AP-seminar 2016

Jules Lamers
​Constructive method for the domain-wall partition function
In this talk I aim to give an accessible introduction to part of the work I did as a PhD at Utrecht University. In the eighties Mills, Robin and Rumsey came with a conjecture in combinatorics, for the number of so-called alternating-sign matrices (ASMs) for any size L x L. About a decade later Zeilberger found a proof, which was soon followed by a much shorter proof by Kuperberg using wisdom from statistical physics.
I will explain how to get from ASMs to what we need from statistical physics, then review work by Korepin and Izergin that was crucial for Kuperberg's proof, and finally outline an alternative to the approach of Korepin--Izergin that was devised by Galleas and further developed by Galleas and myself.
The talk is aimed at a general analysis audience.

Heiko Gimperlein, Heriot Watt University, Edinburgh

In everyday language, this talk studies the question about the optimal shape and location of a thermometer of a given volume to reconstruct the temperature distribution in an entire room. For random initial conditions, this problem was considered by Privat, Trelat and Zuazua (ARMA, 2015). We introduce heat packet decompositions from microlocal analysis in the study of optimal design problems and show that they allow to remove the randomisation assumption and any geometric restrictions on the domain. Analytically, we obtain quantitative estimates for the wellposedness of an inverse problem, in which one determines the solution in the whole domain from its restriction to a subset of given volume. We conclude that there exists a unique optimal such subset, that it is semi-analytic and can be approximated by solving a sequence of finite-dimensional optimization problems.

The talk is aimed at a general analysis audience.

Martin Hallnäs

​On the spectra of complex Lame operators
​ The Lame operators are Schrodinger operators on the real line with periodic potentials given by the elliptic Weierstrass P-function, depending on one positive integer m and a choice of period lattice. For suitable lattices the potential is real-valued, regular and periodic; and the corresponding operators are the simplest examples of so-called finite-gap operators: their spectra have a band structure with precisely m gaps. I will discuss some recent results on the spectra of Lame operators for generic period lattices when the potential is complex-valued. I will concentrate on the simplest case m=1 and a number of examples will be presented. The talk is based on recent joint work with William Haese-Hill and Alexander Veselov.

Edward McDonald, UNSW

Quantum Differentiability and Classical Differentiability
There are several notions of smoothness and differentiability in noncommutative geometry, and one in particular is given by the spectral asymptotics of operator theoretic "one-forms". The relationship between this notion of noncommutative differentiability and classical differentiability is nontrivial and not entirely understood. I will give an exposition of recent results exploring this connection, in particular focusing on how to characterise classical Sobolev spaces in a noncommutative way. Furthermore noncommutative examples are explored, in particular the quantum torus and the Moyal plane.

Benjamin Lees

Néel order using a random loop model
We consider the general spin-1 SU(2) invariant Heisenberg model with a two-body interaction. A related random loop model is introduced that allows to prove Néel order for certain parameters of the model. This order is equivalent to the occurrence of infinite loops which are expected to have a Poisson Dirichlet structure.

Bent Orsted, Aarhus Universitet

Poisson transforms and elliptic equations
Some elliptic boundary value problems considered by Caffarelli and Silvestre may be studied using representation theory of some semisimple Lie groups. We show how in particular some integral transforms related to symmetry breaking play a role and how Sobolev spaces corresponding to so-called complementary series representations arise naturally. This lecture is based on joint work with Jan Möllers and Genkai Zhang.

Juhani Koivisto, University of Southern Denmark, Odense

Amenability and metric measure spaces
A hyperbolic metric measure space satisfies a global Sobolev inequality, while the Heisenberg group satisfies a suitably weighted global Sobolev inequality. From the point of view of large scale geometry this reduces to the observation that the first space is non-amenable while the second is amenable. After introducing these notions I discuss how the weights in the inequalities are determined by controlled coarse homology at the level of metric spaces, and the connection to measure spaces is given by the local weak (1,1)-Poincaré inequality. In particular this gives a new characterisation of amenability for metric measure spaces with applications to p-harmonic functions for spaces that support a local weak (1,1)-Poincaré inequality.

Kristoffer Lindensjö, Stockholm University

Time-inconsistent stochastic control - a (classical) PDE characterization
Time-inconsistent stochastic control is a game-theoretic generalization of standard stochastic control, with well-known applications in economics and finance. One of the most important results of standard (time-consistent) stochastic control is the characterization of the optimal control and the (optimal) value function as the solution to a (deterministic) PDE known as the Hamilton-Jacobi-Bellman equation (HJB). Naturally, one would hope that time-inconsistent stochastic control problems offer a similar possibility. Indeed, Björk, Khapko & Murgoci (2016) introduce a system of PDEs, the extended HJB system, and prove a verification theorem saying that if the extended HJB system has a (classical) solution then this solution is in fact the equilibrium value function and the equilibrium control. The main result of the present paper is that we (under the assumptions of the existence of an equilibrium control and ad hoc regularity) show that the equilibrium value function and the equilibrium control are a (classical) solution to the extended HJB system. The setting is that of a general (Markovian) Itô diffusion, and a general time-inconsistent bequest function.

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