**Speaker:** Guillermo Carrión Santiago (Universidad de Málaga)

**Title:** Construcció plus relativa

**Date:** 24/5/2024

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** La construcció plus per a homologia entera va ser popularitzada per Quillen als anys 70 com a un mètode per a calcular la teoria K d'un anell. Consisteix en transformar un espai X adjuntant cel·les de dimensió 2 i 3 per a matar un subgrup perfecte H del seu grup fonamental però sense modificar la seva homologia. En el cas que H sigui el subgrup perfecte maximal, la construcció plus resulta ser una nul·lificació homològica. Arran d'aquesta propietat, es generalitza aquesta construcció per a qualsevol teoria d'homologia, però amb la pèrdua de poder escollir el subgrup H. Recentment, Broto, Levi i Oliver, defineixen una construcció plus a la Quillen, adjuntant cel·les de dimensió 2 i 3, per a homologia amb coeficients en un anell R relativa a un subgrup (fortament) R-perfecte qualsevol.
En aquesta xerrada, veurem que malgrat que la proposta de BLO no es pot descriure com a una nul·lificació, podem definir la construcció plus relativa com a una nul·lificació homològica a la categoria d'aplicacions entre espais.

**Speaker:** Guille Carrión Santiago (Universidad de Málaga)

**Title:** Construcció plus relativa.

**Date:** 24/5/2024

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** La construcció plus per a homologia entera va ser popularitzada per Quillen als anys 70 com a un mètode per a calcular la teoria K d'un anell. Consisteix en transformar un espai X adjuntant cel·les de dimensió 2 i 3 per a matar un subgrup perfecte H del seu grup fonamental però sense modificar la seva homologia. En el cas que H sigui el subgrup perfecte maximal, la construcció plus resulta ser una nul·lificació homològica. Arran d'aquesta propietat, es generalitza aquesta construcció per a qualsevol teoria d'homologia, però amb la pèrdua de poder escollir el subgrup H. Recentment, Broto, Levi i Oliver, defineixen una construcció plus a la Quillen, adjuntant cel·les de dimensió 2 i 3, per a homologia amb coeficients en un anell R relativa a un subgrup (fortament) R-perfecte qualsevol.
En aquesta xerrada, veurem que malgrat que la proposta de BLO no es pot descriure com a una nul·lificació, podem definir la construcció plus relativa com a una nul·lificació homològica a la categoria d'aplicacions entre espais.

**Speaker:** Thomas Jan Mikhail

**Title:** Type Theory for the Working Mathematician (part 2)

**Date:** 3/5/2024

**Time:** 12:00

**Web:** http://mat.uab.cat/ ~topalg

**Abstract:** In this talk we pick up where we left off last time (12/04/2024) and begin by first sketching out the relationship between type theories and logic, known as the Curry-Howard correspondence. In the remainder of the talk we will go through some applications, exemplifying the utility of type theory as a tool for proving statements in categories. Depending on the time these may include algebraic theories, topoi and homotopy type theory.

**Speaker:** Thomas Jan Mikhail (UAB)

**Title:** Categorical Logic for the Working Mathematician

**Date:** 12/4/2024

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** According to the Curry-Howard-Lambek correspondence, logics, type theories and structured categories are intimately related. In its cleanest form, the relation between the latter two can be realized in terms of an adjunction, given by the internal language functor and the syntactic category functor. One way of understanding this relation is that type theories provide us with a convenient language for proving things about categories by presenting free structures in a particular way. This is the narrative undertaken by Shulman in his draft 'Categorical Logic from a Categorical Point of View' of which I will give an overview. In some sense this can be taken as an answer to the question of why anyone using category theory might also want to learn type theory.

**Speaker:** Thomas Jan Mikhail (UAB)

**Title:** Type Theory vs Category Theory

**Date:** 12/4/2024

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** According to the Curry-Howard-Lambek correspondence, logics, type theories and structured categories are intimately related. In its cleanest form, the relation between the latter two can be realized in terms of an adjunction, given by the internal language functor and the syntactic category functor. One way of understanding this relation is that type theories provide us with a convenient language for presenting free structured categories. This is the narrative undertaken by Shulman in his draft 'Categorical Logic from a Categorical Point of View' of which I could give an overview. In some sense this approach can be taken as answer the question of why anyone using category theory might also want to learn type theory.

**Speaker:** Wolfgang Pitsch (UAB)

**Title:** Quillen's conjecture: first steps into the solvable case, II

**Date:** 15/3/2024

**Time:** 12:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** In this talk we will explain the Cohen-Macaulay properties of the poset for some extensions of a solvable group by uniquely divisible groups. This is the key result to prove Quillen's conjecture for finite solvable groups.

**Speaker:** Wolfgang Pitsch (UAB)

**Title:** Quillen's conjecture: first steps into the solvable case

**Date:** 1/3/2024

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** In this talk we will explain the Cohen-Macaulay properties of the poset $\mathcal{A}_p(G)$ for some extensions of a $p$-solvable group by uniquely $p$-divisible groups. This is the key result to prove Quillen's conjecture for finite solvable groups.

**Speaker:** Gabriel Martínez de Cestafe Pumares (UAB)

**Title:** Quillen´s conjecture via finite topological spaces

**Date:** 9/ 2/2024

**Time:** 12:15

**Web:** http://mat.uab.cat/ ~topalg

**Abstract:** Given a finite group $G$ and a prime $p$, Quillen studied the poset $S_p(G)$ of non-trivial $p$-subgroups of $G$ from an homotopical point of view. For this, he considered the geometric realization of the order complex of $S_p(G)$ and conjectured that it is contractible if and only if $G$ has a non-trivial normal $p$-subgroup. In this talk, we will follow an alternative approach, which exploits the close relation between the notions of finite poset and finite $T_0$ topological space. This will allow us to view $S_p(G)$ itself as a topological space and to reformulate Quillen´s conjecture in a purely homotopical language.

**Speaker:** Jesper M. Møller (Universitat de Copenhaguen)

**Title:** The non-generating and the Quillen simplicial complex of a finite group

**Date:** 12/1/2024

**Time:** 12:15

**Web:** http://mat.uab.cat/~topalg

**Abstract:** The Quillen complex of a finite group $G$ is a $G$-simplicial complex simple homotopy equivalent to the order complex of the poset of nontrivial $p$-subgroups. The non-generating complex is a $G$-collapsible simplicial complex containing the Quillen complex as a subcomplex. The orbit $CW$-complexes are contractible. The $f$-vectors are unimodal and even, in most cases, $\log$-concave. The Quillen conjecture can be formulated simplicially in terms of the Quillen simplicial complex.
The non-generating and the Quillen simplicial complex are associated to subgroup posets. There are similar simplicial complexes associated to coset posets.

**Speaker:** Carles Broto (UAB)

**Title:** Homotopy properties of the poset of non-trivial $p$-subgroups of a group V

**Date:** 15/12/2023

**Time:** 12:15

**Web:** http://mat.uab.cat/~topalg

**Abstract:** Given a map of posets $f\colon X\to Y$, we analyse a spectral sequence converging to the homology of $Y$. This allows one to show that certain posets are Cohen-Macaulay.

**Speaker:** Natàlia Castellana

**Title:** Homotopy properties of the poset of non-trivial $p$-subgroups of a group II

**Date:** 10/11/2023

**Time:** 12:15

**Web:** http://mat.uab.cat/~topalg

**Abstract:** In this talk we follow the study of the homotopy properties of the simplicial complex associated to the poset of non-trivial $p$-subgroups of a group. Bouc proved the equivalence to the subposet of the Bouc family of subgroups. In the work of Thévenaz and Webb, taking into account the action of the group by conjugation, it is shown that all these inclusions induce $G$-homotopy equivalences.

**Speaker:** Roger Bergadà (UAB)

**Title:** Homotopy properties of the poset of non-trivial $p$-subgroups of a group I

**Date:** 3/11/2023

**Time:** 12:15

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** In this talk we review the first part of Quillen´s paper "Homotopy properties of the poset of non-trivial $p$-subgroups of a group" where the author describes homotopy properties of the simplicial complex associated to the poset of non-trivial $p$-subgroups of a group. One of the main results is the equivalence to the subposet of non-identity elementary abelian $p$-subgroups.

**Speaker:** Thomas Jan Mikhail (UAB)

**Title:** Lawvere's Fixed Point Theorem and its Applications

**Date:** 27/10/2023

**Time:** 12:15

**Web:** http://mat.uab.cat/~topalg

**Abstract:** In 1969 Lawvere published a paper called \emph{Cartesian Closed Categories and Diagonal Arguments}. In this paper, he identifies Cartesian closed categories as a suitable general framework unifying known diagonal arguments. He spells out a fixed point theorem with a remarkably short proof (especially when using the internal language of CCCs) and of which the contrapositive may be interpreted as an abstract diagonal argument. Applications include Cantor's diagonal argument (in any elementary topos in fact), the halting paradox, Russell's paradox as well as Gödel's (first) incompleteness theorem.

**Speaker:** Carles Broto (UAB)

**Title:** Quillen´s conjecture on subgroup complexes, I

**Date:** 20/10 /2023

**Time:** 12:15

**Web:** http://mat.uab.cat/ ~topalg

**Abstract:** We will state the conjecture and discuss what is known about it.

**Speaker:** Luca Pol (University of Regensburg)

**Title:** Local Gorenstein duality in chromatic group cohomology

**Date:** 23/9/2021

**Time:** 10:30

**Web:** http://mat.uab.cat/~topalg

**Abstract:** Many algebraic definitions and constructions can be made in a derived or homotopy invariant setting and as such make sense for ring spectra. Dwyer-Greenlees-Iyengar (followed by Barthel-Heard-Valenzuela) showed that one can make sense of local Gorenstein duality for ring spectra. In this talk, I will show that cochain spectra C*(BG;R) satisfy local Gorenstein duality surprisingly often, and explain some of the implications of this. When R=k is a field this recovers duality properties in modular representation theory conjectured by Benson and later proved by Benson-Greenlees. However, the result also applies to more exotic coefficients R such as Lubin-Tate theories, K-theory spectra or topological modular forms, showing that chromatic analogues of Benson´s conjecture also hold. This is joint work with Jordan Williamson.

**Speaker:** Paolo Salvatore

**Title:** Multi-simplicial operations, equivariance and effective homology

**Date:** 28/6/2021

**Time:** 16:00

**Abstract:** We define a cup product on multi-simplicial cochain complexes, and more generally an E-infinity algebra structure. We then show how to apply this, together with the equivariance with respect to a group action on the complexes, to improve substantially the effectiveness of the algorithms appearing in formality problems.

**Speaker:** Rune Haugseng (Trondheim)

**Title:** The universal property of bispans

**Date:** 10/1/2020

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:**

**Speaker:** Luis Javier Hernández Paricio (Universidad de La Rioja)

**Title:** Endomorphisms of the Hopf fibration and numerical methods

**Date:** 13/11/2019

**Time:** 10:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** We have developed and implemented in Julia language a collection of algorithms for the iteration of a rational function that avoids the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of a homogeneous pair of bivariate polynomials.
This homogeneous pair induces in a canonical way a self-map of the pointed Hopf fibration. Moreover, if the homogenous pair is irreducible, we also have a self-map of the standard Hopf fibration. We study the points of indeterminacy evaluating a canonical map associated with a homogeneous pair on the orbit of a point of the Riemann sphere.
These algorithms can be applied to any numerical method that builds a rational map to find the roots of an univariate polynomial equation. In particular with these procedures we analyze the existence of multiple roots for the Newton method and the relaxed Newton method.
This project is being developed together with J.I. Extremiana, J. M. Guti\'errez and M. T. Rivas (University of La Rioja).

**Speaker:** Guillem Sala (UPC)

**Title:** Topological Cyclic Homology and L-functions

**Date:** 25 /10/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/ ~topalg

**Abstract:** It has already been noted in the past that there is a deep connection between number theory, algebraic geometry and algebraic topology. An example of this was Grothendieck's proof of the rationality part of the Weil conjectures, where he provided an étale cohomological interpretation of the Hasse-Weil zeta function for "nice" varieties over finite fields.
The goal of this talk is to follow the work of Lars Hesselholt and extend this result to the realm of homotopy theory, providing a cohomological interpretation of the Hasse-Weil zeta function using the cohomology associated to a certain spectrum, namely the Topological Periodic Cyclic Homology spectrum.

**Speaker:** Michelle Strumila (University of Melbourne)

**Title:** Infinity operads and their generalisations

**Date:** 9/9/2019

**Time:** 15:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** Infinity categories are a way of taking categories up to homotopy. This talk is about how this can be extended to infinity operads, along with generalisations to the non-directional and higher genus cases.

**Speaker:** Assaf Libman (University of Aberdeen)

**Title:** Selfmaps of equivariant spheres

**Date:** 6/9/2019

**Time:** 12:00

**Web:** http:// mat.uab.cat/~topalg

**Abstract:** We describe a stabilization property of the homotopy groups of space of equivariant self maps of spheres with action of a finite group G. This result gives an extension (of a special case of) tom-Dieck's splitting theorem to incomplete universes.

**Speaker:** Eva Belmont (Northwestern University)

**Title:** The motivic Adams spectral sequence

**Date:** 6/9/ 2019

**Time:** 11:00

**Web:** http://mat.uab.cat/ ~topalg

**Abstract:** The Adams spectral sequence is one of the main tools for computing stable homotopy groups of spheres. In this talk, I will give an introduction to the Adams spectral sequence in motivic homotopy theory over C and over R, and describe some connections with classical and C_2-equivariant homotopy theory. I will describe joint work with Dan Isaksen to compute the motivic Adams spectral sequence over R and obtain applications to the Mahowald invariant.

**Speaker:** Mark Penney (MPIM Bonn)

**Title:** Dijkgraaf-Witten invariants of 2-knots

**Date:** 26/7/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** The aim of this talk is to introduce a family of invariants of 2-knots which generalize the Dijkgraaf-Witten (DW) knot invariants. I will begin with a casual review of the DW knot invariants, making connections to Fox n-colourings in the process. The naive generalization to 2-knots yields much weaker invariants and so I will discuss a homotopy-theoretic generalization which has the potential to yield finer results.

**Speaker:** Philip Hackney

**Title:** Right adjoints to operadic restriction functors

**Date:** 18/7/2019

**Time:** 09:30

**Web:** http://mat.uab.cat/~topalg

**Abstract:** If f : P -> Q is a morphism of operads, then there is a restriction functor from Q-algebras to P-algebras. This restriction functor generally admits a left adjoint. This restriction may or may not admit a right adjoint: if G -> H is a group homomorphism, then the forgetful functor from H-sets to G-sets has a right adjoint, while there is no right adjoint to the functor from commutative algebras to associative algebras. In this talk, we provide a concise necessary and sufficient condition for the existence of a right adjoint to the restriction functor, phrased in terms of the operad map f. We give a simple formula for this right adjoint, and examine the criterion in special cases. All of this is applicable over quite general ground categories. (Joint work with Gabriel C. Drummond-Cole)

**Speaker:** Matt Feller (University of Virginia)

**Title:** New model structures on simplicial sets

**Date:** 5/7/2019

**Time:** 12:00

**Web:** http:// mat.uab.cat/~topalg

**Abstract:** In the way Kan complexes and quasi-categories model up-to-homotopy groupoids and categories, can we find model structures on simplicial sets which give up-to-homotopy versions of more general objects? We investigate this question, with the particular motivating example of 2-Segal sets. Cisinski's work on model structures in presheaf categories provides abstract blueprints for these new model structures, but turning these blueprints into a satisfying description is a nontrivial task. As a first step, we describe the minimal model structure on simplicial sets arising from Cisinski's theory.

**Speaker:** Marc Stephan (Max Planck Institute for Mathematics, Bonn)

**Title:** A multiplicative spectral sequence for free p-group actions

**Date:** 24/5/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** Carlsson conjectured that if a finite CW complex admits a free action by an elementary abelian p-group G of rank n, then the sum of its mod-p Betti numbers is at least $2^n$. In 2017, Iyengar and Walker constructed equivariant chain complexes that are counterexamples to an algebraic version of Carlsson´s conjecture. This raised the question if these chain complexes can be realized topologically by free G-spaces to produce counterexamples to Carlsson´s conjecture. In this talk, I will explain multiplicative properties of the spectral sequence obtained by filtering the mod-p cochains of a space with a free p-group action by powers of the augmentation ideal and deduce that the counterexamples can not be realized topologically. This is joint work with Henrik Rüping.

**Speaker:** Sune Precht Reeh (BGSMath-UAB)

**Title:** A formula for p-completion by way of the Segal conjecture

**Date:** 10/5/2019

**Time:** 10:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** A variant of the Segal conjecture (theorem by Carlsson) gives a correspondence between homotopy classes of stable maps from BG to BH and the module of (G,H)-bisets that are H-free and where the module is completed with respect to the augmentation ideal I(G) in the Burnside ring of G. The details of this correspondence change depending on whether you add a disjoint basepoint to BG, BH, or both, and it is also not a priori clear what algebraic consequences the I(G)-adic completion has for the module of (G,H)-bisets. Separately, we have the functor of p-completion for spaces or spectra. We can apply p-completion to each classifying space BG, and according to the Martino-Priddy conjecture (theorem by Oliver) the p-completed classifying space depends only on the saturated fusion system $\mathcal F_p(G)$ of G at the prime p.
Saturated fusion systems also have modules of bisets, and so it is not unreasonable to ask how p-completion interacts with the Segal conjecture: Suppose we are given a (G,H)-biset, we can interpret the biset as a stable map from BG to BH. Apply p-completion to get a stable map from $B\mathcal F_p(G)$ to $B\mathcal F_p(H)$. By the Segal conjecture for fusion systems, that stable map corresponds to an $ (\mathcal F_p(G), \mathcal F_p(H))$-biset -- up to p-adic completion. Which $(\ mathcal F_p(G), \mathcal F_p(H))$-biset do we get?
This innocent question was the starting point for a joint paper with Tomer Schlank and Nathaniel Stapleton, and in my talk I will give an overview of all the categories involved and how they fit together with functors. If time permits, we will even see how p-completion and fusion systems can help us understand the I(G)-adic completion for any finite group -- and I suppose we might even consider that "a formula for the Segal conjecture by way of p-completion".

**Speaker:** Matthew Gelvin (Bilkent University, Ankara)

**Title:** Fusion-minimal groups

**Date:** 26/4/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** Every saturated fusion system $ \mathcal{F}$ on the $p$-group $S$ has an associated collection of characteristic bisets. These are $(S,S)$-bisets that determine $\mathcal{F}$, and are in turn determined by $\mathcal{F}$ up to a more-or-less explicit parameterization. In particular, there is always a unique minimal $\mathcal{F} $-characteristic biset, $\Omega_\mathcal{F}$. If $G$ is a finite group containing $S$ as a Sylow $p$-subgroup and realizing $\mathcal{F}$, then $G$ is itself, when viewed as an $(S,S)$-biset, $\mathcal{F}$-characteristic. If it happens that $_SG_S=\Omega_\mathcal{F}$ is the minimal biset for its fusion system, we say that $G$ is \emph{fusion-minimal}.
In joint work with Sune Reeh, it was shown that any strictly $p$-constrained group (i.e., one that satisfies $C_G(O_p(G))\leq O_p(G)$) is fusion minimal. We conjecture that converse implication holds. In this talk, based on joint work with Justin Lynd, we prove this to be the case when $p$ is odd and describe the obstruction to a complete proof. Along the way, we will draw a connection with the module structure of block algebras and how this relates to the question at hand.

**Speaker:** Joshua Hunt (University of Copenhagen)

**Title:** Lifting G-stable endotrivial modules

**Date:** 12/4/2019

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**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.

**Speaker:** Antonio Díaz (Universidad de Málaga)

**Title:** Fusion systems for profinite groups

**Date:** 29/3/2019

**Time:** 09:00

**Web:** http://mat.uab.cat/~topalg

**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 M. Møller (Københavns Universitet)

**Title:** Counting $p$-singular elements in finite groups of Lie type

**Date:** 25 /1/2019

**Time:** 11:00

**Web:** http://mat.uab.cat/web/ligat/

**Abstract:** Let $G$ be a finite group and $p$ a prime number. We say that an element of $G$ is $p$-singular if its order is a power of $p$. Let $G_p$ be the 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

**Date:** 18/1/2019

**Time:** 11:00

**Web:** http://mat.uab.cat /~topalg

**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

**Abstract:** We start with a brief recollection on the algebraic K-theory of exact categories. The first goal is to introduce a new simplicial delooping of it, which consists of higher dimensional analogues of Waldhausen's S-construction, where short exact sequences are replaced by longer extensions. Our main theorem provides a precise relation to the original Waldhausen construction, and in particular implies the delooping statement. Another consequence is the generalisation of one of the main results of Dyckerhoff-Kapranov's work on higher Segal spaces, which concerns the fibrancy properties of the (higher) S-construction; this is the first example of (> 2)-Segal objects explicitly appearing in the literature.

**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.

**Speaker:** Nitu Kitchloo (Johns Hopkins University)

**Title:** Stability for Kac-Moody Groups

**Date:** 20/7/2018

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** In the class of Kac-Moody groups, one can extend all the exceptional families of compact Lie groups yielding infinite families $(E_n, F_n, G_n)$, as well as other infinite families. We will show that these exceptional families stabilize in a homotopical sense and that the (co)homology of their classifying spaces is torsion free for all but a finite set of primes that is determined by the family (and not the individual groups in the family).

**Speaker:** Marithania Silvero (BGSMath-UB)

**Title:** Strongly quasipositive links and Conway polynomial

**Date:** 20/7/2018

**Time:** 10:45

**Web:** http://mat.uab.cat/~topalg

**Abstract:** Strongly quasipositive links are those links which can be seen as closures of positive braids in terms of band generators. We give a necessary condition for a link with braid index 3 to be strongly quasipositive, by proving that they have positive Conway polynomial (that is, all its coefficients are non-negative). We also show that this result cannot be extended to a higher number of strands, as we provide a strongly quasipositive braid on 5 strands whose closure has non-positive Conway polynomial.

**Speaker:** David Spivak (MIT)

**Title:** A higher-order temporal logic for dynamical systems

**Date:** 6/7/2018

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** We consider a very general class of dynamical systems---including discrete, continuous, hybrid, deterministic, nondeterministic, etc.---based on sheaves. We call these sheaves behavior types: they tell us the set of possible behaviors over any interval of time. A machine can be construed as a wide span of such sheaves, and these machines can be composed as morphisms in a hypergraph category. The topos of sheaves has an internal language, which we use as a new sort of higher-order internal logic for talking about behaviors. We can use this logic to prove properties about a composite system of systems from properties of the parts and how they are wired together.

**Speaker:** Rémi Molinier (Université de Grénoble)

**Title:** Cohomology with twisted coefficients of linking systems and stable elements

**Date:** 8/6/2018

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** A theorem of Boto, Levi and Oliver describes the cohomology of the geometric realization of a linking system, with trivial coefficients, as the submodule of stable elements in the cohomology of the Sylow. When we are looking at twisted coefficients, the formula can not be true in general as pointed out by Levi and Ragnarsson but we can try to understand under which condition it holds. In this talk we will see some conditions under which we can express the cohomology of a linking system as stable elements.

**Speaker:** Thomas Wasserman (Oxford)

**Title:** A Reduced Tensor Product of Braided Fusion Categories containing a Symmetric Fusion Category

**Date:** 1/6/2018

**Time:** 12:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** In this talk I will construct a reduced tensor product of braided fusion categories containing a symmetric fusion category $\mathcal{A}$. This tensor product takes into account the relative braiding with respect to objects of $\mathcal{A}$ in these braided fusion categories. The resulting category is again a braided fusion category containing $\mathcal{A}$. This tensor product is inspired by the tensor product of $G$-equivariant once-extended three-dimensional quantum field theories, for a finite group $G$.

**Speaker:** Jérôme Los

**Title:** Sequences in the mapping class group: some convergence/ divergence questions

**Date:** 23/5/ 2018

**Time:** 16:00

**Web:** http://mat.uab.cat/ ~topalg

**Abstract:**

**Speaker:** Jérôme Los (CNRS-Univ. Marseille)

**Title:** Sequences in the mapping class group: some convergence/ divergence questions

**Date:** 23/5/2018

**Time:** 16:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:**

**Speaker:** Mark Weber

**Title:** Feynman categories as operads

**Date:** 23/5/2018

**Time:** 15:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** In various papers of Kaufmann and Ward, the notion of "Feynman
category" is introduced as a generalisation of "coloured
symmetric operad", and then developed further. In this talk it
will be explained that in fact Feynman categories and coloured
symmetric operads are the same things, in that one can set up a
biequivalence between 2-categories whose objects are these
structures. Moreover, this biequivalence induces equivalences
between the corresponding categories of algebras. Thus Feynman
categories are not really "new", but rather are an interesting
alternative point of view on coloured symmetric operads.
(Joint work with Michael Batanin and Joachim Kock)

**Speaker:** Mark Weber

**Title:** Feynman categories as operads

**Date:** 23/5 /2018

**Time:** 15:00

**Web:** http://mat.uab.cat/ ~topalg

**Abstract:** In various papers of Kaufmann and Ward, the notion of "Feynman category" is introduced as a generalisation of "coloured symmetric operad", and then developed further. In this talk it will be explained that in fact Feynman categories and coloured symmetric operads are the same things, in that one can set up a biequivalence between 2-categories whose objects are these structures. Moreover, this biequivalence induces equivalences between the corresponding categories of algebras. Thus Feynman categories are not really "new", but rather are an interesting alternative point of view on coloured symmetric operads.

**Speaker:** Alex Cebrian (UAB)

**Title:** A simplicial groupoid for plethystic substitution

**Date:** 2/3/2018

**Time:** 11:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** We give a simple combinatorial model for plethystic substitution: precisely, the plethystic bialgebra is realised as the homotopy cardinality of the incidence bialgebra of a simplicial groupoid, obtained from surjections by a construction reminiscent of Waldhausen S and Quillen Q-construction.

**Speaker:** Bob Oliver (Université Paris 13)

**Title:** Recent constructions and theorems on fusion systems due to Michael Aschbacher

**Date:** 23/2/2018

**Time:** 11:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** Fix a prime $p$. The fusion system of a finite group $G$ with respect to a Sylow subgroup $S \in\mathop {\rm Syl}_p(G)$ is the category $\mathcal{F}_S(G)$ whose objects are the subgroups of $S$, and whose morphisms are the homomorphisms induced by conjugation in $G$. More generally, an abstract fusion system over a $p$-group $S$ is a category whose objects are the subgroups of $S$ and whose morphisms are injective homomorphisms between the subgroups that satisfy certain axioms formulated by Lluis Puig and motivated by the Sylow theorems for finite groups.
Starting 10-15 years ago, Michael Aschbacher and some other finite group theorists became interested in fusion systems, hoping that they can be used to help shorten some parts of the proof of the classification of finite simple groups. This has led to many new structures and results such as generalized Fitting subsystems of fusion systems, as well as intersections, central products, and centralizers of normal fusion subsystems. In many cases, these are analogs of basic, elementary structures or operations in finite groups, but are surprisingly difficult to define in the context of fusion systems.

**Speaker:** Sune Precht Reeh (UAB)

**Title:** Constructing a transporter infinity category for fusion systems

**Date:** 21/2/2018

**Time:** 11:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** In this research talk, I will give a tour of the progress I have made in the last two weeks on constructing an infinity category that is supposed to model the transporter category for a fusion system (when given a choice of locality/linking system).
I will explain the construction itself as a category enriched in Kan complexes. I will talk about the results obtained so far, with details as time permits, and I will explain the open problems that I am still working on, including how to adapt this transporter category into a working orbit category.

**Speaker:** Jesper M. Møller (University of Copenhagen)

**Title:** The Alperin weight conjecture, the Knörr-Robinson conjecture, and equivariant Euler characteristics

**Date:** 16/2/2018

**Time:** 11:00

**Web:** http://mat.uab.cat/~topalg

**Abstract:** A topologically biased amateur marvels at the Alperin weight conjecture from different angles without getting anywhere near a solution.