# Sala Seminari (Dip. Matematica)

## Morfismo traccia e deformazioni di coppie (varieta', fascio coerente).

Dato un fascio coerente F su una varieta' liscia proiettiva X, viene determinata l'algebra di Lie differenziale graduata che controlla le deformazioni della coppia (X,F). Viene poi introdotto un morfismo di traccia tra opportuni fasci di operatori differenziali mediante il quale viene esteso un classico risultato di Artamkin sulle deformazioni di fasci.

## Equivariant hyperbolization and applications

Given a closed 3-manifold Y with the action of a finite group G, we show how to find a closely related hyperbolic G-manifold Y’. The two manifolds are related by an invertible equivariant homology cobordism; a cobordism from Y to Y’ is invertible if there is a second cobordism from Y’ to Y such that the union of the two along Y’ is a product cobordism. I will give a collection of applications in 3 and 4-dimensional topology. (Joint work with Dave Auckly, Hee Jung Kim, and Paul Melvin.)

## The compressible Euler equations: Riemann problems and non-uniqueness

## Rohlin’s theorem and non-simply connected 4-manifolds

Most applications of gauge theory in 4-dimensional topology are concerned with simply-connected manifolds with non-trivial second homology. I will discuss the opposite situation, first describing the classical Rohlin invariant for manifolds with first homology = Z and vanishing second homology. I will give an interpretation in terms of a Seiberg-Witten theory, with an unusual index-theoretic correction term. I will discuss recent work with Jianfeng Lin and Nikolai Saveliev giving a new formula for this invariant in terms of monopole homology.

## A weak universality result for the parabolic Anderson model

*Abstract*: We consider a class of nonlinear population models on a two-dimensional lattice which are influenced by a small random potential, and we show that on large temporal and spatial scales the population density is well described by the continuous parabolic Anderson model, a linear but singular stochastic PDE. The proof is based on a discrete formulation of paracontrolled distributions on unbounded lattices which is of independent interest because it can be applied to prove the convergence of a wide range of lattice models. This is joint work with Jörg Martin.

## On corks with low shadow-complexity

A cork is a contractible Stein domain that gives arise to exotic pairs of 4-manifolds. The first example was found by Akbulut. It is known that any two exotic, simply-connected, closed 4-manifolds are related by a cork twist. We show that there are no corks having shadow-complexity zero. We also show that there are infinitely many corks having shadow-conplexity 1 and 2.

## Binomial edge ideals

ABSTRACT:

In the last decades the connections between commutative algebra and combinatorics have been extensively explored. It is interesting to study classes of ideals in a polynomial ring associating with them combinatorial objects, such as simplicial complexes, graphs, clutters or polytopes.

In this talk we are interested in the so-called binomial edge ideals, which are ideals generated by binomials corresponding to the edges of a finite simple graph G. They can be viewed as a generalization of the ideal of 2-minors of a generic matrix with two rows.

## The quantum conditional Entropy Power Inequality

## Chemical reaction networks: deterministic and stochastic models

Chemical reaction networks are mathematical models used in

biochemistry, as well as in other fields. Specifically, the time

evolution of a system of biochemical reactions are modelled either

deterministically, by means of a system of ordinary differential

equations, or stochastically, by means of a continuous time Markov

chain. It is natural to wonder whether the dynamics of the two modelling

regimes are linked, and whether properties of one model can shed light

on the behavior of the other one. In this talk some connections will be