Vorträge in der Woche 06.07.2026 bis 12.07.2026

Dienstag, 07.07.2026: Dr. Moritz Zehnder

Mathematical giftedness: Looking back and looking ahead

The first part of the talk presents the historical development of the concept of giftedness and of theoretical models of giftedness on the basis of a model of mathematical giftedness. Given the central role of learning in the development of giftedness, the second part of the talk focuses on current research concerning the design of learning environments for mathematically gifted students. In addition to providing an overview of the current state of research, the lecture introduces two research projects aimed at generating new insights into both the learning processes of mathematically gifted students and the characteristics of learning environments that support their development.

Donnerstag, 09.07.2026: Killing and homothetic initial data for general hypersurfaces

Dr. Gabriel Sanchez Perez (University of Salamanca)

Donnerstag, 09.07.2026: Two-resolvent local laws with decorrelation in energy space

Oleksii Kolupaiev (IST Austria)

Let $G_1(w_1), G_2(w_2)$ be resolvents of two large random matrices. For a broad class of mean field models it is known that $G_1(w_1)G_2(w_2)$ concentrates around a deterministic matrix, with fluctuations controlled from above by $\eta:=\min\{|\Im w_1|,|\Im w_2|\}$. Such estimates are known as two-resolvent local laws. In this talk, I will discuss both Hermitian and non-Hermitian random matrix ensembles. For each setting, I will present an improved two-resolvent local law whose control parameter captures the dependence on $w_1-w_2$ and other model parameters in an optimal way. I will then discuss several applications of these results, focusing on the recent proof of hyperuniformity for the eigenvalue process of a non-Hermitian matrix with independent identically distributed entries. The talk is based on several recent joint works with G. Cipolloni, L. Erd{\H o}s, J. Henheik and Z. Bao.

Freitag, 10.07.2026: The Nash embedding theorems

Dr. Rodrigo Avalos (Universität Tübingen)

The objective of this talk is to provide a review of J. Nash’s embedding theorems in the context of Riemannian geometry. The conclusion of Nash’s work is that any Riemannian manifold can be isometrically embedded in a Euclidean space of sufficiently large dimension. This result constitutes a cornerstone in Riemannian geometry, closing the long-standing problem of showing that the class of intrinsically defined Riemannian manifolds is the same as that given by the Riemannian submanifolds of Euclidean space. Moreover, historically, these results provided deep connections with the theory of partial differential equations and led to analytic developments with a broad scope of applications. In this setting, the goal for this talk is to contextualise the problem and its historical relevance; review prior results before Nash’s work; stress the difficulties faced by the general problem for smooth metrics; showcase the key ideas from Nash to overcome these difficulties; and comment on recent developments and open problems related to Nash’s theorems. These last developments include the search for the optimal codimensions for the isometric embedding and extensions to semi-Riemannian geometry