Spectral Theory Seminar

Past talks

29 May 2026, 14:15, Classroom 3.06

Gabor Kun (Alfréd Rényi Institute of Mathematics, Budapest)

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.

Bor Plestenjak (University of Ljubljana)

Singular eigenvalue problems

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

Enej Ilievski (University of Ljubljana)

Thermalization in classical systems with discrete phase space

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

Markus Reineke (Ruhr-Uni­ver­si­tät Bo­chum)

Dimension expanders and quiver representations

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.

To see other talks from the 2025/26 semester, please use the link below.

Past semesters:

• 2025/26 Spring semester

• 2025/26 Winter semester

• 2024/25 Spring semester