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Barcelona Algebraic Topology Group

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Friday's Topology Seminar 2016-2017

Speaker: Ramón Flores (Universidad de Sevilla)
Title: Espacios clasificadores de grupos de trenzas.
Place: Seminari C1/366
28th July at 10:00.
En esta charla mostraremos cómo se puede calcular la dimensión del espacio clasificador de los grupos de trenzas respecto de la familia de grupos virtualmente cíclicos. Las herramientas utilizadas incluyen el modelo de Lück-Weiermann de estos espacios, la clasificación de trenzas de Nielsen-Thurston, y resultados homológicos sobre los conmensuradores de los subgrupos cíclicos.

See the calendar for upcoming events.

Speaker: Albert Ruiz (UAB)
Title: On the classification of p-local compact groups over a fixed discrete p-toral group.
Place: Seminari C3b/158
15th September at 12:00
p-local finite groups where defined by Broto-Levi-Oliver as a generalization of finite groups studied at a prime p. Later on, the same authors, defined p-local compact groups as a generalization of compact Lie Groups at a prime p and p-compact groups.
Examples of p-local finite groups which do not correspond to finite groups are known for every prime number p. In the infinite case, very few cases which do not correspond to p-compact groups have been studied. In this talk we will see a classification of p-local compact groups over some special discrete p-toral groups (joint work with Bob Oliver) which include a family of p-local compact groups which are not p-compact groups (joint work with Alex González and Toni Lozano).


Speaker: Carlos Giraldo (Universidad del Rosario, Colombia)
Title: Minimalidad en diagramas de espacios
Place: Seminari C3b1/158
21th July at 10:30.
En esta charla proponemos un concepto de minimalidad en diagramas de espacios indexados por una EI-categoría con un número finito de objetos. Al igual que en el caso clásico podremos relacionar diagramas de fibraciones minimales con haces fibrados y lograr así un teorema de clasificación para diagramas de fibraciones, cuyo espacio base es un diagrama constante. En el caso de base no constante nos enfrentamos a problemas que exigen reformular el concepto de haz fibrado. Mostraremos ( con Carles Broto).

Speaker: Alex González
Title: Fantastic partial groups and where to find them.
Place: Seminari C3b/158
21th July at 12:00.
Explicaré com modelitzar grups p-locals compactes i grups algebraics finits mitjançant localities, i mostraré algunes de les construccions que es poden fer amb aquests models, com per exemple el quocient d'una locality per un subgroup parcial normal. Aquests resultats formen part de la feina realitzada a Kansas State University amb l'Andy Chermak.


Speaker: Xingzhong Xu 
Title: A note on Oliver's p-group conjecture
Place: Seminari C3b/158
30th June at 12:00.
In this talk, I will focus on Oliver's p-group conjecture, and I will discuss some results about this problem

Speaker: David White (Denison University)
Model Categories and the Grothendieck Construction
Place: Seminari C3b
23rd June at 12:00.
I will report on joint work with Michael Batanin studying the homotopy theory of the Grothendieck construction, given a category B and a functor F from B to CAT. From the Grothendieck construction we produce a “horizontal" model structure on the base B and “vertical” model structures on the fibers F(b). I will focus on examples, including pairs (R,A) where R is a (commutative) monoid and A is an R-module, pairs (P,A) where P is a (symmetric or non-symmetric) colored operad and A is a P-algebra, and pairs (T,A) where T is a 2-monad on Cat with rank and A is a T-algebra. I will also discuss how to get a semi-model structure under extremely general conditions. Additionally, we study when these model structures are left proper, and when a weak equivalence in B gives rise to a Quillen equivalence of fibers. Applications include change of rings, rectification of operad-algebras, strictification for categorical structures, and preservation of algebraic structure under left Bousfield localization. I will also explain the relationship of this work to that of Harpaz and Prasma.

Speaker: Rune Haugseng (Københavns Universitet)
Title: Enriched infinity-operads
Place: Seminari C3b
23rd June at 14:30.
I will discuss joint work with Hongyi Chu setting up foundations for a theory of enriched infinity-operads. In cases of interest such as spectral operads and dg-operads this recovers the usual homotopy theory of enriched operads, but is far better behaved - for example, there are natural infinity-categories of algebras over enriched infinity-operads. Time permitting, I will also discuss work in progress describing enriched infinity-operads as monoids in symmetric sequences; this will hopefully be a good setting for understanding Koszul duality beyond characteristic zero.

Title : Combinatorial Dyson-Schwinger equations and polynomial functors
Speaker: Joachim Kock (UAB)
Place: Seminari C3b, Dept. Mat. UAB
Date: Friday 16 June at 12:00.
Abstract: I will explain what are the combinatorial Dyson-Schwinger equations,
first by briefly outlining their role in quantum field theory, and then
by showing that they can be understood in terms of some elementary
category theory. The formulation of the equations by Bergbauer and
Kreimer starts with a Hopf algebra (of trees or graphs), and a
collection of Hochschild 1-cocycles, and one of their main theorems is
that the solution spans a sub Hopf algebra isomorphic to the Faa di
Bruno Hopf algebra (the Hopf algebra dual to composition of formal power
series). The new categorical interpretation starts very abstractly with
a polynomial fixpoint equation of groupoids, nothing more. I will
explain how this data canonically generates trees, Hopf algebras,
Hochschild 1-cocycles, and Faa di Bruno formula. This reveals close
connections with inductive data types in program semantics, which I can
explain if time permits, or perhaps another time.

Speaker: David Spivak (MIT)
An Operadic Approach to Compositionality
Time and place:
Monday 12 June, 15h30, CRM aula petita
In this talk, I'll discuss how operads, algebras,
and especially the maps between them, naturally express a
notion of compositionality. I'll give several examples, both
from pure math and from concrete applications.

Title : Invariants of rational homology 3-spheres.
: Ricard Riba
Place: Seminari C3b
Date: 2nd June at 11:30.
Abstract: In this talk we will give an obstruction of Perron's conjecture which asserts that a certain function is a well defined invariant of rational homology 3-spheres. We will start giving some definitions and general results about the mapping class group.
Next, following the ideas of the paper 'Trivial cocycles and invariants of homology 3-spheres' due to W. Pitsch, we will show a new tool to get invariants of rational homology 3-spheres from a suitable family of 2-cocycles of the second cohomology group of Torelli group mod p. Finally we will give a sketch of the proof of our main theorem.
This is joint work with W.Pitsch.

Title : Incidence bicomodule structure on a bisimplicial groupoid
Speaker: Louis Carlier
Place: Seminari C3b
Date: 2nd June at 12:15.
Investigating a Rota formula connecting Möbius functions on posets given an adjunction, we extend it to locally finite categories and realize that the formula comes from a bicomodule structure.

I will first recall what are the incidence coalgebras and convolution algebras for posets and categories, then explain the formula of Rota and how to extend the proof to categories. We will see that we can express the proof using two mixed convolution products which are compatible. It turns out these exemplify a bicomodule structure induced by any double Segal space which is stable in the sense of Bergner et al., and has CULF augmentations.

Title : An elementary derivation of the Kan model structure
Speaker: Christian Sattler (University of Leeds)
Place: Seminari C3b
Date: 26th May at 12:00.
Abstract: We will sketch an elementary way to establish the classical Kan model
structure on simplicial sets, making use of the glueing construction
developed by Cohen et al. originally developed in the specific setting
of certain cubical sets. In particular, we will not use topological
spaces, minimal complexes, or any concrete model of fibrant replacement
such as Kan’s Ex^infinity functor.

Title : Bivariant elliptic theories
Speaker: David Gepner (Purdue University)
Place: Aula petita CRM
Date: 29th May at 15:30.
Abstract: We consider elliptic (co)homology as a bivariant theory defined on smooth Artin topological stacks. From this point of view, we will see that TMF admits some surprising duality properties. This is joint work with Thomas Nikolaus.

Title : Ascent for stratifications
: Natàlia Castellana
Place: Seminari C3b
Date: 19th May at 12:00.
Abstract: The main result in the paper 'Stratifying the derived category of cochains on BG for G a compact Lie group' by Benson and Greenlees states that for a connected compact Lie group G, the derived category D(C^*(BG)) is stratified by the canonical action of H^*(BG). Their proof is based on descent type methods using Quillen stratification to reduce to elementary abelian p-subgroups.
In this talk we present a new strategy based on a new ascent approach to stratification. This proof applies to more examples like classifying spaces of finite fusion systems and p-compact groups. This is a joint work with Tobias Barthel, Drew Heard and Gabriel Valenzuela.

Title : Variedades de conjugación y pureza homológica.
: Wolfgang Pitsch
Place: Seminari C3b
Date: 12th May at 12:00.
Abstract :En esta charla explicaremos el rsultado siguiente debido principalmente a N. Ricka.
Teorema: Una variedad X es de conjugación si y sólo si es homológicamente pura.
Recordemos que una variedad de conjugación es un variedad con una involución en la cual la subvariedad de puntos fijos se comporta homológicamente como la variedad ambiente, salvo por una divisón por 2 de los grados cohomológicos. La pureza homológica es una noción ligada a la estructura de la cohomología equivariante "a la Mackey" y será explicada durante la charla. Esto es un trabajo en colaboración con N. Ricka y J. Scherer

Title : Random thoughts on modular reflection groups and Kac-Moody groups
: Jaume Aguadé
Place: Seminari C3b
Date: 5th May at 12:00.
Abstract : While looking for a large family of Kac-Moody groups K such that the cohomology of BK is accessible to computations, we (Albert Ruiz and me) were learning about reflection groups in prime characteristic. These attemps to compute the cohomology of BK are used as an excuse to talk about modular reflection groups.

Title : Mixed Hodge structure, purity and formality
Speaker: Geoffroy Horel
Place: Aula del IMUB
Date: Friday April 7 at 12:00.

Abstract : It has long been observed that Hodge theory is a powerful tool for proving formality results, both in the operadic setting or in the setting of rational homotopy theory. In this talk, I will explain a "purity implies formality" result in the abstract setting of symmetric monoidal functors. I will then develop some applications to the formality of certain algebraic objects defined in the category of complex schemes, and the formality of morphisms of schemes in the sense of rational homotopy. This is joint work with Joana Cirici.

Date and Place: March 31st 12h, seminar room CRM
M. Palmer (Mathematical Institute of the University of Bonn)
Twisted homological stability, partitioned braid groups and homological representations

Abstract: In this talk we will discuss a "twisted homological stability" result for configuration spaces, and more generally for moduli spaces of disconnected submanifolds of a fixed ambient manifold. When the submanifolds are 0-dimensional, this includes as a special case the (classifying spaces of the) partitioned braid groups of a surface. We will then discuss interesting examples of twisted coefficient systems, including ones related to homological representations of surface braid groups, such as the Lawrence-Krammer-Bigelow representations. (Part of the work presented is joint with TriThang Tran.)

Date and Place: March 14th (Tuesday!) 12h, seminar room C1/028 CRM
Wolfgang Pitsch (UAB)
Variación de volumen para representaciones en SL_n(C).

Abstract: Explicaremos nuestro trabajo reciente con J. Porti en el que calculamos la variación del volumen de una representación del grupo fundamental de una variedad hiperbólica de dimensión 3 en SL_n(C).

Date and Place: February 17th 12h, seminar room C3b/158
Mitsunobu Tsutaya (Kyushu University)
Infiniteness of $A_\infty$-types of gauge groups

Let $G$ be a compact connected Lie group and $B$ be a finite CW complex. The gauge group of a principal $G$-bundle $P\to B$ is the (typically infinite dimensional) group of automorphisms. The problem we consider is how many $A_n$-equivalence classes appear among the gauge groups of the principal bundles for fixed $G$ and $B$, where the $A_n$-equivalence is in the sense of Stasheff. Crabb-Sutherland and the speaker proved that the number of the $A_n$-equivalence classes of the gauge groups is finite if $n<\infty$. The main theorem of this talk states that this finiteness typically breaks down if $B$ is an appropriate sphere and $n=\infty$. In the proof, the theory of unstable algebras plays an important role. This talk is based on a joint work with Daisuke Kishimoto.

Date and Place: February 10th 10:30h, CRM
Andrew Tonks (University of Leicester)
Restriction Species

We examine the close relation between restriction species [Schmitt] and decomposition spaces [Gálvez-Kock-T.]  A restriction species has an associated incidence coalgebra, which is an instance of the general construction of coalgebras from decomposition spaces.  We also introduce a new notion of directed restriction species, also related to decomposition spaces, as presheaves on the category of finite posets and convex inclusions.  Examples of this notion include rooted trees, directed graphs and posets, and the associated incidence algebras include the Butcher--Connes--Kreimer Hopf algebra of rooted trees.

Date and Place: February 10th 12h, CRM
María Ronco (Universidad de Talca, Chile)
Algebraic structures on the faces of graph associahedra

(joint work with S. Forcey and P. Showers) We want to motivate the study of algebraic structures on the space spanned by graph associahedra.

Combinatorial Hopf algebras are conilpotent biagebras whose underlying vector space are spanned by families of combinatorial objects, like trees, maps between finite sets, paths, etc., and whose bialgebra structure is induced by natural operations on these objects. In many examples, combinatorics provide an important tool in the study of free objects for new algebraic theories, as well as in the understanding of the relations between them. In the last years, an important amount of work has been done on generalisations of associahedra, on one hand, and on combinatorial descriptions of operads related to it, on the other side.

Graph associahedra was introduced by M. Carr and S. Devadoss. They defined a convex polytope ${\mathcal K}\Gamma$ of dimension $n$ associated to any finite simple graph $\Gamma$, with $n+1$ nodes. When $\Gamma$ is the complete graph $K_{n+1}$, the polytope ${\mathcal K}K_{n+1}$ they get the usual permutohedron of dimension $n$, while the line graph $L_{n+1}$ gives rise to the associahedron.

Our aim is to introduce algebraic theories on the spaces spanned by the faces of graph associahedra. We show that it is posible to define a coalgebra structure, as well as an assocative product, on the vector space spanned by all faces of graph associahedra, which restricts to well-know combinatorial Hopf algebras when we consider certain special families of graphs. There exists a notion of suspension of graphs which induces associative products on the vector spaces spanned by the faces of previously studied polytopes, as the stellohedra and the pterohedra. On the other hand, we prove that substitution induces another operation on graph associahedra, which gives a generalization of non-symmetric operads.

Date and Place: February 10th 15h, CRM
Mark Webber (Czech Academy of Sciences)
Operads as polynomial 2-monads

Abstract: The construction of a polynomial 2-monad from an operad will be discussed, and we describe the algebras of the 2-monads which then arise. This construction is different from the standard construction of a monad from an operad in that the algebras of our associated 2-monad are the categorified algebras of the original operad. The usual algebras are recovered in a different way as internal algebras, in a sense to be described. The usefulness of this point of view in relating operads to PROPs and to decomposition spaces will be indicated. There is an earlier construction of a polynomial monad in Set from a sigma-free operad, due to Kock and Szawiel-Zawadowski, whose relation to our construction will be explained.

Date and Place: January 27th 11:45h, seminar room C3b/158
Jesper Moller (University of Copenhagen)
Equivariant Euler characteristics of subspace posets


-\chi_r(n,2) & n=1 & n=2 & n=3 & n=4 & n=5 & n=6 \\ \hline
r=1 &   1  &                0  &                0  &                0  &                0  &                0\\
2 &        1  &               1  &               1  &               1  &               1  &               1\\
3 &        1  &               4  &              12  &              32  &              80  &             192\\
4 &        1  &              13  &             101  &             645  &            3717  &           20101\\
5 &        1  &              40  &             760  &           11056  &          140848  &         1657216\\
6 &        1  &             121  &            5481  &          176921  &         4865465  &       121991097\\
7 &         1  &             364  &           38852  &         2742192  &       160444704  &      8459901568\\
8 &         1  &            1093  &          273421  &        41843005  &      5160361501  &    567145627357\\
9 &         1  &            3280  &         1918320  &       633421856  &    163640016032  &  37253059018752\\
10 &       1  &            9841  &        13441361  &      9548966001  &   5145988736049 &2415591291535153


Date and Place: November 25th 12h, CRM room A1
Federico Cantero (Barcelona Graduate School of Mathematics)
Rational homotopy theory of Thom spaces

In this talk we give commutative differential graded algebras that are rational homotopy models of Thom spaces of oriented vector bundles. The construction uses the Euler class and a model of the base of the vector bundle. We will apply it to improve Papadima’s smoothing theory of cohomology classes.

Date and Place: September 30th 10h45, CRM room A1
Rosona Eldred (WWU Münster)
Brave new deRham Complex

In joint work with Bauer, Johnson and McCarthy, we use the recently developed unbased calculus of functors to provide a good model for the deRham complex for commutative ring spectra, conjectured originally by Waldhausen, and show it is equivalent to a cosimplicial model proposed by Rezk. In this talk, I will have time to give a quick introduction to the calculus of functors, say broadly the relationship between the deRham complex and functor calculus, and why the development of the unbased calculus was a necessary step, as well as some future directions. The calculations will be more algebraic than homotopy-theoretic, so should be comfortable for a broad audience.

Date and Place: September 30th 12h00, CRM room A1
Luis Javier Hernández Paricio (Universidad de La Rioja)
Postnikov factorizations for shape and proper homotopy theory

Abstract: (joint work with J. I. Extremiana y M. T. Rivas)
In this talk we present Postnikov sections for Brown-Grossman homotopy groups and for Steenrod homotopy groups in the category of exterior spaces. The homotopy fibre of one of these decompositions is a Elilemberg-Mac Lane with respect one of this family of homotopy groups and it has two possible non-trivial consecutive homotopy groups with respect the other family.
Some applications are given to proper homotopy theory and shape theory.