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)
TBA
TBA
