Vorträge in der Woche 30.06.2025 bis 06.07.2025

Montag, 30.06.2025: On spectral theory of the modular surface

Ksenia Fedosova

VOR-KOLLOQUIUM: The modular surface, formed as a quotient of the upper half-plane by the action of PSL(2,Z), plays a central role in modern number theory and mathematical physics. In this preparatory talk, we introduce the spectral theory of the Laplace operator on the modular surface. We will discuss Eisenstein series, their Fourier expansions, their role in both analysis and number theory, and the connection to the Riemann zeta function.

Montag, 30.06.2025: Inhomogeneous Laplace equation on modular surface

Ksenia Fedosova

Low-energy expansion of scattering amplitudes of elementary particles arising in string theory are often linked to an inhomogeneous Laplace equation. For example, for the IIB string theory in 10+1 dimensions and scattering of 4-gravitons, the equation is the inhomogeneous Laplace equation on the modular surface, where the inhomogeneous part contains products of Eisenstein series. In this talk, we discuss how to solve this partial differential equation. Additionally, from asymptotic behavior of its solutions we obtain exact identities for infinite convolution sums of even divisor functions weighted by Laurent polynomials with logarithms.

Dienstag, 01.07.2025: Derivation of the Effective Dynamics for the Bose Polaron

Siegfried Spruck (Tübingen)

We consider the dynamics of a dense quantum gas consisting of $N$ bosons evolving in $\mathbb{R}^3$ in the presence of an impurity particle in the mean-field scaling with initially high density $\rho$ and large volume $\Lambda$ of the gas. In the initial state of the system almost all bosons are in the Bose-Einstein condensate, with a few excitations. For this system we derive from the microscopic dynamics in the limit of large densities and volumes the effective description by a quantum field theory modelled by the Bogoliubov-Fröhlich Hamiltonian which describes a quasi- particle, the Bose polaron.

Donnerstag, 03.07.2025: Sharp lower bound for the charged Hawking mass in the electrostatic space

Prof. Dr. Benedito Leandro Neto (Universidade de Brasilia)

We prove sharp lower bounds for the charged Hawking mass of stable surfaces in electrostatic space-times in various contexts. An upper bound for the genus of stable surfaces in the electrostatic system is provided. We also study the positivity for the charged Hawking mass of a minimal surface with index one in the electrostatic space-times. A criterion for a CMC surface in the Reissner-Nordstrom deSitter space to be stable is presented.

Donnerstag, 03.07.2025: Voevodsky’s derived category of motives II

Felix Röhrle