Vorträge in der Woche 12.05.2025 bis 18.05.2025

Dienstag, 13.05.2025: Tannakian categories II

Giacomo Gavelli

Dienstag, 13.05.2025: The standard conjectures

Clemens Nollau

Mittwoch, 14.05.2025: Typical Macroscopic Behavior of Large Quantum Systems

Cornelia Vogel (Tübingen)

Donnerstag, 15.05.2025: Geometric inequalities in nonnegative Ricci and substatic manifolds

Prof. Dr. Mattia Fogagnolo (Universität Padua)

I will discuss some recent sharp bounds for mean curvature functionals and isoperimetric quotients holding in manifolds of nonnegative Ricci curvature, and more generally on substatic manifolds, that constitute a class of manifolds naturally arising in mathematical general relativity. The results presented are contained in a number of works in collaboration with Agostiniani, Borghini, Mazzieri and Pinamonti. Time permitting, I will introduce a work in progress about the large scale structure of substatic manifolds in collaboration with Santilli.

Donnerstag, 15.05.2025: A hierarchy of spectral gap certificates for frustration-free spin systems

Kshiti Rai (Leiden University)

Estimating spectral gaps of quantum many-body Hamiltonians is a challenging computational task, even under assumptions of locality and translation-invariance. Here we present a general method for obtaining lower bounds on the spectral gap of frustration-free quantum Hamiltonians. We formulate the gap certification problem as a hierarchy of optimization problems (semidefinite programs) in which the certificate—a proof of a lower bound on the gap—is improved with increasing levels. Our approach encompasses existing finite-size methods, such as Knabe’s bound, as those appear as particular possible solutions in our optimization, which is thus guaranteed to either match or surpass them. We demonstrate the method on 1D spin-chain models where we observe an improvement in the lower bound over existing finite size criteria in both the accuracy as well as the range of parameters in which a gap is detected.