29 May 2026, 14:15, Classroom 3.06
The Hall theorem fails for treeings
Measurable perfect matchings have been intensively studied in the last decade. Most of the theorems use either the expansion of the corresponding measurable graph (Lyons-Nazarov theorem, Banach-Ruziewicz problem) or its hyperfiniteness (measurable and Borel solution to Tarski’s circle squaring problem). However, not many examples are known when there is no measurable perfect matching though the measurable Hall condition holds. Most examples admit two-ended components (i.e., of linear growth): in the hyperfinite case we characterized with Bowen and Sabok graphings without measurable perfect matchings via such obstructions. I explain this characterization and give applications from the measurable circle squaring to the amenable Lyons-Nazarov theorem.
In the second half of the talk I construct for every d ≥ 3 a d-regular acyclic measurably bipartite graphing that admits no measurable perfect matching. This answers a question of Kechris and Marks. A denser version refutes a conjecture of Peled and Gurel-Gurevich on couplings by constructing a permuton that admits atomless sections, but its support does not contain the graph of a pmp [0,1]->[0,1] mapping.
We say that a generalized eigenvalue problem is singular if its characteristic polynomial is identically zero. In this case, eigenvalues are defined as points where the rank of the pencil drops. Similar notions arise for singular polynomial and multiparameter eigenvalue problems.
Singular eigenvalue problems are highly challenging, both from the viewpoints of theory and numerical computation: even the proper definition of eigenvectors may become nontrivial, while achieving accuracy and computational efficiency remains difficult.
We present a simple method for generalized eigenvalue problems based on a rank-completing perturbation of the original problem, together with several open questions and applications leading to singular eigenvalue problems in various forms.
15 May 2026, 14:15, Classroom 3.06
Eigenstate thermalization hypothesis (ETH) has established itself as a central theoretical framework for exhibiting various aspects of thermalization phenomena in quantum many-body systems. Can ETH explain thermalization in classical systems?
8 May 2026, 14:15, Classroom 3.06
We will briefly review some ideas and results from the representation theory of quivers.
We then relate the notion of dimension expanders to quiver representations, and use results on so-called general subrepresentations to derive optimal expansion coefficients.
Building on this, we propose a general definition of expander representations of quivers as a qualitative refinement of so-called slope stability, and prove existence of uniform expander representations for any wild quiver.
17 April 2026, 14:15, Classroom 2.02 | Joint with Mathematics Colloquium
Introduced in 1967 as a model of fluid mechanics, the Benjamin-Ono equation turns out to be a striking example of an infinite-dimensional integrable system. I will review its main properties, including a Lax pair and an explicit representation formula for its solutions, and I will try to explain how they have been recently used to prove a conjecture on the role of soliton solutions in the long-time dynamics on the line.
10 April 2026, 14:15, Classroom 3.06
For any compact set E in the complex plane, the n-th Chebyshev polynomial is the unique monic polynomial of degree n that minimizes the supremum norm on E. It is a classical result in uniform approximation that the n-th root of this minimum norm converges to the logarithmic capacity of the set. However, the finer asymptotic behavior is captured by the so-called Widom factors: the sequence formed by the ratio of the Chebyshev norm to the n-th power of the capacity.
To provide context, we will first briefly review the theory for real subsets. A notable example is the class of period-n sets, where the corresponding Chebyshev polynomials naturally show up in the spectral theory of periodic Jacobi matrices.
Turning to the general case of complex continua (compact, connected sets), we will focus on two fundamental questions regarding the Widom factors. A natural starting point is to ask which continua share the asymptotic behavior of the unit disk, in the sense that their Widom factors converge to 1. Furthermore, we want to understand the extreme behavior of these factors: is it possible for them to become unbounded, and if not, is there an absolute upper bound on how large they can get within the class of continua?
While it is known that the Widom factors remain uniformly bounded for sets with sufficiently smooth boundaries, it remains a major open problem whether there exists any continuum for which they are unbounded. We will discuss the ongoing search for such a set, examine theoretical lower bounds for Chebyshev norms, and explore the specific pathological and fractal geometries that would be required to force the Widom factors to grow indefinitely.
25 March 2026, 14:15, Classroom 2.02 | Joint with Mathematics Colloquium
This is a story how a harmonic analyst may try to penetrate into the area of theoretical computer science (TCS). PAC=probably approximately correct is notion of TCS where one should determine the object (a function, a matrix,...) by making a "small" number of trials. One pays the price: trials are random and so the object will be found only with large probability, another price is that it will be found approximately, with a possible small error. PAC learning is a classical part of TCS, where many beautiful results are known. Not long ago a sudden breakthrough in a classical PAC learning was made by my former student Paata Ivanisvili jointly with Alexandros Eskenazis. The quantum analog is PAC learning of big matrices. This will be the main topic of my talk as it leads to an interesting question à la Remez and Bernstein inequalities. The talk is based on joint results with Lars Becker, Ohad Klein, Joseph Slote and Haonan Zhang.
20 March 2026, 14:15, Classroom 3.06
Abstract in pdf: download
13 March 2026, 14:15, Classroom 3.06
During the last 100 years, studies of optimization problems for eigenvalues of selfadjoint differential operators led to many notable achievements including the famous Rayleigh–Faber–Krahn inequality. Optimization problems involving eigenvalues of nonselfadjoint operators have emerged relatively recently in connection with Physics and Engineering applications. It is planned to give a short overview of spectral optimization for dissipative wave equations and to explain the principally new difficulties that emerge due to the nonselfadjoint character of the related spectral problems.
The main attention will be paid to the problem how to create a lossy resonator that has an eigenvalue as close as possible to the real line under certain fabrication constraints. Since the problem is connected with the optimization of photonic crystal designs for high-Q optical cavities, we consider it in the settings involving 3-dimensional Maxwell systems following the joint research with Matthias Eller and for the 1-dimensional case of TEM-modes in layered structures following the joint paper with Herbert Koch and Ievgen Verbytskyi. The 3-D case occurs to be connected with homogenization and (non-)unique continuation, whereas the 1-D problem admits effective analytic and numerical approaches via optimal control reformulations.
27 February 2026, 14:15, Classroom 3.06
Ornstein–Uhlenbeck operators and semigroups appear in several fields from quantum mechanics to stochastic analysis. We will first review some general properties of these objects in the simplest one-dimensional case. Then we shall focus to the Ornstein–Uhlenbeck operators acting along the edges of a rooted metric tree equipped with a Gaussian-type measure. Extending known results on star graphs, we shall construct two self-adjoint realisations that generate symmetric, analytic Markov semigroups. Both operators have compact resolvent and pure point spectrum with at least linear eigenvalue growth. For regular trees, we can further provide a decomposition into one-dimensional Ornstein–Uhlenbeck operators along the half-lines, leading to refined spectral estimates.
Based on joint work in progress with Sahiba Arora (Hannover), Delio Mugnolo (Hagen), and Abdelaziz Rhandi (Salerno).
20 February 2026, 14:15, Classroom 2.03 | Joint with Mathematics Colloquium
Stefan problem is one of the oldest free-boundary problems modelling phase transitions between liquids and solids. If, for example, the solid phase melts in finite time - this is an instance of singularity formation in the language of partial differential equations (PDE). If the rate of melting is self-similar we speak of Type I singularity formation.
After a brief introduction to the Stefan problem, I will show that for an open set of radial initial data the melting rate is to the leading order NOT given by the self-similar scaling. It is instead of Type II with rates predicted in the pioneering work of Herrero and Velazquez. Time remaining, I will explain what we expect to happen in non-radial setting. Our techniques rely on ideas from spectral theory, dynamical systems and PDE.