Second part of the talk from last week:
An ergodic system (X;B; μ; T) is said to have the weak Pinsker
property if for any ε > 0 one can express the system as the direct
product of two systems with the first having entropy less than ε and
the second one being isomorphic to a Bernoulli system. The problem
as to whether or not this property holds for all systems was open for
more than forty years and has been recently settled in the affirmative
in a remarkable work by Tim Austin.
Let $\Sigma$ be a Riemann surface of genus $g \geq 2$, and p be a point on $\Sigma$. We define a space $S_g(t)$ consisting of certain irreducible representations of the fundamental group of $\Sigma \setminus p$, modulo conjugation by SU(n). This space has interpretations in algebraic geometry, gauge theory and topological quantum field theory; in particular if Σ has a Kahler structure then $S_g(t)$ is the moduli space of parabolic vector bundles of rank n over Σ.
I will discuss the inverse Monge-Ampere flow and its applications to the existence, and non-existence, of Kahler-Einstein metrics. To motivate this discussion I will first describe the classical theory of the Donaldson heat flow on a Riemann surface, and its relationship with the Harder-Narasimhan filtration of an unstable vector bundle.
Abstract: The classical theory of metric Diophantine approximation is very well developed and has, in recent years, seen significant advances, partly due to connections with homogeneous dynamics. Several problems in this subject can be viewed as particular examples of a very general setup, that of lattice actions on homogeneous varieties of semisimple groups. The latter setup presents significant challenges, including but not limited to, the non-abelian nature of the objects under study.
The recent work of Abe--Henniart--Herzig--Vigneras gives a classification of irreducible admissible mod-$p$ representations of a $p$-adic reductive group in terms of supersingular representations. However, supersingular representations remain mysterious largely, and in general we know them very little. Up to date, there are only a classification of them for the group $GL_2 (Q_p)$ and a few other closely related cases.
I will review the Kostant-Souriau geometric quantization procedure for
passing from functions on a symplectic manifold (classical observables)
to operators on a Hilbert space (quantum observables).
With the "half-form correction" that is required in this procedure,
one cannot quantize a complex projective space of even complex dimension,
and one cannot equivariantly quantize the two-sphere nor any symplectic
toric manifold.
I will present a geometric quantization procedure that uses metaplectic-c
A k-dimensional permutation is a (k+1)-dimensional array of zeros
and ones, with exactly a single one in every axis parallel line. We consider the
“number on the forehead" communication complexity of a k-dimensional permutation
and ask how small and how large it can be. We give some initial answers to these questions.
We prove a very weak lower bound that holds for every permutation, and mention a surprising
upper bound. We motivate these questions by describing several closely related problems:
We will give a short review of various topics discussed in the first semester and last Monday and then we'll pick the fruits: namely, we will show how to get groups which are no approximated.
A random linear (binary) code is a dimension lamba*n (0
Much of the interesting information about a code C is captured by its weight vector. Namely, this is the vector (w_0,w_1,...,w_n) where w_i counts the elements of C with Hamming weight i. In this work we study the weight vector of a random linear code. Our main result is computing the moments of the random variable w_(gamma*n), where 0 < gamma < 1 is a fixed constant and n goes to infinity.
This is a joint work with Nati Linial.
Abstract:
The study of elastic membranes carrying topological defects has a longstanding history, going back at least to the 1950s. When allowed to buckle in three-dimensional space, membranes with defects can totally relieve their in-plane strain, remaining with a bending energy, whose rigidity modulus is small compared to the stretching modulus.
Abstract:
I will discuss a family of random processes in discrete time related to products of random matrices (product matrix processes). Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. I will explain that in certain cases product matrix processes are discrete-time determinantal point processes, whose correlation kernels can be expressed in terms of double contour integrals. This enables to investigate determinantal processes for products of ra ndom matrices in
Title: S-machines and their applications
Abstract: I will discuss applications of S-machines which were first introduced in 1996. The applications include
* Description of possible Dehn functions of groups
* Various Higman-like embedding theorems
* Finitely presented non-amenable torsion-by-cyclic groups
* Aspherical manifolds containing expanders
* Groups with quadratic Dehn functions and undecidable conjugacy problem
Title: Random walks on planar graphs
Abstract: We will discuss several results relating the behavior of a random walk on a planar graph and the geometric properties of a nice embedding of the graph in the plane (e.g. a circle packing of the graph). An example of such a result is that for a bounded degree graph, the simple random walk is recurrent if and only if the boundary of the nice embedding is a polar set (that is, Brownian motion misses it almost surely).
No prior knowledge about random walks, circle packings or Brownian motion is required.
The Hodge decomposition theorem is the climax of a beautiful theory involving geometry, analysis and topology, which has far-reaching implications in various fields. I will present the Hodge decomposition in compact Riemannian manifolds, with or without boundary. The non-empty-boundary case is more interesting, as it requires the formulation of an appropriate boundary condition. As it turns out, the Hodge-Laplacian has two different elliptic boundary conditions generalizing the classical Dirichlet and Neumann conditions, respectively.
