The reduced density matrix of a subsystem induces an intrinsic internal dynamics called the « modular flow ». The flow depends on the subsystem and the given state of the total system. It has been subject to much attention in theoretical physics in recent times because it is closely related to information theoretic aspects of quantum field theory. In mathematics, the flow has played an important role in the study of operator algebras through the work of Connes and others.

It is known that the flow has a geometric nature (boosts resp. special conformal transformations) in case the subsystem is defined by a spacetime region with a simple shape. For more complicated regions, important progress was recently made by Casini et al. who were able to determine the flow for multi-component regions for free massless fermions or bosons in 1+1 dimensions.

In this introductory lecture, I describe the physical and mathematical backgrounds underlying this research area. Then I describe a new approach which is not limited to free theories, based in an essential way on two principles: The so-called « KMS-condition » and the exchange relations between primaries (braid relations) in rational CFTs in 1+1 dimensions. A combination of these ideas and methods from operator algebras establish that finding the modular flow of a multi-component region is equivalent to a certain matrix Riemann-Hilbert problem. One can therefore apply known methods for this classic problem to find or at least characterize the modular flow.

 

Mathematicians have long understood periodic trajectories on the square billiard table, which occur when the slope of the trajectory is rational. In this talk, I will explain my joint work with Samuel Lelièvre on periodic trajectories on the regular pentagon, describing their geometry, symbolic dynamics, and group structure. The periodic trajectories are very beautiful, and some of them exhibit a surprising « dense but not equidistributed » behavior.

Quantum tensor networks provide a new language for describing many body systems. They model the entanglement structure of many body wavefunctions, and give a precise description of symmetries such as arising in systems exhibiting topological quantum order. In this talk, an overview will be given of the challenges, prospects and limitations of this approach.

In my talk, I will report on my ongoing collaborating project together with Yifeng Liu, Liang Xiao, Wei Zhang, and Xinwen Zhu, which concerns the rank 0 case of the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin–Selberg motives for GL(n) x GL(n+1). I will state the main results with some examples coming from elliptic curves, sketch the strategy of the proof, and then focus on the key geometric ingredients, namely the semi-stable reduction of unitary Shimura varieties of type U(1,n) at non-quasi-split places.

In order to analyze mathematical and physical systems it is often necessary to assume some form of order, e.g. perfect symmetry or complete randomness. Fortunately, nature seems to be biased towards such forms of order as well. During the second half of the 20th century, a new paradigm of « aperiodic order » was suggested. Instances of aperiodic order were discovered in different areas of mathematics, such as harmonic analysis and diophantine approximation (Meyer), tiling theory (Wang, Penrose) and additive combinatorics (Freiman, Erdös-Szemeredi); after some initial resistance is has now been accepted that aperiodic order also exists in nature in the form of quasicrystals.

Together with Michael Björklund we have proposed a general mathematical framework for the study of aperiodic structures in metric spaces, based on the notion of an « approximate lattices ». Roughly speaking, approximate lattices generalize lattices in the same way that approximate subgroups (in the sense of Tao) generalize subgroups. Approximate lattices in Euclidean space are essentially the « harmonious sets » of Meyer (a.k.a. mathematical quasi-crystals), but there are interesting examples in other geometries, such as symmetric spaces, Bruhat-Tits buildings or nilpotent Lie groups. It turns out that with every approximate lattice one can associate a dynamical system, which replaces the homogeneous space associated with a lattice – thus the study of approximate lattices can be considered as « geometric group theory enriched over dynamical systems ».

In this talk I will (1) give an overview over the basic framework of approximate lattices and geometric approximate group theory; (2) illustrate the framework by formulating Meyer’s theory of harmonious sets in this language; (3) time permitting, discuss some recent structure theory of approximate lattices in nilpotent Lie groups and applications to Bragg peaks in the Schrödinger spectrum of magnetic quasicrystals.

Based on joint works with Michael Björklund (Chalmers), Matthew Cordes (ETH), Felix Pogorzelski (Leipzig) and Vera Tonić (Rijeka).

I will consider the asymptotic behavior of eigenfunctions of the Laplacian on a compact locally symmetric manifold M « in the level aspect », that is as the volume of M tends to infinity. I will formulate a precise conjecture of « Berry type », and describe partial results obtained in a joint work with Miklos Abert and Étienne Le Masson.

A classification will be given of all separable minimal hypersurfaces in ${mathbb R}^{n geq 3}$.Rn≥3

In this talk I will focus on BPS states in supersymmetric field theories with N=2. In this theories one can consider a certain class of supersymmetric line operators. Such operators support a new class of BPS states, called framed BPS states. I will discuss a formalism based on quivers to understand these objects and their properties. Time permitting I will discuss a relation with the theory of cluster algebras.

The Calabi theorem states that for every regular convex cone K in R^n, the Monge-Ampère equation log det F” = 2F/n has a unique convex solution on the interior of K which tends to +infty on the boundary of K. It turns out that this solution is self-concordant and logarithmically homogeneous, and thus is a barrier which can be used for conic optimization. We consider different aspects of this barrier:

affine spheres as level surfaces
metrization of the interior of K by the Hessian metric F”
primal-dual symmetry
interpretation as a minimal Lagrangian submanifold in a certain para-Kähler space form
complex-analytic structure on 3-dimensional cones.

Kitaev’s quantum double models provide a rich class of examples of two-dimensional lattice systems with topological order in the ground states and a spectrum described by anyonic elementary excitations. The infinite volume ground states of the abelian quantum double models come in a number of equivalence classes called superselection sectors. We prove that the superselection structure remains unchanged under uniformly small perturbations of the Hamiltonians. (joint work with Matthew Cha and Pieter Naaijkens)

The study of Anosov representations deals with discrete subgroups of Lie groups that have a nice limit set, meaning that they share the dynamical properties of limit sets in hyperbolic geometry. However, the geometry of these limits sets is different: while limit sets in hyperbolic geometry have a fractal nature (e.g. non-integer Hausdorff dimension), some Anosov groups have a more regular limit set (e.g. $C^1$ for Hitchin representations).
My talk will focus on quasi-Fuchsian subgroups of SO(n,2), and show that the situation is intermediate: their limit sets are Lipschitz submanifolds, but not $C^1$. I will discuss the two main steps of the proof. The first one classifies the possible Zariski closures of such groups. The second uses anti-de Sitter geometry in order to determine the limit cone of such a group with a $C^1$ limit set.
Based on joint work with Olivier Glorieux.