I will prove that the compactly supported cohomology of certain unitary or symplectic Shimura varieties at level Gamma_1(p^infty) vanishes above the middle degree. The key ingredients come from p-adic Hodge theory and studying the Bruhat decomposition on the Hodge-Tate flag variety. I will describe the steps in the proof using modular curves as a toy model. I will also mention an application to Galois representations for torsion classes in the cohomology of locally symmetric spaces for GL_n. This talk is based on joint work in preparation with D. Gulotta, C.Y. Hsu, C. Johansson, L. Mocz, E. Reineke, and S.C. Shih.

I will discuss a relation between conformal blocks, describing kinematics of a CFT, and integrable models of quantum-mechanical particles. I will show how the dependence of blocks on cross-ratios is encoded in equations of motion of a Calogero-Sutherland model and their dependence on conformal dimension and spin of the exchanged operator – in those of a relativistic Calogero-Sutherland model. Both are simultaneously controlled by an integrable connection generalizing 2d Knizhnik-Zamolodchikov equations. I will review how this connection, associated to representations of degenerate double affine Hecke algebra, comes from a q-deformed bispectrally symmetric setting.

Embryonic development is a self-organized process during which cells divide, interact, change fate according to a complex gene regulatory network and organize themselves in a three-dimensional (3D) space. Here, we model this complex dynamic phenomenon in the context of the acquisition of epiblast (Epi) and primitive endoderm (PrE) identities within the inner cell mass (ICM) of the preimplantation embryo in the mouse. The multiscale model describes cell division and biomechanical interactions between cells, as well as biochemical reactions inside each individual cell and in the extracellular matrix. We use the model to study the Epi and PrE lineage development and the appearance of a so-called salt-and-pepper pattern which the two lineages form.

In this talk we discuss modern mathematical models arising from modeling of tumor-immune system interactions. They are described as systems of ordinary differential equations of generalized Lotka-Volterra type. The corresponding vector fields are rational functions of dimension 2 or 3 containing many free parameters. The elementary qualitative theory can be applied to investigate the solutions of these equations. In particular, we derive conditions for global stability of certain equilibrium points using the Lyapunov functions method.

We consider the simple random walk on two types of tilings of the hyperbolic plane. The first by 2π⁄q-angled regular polygons, and the second by the Voronoi tiling associated to a random discrete set of the hyperbolic plane, the Poisson point process. In the second case, we assume that there are on average λ points per unit area.

In both cases the random walk (almost surely) escapes to infinity with positive speed, and thus converges to a point on the circle. The distribution of this limit point is called the harmonic measure of the walk.

I will show that the Hausdorff dimension of the harmonic measure is strictly smaller than 1, for q sufficiently large in the Fuchsian case, and for λ sufficiently small in the Poisson case. In particular, the harmonic measure is singular with respect to the Lebesgue measure on the circle in these two cases.

The proof is based on a Furstenberg type formula for the speed together with an upper bound for the Hausdorff dimension by the ratio between the entropy and the speed of the walk.

This is joint work with P. Lessa and E. Paquette.

I will discuss the physical background of compressible Euler equations, its mathematical difficulties, and some important research problems.

We will discuss the asymptotic behaviour of the critical two-point function for a long-range version of the n-component $|varphi|^4$ model and the weakly self-avoiding walk (WSAW) on the d-dimensional Euclidean lattice with d=1,2,3. The WSAW corresponds to the case n=0 via a supersymmetric integral representation. We choose the range of the interaction so that the upper-critical dimension of both models is $d+epsilon$. Our main result is that, for small $epsilon$ and small coupling strength, the critical two-point function exhibits mean-field decay, confirming a prediction of Fisher, Ma, and Nickel. The proof makes use of a renormalisation group method of Bauerschmidt, Brydges, and Slade, as well as a cluster expansion. This is joint work with Martin Lohmann and Gordon Slade.

I will discuss a method, based on p-adic height pairings, for determining the integral solutions of certain hyperelliptic equations.

The method produces, for a hyperelliptic curve over the rational numbers whose rank equals its genus, an explicit function on the p-adic points on the curve that takes finitely many explicitly determined values on the integral points. The proof goes via p-adic Arakelov theory.

Time permitting I will explain how one can use the generated function to effectively find the integral points and suggest some potential applications.

Meromorphic solutions of soliton equations usually do not fit in the standard spectral transform scheme. We show, that the spectral theory for the corresponding linear problems should be formulated in terms of Pontrjagin spaces – pseudo-Hilbert spaces with a finite number of negative squares. This observation uses the following property: all eigenfucntions of these linear operators with special singularities are meromorphic for all values of spectral parameter.

 

We also discuss a two-dimensional analog of this property.

 

Associated to a closed hyperbolic surface is its length spectrum, the set of the lengths of all of its closed geodesics. Two surfaces are said to be isospectral if they share the same length spectrum.

The talk will be about the following questions and how they relate:

How many questions do you need to ask a length spectrum to determine it?
How many different surfaces can be isospectral to a surface of a given genus?

The approach to these questions will include finding adapted coordinate sets for moduli spaces and exploring McShane type identities.

« Seminar on homological algebra »

 

A Maurer-Cartan (MC) element in a differential graded (dg) algebra A is an odd element x satisfying the equation dx+x2=0. The group of invertible elements of A acts on MC element by gauge transformations:  g(x):=gxg-1-dgg-1.  MC elements are an abstraction of the notion of a flat connection and are fundamental in many problems of homological algebra, deformation theory, differential geometry etc.

 

There is a notion of a (Sullivan) homotopy of MC elements: two such are homotopic if they could be extended to a family over the de Rham algebra on the interval R[x,dx]. A fundamental result (over 40 years old) due to Schlessinger and Stasheff (SS) states that (under certain assumptions) two MC elements are gauge equivalent if an only if they are homotopic.

 

There is also another notion of homotopy of MC elements, based on the singular cochain complex of the interval, and a corresponding SS type theorem.

 

 

The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been predicted by Nienhuis that for $0le nle 2$ the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_c(n)=1/sqrt{2+sqrt{2-n}}$. For $0<nle 2$, the transition line has been further conjectured to separate a regime with short loops when $x<x_c(n)$ from a regime with macroscopic loops when $xge x_c(n)$.

 

In this talk we will prove that for $nin [1,2]$ and $x=x_c(n)$ the loop $O(n)$ model exhibits macroscopic loops. A main tool in the proof is a new positive association (FKG) property shown to hold when $n ge 1$ and $0<xlefrac{1}{sqrt{n}}$. This property implies, using techniques recently developed for the random-cluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (box-crossing property). We develop a `domain gluing' technique which allows us to employ Smirnov's parafermionic observable to rule out the first alternative when $x=x_c(n)$.