KTH Department of Mathematics   |   Stockholm Mathematics Centre (SMC)   |   SU Department of Mathematics

Session in Mathematical Physics
at the 27th Nordic Congress of Mathematicians, celebrating the 100th anniversary of Institut Mittag-Leffler

Session organizers
Douglas Lundholm (KTH Royal Institute of Technology)
Jan Philip Solovej (Copenhagen University)

Confirmed speakers
Farrokh Atai (KTH Royal Institute of Technology)
Thomas Bäckdahl (Chalmers University of Technology)
Matthias Christandl (Copenhagen University)
Horia Cornean (Aalborg University)
Jan Derezinski (University of Warsaw)
Tobias Ekholm (Institut Mittag-Leffler and Uppsala University)
Maria Esteban (CNRS and University Paris-Dauphine)
Magnus Goffeng (University of Gothenburg)
Arne Jensen (Aalborg University)
Thordur Jonsson (University of Iceland)
Antti Kupiainen (University of Helsinki)
Hans Ringström (KTH Royal Institute of Technology)

Schedule
The following is a preliminary schedule for the session:

 Thursday 17 March Friday 18 March Saturday 19 March 14:00-14:45 Antti Kupiainen Hans Ringström Arne Jensen 14:45-15:30 Maria Esteban Thomas Bäckdahl Thordur Jonsson Pause 16:00-16:45 Matthias Christandl Tobias Ekholm Farrokh Atai 16:45-17:30 Horia Cornean Jan Derezinski Magnus Goffeng

Abstracts

Speaker: Farrokh Atai (KTH Royal Institute of Technology)
Title: Anyons and the deformed Calogero-Sutherland model
Abstract:
In this talk I discuss the collective field theory description of the celebrated Calogero-Sutherland (CS) model, based on conformal field theory, and its relation to quasi-particles known as anyons. I show that previously known results, in this context, naturally generalize to the deformed CS model that was introduced by Chalykh, Feigin, Veselov, and Sergeev.

The deformed CS model is a quantum integrable model with arbitrary numbers of two (different) types of strongly interacting particles and is a natural mathematical generalization of the CS model. Contrary to the CS model, there is no satisfactory interpretation of the deformed CS model as a quantum many-body system. As will be discussed, our results provide a natural application of the deformed CS model in Wen's effective field theory of the fractional quantum Hall effect, where the different particles corresponds to electron and quasi-hole excitations.

The talk is based on joint work with E. Langmann (KTH).

Speaker: Thomas Bäckdahl (Chalmers University of Technology)
Title: Spacetime structure and computer algebra
Abstract:
In this talk, I will present some some computer algebra methods to analyse the structure of a spacetime manifold by constructing conservation laws and symmetry operators. In particular, we study structures of the Kerr spacetime describing a stationary black hole so we can develop tools to prove decay of linear and non-linear fields propagating on that manifold.

Speaker: Matthias Christandl (Copenhagen University)
Title: Nondeterministic multiparty quantum communication
Abstract:
We study nondeterministic multiparty quantum communication with GHZ-type broadcasts, or, equivalently, with preshared GHZ-entanglement and communication by message passing. We show that, with number-in-hand classical inputs, the communication complexity of a Boolean function in this communication model equals the logarithm of the support rank of the corresponding tensor, whereas the 'approximation' complexity in this model is characterized by the border support rank. As a first application, we prove a log-rank conjecture posed by Villagra et al. for nondeterministic multiparty quantum communication without preshared entangle- ment. Second, we study the communication complexity of the graphwise equality problem. For a cycle graph, the complexity of this communication problem is closely related to the complexity of the computational problem of multiplying matrices, or more precisely, to the tensor rank of the iterated matrix multiplication tensor. We exhibit a nontrivial protocol for the three-player and five-player cyclic equality problem, and we show how Young flattenings yield nontrivial lower bounds on the border support rank of the iterated matrix multiplication tensor.

Speaker: Horia Cornean (Aalborg University)
Title: Wannier functions, Bloch bundles and topological degree theory
Abstract:
Consider a real analytic and time reversal symmetric family of Bloch projections of rank $N$. We present a constructive proof for the existence of a corresponding Bloch basis, which is both real analytic and periodic with respect to its $d$-dimensional quasi-momenta when $1\leq d\leq 3$ and $N\geq 1$. This basis is also time reversal symmetric, hence the corresponding exponentially localized composite Wannier functions are real. If time permits, we will show what goes wrong in dimension four and higher. This is joint work with G. Nenciu (Bucharest) and I. Herbst (Charlottesville)

Speaker: Jan Derezinski (University of Warsaw)
Title: Mathematical theory of Bogoliubov Hamiltonians
Abstract:
I would like to review the theory of Bogoliubov Hamiltonians, that is, self-adjoint operators on a Fock space, formally defined by expressions quadratic in creation and annihilation operators.

There are several natural varieties of Bogoliubov Hamiltonians, among them "Weyl-quantized", "Wick-quantized" and "renormalized". Their theory is relatively straightforward if the 1-particle space is finite dmensional, but becomes quite sophisticated if the 1-particle space is infinite dimensional.

Bogoliubov Hamiltonians are relevant for various important problems of quantum field theory and many body quantum physics, such as renormalization, existence of vacuum, existence of quantum dynamics, finiteness of vacuum energy.

Speaker: Tobias Ekholm (Institut Mittag-Leffler and Uppsala University)
Title: Chern-Simons theory, topological string, and knot contact homology
Abstract:
We discuss recent progress relating gauge theory and topological string theory associated to knots in 3-manifolds in terms of holomorphic curve theory at infinity. The talk reports on joint work with Aganagic, Ng, and Vafa.

Speaker: Maria Esteban (CNRS and University Paris-Dauphine)
Title: Nonlinear flows, rigidity and eigenvalues of Schrödinger operators
Abstract:
In this talk I will present various recent results obtained in collaboration with J. Dolbeault, A. Laptev and M. Loss where we give optimal estimates for the principal eigenvalue of Schrödinger operators in compact and non compact manifolds. We do it by relating these estimates to interpolation inequalities, for which we can obtain the value of the best constants using nonlinear flows methods. Some rigidity results obtained for positive solutions of nonlinear elliptic equations on those manifolds allow us to study the cases where only the trivial solution exists, which helps to have a better view of the optimal constants and of the optimal potentials. In particular in the case of spheres and cylinders we show that for small enough potentials (the smallness being measured in some L^p norm) the optimal potentials are "trivial" and explicit.

Speaker: Magnus Goffeng (University of Gothenburg)
Title: Spectral flows along compact obstacles for magnetic Schrödinger operators
Abstract:
Spectral flows measures change in the spectrum along a path of self-adjoint operators. It was first studied by Atiyah-Patodi-Singer for Dirac operators, where the relative index of Dirac operators with APS boundary conditions is a spectral flow of Dirac operators on the boundary. In this talk, I will discuss the spectral flow along a change of boundary conditions for the Landau hamiltonian with a compact obstacle in even-dimensional euclidean space. The computation can be reduced to the boundary. Using a Weyl law on the boundary, a peculiar difference between the spectrum of the Dirichlet and the Neumann realisation can be explained. Based on joint work with Elmar Schrohe.

Speaker: Arne Jensen (Aalborg University)
Title: Memory Effects in Mesoscopic Systems
Abstract:
Consider a quantum dot coupled to two semi-infinite one-dimensional leads at thermal equilibrium. We turn on adiabatically a bias between the leads such that there exists exactly one discrete eigenvalue both at the beginning and at the end of the switching procedure. We investigate the dependence of some observables on the switching procedure. For example, the expectation on the final bound state strongly depends on the history of the switching procedure. On the other hand, the contribution to the final steady-state corresponding to the continuous spectrum has no memory, and only depends on the initial and final values of the bias.

Joint work with H. Cornean (Aalborg) and G. Nenciu (Bucharest).

Speaker: Thordur Jonsson (University of Iceland)
Title: Exponential bounds on the number of 3-dimensional causal triangulations
Abstract:
We discuss the problem of bounding the number of distinct triangulations of manifolds with a given topology. We describe recent results in 3 and 4 dimensions, in particular the exponential bound on 3-dimensions causal triangulations.

Speaker: Antti Kupiainen (University of Helsinki)
Title: Constructive Conformal Field Theory
Abstract:
Liouville Conformal Field Theory is a basic building bloc of 2d gravity which is the scaling limit of discrete random surfaces. We review the physical theory of 2d gravity and then present a probabilistic construction of the Liouville Conformal Field Theory. The construction involves the theory of multiplicative chaos which is also reviewed.

Speaker: Hans Ringström (KTH Royal Institute of Technology)
Title: On the cosmic no-hair conjecture in the Einstein-Vlasov setting
Abstract:
The standard starting point in cosmology is the assumption of spatial homogeneity and isotropy. However, it is preferable to prove that solutions generally isotropise and that the spatial variation (as seen by observers) becomes negligible. This is expected to happen in the presence of a positive cosmological constant; in fact, solutions are in that case expected to appear like the de Sitter spacetime to observers at late times. The latter expectation goes under the name of the cosmic no-hair conjecture. In the talk, we present a result (based on joint work with Håkan Andréasson) concerning a class of spacetimes (T^3-Gowdy, in the Einstein-Vlasov setting) whose members are neither spatially homogeneous nor isotropic, but which all satisfy the cosmic no-hair conjecture. Moreover, we demonstrate that the members of this class are future stable under general perturbations (without symmetries), and that the perturbed solutions satisfy the cosmic no-hair conjecture.