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Friday's Topology Seminar 2015-2016

This semester we will read Benson-Greenlees paper "Stratifying the derived category of cochains on BG for G a compact Lie group".

Our aim will be study the results and see if we can generalize them to the p-compact case. The tentative schedule is the following:

1) February 19th. W. Pitsch. Overview
2) March 4th. L. Carlier. Tensor triangualted categories
3) April 1st. T. Lozano. p-compact groups
4) April 15th. W. Pitsch. Stratification I
5) April 29th. W. Pitsch Stratification II
6) May 6th. J. Kock Pointfree Topology on Spec R and support I
7) May 13rd. J. Kock Pointfree Topology on Spec R and support II
8) May 20th W. Pitsch Some results on D(R) for R a ring spectrum
9) June 3rd N. Castellana Noetherianity in cohomology
10) July 1st N. Castellana Chouinard's theorem for finite p-local groups and p-compact groups

Date and place: Fridays 12h00-13h00, CRM Room A1.

Date and Place: July 15th 12h00, CRM room A1
Nicolas Ricka (Wayne State University)
The stable Picard group of motivic A(1)

Abstract: Let A be the modulo 2 Steenrod algebra, and A(1) be the subalgebra of A generated by the two first Steenrod squares. As the cohomology of the real K-theory spectrum is A//A(1), the structure of the category of A(1)-modules is of particular importance for ko-(co)homology computations.
In this talk, I will talk about a determination of the Picard group of the stable category of modules over a motivic version of A(1). I will then discuss some possible applications in the A^1-stable motivic category, and its version of the connective real K-theory spectrum.


Date and Place: July 11th 14h30, CRM room A1
Darwin Gutiérrez (Cinvestav, México)
Complejidad topológica superior de variedades de banderas

Abstract: En esta charla analizaremos el concepto de complejidad topológica superior (TC_s) que es una generalización  de la complejidad topológica de Farber, y veremos como las cotas que brindan el zero cup length y la dimensión del espacio sobre TC_s  nos dan valores de este invariante homotópico para familias infinitas de variedades de banderas.

Además veremos un estudio que generaliza el resultado clásico TC(RP^n)=Inm(RP^n) para cada n distinto de 3,5,7 para TC_s  en condiciones particulares.


Regular seminar:

Date and Place: July 8th 12h00, CRM room A1
Marcelo FIORE (Cambridge Univ.)
The Calculus of Generalised Species of Structures

Abstract: Generalised species of structures are a categorical analog of exponential generating functions in which operations on exponential generating functions (eg substitution, addition, multiplication, differentiation) translate into a combinatorial calculus.  Drawing analogies from analysis, linear algebra, (linear) logic, and (stable) domain theory, I will give an introduction to the categorical calculus of generalised species presenting its differential and cartesian closed structures.

Extraordinary session:

Date and Place: June 9th 12h00, CRM room A1
Michael Batanin (Macquarie Univeristy)
A less known Deligne's conjecture

Abstract: A now famous Deligne's conjecture on Hochschild cochains (proved
by many authors) asserts that this complex is naturally a
E_2-algebra i.e. an algebra of an operad weakly equivalent to
the little 2-disk operad.

In 1992 Alexey Davydov introduced a construction of a
deformation complex of a monoidal functor which reminds the
classical Hochschild complex but in many respect is a very
different creature.  For example, you can apply this
construction to the identity functor.  The corresponding
cohomology classify obstructions for infinitesimal deformations
of the associator of a tensor category (like classical
Hochschild cohomology classify obstructions for infinitesimal
deformations of the multiplication of an associative algebra).
So, in some sense, this is a second order or categorification of
the Hochschild cohomology.  Deligne (in a letter to Davydov in
1993) suggested a simple algebraic condition called
n-commutativity for a cosimplicial complex of associative
algebras and conjectured that an n-commutative complex has a
natural E_{n+1}-algebra structure.  It can be easily proved that
in general Davydov's complex is a 1-commutative complex in
Deligne's sense but the deformation complex of an identity
functor is 2-commutative.

Regular seminar:

Date and Place: June 3rd 12h00, CRM room A1
Natàlia Castellana (UAB)
Noetherianity in cohomology

Abstract: We will show why various cohomologies of interest (p-compact group, p-local groups) have noetherian cohomology. The proofs rest on nice properties of various transfers in these different settings.

Date and Place: May, 27th 12h00, CRM room A1
Jean Barge (Prof. Honoraire à Univ. Grenoble)
Cancellation of projective modules old and new

Date and Place: May, 20th 12h00, CRM room A1
Wolfgang Pitsch (UAB)
Some results on D(R) for R a ring spectrum

Date and Place: May, 6th 12h00, CRM room A1
Joachim Kock (UAB)
Pointfree Topology on Spec R and support

Abstract: We will present the basics of pointfree approach to support the basics

Date and Place: April, 26th 2016, 12h00, CRM room A2 (first floor)
Antonio Díaz (Universidad de Málaga)
On Quillen's conjecture for p-solvable groups

Abstract: Quillen's conjecture on the p-subgroups complex was figured out for solvable and p-solvable groups in the 80's and 90's thanks to the work of Quillen, Aschbacher, Smith and Alperin. Here we present a fresh new geometric approach to the subject. This includes an asymptotic version for the p-solvable case that does not use the Classification of the Finite Simple Groups. The methods are potentially applicable to the general case of the conjecture.

Date and Place: April, 29th 2016, 12h00, CRM room A1
Wolfgang Pitsch (UAB)
Stratification of triangulated categories II

Abstract: Presentaré un esbozo de la teoría de estratificación de categorías trianguladas R-lineales, R un anillo noeteriano, segun Benson-Iyengar-Krause.

Date and Place: April, 15th-29th 2016, 12h00, CRM room A1
Wolfgang Pitsch (UAB)
Stratification of triangulated categories

Abstract: Presentaré un esbozo de la teoría de estratificación de categorías trianguladas R-lineales, R un anillo noeteriano, segun Benson-Iyengar-Krause.

Date and Place: April, 8th 2016, 12h00, CRM room A1
Tom Leinster (Universiry of Edinburgh)
The categorical origins of entropy

Abstract: Entropy is fundamental to many parts of mathematics (including dynamical systems, information theory and probability theory) as well as many branches of applied science, but it is less often considered in topology and algebra.  However, I will show that the concept of Shannon entropy is present in the heartlands of pure mathematics, whether we like it or not.

Specifically, I will describe a categorical machine which, when given as input the concepts of topological simplex and real number, produces as output the concept of Shannon entropy.  The most important component of this machine is the notion of "internal algebra" in an algebra for an operad (generalizing the notion of monoid in a monoidal category).  The resulting characterization of Shannon entropy can be stated with no categorical language, giving a simple and entirely elementary characterization.  This last theorem is joint work with John Baez and Tobias Fritz (arXiv:1106.1791).

Date and Place: April, 1st, 2016, 11h00, CRM Room A1
Toni Lozano (UAB)
p-compact groups
This will be a review of the theroy of p-compact groups.

Date and Place: March, 11th, 2016, 11h00, CRM Room A1

Speaker: Mark Weber  (Macquarie University)

Title: Vines and internal algebras

Abstract: A vine is a generalisation of a braid in which strings are allowed to merge. The category of natural numbers and vines between them is a canonical object -- it is the universal braided monoidal category containing a commutative monoid. In this talk we describe a conceptual approach to verifying this universal property, and describe the general theory of internal algebras of which this is an instance. Other examples relating to the study of transferred model structures on categories of operads, and the study of E_n operads will also be discussed.


Date and Palce: March, 11th, 2016, 12h00, CRM Room A1

Speaker: Norio Iwase (Kyushu University)

Title: Diffeological Spaces

Abstract: The idea of a space with smooth structure is a generalization of an idea of a manifold. K.T.  Chen introduced such a space as a differentiable space in his study of a loop space to employ the idea of iterated path integrals. Following the pattern established by Chen, J.M. Souriau introduced his version of a space with smooth structure, which is called a diffeological space. These notions are strong enough to include all the topological spaces. However, if one tries to show de Rham theorem, he must encounter a difficulty to obtain a partition of unity and thus the Mayer-Vietoris exact sequence in general. I will introduce a new version of differential forms to obtain a partition of unity, the Mayer-Vietoris exact sequence and a version of de Rham theorem in general. In addition, if we restrict ourselves to consider only CW complexes, we obtain de Rham theorem for a genuine de Rham complex, and hence the genuine de Rham cohomology coincides with the ordinary cohomology for a CW complex.

Date and place: March, 4th,  2016. CRM Room A1 12h00
Speaker : Louis Carlier

Tensor triangulated Categories
Abstract: This will be a reminder of some basic properties of tensor triangulated categories.

Date and place: February 26th,  2016. CRM Room A1
Speaker : Celeste Damiani
(Univ. Caen)
Alexander invariants of ribbon tangles and circuit algebras.

Abstract: Ribbon 2-knotted objects are locally flat 2-dimensional submanifolds of $\mathbb{R}^4$ that
bound immersed 3-manifolds with only ribbon singularities. They appear as topo-
logical realizations of welded knotted objects, where welded knot theory is a quo-
tient of virtual knot theory. We consider colored ribbon tangles and cobordisms,
and observe that the action of cobordisms on tangles endows the set of colored
ribbon tangles with a structure of circuit algebra $Tan_\mu$over the cobordisms operad.
We construct a morphism of $Tan_\mu$ to the algebra of modules over a Laurent poly-
nomial ring. Moreover, we use the diagrammatic representation of ribbon tangles
through welded diagrams to describe a purely combinatorial model for our invari-
ant, that extends a construction by Archibald and Bar-Natan. When restricted
to tangles without virtual crossings, the invariant coincides with a functor intro-
duced by Bigelow, Cattabriga and Florens. In particular, when restricted to colored
oriented braids, it coincides with exterior powers of Burau-Magnus representation.


Date and place: February 19th,  2016. CRM Room A1
Speaker : Wolfgang Pitsch

Overview of Benson-Greenlees paper.
Abstract: We will give an overview of Benson-Greenlees paper.

Date and place: January 29,  2016. CRM

Speaker : Jesper Møller (University of Copenhagen)

Title: Equivariant Euler characteristics of partition posets

Abstract: We discusss the equivariant Euler charachteristics of the $\Sigma_n$-poset of partitions of the $n$-set.

Date and place: January 22,  2016. CRM
Speaker : Bob Oliver
(Université Paris 13)
The role of tameness in Aschbacher's program
Abstract: I plan to start with some general remarks on Aschbacher's program for classifying certain simple 2-fusion systems, with focus on how he analyzes centralizers of involutions. In particular, I'll describe the role played by tameness when carrying out this reduction process.

Date and place: January 15,  2016. CRM
Speaker : Ran Levi
(University of Aberdeen)
Groups of unstable Adams operations on p-local compact groups
Abstract: A p-local compact group is an algebraic object modelled on the homotopy theory associated with p-completed classifying spaces of compact Lie groups and p-compact groups. In particular p-local compact groups give a unified framework in which one may study p-completed classifying spaces from an algebraic and homotopy theoretic point of view. Like connected compact Lie groups and p-compact groups, p-local compact groups admit unstable Adams operations - self equivalences that are characterised by their cohomological effect. Unstable Adams operations on p-local compact groups were constructed in a previous paper joint with  F. Junod and A. Libman. In this talk we consider groups of unstable operations from a geometric and algebraic point of view. We give a precise description of the relationship between algebraic and geometric operations, and show that under some conditions unstable Adams operations are determined by their degree.
We also examine a  particularly well behaved subgroup of operations.

Date and place: January 8,  2016. CRM
Speaker :
Ismar Volic (Wellesley College)
Embedding spaces and calculus of functors
Abstract: Manifold calculus of functors and related
homotopy-theoretic techniques have in recent years been applied with
great success to spaces of embeddings, most notably knot and link
spaces.  During this time, some interesting problems arose as a
consequence.  After describing the general setup of manifold calculus
of functors and how it specializes to knots and links, I will describe
some of these problems relating to certain spaces of diagrams and
trees, homology of link spaces, and subspace arrangements.  Time
permitting, I will make some general comments about how functor
calculus might provide a different point of view on some recent
results surrounding the Tverberg Conjecture.

Date and place: December 18,  2015. CRM
Speaker :
Carlos Giraldo
Minimal Fibrations that are not Fibre Bundles
By using the structure of simplicial cofibrantly generated model category over the category S^C of C-diagrams of simplicial sets (where C is a small category), we formulate one definition of minimality for C-diagrams that are free. When C is an EI-category with a finite number of objects, it is possible to show that any free C-diagram X has a minimal model with good properties. Using this tool, we are able to classify fibrations
in S^C whose base space is a constant diagram. Moreover, when the category C is a rooted tree, it is possible to classify fibrations in S^C whose base space is an arbitrary C-diagram.
This is a joint work with Carles Broto and Ramón Flores.

Date and place: October 30,  2015. 12:00. CRM Aula petita.
Speaker :
Alejandro González (Kansas State University)
Aproximacions finites de grups p-locals compactes.
Abstract: Els grups p-locals compactes són estructures algebraiques que modelen espais classificadors (p-completats) de grups finits, grups de Lie compacts i grups p-compactes, entre altres exemples. En aquesta xerrada presentaré una versió p-local d'un resultat de Friedlander i Mislin que relaciona espais classificadors de grups de Lie compactes amb telescopis d'espais classificadors de grups finits. Aquest resultat permet deduir  una versió p-local del teorema d'elements estables per calcular la cohomologia mòdul p d'un espai classificador, entre altres resultats que comentaré si hi ha temps.

Date and place: October 16,  2015. 12:00. CRM Aula A1.
Speaker :
Sejong Park (National University of Ireland Galway)Title: Sharpness conjecture for p-local finite groups and Mackey functors
Abstract: Sharpness of various homology decompositions of classifying spaces of finite groups has been studied extensively, especially by Dwyer. In particular, subgroup decomposition with respect to centric subgroups is sharp for finite groups, but an analogous statement for p-local finite groups is still open. I will present some confirmed cases and explain ingredients of proof. In particular, it will be noted that sharpness can be seen in terms of general Mackey functors, not just the cohomology functor. This is a joint work with Antonio Díaz and Radu Stancu.