Seminário de Álgebra e Topologia 

In this informal talk, motivated by recent work in progress on sheaves in the context of noncommutative spaces, I shall examine some of the less known aspects of sheaves. On the one hand, I want to consider the way in which the fibre space and the functorial ways of defining sheaves adapt to allow one to consider sheaves, not just of sets, or groups, or rings, but of Banach spaces and C*-algebras. On the other hand, but closely linked with this approach, I want to recall the alternative way of considering sheaves as local sets, as it was developed by Higgs, and then by Fourman and Scott, from the Boolean-valued sets introduced by Scott as an alternative approach to proving the independence of the Continuum Hypothesis. All of which begins to indicate the way in which this may be extended to the noncommutative context of quantal sets over an involutive quantale, at least in the case of the quantales obtained by taking the spectrum of a C*-algebra, and of the quantales introduced by Resende in characterising localic étale groupoids.

In 1898 Hurwitz posed the problem of determining the possible dimensions for certain kinds of 'sums-of-squares' formulas. This problem arose as a generalization of the now classical '1,2,4,8-theorem' concerning the normed division algebras over the real numbers. While Hurwitz's problem is completely elementary, it is still wildy unsolved. I will describe an old cohomological approach to this problem (originally due to Hopf), and explain some recent advances using motivic cohomology and algebraic K-theory.

In this talk we will first survey basic aspects of the Brauer group and Taylor's "bigger Brauer group" both in an algebraic and topological contexts. We then proceed to present the connection between these notions and twisted forms of K-theory. In the topological context, these forms of twisted K-theory have appeared in the work of Witten and in the study of orbifolds in mathematical physics. We relate the Brauer group of the real numbers with two distinct equivariant forms of K-theories and corresponding equivariant cohomology theories. Finally, using the formalism of motives and recent work of Voevodsky, we propose to develop analogous twisted forms of algebraic K-theory and motivic cohomology for schemes over a base scheme S. These theories should be indexed by the (bigger) Brauer group of S.

Kapranov e Voevodsky (1991) propuseram uma teoria de representações categóricas lineares de categorias monoidais. Neuchl (1997) provou que as representações categóricas de uma dada categoria monoidal são os objectos de uma 2-categoria monoidal, em que há "1-" e "2-intertwiners" também. Em colaboração com John Barrett (Nottingham) investiguei a teoria das representações categóricas de grupos categóricos, que é ligeiramente mais simples. Conseguimos determinar completamente a 2-categoria monoidal das representações categóricas de qualquer grupo categórico discreto. Na minha apresentação falarei principalmente do caso concreto do grupo categórico correspondente a um produto semi-directo de grupos discretos. Mostrarei que a 2-categoria das representações categóricas deste tipo de grupo categórico enquadra, de uma forma muito natural, todos os elementos bem conhecidos da teoria das representações lineares de produtos semi-directos de grupos discretos.

Uma operação de Mal'cev é uma operação ternária $p(x,y,z)$ tal que $p(x,x,y)=y$ e $p(x,y,y)=x$. O caso mais conhecido é o dos grupos, onde $p(x,y,z)=x-y+z$. Numa teoria que tem uma operação de Mal'cev e uma única constante $0$, é possível obter uma caracterização dos subobjectos normais (os núcleos) que generaliza ambos os casos dos subgrupos normais e o dos ideais dum anel.

 
1. Comparação de várias dimensões em Teoria de Módulos: dimensão de espaços vectoriais e módulos livres, comprimento, condições de cadeia, dimensão de Goldie, dimensão de Krull.
2. Vantagens de abordar estes conceitos através da Teoria de Reticulados Modulares.
3. Aplicações.

Topological Quantum Field Theories (TQFT's) are a toy model for a full blown theory of quantum gravity, associating in a functorial way linear spaces to manifolds of dimension n and linear maps to cobordisms. After a few definitions we go about their actual construction via state sum models, focusing on the 1 + 1 situation, conceptually and technically simpler although relatively uninteresting, making reference to how one can extend the constructions to higher dimensions. Along the way we will talk about some work of the author on a particular extension of TQFT's, Homotopical Quantum Field THeories (HQFT's) which can be seen as a toy model of quantum gravity coupled to matter.

I will discuss recent work (joint with G. Tian) on CM stability and its relationship to Mumfords' Geometric Invariant theory. In particular we have identified the polarisation defining the CM stability with a "generalised chow form". Open problems and further directions will be discussed. The talk should be accesible to graduate students who have some knowledge of basic complex algebraic geometry.

Trying to discover a unifying theory connecting the noncommutative geometry of schematic algebras to the classical basis for quantum mechanics and the noncommutative geometry in terms of C*-algebras, we develop noncommutative topology both in the analytic sense or in the sense of Grothendieck topology. The relation with the lattice of linear closed subspaces of a Hilbert space H is indicated. Function theory in a quaternion variable viewed as two noncommuting complex variables is developed and leads to noncommutative Riemann manifolds and other examples of noncommutative manifolds.

There is a general construction of ''blowing-up'' a manifold $W$ along a submanifold $V$. This construction has applications both in algebraic geometry and in symplectic topology. In this talk we show that the rational homotopy type of such a blow-up is completely determined by the rational homotopy class of the embdedding and the Chern classes of its normal bundle, at least when $\dim (W)\ge 2\dim (V)+4$. In fact a computable model of the rational homotopy type of the blow-up $\tilde{W}$ can be explicitely described. This construction gives many examples of non-formal simply-connect symplectic manifolds.

We study the interplay between the integral closedness - or even the normality - of an $m$-primary ideal $I$ and conditions on the Hilbert-Samuel coefficients of $I$. We relate these properties to the Cohen-Macaulayness of the associated graded ring of $I$.

The purpose of this talk is to provide an introduction to the main techniques and results of rational homotopy theory. No original results will be presented.