Monday, September 23th
🎉 08:55-09:00 Opening remarks
09:00-09:50 Philippe Biane: Laver tables and combinatorics (slides)
I will present Laver tables which are combinatorial objects with a very simple definition yet a very complex and fascinating behaviour. In short a Laver table is a left distributive system in which the right multiplication by some special element induces a circular permutation. I will of course define everything in the talk. I will describe results from the 90's when they were defined as well as more recent work describing the statistical and the extremal behaviour of these objects. There is some connection with partial permutations, braid groups and integer partitions which are related to the subject of the conference.
🍵 09:50-10:30 Coffee break
10:30-11:20 Vladimir Retakh: Noncommutative Catalan numbers: orthogonality, clusters, and beyond
We show that noncommutative Catalan numbers previously are moments of a recursion matrix for a certain orthogonal basis of noncommutative polynomials in one variable. This allows us to investigate both commutative and noncommutative orthogonality to any lower triangular recursion matrix and express all coefficients of the involved polynomials via (generalized) cluster variables of a relevant cluster structure. Joint work with Arkady Berenstein and Michael Gekhtman.
11:30-12:20 Cesar Cuenca: Random alpha-partitions at high and low temperatures (slides)
I will discuss the global-scale behavior of discrete versions of Boltzmann distributions of random Young diagrams with a fixed number of boxes. For the Jack Plancherel measure, the limit shape in the high and low temperature regimes is an infinite staircase with corners determined by the roots of the Bessel function, while in general, the limit shape is reduced to finding the solution to a moment problem. I will also hint at different versions of random Young diagrams and random matrix eigenvalue ensembles that depend on the inverse temperature parameter alpha, as well as connections with the theory of Free Probability. Most of the talk is based on a joint project with Maciej Dolega and Alex Moll.
🍽️ 12:30-14:30 Lunch time
14:30-15:20 Evgeny Strahov: Big wreath products (slides)
I will discuss two problems that arise in representation theory of wreath products of finite groups with the infinite symmetric group (big wreath products). The first problem concerns a description of all central probability measures on an analogue of the space of virtual permutations. The second problem is that of harmonic analysis on big wreath problems.
🍵 15:20-16:00 Coffee break
16:00-16:50 Alice Guionnet: Large deviations for theta-Bernoulli Random Walks
Discrete Theta-ensembles generalize the distribution of random tiling in the same way that beta-ensembles generalize that of Gaussian matrices. Theta non-intersection Bernoulli random walks generalize Dyson Brownian motion. In this talk, I will discuss large deviations for these objects, with applications to the asymptotics of Jack and Macdonald polynomials. The techniques are based on the analysis of the so-called Nekrasov's equations. This talk is based on a joint work with Jiaoyang Huang.
🥂 17:00-18:00 Reception
Tuesday, September 24th
09:00-09:50 Mikhail Sodin: Fourier Interpolation and Uniqueness (slides)
I will begin with a beautiful and surprising Fourier interpolation formula discovered by Danilo Radchenko and Marina Viazovska. Their formula enables the reconstruction of a function using its values and the values of its Fourier transform on a particular sequence of points. It yields, probably, the first example of Fourier uniqueness pairs. Their work relied on the theory of modular forms. Then I will shift to an analytic approach to the detection and study of Fourier uniqueness and non-uniqueness pairs, as well as interpolation formulas. The talk is based on ongoing work with Fedor Nazarov and Aleksei Kulikov (a part of which can be found in arXiv:2306.14013).
🍵 09:50-10:30 Coffee break
10:30-11:20 Margit Rösler: Limit transitions for special functions associated with root systems (slides)
Within the framework of Dunkl theory, Bessel functions and hypergeometric functions associated with root systems generalize the spherical functions of Riemannian symmetric spaces. They carry certain continuous parameters, the multiplicities, which take only few specific values in the group cases. After some general background, we shall discuss limit transitions for such functions, related to root systems of type A and type B, in different settings: as certain multiplicities tend to infinity, but the rank stays fixed, and also for the case where the rank tends to infinity. For multiplicities related to group cases, these results have an interpretation in the context of asymptotic harmonic analysis in the sense of Olshanski.
11:30-12:20 Michael Voit: Free limits for some Calogero-Moser-Sutherland particle processes (slides)
From a probabilistic point of view, CMS-models are diffusions $(X_t=(X_{t,1},\ldots,X_{t,N}))_{t\ge0}$ (partially with additional jumps) on Weyl chambers or alcoves in $\mathbb R^N$ or directly on $\mathbb R^N$ which describe the movements of $N$ particles. This includes Dyson Brownian motions, Dunkl-Bessel processes, Dunkl processes as well as multivariate Jacobi processes and Heckman-Opdam processes. We present several almost sure limit results for the empirical measures $$\frac{1}{N}(\delta_{X_{t,1}}+\ldots+\delta_{X_{t,N}})$$ for $N\to\infty$ and $t>0$ under suitable initial conditions. It is classical that for Dyson Brownian motions semicircle distributions appear. We show that in the noncompact Weyl-group invariant Heckman-Opdam cases of type A (i.e., for the noncompact multiplicative free Brownian motions of Biane) in suitable coordinates free sums of semicircle and uniform distributions appear. Moreover, for some Dunkl processes of type B, we present explicit examples where quartercircle distributions at time $t=0$ are deformed into semicircle distribution for $t=\infty$. The talk is based on joint work with Martin Auer and Jeannette Woerner.
🍽️ 12:30-14:30 Lunch time
14:30-15:20 Ashkan Nikeghbali: The stochastic zeta function
The stochastic zeta function, as named by Valko and Virag, is a random holomorphic function whose zeros is a sine-beta point process. The first appearance of this function was in a joint work with Chhaibi and Najnudel, in relation to problems from analytic number theory and in works by Borodin, Olshanski and Strahov. In the first part of the talk, I will emphasise the impact of ideas and results by Olshanski (but also Borodin, Kerov, Neretin and Vershik) which led to the natural introduction of this remarkable random holomorphic function. In the second part, I will introduce my new work with Najnudel which introduces new ideas and which provides a new unified framework to obtain universal convergence of stochastic zeta like functions for many ensembles of random matrices and point processes (spectrum). I will also mention some new insights on the value distribution of the Riemann zeta function and linear combination of L-functions.
🍵 15:20-16:00 Coffee break
16:00-16:50 Yurii Neretin: Infinite symmetric groups and cobordisms of triangulated surfaces
Let $G$ be the product of three copies of the infinite symmetric group $S(\infty)$. The diagonal subgroup $K$ of $G$ and stabilizers $K(n) ⊂ S(\infty)$ of the first $n$ points. We show that double coset spaces $K(m) \ G/K(n)$ form a category and this category acts in a natural way in unitary representations of $G$. This category admits a description in terms of concatenation of cobordisms of colored triangulated surfaces and the set $K(0) \ G/K(0)$ is in one-to-one correspondence with Belyi data.
Wednesday, September 25th
09:00-09:50 Alexander Gnedin: Cross Modality of the Extended Binomial Sums (slides)
For a family of probability functions (or a probability kernel), cross modality occurs when every likelihood maximum matches a mode of the distribution. This implies existence of simultaneous maxima on the modal ridge of the family. The paper explores the property for extended Bernoulli sums, which are random variables representable as a sum of independent Poisson and any number (finite or infinite) of Bernoulli random variables with variable success probabilities. We show that the cross modality holds for many subfamilies of the class, including power series distributions derived from entire functions with totally positive series expansion. A central role in the study is played by the extended Darroch's rule \cite{Darroch, Pitman}, which originally localised the mode of Poisson-binomial distribution in terms of the mean. We give different proofs and geometric interpretation to the extended rule and point at other modal properties of extended Bernoulli sums, in particular discuss stability of the mode in the context of a transport problem.
🍵 09:50-10:30 Coffee break
10:30-11:20 Alexander Bufetov: Gaussian multiplicative chaos for the sine-process
11:30-12:20 Alexander Molev: Quantum Casimir elements and higher Capelli identities (slides)
We will consider central elements in the quantized enveloping algebra of type A. A family of such elements (quantum Gelfand invariants) was constructed in a landmark paper by Reshetikhin, Takhtadzhyan and Faddeev. We will calculate the eigenvalues of the invariants acting in irreducible highest weight representations. We will also construct q-immanants which are central elements of the quantized enveloping algebra parameterized by arbitrary Young diagrams, analogous to Okounkov's quantum immanants. In contrast to the general linear Lie algebra case considered by Okounkov and Olshanski, their eigenvalues coincide with the classical (non-shifted) Schur polynomials. We also prove q-analogues of the higher Capelli identities and derive Newton-type identities. This is joint work with Naihuan Jing and Ming Liu.
⛲ Free Afternoon
Thursday, September 26th
09:00-09:50 Patrik Ferrari: TASEP with dynamics restricted by moving wall (slides)
We study the totally asymmetric simple exclusion process on Z with the step initial condition and with the presence of a rightward-moving wall that prevents the particles from jumping. We determine the asymptotic law of a tagged particle when its law is influenced around one or more macroscopic times.
🍵 09:50-10:30 Coffee break
10:30-11:20 Leonid Petrov: Grothendieck shenanigans (slides)
The story started with a problem of Stanley from his 2017 paper "Some Schubert shenanigans": what is the asymptotically maximal value of a principal specialization of a Schubert polynomial S_w, and what does the corresponding permutation w look like? While this question is still out of reach, we investigate its variant for nonsymmetric Grothendieck polynomials. We analyze the asymptotic behavior of random permutations whose probabilities are proportional to the Grothendieck polynomials using TASEP and tools from integrable probability. The typical behavior of these permutations is described by a specific permuton (a continuous analog of a permutation). We also examine a family of atypical layered permutations that achieve the same asymptotically maximal weight. Based on a joint work with Morales, Panova, and Yeliussizov (https://arxiv.org/abs/2407.21653).
11:30-12:20 Valentin Féray: Up-down chains and scaling limits: construction of permuton- and graphon-valued diffusions (slides)
We study some simple Markov chains on permutations and graphs defined by an up-down procedure. Namely, one step of the chain consists in first duplicating a random element of the permutation or a random vertex of the graph (up-step), and then deleting another random element/vertex (down-step). We prove that these chains converge in the large size limit and after an appropriate time rescaling to Feller diffusions on the space of permutons and graphons, respectively. We also obtain an explicit formula for the separation distance between the distribution of the chains after n steps, excluding the appearance of a cutoff phenomenon. Our approach works in a more general setting; it is based on commutation relations between the up and down operators, and is inspired by works of Fulman, Olshanski and Borodin--Olshanski on the space of partitions and the Thoma simplex. I will not assume any previous knowledge of permutons, graphons, Feller diffusions, separation distances, cutoffs, ... Joint work with Kelvin Rivera-Lopez, Gonzaga University.
🍽️ 12:30-14:30 Lunch time
14:30-15:20 Mariya Shcherbina: Supersymmetric approach to the analysis of random band matrices (slides)
We discuss the application of the SUSY approach to the analysis of local eigenvalue statistics of Hermitian and non Hermitian random band matrices. The method allows us, in particular, to obtain integral representations for correlation functions of characteristic polynomials. In 1D case the application of the transfer operator approach to these representations allows us to analyze the mechanism of transition between “localized” and “delocalized” spectral behaviour.
🍵 15:20-16:00 Coffee break
16:00-16:50 Natasha Rozhkovskaya: Generating functions of action of the W-type Lie algebra on symmetric functions
We consider a W-type Lie algebra which is the central extension of the Lie algebra of differential operators on the circle. It has applications in integrable systems and two-dimensional quantum field theory. We describe the action of this Lie algebra on the basis of Schur symmetric functions of Fock representation through the techniques of formal distributions.
🥘18:00-20:00 Conference Dinner
Friday, September 27th （Move to MPI--Room G3 10--Inselstraße 22）
09:30-10:20 Anna Melnikov: On the complexity of Borel action on orbital varieties in classical Lie algebras
Let G be a classical linear group over ℂ and 𝔤 its Lie algebra. Let B be a Borel subgroup of G and 𝔟 its Lie algebra. Let 𝒪 be a nilpotent orbit of 𝔤 under the adjoint action of G. An orbital variety is a component of 𝒪 ∩ 𝔟. It is naturally stable under the adjoint action of B, but in general an orbital variety does not admit a dense B-orbit. The complexity of Borel action on an orbital variety is the minimal codimension of a B-orbit in an orbital variety. In 𝒪 ∩ 𝔟 there is at least one orbital variety with a dense B-orbit, but the existence of a dense B-orbit does not provide that an orbital variety is spherical, that is a union of the finite number of B-orbits. However we have found the global spherical condition in classical Lie algebras, namely 𝒪 ∩ 𝔟 is the union of the finite number of B-orbits if and only if each its orbital variety admits a dense B-orbit. Another question, which we discuss, is the connection of smooth orbital varieties with orbital varieties with a dense B-orbit via duality. The third question is the partial classification of orbital varieties according to the complexity of B-action. This is a joint work with L. Fresse.
🍵 10:20-11:00 Coffee break
11:00-11:50 Friedrich Knop: Multiplicities and dimensions in enveloping tensor categories
To every finitely powered regular category one can attach a family of symmetric tensor categories, generalizing the calculus of relations. Under certain conditions these categories are semisimple, hence abelian. In thus case we are able to compute the dimensions and the tensor products of simple objects.
12:00-12:50 Andrei Okounkov: Invitation to qq-characters
This will be a basic and informal discussion of Nekrasov's qq-characters. These have already many uses, especially in geometric representation theory, but definitely deserve to be studied and applied more widely.