Colloquium Ical Atom Mailing List

On a closed Riemannian manifold, the Courant nodal domain theorem gives an upper bound on the number of nodal domains of n-th eigenfunction of the Laplacian. In contrast to that, there does not exist such bound on the number of isolated critical points of an eigenfunction. I will try to sketch a proof of the existence of a Riemannian metric on the 2-dimensional torus, whose Laplacian has infinitely many eigenfunctions, each of which has infinitely many isolated critical points. Based on a joint work with A. Logunov and M. Sodin.

Suppose that Y_1, …, Y_N are i.i.d. (independent identically distributed) random variables and let X = Y_1 + … + Y_N. The classical theory of large deviations allows one to accurately estimate the probability of the tail events X < (1-c)E[X] and X > (1+c)E[X] for any positive c. However, the methods involved strongly rely on the fact that X is a linear function of the independent variables Y_1, …, Y_N. There has been considerable interest—both theoretical and practical—in developing tools for estimating such tail probabilities also when X is a nonlinear function of the Y_i. One archetypal example studied by both the combinatorics and the probability communities is when X is the number of triangles in the binomial random graph G(n,p). I will discuss two recent developments in the study of the tail probabilities of this random variable. The talk is based on joint works with Matan Harel and Frank Mousset and with Gady Kozma.

The automorphism group $Aut(\sigma)$ of a subshift $(X,\sigma)$ consists of all homeomorphisms $\phi\colon X\to X$ such that $\phi\sigma=\sigma\phi$. When $(X,\sigma)$ is a shift of finite type, $Aut(\sigma)$ is known to have a rich group structure, and we’ll discuss some background and problems related to the study of $Aut(\sigma)$. Finally, we’ll introduce a certain stabilized automorphism group and outline results which, among other things, provide new cases in which we can distinguish (up to isomorphism) the stabilized groups of certain full shifts. This is joint work with Yair Hartman and Bryna Kra.​

In the early 1970’s, Hindman proved a beautiful theorem in additive Ramsey theory asserting that for any partition of the set of natural numbers into finitely many cells, there exists some infinite set such that all of its finite sums belong to a single cell.

In this talk, we shall address generalizations of this statement to the realm of the uncountable. Among others, we shall present a new theorem concerning the real line which simultaneously generalizes a recent theorem of Hindman, Leader and Strauss, and a classic theorem of Galvin and Shelah.

This is joint work with David Fernandez-Breton.

The billiards in the plane with periodic obstacles are dynamical systems with a simple description but intricate features in their behavior. A specific example, introduced by Paul and Tatania Ehrenfest in 1912, is the so-called “wind-tree” model, where a ball reduced to a point moves on the plane and collides with parallel rectangular scatters according to the usual law of geometric optics.

Natural questions are: does the ball return close to its starting point (recurrence), how fast the ball goes far from it? (diffusion), what is the set of scatters reached by the ball?

These billiards can be modeled as dynamical systems with an infinite invariant measure. The position of the particle can be viewed as a stationary random walk, sum of a stationary sequence of random variables with values in $R^2$, analogous to the classical random walks. For the billiard the increments are the displacement vectors between two collisions, while for the classical random walks the increments are independent random variables.

In the talk, after some general facts about systems with infinite invariant measure, the notions of recurrence and growth (or diffusion) of a stationary random walk will be illustrated by examples, in particular the “wind-tree” model.

Dilation theory is a paradigm for understanding a general class of objects in terms of a better understood class of objects, by way of exhibiting every general object as ``a part of” a special, well understood object. In the first part of this talk I will discuss both classical and contemporary results and applications of dilation theory in operator theory. Then I will describe a dilation theoretic problem that we got interested in very recently: what is the optimal constant $c = c_{\theta,\theta’}$, such that every pair of unitaries $U,V$ satisfying $VU = e^{i\theta} UV$ can be dilated to a pair of $cU’, cV’$, where $U’,V’$ are unitaries that satisfy the commutation relation $V’U’ =e^{i\theta’} U’V’$?

I will present the solution of this problem, as well as a new application (which came to us as a pleasant surprise) of dilation theory to the continuity of the spectrum of the almost Mathieu operator from mathematical physics.

Based on a joint work with Malte Gerhold.

Several functional analogs of fundamental geometric inequalities have appeared in recent decades, beginning with the works of Prekopa and Leindler in the 1970’s. In this talk I will, after discussing the method of functionalization of geometry, present new functional extensions of the Brunn Minkowski inequality and their consequences.

A polytope is called simplicial if all its proper faces are simplices. The celebrated g-theorem gives a complete characterization of the possible face numbers (a.k.a. f-vector) of simplicial polytopes, conjectured by McMullen ’70 and proved by Billera-Lee (sufficiency) and by Stanley (necessity) ’80. The latter uses deep relations with commutative algebra and algebraic geometry. Moving to general polytopes, a finer information than the f-vector is given by the flag-f-vector, counting chains of faces according to their dimensions. Here much less is known, or even conjectured.

I will discuss what works and what breaks, at least conjecturally, when passing from simplicial to general polytopes, or subfamilies of interest.