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, but can also be in the format 2x45 minutes when indicated in the abstract.

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 Gengkai Zhang (gengkai 'at' chalmers.se, Analysis).

Boundary value problems is a well-studied topic. A few years ago Ballmann-Bär introduced a new formalism for studying boundary conditions on self-adjoint first order problems, later extended by Bandara-Bär to general first order problems. The Ballmann-Bär approach relates closely to boundary conditions posed by Atiyah-Patodi-Singer. Using work of Seeley, we extend Bandara-Bär’s work to general elliptic operators on manifolds with boundary and give a complete characterization of which boundary conditions give Fredholm realizations as well as provide a naïve index formula. Based on joint work with Lashi Bandara and Hemanth Saratchandran.

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 Gengkai Zhang (gengkai 'at' chalmers.se, Analysis).

**Coming seminars****2020-10-27 at 13:15**

Magnus Goffeng (Lund)Magnus Goffeng (Lund)

*Elliptic and Fredholm realizations of elliptic operators*Boundary value problems is a well-studied topic. A few years ago Ballmann-Bär introduced a new formalism for studying boundary conditions on self-adjoint first order problems, later extended by Bandara-Bär to general first order problems. The Ballmann-Bär approach relates closely to boundary conditions posed by Atiyah-Patodi-Singer. Using work of Seeley, we extend Bandara-Bär’s work to general elliptic operators on manifolds with boundary and give a complete characterization of which boundary conditions give Fredholm realizations as well as provide a naïve index formula. Based on joint work with Lashi Bandara and Hemanth Saratchandran.

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).

Abstract: How does a colony of ants find the shortest path between its nest and a source of food without any means of communication other than the pheromones each ant leave behind itself? In this joint work with Daniel Kious (Bath) and Bruno Schapira (Marseille), we introduce a new probabilistic model for this phenomenon. In this model, the nest and the source of food are two marked nodes in a finite graph. Ants perform successive random walks from the nest to the food, and ths distribution of the n-th walk depends on the trajectories of the (n-1) previous walks through some linear reinforcement mechanism. Using stochastic approximation methods, couplings with Pólya urns, and the electric conductances method for random walks on graphs, we prove that, in this model, the ants indeed eventually find the shortest path(s) between their nest and the source food.

Abstract: I discuss entanglement entropy of bipartite systems using the von Neumann entropy as measure of entanglement. Some of the most widely studied systems include one-dimensional quantum critical systems, such as quantum spin chains, which in their simplest setting consist of N spins. Of particular interest is the XX spin chain model with zero magnetic field and the study of the von Neumann entropy of the subsystem $P$ of spins on lattice sites $\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$, which can be analysed using certain integral representations. For a single block subsystem, the integral representation involves Toeplitz determinants and the entropy can be calculated using the Fisher-Hartwig asymptotic expansion of these determinants. In this talk, we consider a subsystem that consists of two blocks of spins separated by one spin and compute the mutual information between the two intervals using the Riemann-Hilbert approach. Joint work with György Gehér (Reading University) and Alexander Its (Indiana University-Purdue University Indianapolis). The talk is aimed at a general mathematical audience (familiar with basic complex analysis).

The index theorem for compact manifolds with boundary, established by Atiyah-Patodi-Singer in 1975, is considered one of the most significant mathematical achievements of the 20th century. An important and curious fact is that local boundary conditions are topologically obstructed for index formulae and non-local boundary conditions lie at the heart of this theorem. Consequently, this has inspired the study of boundary value problems for first-order elliptic differential operators by many different schools, with a class of induced boundary operators taking centre stage in establishing non-local boundary conditions.

The work of Bär and Ballmann from 2012 is a modern and comprehensive framework that is useful to study elliptic boundary value problems for first-order elliptic operators on manifolds with compact and smooth boundary. As in the work of Atiyah-Patodi-Singer, a fundamental assumption in Bär-Ballmann is that the induced operator on the boundary can be chosen self-adjoint. All Dirac-type operators, which in particular includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator, are captured via this framework.

In recent work with Bär, we extend the Bär-Ballmann framework to consider general first-order elliptic differential operators by dispensing with the self-adjointness requirement for induced boundary operators. Modulo a zeroth order additive term, we show every induced boundary operator is a bi-sectorial operator via the ellipticity of the interior operator. An essential tool at this level of generality is the bounded holomorphic functional calculus, coupled with pseudo-differential operator theory, semi-group theory as well as methods connected to the resolution of the Kato square root problem. This perspective also paves way for studying non-compact boundary, Lipschitz boundary, as well as boundary value problems in the L^p setting.

**2020-11-04 at 13:15**

Cécile Mailler (Bath)Cécile Mailler (Bath)

*The ants walk: finding geodesics in graphs using reinforcement learning.*Abstract: How does a colony of ants find the shortest path between its nest and a source of food without any means of communication other than the pheromones each ant leave behind itself? In this joint work with Daniel Kious (Bath) and Bruno Schapira (Marseille), we introduce a new probabilistic model for this phenomenon. In this model, the nest and the source of food are two marked nodes in a finite graph. Ants perform successive random walks from the nest to the food, and ths distribution of the n-th walk depends on the trajectories of the (n-1) previous walks through some linear reinforcement mechanism. Using stochastic approximation methods, couplings with Pólya urns, and the electric conductances method for random walks on graphs, we prove that, in this model, the ants indeed eventually find the shortest path(s) between their nest and the source food.

**2020-11-10 at 13:15**

Daniel Ahlberg (Stockholm)Daniel Ahlberg (Stockholm)

**2020-11-17 at 13:15**

Suresh Eswarathasan (Dalhousie, Canada)Suresh Eswarathasan (Dalhousie, Canada)

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

Malin Palö (KTH)Malin Palö (KTH)

**2020-11-01 at 13:15**

Alexey Kuzmin (Chalmers)Alexey Kuzmin (Chalmers)

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

Marcin Lis (Vienna)Marcin Lis (Vienna)

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

Siddhartha Sahi (Rutgers Univ., NJ, USA)Siddhartha Sahi (Rutgers Univ., NJ, USA)

**Previous seminars****2020-03-09 (Monday 1315, MVL15)**

Jani Virtanen (University of Reading)Jani Virtanen (University of Reading)

*Entanglement entropy in quantum spin chain models*Abstract: I discuss entanglement entropy of bipartite systems using the von Neumann entropy as measure of entanglement. Some of the most widely studied systems include one-dimensional quantum critical systems, such as quantum spin chains, which in their simplest setting consist of N spins. Of particular interest is the XX spin chain model with zero magnetic field and the study of the von Neumann entropy of the subsystem $P$ of spins on lattice sites $\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$, which can be analysed using certain integral representations. For a single block subsystem, the integral representation involves Toeplitz determinants and the entropy can be calculated using the Fisher-Hartwig asymptotic expansion of these determinants. In this talk, we consider a subsystem that consists of two blocks of spins separated by one spin and compute the mutual information between the two intervals using the Riemann-Hilbert approach. Joint work with György Gehér (Reading University) and Alexander Its (Indiana University-Purdue University Indianapolis). The talk is aimed at a general mathematical audience (familiar with basic complex analysis).

**2020-03-10**

Lashi Bandara (University of Potsdam)Lashi Bandara (University of Potsdam)

*Boundary value problems for general first-order elliptic differential operators*The index theorem for compact manifolds with boundary, established by Atiyah-Patodi-Singer in 1975, is considered one of the most significant mathematical achievements of the 20th century. An important and curious fact is that local boundary conditions are topologically obstructed for index formulae and non-local boundary conditions lie at the heart of this theorem. Consequently, this has inspired the study of boundary value problems for first-order elliptic differential operators by many different schools, with a class of induced boundary operators taking centre stage in establishing non-local boundary conditions.

The work of Bär and Ballmann from 2012 is a modern and comprehensive framework that is useful to study elliptic boundary value problems for first-order elliptic operators on manifolds with compact and smooth boundary. As in the work of Atiyah-Patodi-Singer, a fundamental assumption in Bär-Ballmann is that the induced operator on the boundary can be chosen self-adjoint. All Dirac-type operators, which in particular includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator, are captured via this framework.

In contrast to the APS index theorem, which is essentially restricted to Dirac-type operators, the earlier index theorem of Atiyah-Singer from 1968 on closed manifolds is valid for general first-order elliptic differential operators.
There are important operators from both geometry and physics which are more general than those captured by the state-of-the-art for BVPs and index theory.
A quintessential example is the Rarita-Schwinger operator on 3/2-spinors, which arises in physics for the study of the so-called delta baryons.
A fundamental and seemingly fatal obstacle to study BVPs for such operators is that the induced operator on the boundary may no longer be chosen self-adjoint, even if the operator on the interior is symmetric.

In recent work with Bär, we extend the Bär-Ballmann framework to consider general first-order elliptic differential operators by dispensing with the self-adjointness requirement for induced boundary operators. Modulo a zeroth order additive term, we show every induced boundary operator is a bi-sectorial operator via the ellipticity of the interior operator. An essential tool at this level of generality is the bounded holomorphic functional calculus, coupled with pseudo-differential operator theory, semi-group theory as well as methods connected to the resolution of the Kato square root problem. This perspective also paves way for studying non-compact boundary, Lipschitz boundary, as well as boundary value problems in the L^p setting.

**2020-05-19 at 10:00**

Jules Lamers (University of Melbourne)

Jules Lamers (University of Melbourne)

*Spin-Ruijsenaars, q-deformed Haldane--Shastry and Macdonald polynomials*

The q-deformed Haldane-Shastry model is a partially isotropic (XXZ-like) long-range spin chain that enjoys quantum-affine (really: quantum-loop) symmetries at finite size. I will start by introducing the model and its special properties. Next I will explain how its hierarchy of commuting Hamiltonians are obtained by 'freezing' the spin-version of Ruijsenaars--Macdonald operators. I will also present an explicit expression for the model's exact eigenvectors, which are of ('pseudo' or 'l-') highest weight in the sense that, in terms of the operators from the monodromy matrix, they are eigenvectors of A and D and annihilated by C. Each has a simple component featuring the 'symmetric square' of the q-Vandermonde times a Macdonald polynomial -- or more precisely its quantum spherical zonal special case. If time permits I will outline how we obtain these eigenvectors using the affine Hecke algebra.

This is based on joint work with Vincent Pasquier and Didina Serban (IPhT CEA/Saclay), arXiv:2004.13210 and ongoing.

**2020-10-20 at 13:15**

Lyudmila Turowska (Chalmers)

Lyudmila Turowska (Chalmers)

*Weighted Fourier algebras and Complexification*

Fourier algebra $A(G)$ of a locally compact group $G$, introduced by Eymard, is one of the favourite objects in abstract harmonic analysis. If $G=\mathbb T$ then $A(\mathbb T)$ is the algebra of continuous function on the circle with absolutely convergent Fourier series. The algebra $A(G)$ has an advantage to be commutative that allows one to examine its Gelfand spectrum, which is known to be topologically isomorphic to $G$; the fact makes a non-trivial connection between Banach algebras and groups. We will discuss a weighted variant of Fourier algebra and show its connection with complexification of the underlying group. For compact groups this was done thanks to abstract complexification due to McKennon [Crelle, 79'] and Cartwright/McMullen [Crelle, 82']. We extended this theory to general locally compact groups and use the model to describe the Gelfand spectrum of weighted Fourier algebras, showing that the latter is a part of the complexification for a wide class of locally compact groups and weights. I shall discuss different examples of weights and determine the spectrum of the corresponding algebras, weighted Fourier algebras of $\mathbb T$, $\mathbb R$ and the Heisenberg group will be of particular interest.

This talk is based on joint work with Olof Giselsson, Mahya Ghandehari, Hun Hee Lee, Jean Ludwig and Nico Spronk. It is aimed at general mathematical audience. All notions related to Banach algebras and locally compact group theory, that are needed for the talk, will be defined.