Giovedì 20 Febbraio 2025, dalle ore 14:00 in Aula 2109 (2° Piano) - Edificio U5, Alessio D'Alì (Politecnico di Milano) e Davide Dameno (Università degli Studi di Milano) terranno i seguenti seminari
Relatore: Alessio D'Alì
Titolo: Symmetry counts: an introduction to equivariant Hilbert and Ehrhart series
Abstract: The Ehrhart series of a lattice polytope P is a combinatorial gadget that counts the number of lattice points of P and of its dilations. The Hilbert series of a simplicial complex S counts how many monomials supported on faces of S exist in each possible degree. The aim of this talk is to introduce equivariant versions of such constructions, where we are not just interested in counting but we also want to record how the action of a finite group affects such collections of lattice points or monomials. Inspired by previous results by Betke-McMullen, Stembridge, Stapledon and Adams-Reiner, we will investigate which extra combinatorial features of the group action give rise to "nice" rational expressions of the equivariant Hilbert and Ehrhart series, and how the two are sometimes related. This is joint work with Emanuele Delucchi.
Relatore: Davide Dameno
Titolo: Riemannian twistor spaces in four-dimensional geometry: an overview and some rigidity results
Abstract: In this talk, we will briefly present some unique features of four-dimensional Riemannian Geometry and their connections with twistor theory: indeed, it is well known that four-dimensional Riemannian manifolds carry many peculiar properties, which give rise to the existence of unique special metrics (e.g. half conformally flat metrics). In their study of self-dual solutions of Yang-Mills equations, Atiyah, Hitchin and Singer adapted the celebrated Penrose’s construction of twistor spaces to the Riemannian context, showing that a Riemannian four-manifold is half conformally flat if and only if its twistor space is a complex manifold: this paved the way for the study of many other characterizations of curvature properties for Riemannian four-manifolds. After giving an overview of the basic properties of twistor spaces in the four-dimensional case, we present some new rigidity results for Riemannian four-manifolds whose twistor spaces satisfy specific vanishing curvature conditions. This is based on joint works with Giovanni Catino and Paolo Mastrolia.
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