30 May 2025, 14:15, Classroom 3.06
A recent breakthrough in random matrix theory is the ability to prove strong convergence using limited combinatorial data, contrasting with the traditionally heavy analytic machinery. This second survey talk focuses on the notion of strong convergence in the setting of permutation groups. In particular, we will examine the proof of strong convergence for the action of the symmetric group of degree n on k-subsets of {1,2,…,n}. The talk is intended to stand independently from the previous one, though it will not revisit the broader motivations behind the topic. Warning! I am far from an expert in the area, thus any comments, questions, or corrections will be very welcome.
23 May 2025, 14:15, Classroom 3.06
The Laplace–Beltrami operator for the (Arakelov-)Bergman metric is an interesting operator in spectral theory on Riemann surfaces. Its spectral theory is in particular relevant to arithmetic geometry. Since the early 90s, there has been great interest in understanding the behavior of the Laplacian and related objects when the underlying Riemann surface degenerates to a singular Riemann surface.
In this talk, we discuss a recent approach which explains the degeneration behavior of some of these objects using analogous objects on graphs.
Based on joint work with Omid Amini (École Polytechnique).
16 May 2025, 14:15, Classroom 3.06
The additive diameter of a set S with a subset X is the smallest number of X-summands needed to reach every element of S. The study of these diameters appears in many diverse settings. After giving a brief survey, we will explore nonabelian versions of this, where richer growth and expansion phenomena emerge. The main focus will be presenting a version of these phenomena for group representations, and showing how this viewpoint unifies some of the other diameter problems. Joint work with Špela Špenko.
9 May 2025, 14:15, Classroom 3.06
A recent breakthrough in random matrix theory is the ability to prove strong convergence using limited combinatorial data, in contrast with the heavy analytic techniques previously required. This survey talk will introduce the notion of strong convergence and review some of its recent applications to random models of graphs, hyperbolic surfaces and surface groups. Time permitting, we will outline the proof of the new method for establishing strong convergence. Warning! I am far from an expert in the area, thus any comments, questions, or corrections will be very welcome.
25 April 2025
Recently some new tools in bi-parameter potential theory have been developed in order to characterize the multiplier space for the holomorphic Dirichlet space on the bi-disc: a strong capacitary inequality and a Mazya-type trace inequality. It is natural to ask how much of classical potential theory can be carried out in this new context. Problems of a basic nature abound, and known techniques seem to be of little help. (I mostly talk of work in collaboration with Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti).
18 April 2025
Utilizing the machinery of algebraic integrability, I will discuss how to compute exact thermodynamic partition functions of classical soliton systems. Particularly, I will consider the paradigmatic case of the anisotropic Landau-Lifshitz theory, an integrable equation describing classical ferromagnetism in one spatial dimension. The equation of state, which physically describes a dense soliton gas, takes the form of a Fredholm integral equation -- the classical counterpart of thermodynamic Bethe Ansatz equations originally obtained for integrable quantum spin chains. Some of the lingering open problems will be highlighted as well.
10 April 2025 (joint with Math. Colloquium)
Loosely speaking, a phase transition occurs when a small change in physical parameters leads to drastic changes of the physical observables. Quite similarly, a bifurcation occurs when a small change of the parameters leads to wildly different behavior of a dynamical system. Clearly there is a strong similarity.
In joint ongoing work with Misha Hlushchanka, we study the behavior of the independence polynomial (the partition function of the hard-core model in statistical physics) on recursively defined sequences of graphs. In this setting there is a clear correspondence between the recursion on the level of graphs, and the iteration of a rational map. By exploiting the dynamical behavior of these maps, the existence or absence of phase transitions can be proved.
4 April 2025
Local order isomorphisms of matrix and operator domains will be discussed. A connection with Loewner's theorem and the fundamental theorem of chronogeometry will be explained. The first one characterizes operator monotone functions while the second one describes the general form of bijective preservers of light-likeness on the classical Minkowski space.
21, 28 March 2025
The answer to the question in the title is well-known to be negative. However, the field is still very active and I'll try to present a (non-expert!) overview of the current state of the art.
14 March 2025
Barry Simon's Wonderland Theorem exhibited the genericity and ubiquity of singular continuous spectrum in the study of Schrödinger operators. In this talk we revisit this issue and in particular the following question: "What is the generic spectral type of a Schrödinger operator with a bounded potential?" Based on recent work of Gorodetski and Kleptsyn, a surprising answer can be given. We will explain this answer, why it is surprising, and how it arises. This is joint work with Artur Avila.