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

An important conjecture in quantum mechanics is that non-interacting, disordered 3d quantum systems should exhibit a localization-delocalization transition as a function of the disorder strength. From a mathematical viewpoint, a lot is known about the localized phase for strong disorder, much less about the delocalized phase at weak disorder. In this talk I will present results for a hierarchical supersymmetric model for a class of 3d quantum systems, called Weyl semimetals. We use rigorous renormalization group methods to compute the correlation functions of the system. In particular, I will report a result about the algebraic decay of the averaged two-point correlation function, compatible with delocalization. Our method is based on a rigorous implementation of RG, reminiscent of the Gawedzki-Kupiainen block spin transformations; the main technical novelty is the multiscale analysis of Gaussian measures with purely imaginary covariances.

Joint work with Luca Fresta and Marcello Porta.

In 1996 Rinat Kashaev introduced a new way to write the star-triangle move (or Yang-Baxter equations) of the Ising model as a single polynomial relation. In 2013 Richard Kenyon and Robin Pemantle understood that this relation could be seen as a kind of spatial recurrence. I will show how the iteration of Kashaev’s recurrence can be related the combinatorics of a loop model with two colors, that was introduced for different reasons by Ole Warnaar and Bernard Nienhuis in 1993. It is also possible to couple this loop model with known relatives of the Ising model: a dimer model and a six-vertex model. Finally I will show a few limit shape phenomena.

I will show that for any integer N, there are only finitely many cuspidal algebraic automorphic representations of GL_m over Q whose Artin conductor is N and whose « weights » are in the interval {0,…,23} (with m varying). Via the conjectural yoga between geometric Galois representations (or motives) and algebraic automorphic forms, this statement may be viewed as a generalization of the classical Hermite-Minkowski theorem in algebraic number theory. I will also discuss variants of these results when the base field Q is replaced by an arbitrary number field.

We describe dynamical systems arising from the classification of locally homogeneous geometric structures on manifolds. Their classification mimics the classification of Riemann surfaces by the Riemann moduli space – the quotient of Teichmüller space by the properly discontinuous action of the mapping class group. However, this action is misleading: mapping class groups generally act chaotically on character varieties. For fundamental examples, these varieties appear as affine cubics, and we relate the projective geometry of cubic surfaces to dynamical properties of the action.

Let Γ be a group acting by isometries on a proper metric space (X,d). The critical exponent δΓ(X) is a number which measures the complexity of this action. The critical exponent of a subgroup Γ'<Γ is hence smaller than the critical exponent of Γ. When does equality occur? It was shown in the 1980s by Brooks that if X is the real hyperbolic space, Γ’ is a normal subgroup of Γ and Γ is convex-cocompact, then equality occurs if and only if Γ/Γ’ is amenable. At the same time, Cohen and Grigorchuk proved an analogous result when Γ is a free group acting on its Cayley graph.
When the action of Γ on X is not cocompact, showing that the equality of critical exponents is equivalent to the amenability of Γ/Γ’ requires an additional assumption: a « growth gap at infinity ». I will explain how, under this (optimal) assumption, we can generalize the result of Brooks to all groups Γ with a proper action on a Gromov hyperbolic space.
Joint work with R. Coulon, R. Dougall and B. Schapira.

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.

 

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.

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.