The seminars take place on Wednesday afternoon, usually at 14:30 Italian time.
November 5th, 2025
Phan Thành Nam (Ludwig-Maximilians Universität München)
Weyl's law: from music to quantum mechanics
I will discuss the spectral theory of Schrödinger operators, ranging from the general question “Can we hear the shape of a drum?” to semiclassical approximations. In particular, I will focus on Weyl’s law and explore several open problems, including the Lieb–Thirring conjecture (on the semiclassical constant), the Pólya conjecture (on the first-order term of Laplacian eigenvalues), Weyl’s conjecture (on the second-order term of Laplacian eigenvalues), and the Hardy–Landau conjecture (related to the Gauss circle problem).
December 17th, 2025
Gian Michele Graf (ETH Zürich)
An elementary derivation of the periodic table of topological matter
Band insulators and superconductors are of topological interest, depending on the dimension of physical space and on their symmetry classes. Within the context of the independent particle approximation, their topological content is summarised by a periodic table (due to Kitaev and precursors) that lists the index groups for each dimension and each of 10 classes. Various derivations of the table have been provided. The talk is about one more, prompted by the striking feature that groups are constant along the diagonals of the table. That observation calls for a corresponding proof, which will be provided by an isomorphism between groups that are diagonal neighbours. The details of the isomorphisms depend on the pair of classes involved. For instance, if the domain of that map relates to a non-chiral class (and hence the codomain to a chiral class in the next lower dimension), the map itself can be understood quite simply by way of an analogy: A real bundle on a circle can be pictured as a strip, either as a Möbius strip or an ordinary one. The isomorphism is the one mapping the bundle to the clutching map that comes from cutting the circle. (Joint work with F. Santi).
January 7th, 2026
Pedro Caro (BCAM)
An inverse problem for data-driven prediction in quantum mechanics
Data-driven prediction in quantum mechanics consists in providing an approximative description of the motion of any particles at any given time, from data that have been previously collected for a certain number of particles under the influence of the same Hamiltonian. The difficulty of this problem comes from the ignorance of the exact Hamiltonian ruling the dynamic. In order to address this problem, Alberto Ruiz and I have formulated an inverse problem consisting in determining the Hamiltonian of a quantum system from the knowledge of the state at some fixed finite time for each initial state. We focus on the simplest case where the Hamiltonian is given by −∆ + V , where the electric potential V is non-compactly supported. During the talk I will present several uniqueness results for time-dependent potentials V = V(t, x) and stationary potentials V = V(x), and the difference between them. Roughly speaking, these results are uniqueness theorems, that explain why the Hamiltonians ruling the dynamics of all quantum particles are determined by the corresponding initial and final states of all these particles. As a consequence, one expects to be able to solve the data-driven prediction problem in quantum mechanics. The theorems I will discuss are the results of collaborations with Alberto Ruiz, and Manuel Cañizares, Ioannis Parissis and Athanasios Zacharopoulos.
February 25th, 2026
Antti Knowles (Université de Genève)
Localization and delocalization in random graphs
A disordered quantum system is mathematically described by a large Hermitian random matrix. One of the most remarkable phenomena expected to occur in such systems is a localization-delocalization transition for the eigenvectors. Originally proposed in the 1950s to model conduction in semiconductors with random impurities, the phenomenon is now recognized as a general feature of wave transport in disordered media, and is one of the most influential ideas in modern condensed matter physics. A simple and natural model of such a system is given by the adjacency matrix of a random graph. In this talk, I review recent results on the localization and delocalization for the Erdös-Renyi model of random graphs. In the first part of the talk, I explain the emergence of fully localized and fully delocalized phases, which are separated by a mobility edge. In the second part of the talk, I explain how optimal delocalization bounds can be obtained using a dynamical Bernoulli flow method. Based on joint work with Johannes Alt, Raphael Ducatez, and Joscha Henheik.
March 25th, 2026
Clotilde Fermanian Kammerer (Université d'Angers)
Semiclassical propagators and systems of PDEs
In this talk, we will describe different phenomena that arise when analyzing systems of coupled semiclassical PDEs. We will discuss approximations of the propagator in the semiclassical limit, methods and example.
April 8th, 2026
Yu Deng (University of Chicago)
Mathematical theory of wave turbulence
The theory of wave turbulence describes the statistical behavior of wave interactions (as in nonlinear dispersive equations) in the "large box" kinetic limit. The central concept of the subject is the wave kinetic equation, which has wide applications in different areas of physics and science. In this talk we will discuss recent progress on derivation of wave kinetic equations for various dispersive models. This is based on joint works with Zaher Hani (University of Michigan).
May 20th, 2026
Marcello Porta (SISSA)
Validity of linear response for gapless many-body quantum systems
In this talk I will discuss the transport properties of interacting fermionic lattice models, at low temperature, exposed to external perturbations slowly varying in space and in time. For weak enough perturbations, the linear response of physical observables is typically computed using Kubo formula, obtained truncating the Duhamel expansion for the time-dependent dynamics at first order. A first nontrivial question is to explicitly compute Kubo formula, and to understand its dependence on the parameters of the model. A second, complementary question is to understand the rigorous validity of Kubo formula, namely to prove that all higher order corrections in the Duhamel expansion are subleading, uniformly in the size of the system. Both questions become particularly hard for gapless systems, which are relevant for describing metals. In this talk, I will present a rigorous derivation of Kubo formula for interacting one-dimensional Fermi systems, in the absence of a spectral gap. Our result provides an explicit expression for the linear response of the current and of the density operators, as well as rigorous quantitative estimates for all higher order corrections. The proof is based on the combination of a number of ingredients: the rewriting of the real-time Duhamel expansion in terms of Euclidean correlation functions; the renormalization group analysis of Euclidean multipoint correlation functions, which provides sharp estimates for such correlations on large scales; the use of conservation laws and of the associated Ward identities, to determine the explicit form of the leading term, and to prove a key cancellation for all higher order terms which reproduces a prediction of bosonization. Based on a joint work with Giuseppe Scola and with Harman Preet Singh.
