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Barcelona Algebraic Topology Group

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Friday's Topology Seminar 2011-2012

July 9th, 2012, 15:30 CRM.

José Cantarero (Stanford University), "El espacio de órbitas de un grupo p-compacto".

Abstract: Para un grupo finito G, el complejo de Brown es el G-poset de cadenas de p-subgrupos no triviales ordenados por inclusión de cadenas, donde G actúa por conjugación. P. Webb conjeturó que el espacio de órbitas de esta acción es contráctil. Esta  conjetura fue demostrada por P. Symonds. Recientemente han aparecido versiones de este resultado para sistemas de fusión y grupos compactos de Lie. En esta charla se discutirá una generalización de este resultado y las técnicas de una demostración que aplica a todos estos casos y que además proporciona un nuevo resultado para grupos p-compactos y espacios de lazos finitos.

June 15th, 2012, 10:00 CRM.

Bob Oliver (Université Paris Nord), "Reduced fusion systems over 2-groups of sectional rank at most four".

Abstract: The sectional rank of a $p$-group is the largest rank of any of its abelian subquotients. One of the steps towards classifying finite simple groups was the 1974 book by Gorenstein and Harada, where they list all finite simple groups whose Sylow 2-subgroup has sectional rank at most four.
I want to describe a similar result, listing all reduced fusion systems over 2-groups of sectional rank at most four. The list contains no surprises: it consists entirely of fusion systems of simple groups on the Gorenstein-Harada list. But the method of proof seems very different, since it is based on studying the different types of essential subgroups which can occur, rather than the centralizers of involutions. This also leads to a different way of organizing the final result, which I hope will be of interest.

June 8st, 2012, 10:00 CRM.

Andy Chermak (Kansas State University), "Partial groups".

June 1st, 2012, 10:00 CRM.

Jesper Moller (Copenhagen University), "Homotopy equivalences between p-subgroup categories".

May 25th, 2012, 10:00 CRM.

Jesper Grodal (Copenhagen University), "F-isomorphism and control of p-fusion".

May 18th, 2012, 10:00 CRM.

Andy Tonks (London Metropolitan University), "L'associaedre unitari".

Resum: Els associaedres d'Stasheff parametritzen les operacions en diverses variables que apareixen de manera natural als espais de llaços d'espais connexos. Formen una opèrada topològica Ass∞ que resol l'opèrada Ass que controla els espais dotats d'una multiplicació associativa. Les cadenes cel·lulars en els associaedres formen una dg-opèrada que controla les àlgebres A∞. A les aplicacions clàssiques, hom suposava que la unitat per a la multiplicació era estricta.
Les unitats llevat d'homotopia per a àlgebres A∞ van ésser introduïdes recentment per Fukaya-Ono-Oh-Ohta en el seu treball sobre interseccions lagrangianes en teoria de Floer. Lyubashenko i Milles-Hirsch han donat descripcions de la corresponent dg-opèrada. En aquesta xerrada presentem la baula perduda: una opèrada cel·lular topològica uAss∞ d'"associaedres untiaris" que dóna una resolució de l'opèrada que controla els monoides topològics i tal que les seves cadenes cel·lulars són precisament la dg-opèrada de Fukaya-Ono-Oh-Ohta.
(treball conjunt amb Fernando Muro, pendent d'aparició a Forum Mathematicum)

May 18th, 2012, 11:20 CRM.

Jean Barge (École Polytechnique), "Syvester's matrices,  transversality's defect , ternary index. Applications".

(Joint work with Jean Lannes.) To any pair of homotopic lagrangiens in a symplectic space we associate à symetric bilinear form defined up to addition of non-degenerate form and wich is non-degenerate iff the two lagrangiens are transversal.

To any triple of lagrangiens we associate an index wich both generalise Wall index and Maslov index.

We will illustrate the use of theses invariants to produce a formula of non addivite for linking forms of 4k-1 dimensional manifold and the relationship with Wall index and not additivity of signatures .

May 4th, 2012, 10:00 CRM.

 

Frank Neumann (University of Leicester), "Homotopy Types of Stacks"
Abstract: I will give an introduction into the theory of topological
stacks and their homotopy types as recently developed by Noohi. As a main
application I will then indicate how you can associate homotopy types to
algebraic stacks over the complex numbers and how to calculate them for
interesting examples like quotient stacks or moduli stacks of principal
G-bundles over an algebraic curve.

April 20th, 2012, 10:00 CRM.
Carles Broto (UAB), "Estructura local d'espais de llaços finits ".
Abstract. Demostrem que tot espai de llaços finit té estructura de grup p-local compacte
per a qualsevol primer p. En altres paraules, admet un p-subgrup de Sylow, que
és un grup discret p-toral, i relacions de conjugació entre els seus subgrups amb
propietats comparables a les dels grups finits. Es tracta d'un treball conjunt amb
Ran Levi i Bob Oliver.

April 13th, 2012, 10:00 CRM.
Michael Joachim (University of Münster), "On the classification of twists in equivariant K-theory for proper discrete actions".

We define the equivariant K-theory twisted by a projective unitary stable bundle and we construct a universal projective unitary stable bundle for proper actions of discrete groups. The results contained extend and generalize results of Atiyah-Segal.

March 23th, 2012, 10:00 CRM.
Wolfgang Pitsch (UAB), "Quasi-morfismos sobre el mapping class groups y TQFT".

Resumen:
En esta charla presentaré parte de mi trabajo con Louis Fuanr (CNRS-Institut Fourier) sobre los quasi-morfismos que aparecen sobre el mapping class group cuando se analizan las representaciones de este que aparecen en las TQFT. Introduciremos los trabajos de Burger et iozzi sobre representaciones Zariski densas y quasi-morfismos sobre los grupos SU(p,q) y como construir concretamente estos objetos a partir de la cohomología contínua de un grupo de Lie.

March 9th, 2012, 10:00 CRM.
André Joyal (Université du Québec, Montreal), "On the operadic bar-cobar duality".

Abstract: We show that the category of operads is enriched
over the monoidal category of cooperads, and that the
latter is closed. We apply this result to the operadic
bar-cobar duality (joint work with Matthieu Anel).

March 2nd, 2012, 10:00 CRM.
Urtzi Buijs (Universitat de Barcelona), "Teoría homotópica de L-infinity-álgebras y
modelos de espacios no conexos
".

Abstract: El término "teoría de deformación'' se refiere al conjunto de cada uno de los procedimientos de deformación y perturbación, que estudian pequeñas variaciones paramétricas de una estructura matemática específica.

Un aspecto unificador, sobre un cuerpo de característica 0, es que todo problema de deformación está gobernado por una álgebra de Lie graduada diferencial (DGL).

Como puso de manifiesto por primera vez D. Quillen en 1969, el tipo de homotopía racional de un espacio X puede describirse completamente en términos puramente algebráicos, a través de la categoría homotópica de DGL's con graduación positiva, dando lugar a la teoría de homotopía racional.

A pesar de que las DGL's que surgen en teoría de deformación no son acotadas, en esta charla trataremos sobre la profunda interacción existente entre ambos campos, desarrollando una teoría homotópica consistente para L-infinity-álgebras no acotadas que se corresponderá con la noción topológica de CW-complejo no conexo.

February 24th, 2012, 10:00 CRM.
Norio Iwase (Kyushu University,Fukuoka), "A-infinity category and its realization".

Abstract: Toward a construction of new algebraic invariants, we introduce some algebraic tools, working in a slightly general setting using an idea of an internal A_\infty category, which are corresponding ideas to nerve construction and realization.

 

February 15th, 2012, 10:00 CRM.
Sarah Whitehouse (University of Sheffield), "Derived A-infinity algebras from the point of view of operads",
Abstract:A-infinity algebras arise whenever one has a multiplication which is "associative up to homotopy". There is an important theory of minimal models which involves studying differential graded algebras (dgas) via A-infinity structures on their homology algebras. However, this only works well over a ground field. Recently Sagave introduced the notion of a derived A-infinity algebra in order to extend the theory of minimal models to a general ground ring. I will put derived A-infinity algebras into the context of operads and show that the operad for derived A-infinity algebras can be viewed as a free resolution of the operad for bidgas, in the same sense that the A-infinity operad is a free resolution of the operad for dgas.
This is joint work with Muriel Livernet and Constanze Roitzheim.

February 10th, 2012, 10:00 CRM.
Fei Xu (UAB), "Cohomology of finite transporter categories".


Abstract:Finite transporter categories come from finite groups and they may be considered as generalizations of groups. Let G be a finite group and P a finite G-poset. We fix an algebraically closed field k whose characteristic is a prime p>0, dividing the order of G. We call the Grothendieck construction on the G-poset P, written as G*P, a transporter category. It is a finite category in the sense that it has finitely many morphisms (and objects). When P is a point with trivial G-action, the transporter category is just G itself. Directly from the definition, we find two natural functors P --> G*P --> G which, upon passing to their classifying spaces, gives rise to the fibration BP --> EG X_G BP --> BG, with the Borel construction as the total space. My interests in transporter categories may be roughly described as providing an algebraic cohomology theory of Borel constructions with local coefficient systems. This is entirely analogous to the group case: one can introduce group cohomology with BG, but algebraists prefer using the group algebra kG because it has much to do with the modular representation theory of G. In our situation, the algebraic counterpart to the Borel construction is the category algebra k(G*P). This algebra has a trivial module \k. It is well known that Ext*(\k,\k)=H*(EG X_G BP, k). We shall call it the ordinary cohomology ring of G*P with coefficients in k. In 1972 Quillen proved that this graded commutative ring is Noetherian, generalizing (half of) Evens-Venkov's theorem on the finite generation of group cohomology. Since finite generation of cohomology often has many useful applications, we would like to add the missing half to Quillen's generalization by showing that Ext*(M,N) is finitely generated over the ordinary cohomology ring, if M and N are finitely generated k(G*P)-modules. If time permits, we may continue to discuss some consequences in support variety theory as well as classifications of thick/localizing subcategories of certain triangulated categories coming from k(G*P).

February 3rd, 2012, 10:00 CRM. (Aquest seminari ha estat suspès)
Andy Tonks (London Metropolitan University), "L'associaedre unitari".


Resum: Els associaedres d'Stasheff parametritzen les operacions en diverses variables que apareixen de manera natural als espais de llaços d'espais connexos. Formen una opèrada topològica Ass∞ que resol l'opèrada Ass que controla els espais dotats d'una multiplicació associativa. Les cadenes cel·lulars en els associaedres formen una dg-opèrada que controla les àlgebres A∞. A les aplicacions clàssiques, hom suposava que la unitat per a la multiplicació era estricta.
Les unitats llevat d'homotopia per a àlgebres A∞ van ésser introduïdes recentment per Fukaya-Ono-Oh-Ohta en el seu treball sobre interseccions lagrangianes en teoria de Floer. Lyubashenko i Milles-Hirsch han donat descripcions de la corresponent dg-opèrada. En aquesta xerrada presentem la baula perduda: una opèrada cel·lular topològica uAss∞ d'"associaedres untiaris" que dóna una resolució de l'opèrada que controla els monoides topològics i tal que les seves cadenes cel·lulars són precisament la dg-opèrada de Fukaya-Ono-Oh-Ohta.
(treball conjunt amb Fernando Muro, pendent d'aparició a Forum Mathematicum)

January 27th, 2012, 10:00 CRM.
Jaume Aguadé (UAB), "Cohomology of Kac-Moody groups over finite fields".
Abstract: I will present some work in progress about the cohomology of Kac-Moody groups defined over finite fields. These discrete groups were defined by Tits and they generalize the discrete groups of Lie type of Chevalley.

January 13th, 2012, 10:00 CRM.
Bob Oliver (Université Paris Nord), "Chermak's proof of the existence of classifying spaces".

November 25th, 2011, 10:00 CRM.
Jérôme Scherer (EPFL, Lausanne): "Nilpotent groups up to homotopy and homotopy nilpotent groups"
Abstract: I will start by explaining how nilpotent groups can be described as algebras over a Lawvere theory. In the homotopy category, this description leads to the notion of nilpotent groups "up to homotopy", which has a very close link with Whitehead products. In a second part I will review the algebraic theory based on Goodwillie calculus which Biedermann and Dwyer constructed to define homotopy nilpotent groups. Comparison with other forms of nilpotency in homotopy theory shows this is the best way to approach this concept.

November 18th, 2011, 10:00 CRM.
Natàlia Castellana (UAB): "The BZ/p-homotopy theory of classifying spaces"
Abstract: The notion of A-homotopy was introduced by E. Dror farjoun for an arbritary connected space A. This space A ans its suspensions play the same role as spheres in classical homotopy theory. The notion of CW-complex is replaced then by the notion of being A-cellular. Fix a primer p. If we are interested in describing the p-primary part of X through its A-homotopy theory, some choices are clear. The case in which A is a Moore space was studied by Bousfield and Rodriguez-Scherer. In this work we consider the Eilenberg-MacLane space K(Z/p,1) and we restrict our attention to the study of classifying spaces of compact Lie groups. This is a joint work with Ramón J. Flores.

November 11th, 2011, 10:00 CRM.
Carles Casacuberta (UB): "Propietats de clausura de la gent local"
Resum: Mostrarem un exemple d'una localització a la categoria homotòpica dels espectres tal que la classe dels espectres locals no és tancada per desuspensió. Aquest exemple és interessant perquè no pot ser una f-localització; és a dir, no prové de cap functor coaugmentat en una categoria de models.

November 4th, 2011, 10:00 CRM.
Javier Gutiérrez (UB): "Una generalización del teorema de Okhawa para categorías de modelos combinatorias"
Resumen: El teorema clásico de Ohkawa nos dice que en la categoría homotópica de los espectros hay solo un conjunto de clases de Bousfield. Este hecho ha sido estudiado y generalizado a otras categorías trianguladas por Neeman, Iyengar-Krause y Dwyer-Palmieri. En esta charla, presentaré una generalización del teorema de Ohkawa en el contexto de las categorías de modelos combinatorias. Este es un trabajo conjunto con C. Casacuberta y J. Rosický.

October 28th, 2011, 10:00 CRM.
Carles Broto (UAB): "Self-equivalences of p-completed classifying spaces of finite groups of Lie type"
Abstract: We discuss tameness of saturated fusion systems and its relation to the existence of exotic systems. Then, we explore the case of fusion systems of finite groups of Lie type. In the cross-characteristic case we show that these fusion systems are all tame. We will also discuss the equicharacteristic case. This is joint work in progress with Jesper Møller and Bob Oliver. In particular, we aim at determine the self-equivalences of the p-completed classifying spaces of these groups.

October 14th, 2011, 09:30 CRM.
Imma Gálvez Carrillo (UPC): "Groupoids, and Faà di Bruno formulae for Green functions"
Abstract: Van Suijlekom recently discovered that the Green functions in the Hopf algebra of Feynman graphs satisfy a version of the classical Faà di Bruno formula for substitution of power series. In this talk we will revise that classical formula and several of its combinatorial incarnations and will show how the theory of groupoids can be used to give a very conceptual proof of the Faà di Bruno formulae for Green functions in the bialgebra of trees. In this framework a Green function is the homotopy cardinality of a groupoid and the Faà di Bruno formula is shown to be essentially an equivalence of groupoids (joint work with J Kock, A Tonks).

October 7th, 2011, 9:30 CRM.
Joachim Kock (UAB): "Symmetries of graphs and trees"
Abstract: In quantum field theory, trees serve to express nestings of Feynman graphs. Important aspects of the combinatorics of renormalisation are captured nicely by the Connes-Kreimer Hopf algebra of rooted trees. However, the isolated graphs do not have a physical meaning: the meaning is carried rather by the Green functions, which are sums of graphs weighted by their symmetry factors. Now the trees of Connes and Kreimer do not capture anything about symmetries of graphs. I will explain how this can be 'fixed' by using instead operadic trees, or more precisely P-trees for certain polynomial endofunctors P defined over groupoids.

September 30th, 2011, 9:30 CRM
Alexandre Turull (University of Florida): "Representacions de grups finits i endoisomorfismes de mòduls"
Resum: En aquest seminari no suposarem familiaritat amb les propietats bàsiques de les representacions dels grups finits més enllà del que es veu a la carrera de matemàtiques. Donats un grup finit G i un cos F, les representacions de G sobre el cos F es poden mirar com a les representacions de l'àlgebra de grup FG. Quan variem el cos F anem a subgroups de G, això dóna lloc a famílies de categories de representacions. Tenim interès especial en els casos on F és petit, i també què passa quan varia la característica d'F. La informació que proporciona el tenir un morfisme de grups suprajectiu \pi:G→G s'anomena teoria de Clifford. En aquest cas, donat un FG-mòdul topològic M definirem una G-àlgebra d'endomorfismes End(M). Usant aquest objecte, definirem endoisomorfismes entre dos mòduls M1 i M2 sobre grups G1 i G2, on tenim donats per a G1 i G2 morfismes suprajectius de grup a G−−. Veurem com M1 i M2 produeixen naturalment categories de representacions per a molts subgrups i molts cossos, i cada endoisomorfisme de M1 a M2 determina un isomorfisme entre aquestes categories de manera única.