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Barcelona Algebraic Topology Group

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Friday's Topology Seminar 2018-19

Speaker: Joshua Hunt (University of Copenhagen)
Title: Lifting G-stable endotrivial modules
Place: Room Seminar C3b
Date: Friday April 12, 12h-13h

Abstract: Endotrivial modules of a finite group G are a class of modular representations that is interesting both because endotrivial modules have enough structure to allow us to classify them and because such modules give structural information about the stable module category of G. They form a group T(G) under tensor product, and Carlson and Thévenaz have classified the endotrivial modules of a p-group. We examine the restriction map from T(G) to T(S), where S is a Sylow p-subgroup of G, and provide an obstruction to lifting an endotrivial module from T(S) to T(G). This allows us to describe T(G) using only local information and to provide a counterexample to some conjectures about T(G). This is joint work with Tobias Barthel and Jesper Grodal.

See the calendar for upcoming events.

Speaker: Antonio Díaz (Universidad de Málaga)
Title: Fusion systems for profinite groups
Place: Room Seminar C3b
Date: Friday March 29, 10h-11h

Abstract: For both finite groups and compact Lie groups, there exist algebraic structures that encode their fusion patterns as well as their classifying spaces at a given prime. In this talk, I will introduce similar ideas for profinite groups and, in particular, for compact p-adic analytic groups. In particular, we will study classifying spaces and stable elements theorem for continuous cohomology. We will provide some concrete continuous cohomology computations. This is an ongoing joint work with O. Garaialde, N. Mazza and S. Park.

Speaker: Jesper Moller (University of Copenhagen)
Title: Counting p-singular elements in finite groups of Lie type
Place: Room Seminar C3b
Date: Friday January 25, 12h-13h

Abstract: Let $G$ be a finite group and $p$ a prime number. We say that an element of $G$ is $p$-singular of its order is a power of $p$. Let $G_p$ be the {\em set\/} of $p$-singular elements in $G$, i.e. the union of the Sylow $p$-subgroups of $G$. In 1907, or even earlier, Frobenius proved that $|G|_p \mid |G_p|$: The number of $p$-singular elements in $G$ is divisible by the $p$-part of the order of $G$. The number of $p$-singular elements in a symmetric group is known. In this talk we discuss the number of $p$-singular elements in a finite (untwisted) group of Lie type in characteristic $p$.
The situation in the cross-characteristic case will maybe also be considered.

 


Speaker: Letterio Gatto (Politecnico di Torino)
Title: Hasse-Schmidt Derivations on Exterior Algebras and how to use them
Place: Room Seminar C3b
Date: Friday January 18, 12h-13h

Abstract: In the year 1937, Hasse & Schmidt introduced the so-called higher derivations in Commutative Algebra, to generalize the notion of Taylor polynomial to positive characteristic. Exactly the same definition can be phrased in the context of exterior algebras, by replacing the ordinary associative commutative multiplication by the wedge product. Hasse-Schmidt derivations on exterior algebras embody a surprisingly rich theory that candidates itself to propose a unified framework for a number of theories otherwise considered distincts, such as, e.g., (quantum, equivariant) Schubert Calculus for complex Grassmannians. In the talk we shall focus on one of the simplest but most powerful tools of the theory, the integration by parts formula. It will enable us to guess the shape of the vertex operators arising in the representation theory of certain infinite dimensional Lie algebras. In spite of the fancy vocabulary used in the abstract, the talk shall be entirely self-contained and no special prerequisite, but elementary multi-linear algebra, will be required.

Speaker: Branislav Jurco (Charles University)
Title: Quantum L-infinity Algebras and the Homological Perturbation Lemma
Date: 17/9/2018
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: Quantum homotopy Lie algebras are a generalization of homotopy Lie algebras with a scalar product and with operations corresponding to higher genus graphs. We construct a minimal model of a given quantum homtopy Lie algebra algebra via the homological perturbation lemma and show that it is given by a Feynman diagram expansion, computing the effective action in the finite-dimensional Batalin-Vilkovisky formalism. We also construct a homotopy between the original and this effective quantum homotopy Lie algebra.
Speaker: Thomas Poguntke (Bonn)
Title: Higher Segal structures in algebraic K-theory
Date: 14/9/2018
Time: 12:00
Web: http://mat.uab.cat/~topalg

Speaker: Louis Carlier (UAB)
Title: Hereditary species as monoidal decomposition spaces
Date: 7/9/2018
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: Schmitt constructed an important family of combinatorial bialgebras from what he called hereditary species: they are combinatorial structures with three different functorialities. The species of simple graphs is an example. These bialgebras do not fit into the standard theory of incidence algebras of posets or categories. We show Schmitt's hereditary species induce decomposition spaces, the more general homotopical framework for incidence algebras and Möbius inversion introduced recently by Gálvez, Kock, and Tonks, and we show that the bialgebra associated to a hereditary species is the incidence bialgebra of the corresponding monoidal decomposition space.