This website is devoted to the Spectral Theory Seminar that we are running at the Department of Mathematics, University of Ljubljana.
We meet on Fridays at 14.15 in Classroom 3.06. The talks are streamed to Zoom.
Organizers: Roman Bessonov, Urban Jezernik, Aleksey Kostenko
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18 December 2025, 15:15, Classroom 3.05
This talk provides a leisurely introduction to the ideas, methods, and theory of self-similar groups. The class of self-similar groups consists of recursively defined automorphisms of rooted k-regular trees, but they can also be described using automata or wreath products. These groups also arise as the iterated monodromy groups of rational endomorphisms of the Riemann sphere, where the limiting Schreier graphs of the action on the tree’s level sets yield intriguing fractal sets. Consequently, self-similar groups contribute a wealth of exotic examples to the 'algebraic zoo.' We will also introduce and briefly discuss amenability, a group property that generalizes the concept of finiteness. We are particularly interested in the following characterization: a group is amenable if it admits a random walk with trivial behavior at infinity. Because random walks on self-similar groups are themselves self-similar, we can utilize this recursive structure to analyze their asymptotic behavior. We will show using these ideas that the iterated monodromy group IMG(z^2−1) (the so-called Basilica group) is amenable. Time permitting, we will discuss extending this result to all self-similar groups generated by automata of linear growth.
18.12 (Thursday!): Miloš Puđa (Ljubljana)
09.01: Aljaž Zalar (Ljubljana)
16.01: Eugene Stepanov (Pisa & St.Petersburg)
23.01: Artur Nicolau (Barcelona)
12 December 2025, 14:15, Classroom 3.06
I will discuss spectral properties of bounded self-adjoint Hankel operators H, realised as integral operators on the positive semi-axis, that commute with dilations by a fixed factor. In analogy with the spectral theory of periodic Schroedinger operators, the Hankel operators H of this class admit the Floquet-Bloch decomposition, which represents H as a direct integral of certain compact fiber operators. As a consequence, operators H have band spectra (the spectrum of H is the union of disjoint intervals). A striking feature of this model is that flat bands (i.e. intervals degenerating into points, which are eigenvalues of infinite multiplicity) may co-exist with non-flat bands; I will discuss some simple explicit examples of this nature. The spectral analysis of this class of Hankel operator is based on the theory of elliptic functions; if time permits, I will explain this connection. This is joint work with Alexander Sobolev (University College London).
05 December 2025, 14:15, Classroom 3.06
We will discuss harmonic analysis techniques to bound the one-collision operator, which appears as the linear term in the Duhamel expansion for evolution equations. The applications to the scattering problem and the energy transfer will be discussed.
28 November 2025, 14:15, Classroom 3.06
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
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
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
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
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
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
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.