Grassmannians and flag varieties are important moduli spaces in algebraic geometry. Quiver Grassmannians are generalizations of these spaces arise in representation theory as the moduli spaces of quiver subrepresentations. These represent arrangements of vector subspaces satisfying linear relations provided by a directed graph.
  The methods of tropical geometry allow us to study these algebraic objects combinatorially and computationally. We introduce matroidal and tropical analoga of quivers and their Grassmannians obtained in joint work with Alessio Borzì and separate joint work in progress with Giulia Iezzi; and describe them as affine morphisms of valuated matroids and linear maps of tropical linear spaces.

In this talk, the relative entropy between states of the CAR algebra will be considered. One of the states (the "reference state") is a KMS state with respect to a 1-parametric automorphism group induced by a unitary group on the 1-particle Hilbert space, and the other is a multi-excitation state relative to the reference state. In the case that the reference state is quasifree, a compact formula for the relative entropy can be derived. The results are taken from joint work with Stefano Galanda and Albert Much (MPAG 26 (2023) 21; arXiv:2305.02788 [math-ph]). Time permitting, results on work in progress (with Harald Grosse and Albert Much) will be mentioned on the relative entropy for coherent states of the Rieffel-Moyal deformed quantized Klein Gordon field on algebras of wedge regions on Minkowski spacetime.
Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

Let C be a smooth complex curve of genus 2. We construct double Kodaira fibrations with small signature as (branched) Galois covers of C × C, whose Galois group is extra-special of order 32. This is based on joint papers with A. Causin and P. Sabatino.

Given a 2-dimensional closed surface, we will show that the Moser-Trudinger functional has critical points of arbitrarily high energy. Since the functional is too critical to directly apply to it the known variational methods (in particular the Struwe monotonicity trick), we will approximate it by subcritical ones, which in fact interpolate it to a Liouville-type functional from conformal geometry. Hence our result will also unify and give common results for these two apparently unrelated problems. This is a joint work with F. De Marchis, A. Malchiodi and P-D. Thizy.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

The aim of this talk is to provide a uniform and intuitive description of the cohomology ring of arrangement complements. We introduce complex hyperplane arrangements and state the Orlik-Solomon theorem (1980). Then, we describe the real case and the Gelfand-Varchenko ring (1987). We define toric arrangements and present their cohomology ring (De Concini, Procesi (2005) and Callegaro, D'Adderio, Delucchi, Migliorini, and P. (2020)). Finally, we show a new technique to prove the Orlik-Solomon and De Concini-Procesi relations from the Gelfand-Varchenko ring. The technique applied to abelian arrangements provides a presentation of their cohomology. This is work in progress with Evienia Bazzocchi and Maddalena Pismataro. This talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).

The Leray-Serre and the Eilenberg-Moore spectral sequence are fundamental tools in algebraic topology for computing cohomology. We describe the relationship between these two spectral sequences when both of them share the same abutment. There exists a joint tri-graded refinement of the Leray-Serre and the Eilenberg-Moore spectral sequence. This refinement involves two more spectral sequences which abut to the initial terms of the Leray-Serre and the Eilenberg-Moore spectral sequence, respectively. We show that one of these always degenerates from its second page on and that the other one satisfies a local-to-global property: it degenerates for all possible base spaces if and only if it does so when the base space is contractible. When the preludes degenerate early enough, they appear to echo Deligne's decalage machinery, but in general, this is an illusion. We will discuss several principal fibrations to illustrate the possible cases and give applications, in particular, to Lie groups, group extensions, and torus bundles. This talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006).

Conformal field theory can be defined using the associativity and the commutativity of the product of quantum fields (operator product expansion). An important difference between conformal field theory and classical commutative associative algebra is "the divergence" arising from the product of quantum fields, a difficulty that appears in quantum field theory in general. In this talk we will explain that in the two-dimensional case this algebra can be controlled by "the representation theory" of a vertex operator algebra and that the convergence of quantum fields is described by the operad structure of the configuration space.

It is a classical problem in algebraic geometry, dated back to Riemann, to characterize Jacobians of smooth projective curves among all principally polarized abelian varieties. In 2008, Casalaina-Martin proposed a conjecture in terms of singularities of theta divisors. In this talk, I will present a partial solution of this conjecture using Hodge theory and D-modules. We also show that this conjecture can be deduced from a conjecture of Pareschi and Popa on GV sheaves and minimal cohomology classes.

  Arisen in Algebraic Topology to model the up-to-homotopy associative algebra structure of loop spaces, operads can be thought of as collections of *spaces* of n-ary operations together with composition laws between them. We talk about oo-operads when these operations can be composed only 'up-to-homotopy'. When the n-ary operations organise into actual topological spaces/simplicial sets, several equivalent models for the homotopy theory of oo-operads have been developed. Of our interest is Weiss and Moerdijk's approach, where a certain category of trees replaces the simplex category, and oo-categorical methods are generalized to the operadic context. However, while the theory is well developed in the topological case, very little is known for what it concerns oo-operads enriched in chain complexes ('linear'). In this talk, we explain how the tree-like approach can be applied to the linear case. We discuss the combinatorics of trees and a Segal-like condition which allows to define linear oo-operads as certain coalgebras over a comonad. Then, by considering a category of 'trees with partitions', we realize linear oo-operads as a full subcategory of a functor category

  In a renowned series of papers, Etingof and Kazhdan proved that every Lie bialgebra can be quantized, answering positively a question posed by Drinfeld in 1992. The quantization is explicit and "universal", that is it is natural with respect to morphisms of Lie bialgebras. A cohomological construction of universal quantizations has been later obtained by Enriquez, relying on the coHochschild complex of a somewhat mysterious cosimplicial algebra. In this talk, I will review the realization of Enriquez' algebra in terms of "universal endomorphisms" of a Drinfeld-Yetter module over a Lie bialgebra, due to Appel and Toledano Laredo, and present a novel combinatorial description of its algebra structure. This is a joint work with A. Appel.