck group of coherent sheaves, with applications to Riemann-Roch and degree formulas (used in the proof of the Bloch-Kato conjecture). With R. Pandharipande, we have given a simple description of the relations defining algebraic cobordism, the so-called double point cobordism; we will discuss applications this has had to Donaldson-Thomas theory. Finally, we will discuss the relation of our geometric theory with a more sophisticated version defined using motivic homotopy theory.

##### Characters of finite Chevalley groups and categorification

An important branch of representation theory studies representations of reductive groups over finite fields, such as $GL(n,F_q), Sp(2n,F_q)$ etc. A deep theory due mostly to Lusztig and Shoji provides a classification of irreducible representations and a formula for their characters in terms of certain algebro-geometric objects called character sheaves. In a joint work with M. Finkelberg and V. Ostrik we establish some new nice features of the geometric objects, motivated by an attempt to find a conceptual explanation for the beautiful but somewhat mysterious results of Lusztig and Shoji.

##### The classical and quantum geometry of polyhedral singularities and their resolutions

Let $G$ be a finite subgroup of $SO(3)$. Such groups admit an ADE classification: they are the cyclic groups, the dihedral groups, and the symmetries of the platonic solids. The singularity $C^3/G$ has a natural Calabi-Yau resolution $Y$ given by Nakamura's $G$-Hilbert scheme. The classical geometry of $Y$ (its cohomology) can be described in terms of the representation theory of $G$. The quantum geometry of $Y$ (its quantum cohomology) can be described in terms of $R$, the ADE root system associated to $G$. This leads to an interesting family of algebra structures on the affine root lattice of R. Other aspects of the "quantum geometry" of $Y$ and $C^3/G$ (namely their Gromov-Witten and Donaldson-Thomas theories) are also governed by the root system $R$.

##### Paint-by-numbers: pattern formation in two-dimensional sheets of cells

One of the basic mechanisms responsible for the formation of three-dimensional organs relies on the regulated folding of epithelia (two-dimensional sheets of cells). This process is driven by the spatially nonuniform and dynamic distribution of multiple chemical components (products of gene expression) across the epithelia that prepare for folding. Some of the key questions in this class of biological problems are related to the total number of involved genes, the diversity and dynamics of their expression patterns, and the mechanisms of pattern formation. I will present the results of our experimental and computational work that explores these questions during the formation of an elaborate three-dimensional structure (the fruit fly eggshell).