Talks and Abstracts (tentative)

Nalini Anantharaman: From quantum chaos to random hyperbolic surfaces

A hundred years ago, Einstein wondered about quantization conditions for classically ergodic systems. However, a mathematical description of the spectrum of Schrödinger operators associated to ergodic classical dynamics is still missing today. The main paradigm where these questions are studied in mathematics concerns the spectrum of the laplacian on a hyperbolic manifold. I will review the 3 main conjectures in quantum chaos, asking to show universal behaviour for eigenvalues and eigenfunctions of these operators.

The explosion of recent results on the universality of the spectrum of random graphs gives the hope that introducing randomness may help to make progress understanding the issues of quantum chaos.

I will, in particular, introduce models of random hyperbolic surfaces, discuss Mirzakhani's work and a few recent results concerning the spectrum of the laplacian of random hyperbolic manifolds.

Mario Berta: Entropy for multipartite quantum states

From classical to quantum entropy inequalities.

Serena Cenatiempo: Low-Temperature Phases of Dilute Bose Gases

Dilute Bose gases are unique quantum systems that exhibit a fascinating low-temperature phase known as a Bose–Einstein condensate. In this talk, we will discuss the challenge of developing rigorous mathematical models that explain how this macroscopic phase emerges from the system’s underlying microscopic description, with a focus on advances made in the field over the past two decades.

Martin Fraas: On Fractions in Fractional Quantum Hall Effect

The quantum Hall effect refers to the behavior of Hall (transversal) conductance at low temperatures. As observed in early 80's, whenever the material is insulating, the Hall conductance has a fractional value. The Hall conductance cannot take any fractional value, for example it can take value 1/3 but not 1/2. Over the years theoretical and mathematical physicist developed many theories aimed to explain which values are allowed. In this talk, I will first give an overview of various approaches and then focus on what is known for quantum lattice models. 

Rupert Frank: Weyl’s law for the eigenvalues of the Laplacian

A classical topic in spectral theory is Weyl’s law describing the asymptotics of the eigenvalues of the Laplacian on a bounded open set. We are interested in these asymptotics in low regularity situations. Both in the Dirichlet and in the Neumann case we show two-term asymptotics for Riesz means of any positive order under the assumption that the boundary is Lipschitz continuous. For convex sets we obtain universal, nonasymptotic bounds. Tools in our proof are universal heat kernel bounds, as well as Tauberian Remainder Theorems.

Ludovico Lami: On the ultimate limits of entanglement testing and the generalised quantum Stein’s lemma

The task of entanglement testing consists in discriminating a given entangled state from the set of all separable (i.e. un-entangled) states. The ultimate limits of this task, as measured by the largest type-1 and type-2 error exponents that are achievable with arbitrary measurements, can be calculated with the help of two key results in entanglement theory, the generalised Sanov theorem and the generalised quantum Stein’s lemma, respectively. I will survey the recent proofs of these statements in [Lami/Berta/Regula, arXiv:2408.07067, 2024] and [Lami, IEEE Trans. Inf. Theory, 71(6), 2025], highlighting their connections with the theory of entanglement manipulation and the notion of asymptotic reversibility.

Amanda Young: A bulk gap in the presence of edge states for a truncated Haldane pseudopotential

We discuss the bulk gap for a truncated 1/3-filled Haldane pseudopotential for the fractional quantum Hall effect in the cylinder geometry. In the case of open boundary conditions, a lower bound on the spectral gap (which is uniform in the volume and particle number) accurately reflects the presence of edge states, which do not persist into the bulk. A uniform lower bound for the Hamiltonian with periodic boundary conditions is also obtained. Both of these bounds are proved by identifying invariant subspaces to which spectral gap and ground state energy estimating methods originally developed for quantum spin Hamiltonians are applied. Customizing the gap technique to the invariant subspace, however, we are able to avoid the edge states and establish a more precise estimate on the bulk gap in the case of periodic boundary conditions. The same approach can also be applied to prove a bulk gap for the analogously truncated Haldane pseudopotential with maximal half filling, which describes a strongly correlated system of spinless bosons in a cylinder geometry. This is based of joint work with S. Warzel.