Algebra Seminar 

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.

I will outline how to determine explicitly the action of the various geometric and arithmetic Frobenius morphisms on the l-adic cohomology of the moduli stack of vector bundles with fixed rank and degree on a smooth projective algebraic curve of positive characteristic. If time permits I will indicate how to prove the Weil Conjectures for this stack. This is joint work in progress with U. Stuhler (Goettingen).

Let $p$ be an odd prime. We study simply connected finite complexes having $p$-torsion free homology which are rationally equivalent to a product of odd dimensional spheres (=rational $H$-spaces). We ask whether the p-localization is an $H$-space. If the rank $r\le p-2$, an answer has been available for about 25 years. In particular, if the $p$-localization has a retractile generating complex, then the p-localization is an $H$-space. When the rank $r\ge p-1$, the statement is not necessarily true. Recent work (with L. Fernandez-Suarez, M. Mimura and J. Wu) has gained information for the case $r=p-1$ with detailed results for $r=2$, $p=3$ in terms of classical homotopy invariants.

We describe the main result for $r=2$, $p=3$. Let $\alpha \in {\pi }_{q-1}(S^n)$ with $n,q$ odd and localization at $3$ understood. Let $C=S^k{\cup }_{\alpha }{e}^q$ be the two cell complex with attaching map $\alpha$. Let $w\in {\pi }_{3n}{S}^n$ generate the kernel of the double suspension map ($\simeq Z/3Z$). Let $j:S^n\to C$ be the inclusion of the bottom cell. The space $\Lambda (C)$, with $C$ retractile and rationally equivalent to $S^n\times S^q$, exists. It is a $3$-local $H$-space if and only if $j\circ w\circ D(\alpha )=0$ where $D(\alpha ):{\Sigma }^{n+q-1}C\to C$ is a certain map constructed from a splitting of $\Sigma C\wedge C$.

It is well known that the internal sup-lattices in the topos of sheaves on a locale are precisely the modules on that locale. I shall show a generalization of this result to the case of ordered sheaves on a quantaloid. A quantaloid is the "many-object version" of a quantale (which is more-or-less a "non-commutative locale"), and a module on a quantaloid is the obvious generalization of the common notion of module on a quantale. On the other hand, an ordered sheaf on a quantaloid should be thought of as an ordered set in a universe governed by a logic whose truth values are the arrows of the quantaloid. This subject thus has strong links with non-commutative topology, (linear or rather "dynamic") logic, order theory, and (enriched) category theory. I shall try my best to avoid technicalities and concentrate rather on getting across the main ideas.

We prove that if $M$ is a CW-complex, then the homotopy type of the skeletal filtration of $M$ does not depend on the cell decomposition of $M$ up to wedge products with $n$-disks $D^n$, when they are given their natural CW-decomposition with unique cells of order $0$, $(n-1)$ and $n$; a result resembling J.H.C. Whitehead's work on simple homotopy types. From the Colimit Theorem for the Fundamental Crossed Complex of a CW-complex (due to R. Brown and P.J. Higgins), follows an algebraic analogue for the fundamental crossed complex $\Pi (M)$ of the skeletal filtration of $M$, which thus depends only on the homotopy type of $M$ (as a space) up to free product with crossed complexes of the type $D^n\doteq \Pi (D^n),n\in N$. This expands an old result (due to J.H.C. Whitehead) asserting that the homotopy type of $\Pi (M)$ depends only on the homotopy type of $M$. We use these results to define a homotopy invariant $I_A$ of CW-complexes for any finite crossed complex $A$. We interpret it in terms of the weak homotopy type of the function space $\mathrm{TOP}((M,*),(|A|,*))$, where $|A|$ is the classifying space of the crossed complex $A$.

We will give an overview of some of the proofs which exist for the theorem of Ax (1969). The theorem states that a morphism of an algebraic variety to itself which is injective is also surjective. It can be proved also that it is an automorphism. In particular we will present a proof which uses model theory. We will also present an analogous theorem for cellular automata.

If $A$ is a DGA over a field and $X$ is a module over $H_{*}(A)$ one can ask whether $X=H_{*}(M)$ for some DG module $M$. If this is the case one says that $X$ is realizable. There are several obstruction theories to answer the question of realizability. I will explain the equivalence between four such obstruction theories, namely, between the obstructions to
(i) Finding an $A_{\infty }$ module structure on $X$,
(ii) Finding a Postnikov system for $X$ in the derived category of $A$-modules,
(iii) Realizing the simplicial bar resolution for $X$ in the category of $A$-modules,
(iv) Realizing the bar complex for $X$ in the category of $A$-modules.
Joint work with Sharon Hollander (Hebrew University).

Utilizando as ferramentas de álgebra homotópica de Quillen, construímos "o invariante aditivo universal". Entendemos como tal um functor da categoria das pequenas dg-categorias com valores numa categoria aditiva, que inverte os dg-functores de Morita, transforma as decomposições semi-ortogonais de Bondal-Orlov em somas directas e é universal em relação a estas propriedades.

If $X$ is the complement of a projective hypersurface, then the nilpotent completion of its fundamental group is isomorphic to the nilpotent completion of the holonomy Lie algebra of $X$, as shown by Kohno. In the particular case where $X$ is the complement of a hyperplane arrangement $A$, the ranks ${\phi }_k$ of the lower central series quotients of ${\pi }_1(X)$ are well-known in only two very special cases: if the arrangement $A$ is hypersolvable (a linear slice of an arrangement which admits a sequence of linear fibrations), or if the the holonomy Lie algebra decomposes in degree greater or equal to $2$ as a direct product of local components. In this talk we show how to use the holonomy Lie algebra to obtain an explicit combinatorial formula for the ranks ${\phi }_k$, whenever $A$ is a graphic arrangement. This formula generalizes Kohno's result for braid arrangements, and provides the first instance of a lower central series formula for a large class of arrangements which are not decomposable or fiber-type.

Given a perfect complex $A$ in the derived category of a ring one can define the categories of $A$-torsion (respectively $A$-complete modules). If $A=Z/p$ and the ground ring is the integers these turn out to be the complexes which are quasi-isomorphic to complexes with $p$-torsion homology (respectively $p$-complete homology). I will explain how one can use derived Morita theory to establish an equivalence between the triangulated categories of torsion and complete modules. I will then explain how Dwyer and Greenlees' use these ideas to interpret local cohomology as celullar approximation in the derived category of $R$-modules (and local homology as Bousfield localization).

Given a perfect complex $A$ in the derived category of a ring one can define the categories of $A$-torsion (respectively $A$-complete modules). If $A=Z/p$ and the ground ring is the integers these turn out to be the complexes which are quasi-isomorphic to complexes with $p$-torsion homology (respectively $p$-complete homology). I will explain how one can use derived Morita theory to establish an equivalence between the triangulated categories of torsion and complete modules. I will then explain how Dwyer and Greenlees' use these ideas to interpret local cohomology as celullar approximation in the derived category of $R$-modules (and local homology as Bousfield localization).

I will explain Dugger and Shipley's result that an equivalence between derived categories of rings implies a Quillen equivalence between the model categories of chain complexes and hence an isomorphism between the algebraic K-theory of the rings. I will assume an acquaintance with the basics of model categories and algebraic K-theory.

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.