Vorträge in der Woche 23.10.2023 bis 29.10.2023

Mittwoch, 25.10.2023: The Hessian correspondence of hypersurfaces of degree 3 and 4

Javier Sendra-Arranz (Tübingen)

Let X be a hypersurface in a n-dimensional projective space. The Hessian map is a rational map from X to the projective space of symmetric matrices that sends a point p to the Hessian matrix of the defining polynomial of X evaluated at p. The Hessian correspondence is the map that sends a hypersurface to its Hessian variety; i.e. the Zariski closure of its image via the Hessian map. We study this map for the cases of hypersurfaces of degree 3 and 4. We prove that, for degree 3 and 4 the Hessian correspondence is birational, apart from degree 3 and d=1, where it is of degree two. Moreover, we provide effective algorithms for recovering a hypersurface from its Hessian variety, for degree 3 and any n, and for degree 4 and n even.

Donnerstag, 26.10.2023: The Positive Mass Theorem for Asymptotically Flat Graphs in Euclidean Space and its Stability

Florian Wendt (Universität Tübingen)

Smooth functions with certain decay properties at infinity are called asymptotically flat functions. The graph of such a function with the induced metric from Euclidean space is an asymptotically flat Riemannian manifold in the usual sense and called an asymptotically flat graph in Euclidean space. We prove the Positive Mass Theorem for the class of asymptotically flat graphs in Euclidean space in all dimensions including the rigidity case. For the proof we follow the ideas of Lam for the nonnegativity of the ADM-mass and of Huang and Wu for the rigidity case. Also, we extract from the technique of the proof a scalar curvature result, which applies for the graph of a smooth function with certain decay properties embedded either in Euclidean space or in Minkowski space. Next, we give a brief introduction into the theory of currents. For these currents Federer and Fleming have defined a norm, called the flat norm. We show a concept how to interpret an asymptotically flat graph in Euclidean space as a current, which allows us to measure the distance between two asymptotically flat graphs in Euclidean space in the flat norm. We prove a stability result with respect to that distance for the class of asymptotically flat graphs in Euclidean space. For this result we follow the approach of Huang and Lee

Donnerstag, 26.10.2023: Applying codes for ordinary and delay differential equations to problems with distributed delays

Nicola Guglielmi (Gran Sasso Science Institute, L'Aquila)

Donnerstag, 26.10.2023: Hodge theory and Hyperkahler manifolds

Angel Rios (Paris Saclay)

K3 surfaces are special types of algebraic surfaces that play a central role in several areas of mathematics, including algebraic geometry, arithmetic geometry, and mathematical physics. The celebrated Torelli theorem of Pyatetskii-Shapiro-Shafarevich-Burns-Rapoport states that two K3 surfaces can be distinguished by their weight-two Hodge structure together with the intersection product; henceforth a K3 surface can be effectively encoded by discrete data, bringing a particularly rich connection with the theory of lattices. For the higher dimensional analogues of K3 surfaces, that is Hyperkahler manifolds, there is still a Torelli theorem, but with several caveats. In this talk I will speak about how to exploit this theorem in order to decide the "shape" of a given Hyperkahler manifold. This is joint work with Benedetta Pirodi (Milano).