Seminário de Álgebra 

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.

There is a canonical desingularization of the Unipotent variety of ${\mathrm{SL}}_n$, whose fibres can be identified with the variety of fixed flags under the action of the corresponding unipotent element. If this unipotent element is of type $(p,q)$, meaning its Jordan canonical form has two blocks of sizes $p>q$, then we will see that the irreducible components of the fiber have a very simple geometrical description and use this description to compute the corresponding Chow Motives.

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.

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.

In the 1950s, Serre introduced localization at primes into algebraic topology as a way of isolating the study of primary information about homotopy groups. In the 1960s, various authors including Sullivan, Quillen, Kan, and Bousfield realized that the localization of modules in commutative algebra has an analogue in algebraic topology which amounts to replacing a space by a new space in which the homology and homotopy groups have been localized. Since this new procedure applied to spaces it enabled the construction of interesting spaces which exhibited desirable phenomena in homotopy or homology. In the 1980s, Dror-Farjoun and Bousfield studied a generalization of this which also included a procedure to complete homotopy groups, construction which had previously seemed very different from localization. This talk will describe localization and completion in its various forms and some surprising consequences that they have when combined with Miller's solution to the Sullivan conjecture.

Unprojection is an effort, initiated by Miles Reid, to develop an algebraic language for the study of birational geometry. Algebraically, unprojection constructs more complicated rings from simpler ones, while geometrically unprojection is a method to invert projections and to perform birational 'contractions' of divisors. The talks will be introductory and will focus on foundational and computational aspects of unprojection, and also to applications to algebraic geometry.

Unprojection is an effort, initiated by Miles Reid, to develop an algebraic language for the study of birational geometry. Algebraically, unprojection constructs more complicated rings from simpler ones, while geometrically unprojection is a method to invert projections and to perform birational 'contractions' of divisors. The talks will be introductory and will focus on foundational and computational aspects of unprojection, and also to applications to algebraic geometry.