Seminário de Álgebra e Topologia 

The original McKay correspondence, due to John McKay in the late 1970s, relates for a finite subgroup $G$ of $\mathrm{SL}(2,C)$ the geometry of the minimal resolution of singularities of the quotient $C^2/G$ with the representation theory of $G$. The talk will be introductory, and will try to discuss both classical and modern approaches to the topic.

The concept of smallness in homotopy theory generalizes the concept of compactness from classical topology. However, there are two possible generalizations of this notion: one is used in model category theory, while the other one is used in the realm of triangulated categories. The relation between these two concepts remained mysterious for a long time. Mark Hovey has shown in his book on Model categories that smallness in a stable finitely generated model category implies smallness in its homotopy category. Recently Rosicky generalized this result to combinatorial model categories. In this talk we will exhibit an example of a model category Quillen equivalent to the category of spaces with the following property: every homotopy type has a countably small representative. In particular, smallness in this model category does not translate into smallness in the homotopy category. Our example stems from work on enriched Brown representability. Connections with homotopy calculus and orthogonal calculus will also be discussed.

We will give an overview of the relevant definitions, including the notions of critical and $F$-essential subgroups of a given 2-group $S$. Then we will present a systematic procedure to find those subgroups of $S$ and how to determined all nonconstrained centerfree fusion systems over $S$, up to isomorphism, using that information. We will finish the talk by applying those methods to some examples.

To quantify something means to compare it with something else which is presumed to be more fundamental. In my talk I will compare taking extensions by fibrations to the operations of homotopy push-out. I will reformulate the nilpotence theorem of Devinats-Hopkins-Smith in these terms. The aim is to give an overview of what is known and what is not about Dror Farjoun's cellularity of topological spaces.

Let X be the moduli of SL(3,C) representations of a free group; that is the character variety. We determine minimal generators of the coordinate ring of X for any rank free group. This at once gives explicit global coordinates for X and determines the dimension of the moduli's minimal affine embedding. In this talk we present the minimal generators and discuss the constructive methods employed to establish the minimal generating set.

For a pointed cosimplicial space $X$, Bousfield and Kan constructed a pointed space $\mathrm{Tot}X$, which is analogous to the geometric realization of a simplicial space, and developed a spectral sequence abutting to the homotopy groups of $\mathrm{Tot}X$. In addition they showed that this spectral sequence supports a multiplicative pairing. In this talk I wish to present an analogous property for pointed simplicial spaces as well as discussing a cap product pairing for cosimplicial and simplicial pointed spaces and their respective homotopy spectral sequences.

For a pointed cosimplicial space $X$, Bousfield and Kan constructed a pointed space $\mathrm{Tot}X$, which is analogous to the geometric realization of a simplicial space, and developed a spectral sequence abutting to the homotopy groups of $\mathrm{Tot}X$. In addition they showed that this spectral sequence supports a multiplicative pairing. In this talk I wish to present an analogous property for pointed simplicial spaces as well as discussing a cap product pairing for cosimplicial and simplicial pointed spaces and their respective homotopy spectral sequences.

Given a real variety $X$, we use methods of equivariant topology to introduce integral Deligne cohomology groups $H_{𝒟}^{n,p}X$. These groups are related to the bigraded Bredon cohomology $H^{n,p}X(ℂ)$ of the $\mathrm{Gal}(ℂ/ℝ)$-space $X(ℂ{)}^{\mathrm{an}}$ in the same manner that the Deligne cohomology $H_{𝒟}^{n,p}{X}_{ℂ}$ of the complex variety $X_{ℂ}$ is related to its singular cohomology. These groups are natural recipients of cycle maps from motivic cohomology and introduce variants of intermediate Jacobians and refined Abel-Jacobi maps. Amongst the examples discussed will be the case where $X=\mathrm{Spec}(K\otimes ℝ),$ where $K$ is a number field, and a geometric interpretation of $H_{𝒟}^{\mathrm{2,2}}X$. In the case of number fields, we provide a homomorphism from the Milnor $K$-theory of $K$ to the “diagonal part” of Deligne cohomology which is an isomorphism away from dimension $2$ and relate change of coefficients homomorphism with classical regulators in algebraic number theory. This is joint work with Pedro F. dos Santos.

The talk will outline basic constructions and properties of cellular approximation, mostly in homological algebra and group theory. Cellular approximation attempts to examine spaces or groups by constructing "approximations" using a chosen group or space such as a sphere, a given finite group, or a given chain complex. Examples are taking the subgroup generated by torsion elements, taking the canonical central extension or taking the usual CW-approximation for spaces. It turns out that very general constructions can be presented as cellular approximations. One gets a functor $X\to {\mathrm{cell}}_{A}X$ which has interesting properties: For example they alway turn a finite group into a finite group and the same for nilpotent groups and spaces. Sometimes this can help the study of more complicated spaces and chain complexes. An example, is the (K-theoretical) chain complex of a fiber of a given map, or the chain complex of the homotopy fixed points of a group action on a space.

Hopf spaces were introduced in the 50s by Serre in order to understand Lie groups from a homotopical point of view. In this talk I will give some highlights of the subject, give basic examples and applications. I will then report on recent work with N. Castellana and J. A. Crespo on structure theorems for "large" Hopf spaces.

A $p$-local finite group consists of a fusion system — a category which models the fusion (conjugacy) relations in a finite group — together with enough extra structure to determine an associated classifying space. Also, this classifying space has many of the homotopy theoretic properties of the $p$-completed classifying space of a finite group $G$. A $p$-local compact group is a similar structure, but modelled on fusion relations in a compact Lie group or a $p$-compact group. We will describe in more detail the definition and basic properties of $p$-local compact groups, and also their relation with $p$-completed classifying spaces of compact Lie groups, $p$-compact groups, and certain infinite discrete groups. We then discuss some of the open questions which arise in this subject, such as whether there is a natural definition of connected components. All of this is joint work with Carles Broto and Ran Levi.

We will explain how to use mathematical logic in order to of classifying the complexity of a mathematical statement and at what level of complexity there are natural statements which are undecidable, for example, morphisms of uncountable algebras. Also, we will define basic objects of set theory and explain the reason why there are statements which are undecidable in a natural way and why undecidability is very different from ignorance.

We will explain how to use mathematical logic in order to of classifying the complexity of a mathematical statement and at what level of complexity there are natural statements which are undecidable, for example, morphisms of uncountable algebras. Also, we will define basic objects of set theory and explain the reason why there are statements which are undecidable in a natural way and why undecidability is very different from ignorance.

We give a complete description of the bigraded Bredon cohomology ring of smooth projective real quadrics, with coefficients in the constant Mackey functor $\mathbb{Z}$. These invariants are closely related to integral motivic cohomology ring, which is still not known for these varieties.

I will introduce the notions of fusion and linking systems, and define extensions by a normal p-group. These normal extensions can be exotic (i.e. its associated fusion system is not the fusion system of a finite group), and I will give such an example.

$o$-minimal geometry is a model theoretic (logic) generalization of real algebraic and subanalytic geometry. $o$-minimality introduces new tools and provides a uniform treatment of these classical theories. In this talk we will explain how to generalize Delf’s sheaf cohomology in real algebraic geometry to arbitrary $o$-minimal structures.

Given $R$-modules $k$ and $M$, a $k$-cellular approximation to $M$ is the "closest approximation" of $M$ that can be built from $k$ using homotopy colimits. The results of Dwyer, Greenlees and Iyengar show the target of the Eilenberg-Moore cohomology spectral sequence for a fibration has a natural interpretation as a certain $k$-cellular approximation. I will introduce the concept of cellular approximations and show how they can be applied to give new proofs for known convergence results of the Eilenberg-Moore spectral sequence and generalize another.