Colloquium Ical Atom

We study random hyperbolic planar graphs by using their circle packing embedding to connect their geometry to that of the hyperbolic plane. This leads to several results: Identification of the Poisson and geometric boundaries, a connection between hyperbolicity and a form of non-amenability, and a new proof of the Benjamini-Schramm recurrence result. Based on works with subsets of Omer Angel, Martin Barlow, Ori Gurel-Gurevich, Tom Hutchcroft and Gourab Ray.

Improving upon theorems of Hilbert, Schur, and others, and establishing a longstanding conjecture, Hindman proved that, for each finite coloring of the natural numbers, there is an infinite set such that all finite sums of elements from the set have the same color. Galvin and Glazer used the algebraic and topological structure of the set of ultrafilters (to be defined in the lecture) to provide a very clear and elegant proof of Hindman’s Theorem. This soon became the leading method for establishing coloring theorems in arithmetic and related fields.

We will survey the Galvin-Glazer method and proof, and indicate a surprising recent discovery, that Hindman’s theorem is a special (in a sense, degenerate) case of a theorem about open covers of topological spaces with a property introduced by Karl Menger. The proof uses, in addition to extensions of the Galvin-Glazer theory, infinite games and selection principles.

The talk will be aimed at a general mathematical audience. In particular, we do not assume familiarity with any of the concepts mentioned above. The price is that we will not provide proof details; these are too subtle and laborious for a colloquium talk. The emphasis will be on the introduction to this beautiful connection.

The subject of my talk will be the following topics: 1. Classical and non-commutative Brownian motions with realization on some Fock spaces.

1. q-Brownian motions,q-CCR relations (a a- qa a =1) and theta functions of Jacobi.

2. Meixner-Levy processes and relation to the free Levy processes.

4.Factorial q-von Neumann algebras and ultracontractivity of q-Ornstein-Uhlenbeck semigroup.

The subject of this talk is the analysis of discrete distributions of masses that have a pure point spectrum. A new peak of interest in this subject has appeared after the experimental discovery of quasicrystalline materials in the middle of 80’s. I will present the relevant background and discuss some recent results obtained in joint work with Alexander Olevskii.

Theoretical models of disordered materials lead to challenging mathematical problems with applications to random combinatorial problems and coding theory. The underlying structure is that of many discrete variables that are strongly interacting according to a mean field model determined by a random sparse graph. Focusing on random finite graphs that converge locally to trees we review recent progress in validating the `cavity’ prediction for the limiting free energy per vertex and the approximation of local marginals by the belief propagation algorithm. This talk is based on joint works with Anirban Basak, Andrea Montanari, Allan Sly and Nike Sun.

Given an n-dimensional Riemannian manifold endowed with a probability density, we are interested in studying its isoperimetric, spectral and concentration properties. To this end, the Curvature-Dimension condition CD(K,N), introduced by Bakry and Emery in the 80’s, is a very useful tool. Roughly put, the parameter K serves as a lower bound on the weighted manifold’s “generalized Ricci curvature”, whereas N serves as an upper bound on its “generalized dimension”. Traditionally, the range of admissible values for the generalized dimension N has been confined to [n,infty]. In this talk, we present some recent developments in extending this range to N < 1, allowing in particular negative (!) generalized dimensions. We will mostly be concerned with obtaining sharp isoperimetric inequalities under the Curvature-Dimension condition, identifying new one-dimensional model-spaces for the isoperimetric problem. Of particular interest is when curvature is strictly positive, yielding a new single model space (besides the previously known N-sphere and Gaussian measure): the sphere of (possibly negative) dimension N<1, which enjoys a spectral-gap and improved exponential concentration.

Time permitting, we will also discuss the case when curvature is only assumed non-negative. When N is negative, we confirm that such spaces always satisfy an N-dimensional Cheeger isoperimetric inequality and N-degree polynomial concentration, and establish that these properties are in fact equivalent. In particular, this renders equivalent various weak Sobolev and Nash inequalities for different exponents on such spaces.

An expert provides probabilistic predictions about a sequence of future outcomes (for example, an outcome can be the daily price of a stock and the expert provides the distribution of the price). An inspector reviews the predictions made by the expert and the observed outcomes and applies some test to decide on the validity of the expert’s predictions. The expert testing literature asks whether there exists some test that distinguishes a “true expert”, who provided the correct predictions from a “charlatan”, who concoct predictions strategically to pass the test. I will give a survey of the literature, heavily biased towards my own papers.

The story of Pythagorean triples is an ancient one, as the name suggests. We are looking for triples (a, b, c) of positive integers that are the sides of a right angled triangle; namely they satisfy the Pythagorean Theorem. In this talk I will explain how to find all Pythagorean triples (reduced and ordered) with a given hypotenuse c. The method is simple and constructive. For instance, we will be able to find (by hand) the only triple with hypotenuse 289, and the only two triples with hypotenuse 85.

The relation between Pythagorean triples and complex numbers, prime numbers and Gauss integers is well-known. What might be new in the talk (but I can’t vouch for it) is the connection to abelian group theory.

p-adic fields have been introduced by number theorists for arithmetic purposes. Such a field is complete with respect to an absolute value with some strange behaviour: for example, every closed ball with positive radius is open, and every point of such a ball is a center. Because of those properties, to develop a relevant geometric theory over p-adic fields is non-trivial: one can not naively mimic what is done in real or complex geometry, and one has to use a more subtle approach. In this talk we will present that of Berkovich. His main idea consists in “adding a lot of points to naive p-adic spaces” in order to get good topological properties, like local compactness or local path-connectedness. After having given the basic definitions, we will investigate some significant examples, and give a survey of some of the (numerous) applications of the theory has had in various areas (spectral theory, dynamics, algebraic and arithmetic geometry…).