Spectral Theory Seminar

Past talks

28 November 2025, 14:15, Classroom 3.06

Marko Žnidarič (University of Ljubljana)

Pseudospectra

In physics as well as elsewhere one often relies on spectral properties to infer the long-time behavior. For instance, for equilibrium physics at low temperatures it is the energy difference between the ground and the first excited state that matters, for nonequilibrium physics described by Lindblad equation, or Markovian processes, it is the spectral gap. It turns out that when dealing with non-normal linear operators sometimes the spectrum can be irrelevant, even more, it can lead to incorrect conclusions. What can be of use in such situations is a pseudospectrum. I will introduce the concept of a pseudospectrum and show few examples of its use: transfer matrices in 2D quadratic systems, entanglement in random circuits, and certain partial differential equations.

21 November 2025, 14:15, Classroom 3.06

Aleksey Kostenko (University of Ljubljana)

On infinite-peakon solutions of the Camassa-Holm equation

We study a class of conservative solutions to the Camassa-Holm equation on the line by exploiting the classical moment problem (in the framework of generalized indefinite strings) to develop the inverse spectral transform method. In particular, we identify explicitly the solutions that are amenable to this approach, which include solutions made up of infinitely many peaked solitons (peakons). We determine which part of the solution can be recovered from the moments of the underlying spectral measure and provide explicit formulas. We show that the solution can be recovered completely if the corresponding moment problem is determinate, in which case the solution is a (potentially infinite) superposition of peakons. However, we also explore the situation when the underlying moment problem is indeterminate.

Based on joint work with X.-K. Chang and J.Eckhardt.

14 November 2025, 14:15, Classroom 3.06

Alexander Glazman (Innsbruck)

2D Potts model at a transition point

The q-state Potts model is a distribution on all possible colouring's of vertices of a box [-N,N]×[-N,N] in q colours that depends on a temperature T. It is classical that the model undergoes a phase transition at some T_c: when T<T_c, there is a giant monochromatic component (ordered regime); when T>T_c, there is exponential decay of correlations (disorder). 

In this talk, we discuss the behaviour at T_c which is known to depend on q. When q ≤ 4, one should see random fractals in the limit (proven only at q=2 by Smirnov et al) and we explain a connection to random surface models via a Fourier transform. When q>4, we show convergence of interfaces between ordered and disordered parts to Brownian bridges. In particular, we establish the wetting phenomenon conjectured by Bricmont and Lebowitz in 1987.

Based on joint works with M. Dober and S. Ott, and with P. Lammers.

24 October 2025, 14:15, Classroom 3.06

Artem Dudko (Warsaw)

On spectra of self-similar groups and graphs

Studying spectra of self-similar groups and related questions lead to many important discoveries. Self-similar groups contain examples of groups solving various important open questions, including the  Grigorchuk group (the first example of finitely generated groups of intermediate growth), the Basilica group (first example of amenable but not subexponentially amenable groups), and the Lamplighter group (a counterexample to a strong Atiyah conjecture). In this talk I will give an introduction to self-similar groups, discuss methods of studying their spectral properties and spectral properties of associated graphs, and present some results on this topic joint with Rostislav Gigorchuk.

17 October 2025, 14:15, Classroom 3.06

Benjamin Eichinger (Lancaster University)

Universality beyond convergence

In this talk, I will survey recent advances in the study of universality limits of orthogonal polynomials. I will discuss cases where the Christoffel–Darboux kernel admits a power-law scaling limit. Such universality limits typically arise in the bulk or at the edge of the spectrum. However, we show that at accumulation points of spectral gaps, the scaling behavior can be quite different. In particular, we discuss scaling limits where there is not a unique limiting kernel, but rather a full limit cycle. Balanced measures on real Julia sets of arbitrary expanding polynomials provide natural examples of this type of scaling behavior.

This talk is based on joint works with Milivoje Lukić, Brian Simanek, Harald Woracek and Peter Yuditskii.

10 October 2025, 14:15, Classroom 3.06

Roman Bessonov (University of Ljubljana)

An introduction to universality for orthogonal polynomials

This talk is an introduction to the concept of univerasilty in the theory of orthogonal polynomials. We will discuss basic objects and illustrate the usage of universality with one particular example: the nonlinear Carleson problem for Golinski-Ibragimov weights. Recent advances and the current state of art in universality will be presented in the forthcoming talk by Benjamin Eichinger.

3 October 2025, 14:15, Classroom 3.06

Tomaž Prosen (University of Ljubljana)

Reversible cellular automata: Minimal models of solvable statistical mechanics

I will discuss  reversible cellular automata in 1+1 dimensions which describe deterministic interacting particle systems, and present several nontrivial examples that show certain aspects of integrability. Most notable is the reversible Rule 54, for which one can obtain exact matrix product expressions for probability state vectors in various setups, ranging from nonequilibrium steady state of the system coupled to markovian stochastic boundaries (and completely diagonalizing the corresponding markov chain) to time-dependent statistical ensembles.
I will present also two deformations which turn the model into either quantum or stochastic cellular automaton. Both deformations, being parametrixed by an arbitrary unitary or stochastic 2x2 matrix, respectively, exhibit some features of integrability. The later (stochastic version) can be interpeted as a lattice discretization of deformed polynuclear growth model. Depending on the remaining time,  I will discuss the construction of non-trivial conservation laws of the quantum deformation and/or the matrix product form of the steady state of the boundary driven stochastic deformation. 

• 2024/25 Spring semester