The seminar is joint for the division of Analysis and Probability and its main themes are Mathematical Physics, Probability Theory and Harmonic and Functional Analysis. The seminar is encouraged to be aimed at a broader audience, but it may be of a more specialised nature when indicated in the last line of the abstract.
The talks in the seminar are usually 60 minutes, including questions.

At the moment, the seminar takes place via Zoom. See schedule below for links to Zoom meetings.

Should you have any questions or suggestions, feel free to email one of the organizers Jakob Björnberg (jakobbj 'at' chalmers.se, Probability Theory), Erik Broman (broman 'at' chalmers.se, Probability Theory) or Genkai Zhang (genkai 'at' chalmers.se, Analysis).

Should you have any questions or suggestions, feel free to email one of the organizers Jakob Björnberg (jakobbj 'at' chalmers.se, Probability Theory), Erik Broman (broman 'at' chalmers.se, Probability Theory) or Genkai Zhang (genkai 'at' chalmers.se, Analysis).

**Coming seminars****2021-01-26 at 13:15**

Pál Galicza (Budapest)

Pál Galicza (Budapest)

*Sparse reconstruction in spin systems*

We consider a sequence of transitive spin systems on the sequence of finite graphs $G_n(V_n, E_n)$ and we are investigating the following question: Is there a non-degenerate sequence of transitive Boolean functions $f_n: \{-1,1\}^{V_n} \longrightarrow \{-1,1\}$ such that there is some sequence of subset $U_n$ of the vertex set, such that $|U_n|=o(V_n)$ but knowing the spins in $U_n$ give significant information on the value of $f_n$. Our main focus are product and Ising measures. We start by showing that no sparse reconstruction is possible for product measure. As for the Ising model it turns out that for supercritical and critical Ising measures on any graph sequence, magnetisation can be reconstructed from some small subsets of spins. For the subcritical case we present some results suggesting that no such reconstruction is possible. In particular, we show using information theoretical methods that there is no sparse reconstruction for the sequence of subcritical Curie-Weiss models. Reducing no sparse reconstruction to rapid mixing of certain block dynamics we prove the same for the Ising model on the sequence $\mathbb{Z}^2_n$ . Joint work with Gábor Pete.

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**2021-02-02 at 16:15**

Simon Larson (Caltech)

Simon Larson (Caltech)

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**2021-02-09 at 13:15**

Diane Holcomb (KTH)

Diane Holcomb (KTH)

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**2021-02-17 at 15:15**

Olof Sisask (Stockholm)(Joint with Number Theory -- hence the unusual time and day)

Olof Sisask (Stockholm)

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**2021-02-23 at 13:15**

Noam Berger (Munich)

Noam Berger (Munich)

*Harnack inequalities for difference equations with random balanced coefficients.*

We consider difference equations with balanced i.i.d. coefficients which are not necessarily elliptic, and prove a (large scale) elliptic Harnack inequality with an optimal Harnack constant for non-negative solutions of such equations. We then turn to prove a parabolic Harnack inequality in the same setting, and prove it under the additional condition of a (relatively mild) growth condition. We show by example that the growth condition is necessary. I will start the talk with a general background on Harnack inequalities and their significance in (analysis and) probability theory, then I will sketch the main ingredients of the proofs and then I will show a few applications.

The talk is based on joint works with Moran Cohen, David Criens, Jean-Dominique Deuschel and Xiaoqin Guo.

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**2021-03-02 at 13:15**

Jan Felipe van Diejen (Universidad de Talca, Chile)

Jan Felipe van Diejen (Universidad de Talca, Chile)

*Quantum eigenfunctions and bispectral duality for Inozemtsev's Toda chain with boundary interactions*

In the late nineteen eighties, Inozemtsev observed by means of an explicit Lax-pair representation that the Toda chain remains integrable when coupled to a Pöschl-Teller potential. Previous boundary potentials of Morse type that were known to preserve the integrability of the Toda chain arise in his picture as limiting cases. In this talk we report on some recent results in collaboration with Erdal Emsiz concerning the eigenfunctions of the quantum version of Inozemtsev's Toda chain, with an emphasis on bispectrality.

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**2021-03-09 at 13:15**

Irina Pettersson (Chalmers)

Irina Pettersson (Chalmers)

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**2021-03-16 at 13:15**

Gabor Pete (Budapest)

Gabor Pete (Budapest)

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**2021-03-30 at 13:15**

Henna Koivusalo (Bristol)

Henna Koivusalo (Bristol)

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**2021-04-06 at 13:15**

Eusebio Gardella (Munster)

Eusebio Gardella (Munster)

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**Previous seminars**

**2020-11-24 at 13:15**

Malin Palö (KTH)

Malin Palö (KTH)

*Wilson loop expectations in Abelian lattice gauge theories, with and without a Higgs field*

Abstract: Lattice gauge theories have since their introduction been used to predict properties of elementary particles, as approximations of Yang-Mills gauge theory. However, many of these predictions are not rigorous. Moreover, it is not clear how to obtain a limit as the lattice spacing tends to zero even in the simplest cases. A natural first step in this direction would be to attempt to understand the properties of so-called Wilson loops. In this talk, I will introduce Abelian lattice gauge theories from a probabilistic perspective and discuss some of its properties and natural observables. Also, some interesting tools which can be used to study these models will be mentioned. In particular, I will present recent results on the expected value of Wilson loops in $\mathbb{Z}_4$. This talk is based on joint work with Jonatan Lenells and Fredrik Viklund.

**2020-12-01 at 13:15**

Alexey Kuzmin (Chalmers)

Alexey Kuzmin (Chalmers)

*Reflections on the definition of the fundamental group of a C*-algebra*

Recently several approaches to the definition of the fundamental group of a C*-algebra have been made. Unfortunately they are either ad-hoc and work for a restricted class of C*-algebras, or they lack good functorial properties which one would expect to have. In this talk I will propose an alternative definition of the fundamental group influenced by the definition of the fundamental group of a scheme in algebraic geometry.

**2020-12-08 at 13:15**

Marcin Lis (Vienna)

Marcin Lis (Vienna)

*The monomer-dimer model and the Neumann Gaussian Free field*

The classical dimer model is a uniform probability measure on the space of perfect matchings of a graph, i.e., sets of edges such that each vertex is incident on exactly one edge.

In two dimensions, one can define an associated height function which naturally models a ‘’uniform'' random surface (with specified boundary conditions). Moreover the model can be solved exactly which in particular means that its correlations are given by the entries of the inverse Kasteleyn matrix. This exact solvability was the starting point for the breakthrough work of Kenyon who proved, already 20 years ago, that the scaling limit of the height function in bounded domains approximated by the square lattice with vanishing mesh is the Dirichlet (or zero boundary conditions) Gaussian free field. This was the first mathematically rigorous example of conformal invariance in planar statistical mechanics.

In this talk, I will focus on a natural modification of the model where one allows the vertices on the boundary of the graph to remain unmatched. This is the so-called monomer-dimer model (or dimer model with free boundary conditions) (in our case the presence of monomers is restricted to the boundary). This modification complicates the classical analysis in several ways and I will discuss how to circumvent the arising obstacles. In the end, the main result that we obtain is that the scaling limit of the height function of the monomer-dimer model in the upper half-plane approximated by the square lattice with vanishing mesh is the Neumann (or free boundary conditions) Gaussian free field.

This is based on joint work with Nathanael Berestycki (Vienna) and Wei Qian (Paris).

**2020-12-15 at 13:15**

Siddhartha Sahi (Rutgers Univ., NJ, USA)

Siddhartha Sahi (Rutgers Univ., NJ, USA)

*Positivity of interpolation polynomials*

Abstract: The interpolation polynomials of the title are a family of inhomogeneous symmetric polynomials that were defined by the speaker in 1994 and are characterized by rather simple vanishing properties.

In 1996 F. Knop and the speaker showed that their top homogeneous parts are Jack polynomials; for this reason these polynomials are sometimes called interpolation Jack polynomials, or shifted Jack polynomials, or even Knop-Sahi polynomials.

We prove the main conjecture of Knop-Sahi which asserts that, after a suitable normalization, the interpolation polynomials have positive integral coefficients. This result generalizes Macdonald's conjecture for Jack polynomials that was proved by F. Knop and the speaker in 1997.

This is joint work with Y. Naqvi and E. Sergel.

**2021-01-12 at 13:15**

Alexander Holroyd (Bristol)

Alexander Holroyd (Bristol)

*Matching Random Points*

What is fairness, and to what extent is it practically achievable? I’ll talk about a simple mathematical model under which one might hope to understand such questions. Red and blue points occur as independent homogeneous Poisson processes of equal intensity in Euclidean space, and we try to match them to each other. We would like to minimize the sum of a some function (say, a power, gamma) of the distances between matched pairs. This does not make sense, because the sum is infinite, so instead we satisfy ourselves with minimizing *locally*. If the points are interpreted as agents who would like to be matched as close as possible, the parameter gamma encodes a measure of fairness – large gamma means that we try to avoid occasional very bad outcomes (long edges), even if that means inconvenience to others – small gamma means everyone is in it for themselves.

In dimension 1 we have a reasonably complete picture, with a phase transition at gamma=1. For gamma<1 there is a unique minimal matching, while for gamma>1 there are multiple matchings but no stationary solution. In higher dimensions, even existence is not clear in all cases.

**2021-01-19 at 13:15**

Masatoshi Noumi (Kobe/KTH)

Masatoshi Noumi (Kobe/KTH)

*Eigenfunctions for the elliptic Ruijsenaars difference operators*

On the basis of a collaboration with Edwin Langmann (KTH) and Junichi Shiraishi (Tokyo), I report recent progresses in understanding the joint eigenfunctions for the commuting family of elliptic Ruijsenaars difference operators. After reviewing some basic known facts regarding the Macdonald-Ruijsenaars operators in the trigonometric case, I propose two classes of joint eigenfunctions for the elliptic Ruijsenaars operators:

(A) symmetric eigenfunctions around the torus that deform Macdonald polynomials, and

(B) asymptotically free eigenfunctions in a certain asymptotic domain.