Pagina 12

24/11/23Seminario16:0017:001201 Dal PassoFrank NeumannU PaviaPreludes to the Eilenberg-Moore and the Leray-Serre spectral sequences

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).
24/11/23Seminario14:3015:301201 Dal PassoRoberto PagariaU BolognaCohomology ring of arrangement complements

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).
22/11/23Seminario16:0017:001201 Dal PassoYuto MoriwakiRiken (Wako, Japan)
Operator algebras seminar
Operator product expansion in two dimension conformal field theory

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.
21/11/23Seminario14:3016:001101 D'AntoniRuije YangHumboldt University, BerlinThe geometric Riemann-Schottky problem and Hodge theory

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.
17/11/23Seminario16:0017:001201 Dal Passo
Università di Milano "Bicocca"
Algebra & Representation Theory Seminar (ARTS)
"Universal quantizations and the Drinfeld-Yetter algebra"

  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.
17/11/23Seminario14:3015:301201 Dal Passo
Francesca PRATALI
Université Sorbonne - Paris Nord
Algebra & Representation Theory Seminar (ARTS)
"A tree-like approach to linear infinity operads"
N.B.: This talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)

  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
15/11/23Seminario15:0016:301101 D'AntoniLorenza GuerraU Roma Tor VergataOn the mod p cohomology of complete unordered flag manifolds in C^n and R^n.

Flag manifolds are topological spaces parametrizing nested subspaces in a fixed vector space. On the complete flag manifold of C^n and R^n there is a natural action of the symmetric group on n letters. In this talk I will describe the cohomology of the quotient space of this action with coefficients in prime fields of positive characteristic. After recalling the basic definitions and providing some motivation, I will recall some algebraic and combinatorial properties of the cohomology of extended symmetric powers of topological spaces. I will then apply them to the classifying spaces of wreath products and use some spectral sequence argument to determine the desired cohomology. If enough time remains, I will briefly hint at a connection with E_n operads and Atiyah and Sutcliffe’s conjecture on the geometry of point particles.
14/11/23Seminario14:3016:001101 D'AntoniErnesto MistrettaUniversità di PadovaVector Bundles, Parallelizable manifolds, Fundamental groups

We will show how some basic questions about semiampleness of vector bundles can be interpreted in a geometric way. In particular we will distinguish between two non equivalent definitions of semiampless appearing in the literature, and give a geometric interpretation considering the holomorphic cotangent bundle. We will generalize these examples obtaining a biholomorphic characterisation of abelian varieties and their quotients (called hyperelliptic varieties). In order to achieve a similar biholomorphic characterisation of parallelizable compact complex manifolds and their quotients, we will consider another basic question about semiample vector bundles. Time permitting, we will conclude with a question on fundamental groups of manifolds with semiample cotangent bundle. Part of this work is in collaboration with Francesco Esposito.
07/11/23Seminario15:0016:001101 D'AntoniAnne MoreauLaboratoire de Mathématiques d'OrsayFunctorial constructions of double Poisson vertex algebras

To any double Poisson algebra we produce a double Poisson vertex algebra using the jet algebra construction. We show that this construction is compatible with the representation functor which associates to any double Poisson (vertex) algebra and any positive integer a Poisson (vertex) algebra. We also consider related constructions, such as Poisson and Hamiltonian reductions. This allows us to provide various interesting examples of double Poisson vertex algebras, in particular from double quivers. This is a joint work with Tristan Bozec and Maxime Fairon.
07/11/23Seminario14:0015:001101 D'AntoniEmanuele MacrìLaboratoire de Mathematiques d'OrsayDeformations of stability conditions

Bridgeland stability conditions have been introduced about 20 years ago, with motivations both from algebraic geometry, representation theory and physics. One of the fundamental problem is that we currently lack methods to construct and study such stability conditions in full generality. In this talk I would present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari and Zhao. As application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces.

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