# Seminars

## TODAY

L functions from fermionic traces of local ${\mathbb P}^2$

**Speaker:**

*Veronica Fantini*- IHES

**Seminar Geometry, Arithmetic and Differential Equations of Periods (GADEPs)**

**Abstract: **In the framework of the Topological String/Spectral Theory correspondence, fermionic spectral traces conjecturally encode non perturbative contributions of the total string grand potential for local Calabi--Yau 3-folds. The resurgence structure of the first fermionic spectral trace of local P^2 in the strong and weak coupling regime have a rich arithmetic structure. In particular, the Stokes constants are the coefficients of two L functions and their generating functions are quantum modular forms. This talk is based on a joint project with C. Rella arXiv:2404.10695, and 2404.11550. meet.google.com/hzq-vads-ntx

## UPCOMING

Some applications of Poincaré limit theorem to rigidity problems in Gaussian half-space.

**Speaker:**

*Levi Lopes de Lima*- UFC

**Differential Geometry**

**Abstract: **Abstract: A famous result in Probability Theory, usually known as "Poincaré limit theorem", states that the standard Gaussian measure can be arbitrarily approximated by orthogonal projections of uniformly distributed measures on certain spheres of sufficiently large radius and dimension. Appropriate interpretations of this remarkable convergence (which lies at the heart of the celebrated "concentration of measure phenomenon") open up the possibility, already explored in other occasions, of transplanting geometric problems and techniques from these spheres to their "Poincaré limit", which turns out to be the ubiquitous Gaussian space. In this talk I will explain how this heuristics applies to certain Pohozhaev and Reilly type identities (established in the spherical setting by Ciraolo-Vezzoni and Qiu-Xia, respectively), so that, as a consequence, the corresponding rigidity results for domains in a Gaussian half-space, including an Alexandrov-type soap bubble theorem, can be verified without much difficulty. Based on arXiv:2409.03554.

Brody lemma and algebraic hyperbolicity

**Speaker:**

*Misha Verbitsky*- IMPA

**Geometric structures on manifolds**

**Abstract: **Fix a complete metric of constant negative curvature 1 on a disk $D$. Pseudometric on a set $S$ is a function $S\times S\to {\Bbb R}^{\geq 0}$ which satisfies the same axioms as metric with exception of strict positivity. Kobayashi pseudometric on a complex manifold $M$ is the largest pseudometric such that any holomorphic map $D \to M$ is 1-Lipschitz. A manifold is called Kobayashi hyperbolic if its Kobayashi pseudometric is strictly positive. Brody lemma claims that any compact Kobayashi non-hyperbolic manifold $M$ contains an entire curve, that is, a non-constant holomorphic map ${\Bbb C} \to M$. I will prove Brody lemma; the same argument relates hyperbolicity to algebraic hyperbolicity, showing that any Kobayashi hyperbolic manifold is algebraically hyperbolic.