Algebra Seminar 

07/07/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Inês Henriques, University of California at Riverside

Free resolutions over quasi complete intersections

We will introduce a class of homomorphisms of commutative Noetherian rings, which strictly contains the class of locally complete intersection homomorphisms, while sharing many of its remarkable properties. This is joint work with L. L. Avramov and L. M. Sega.

16/06/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Paulo Lima-Filho, Texas A&M

Explicit examples of regulator maps

We will present a simple construction of cocycles representing certain generators in motivic cohomology, using Voevodsky's motivic complexes. This will motivate a novel construction of the regulator map from motivic to Deligne cohomology of complex algebraic varieties. In particular, this will exemplify the importance of the identity $\pi^2/6 = \sum_{n\geq 1} 1/n^2$. This is joint work with Pedro F. dos Santos and James Lewis.

31/05/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Dan Christensen, University of Western Ontario

Categories of smooth spaces

I will describe some categories of "smooth spaces" which generalize the notion of manifold. The generalizations allow us to form smooth spaces consisting of subsets and quotients of manifolds, as well as loop spaces and other function spaces. In more technical language, these categories of smooth spaces are complete, cocomplete and cartesian closed. I will give examples, discuss possible applications and explain why it would be useful to be able to do homotopy theory in a category of smooth spaces. Talk slides

12/05/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Rui Reis, University of Aberdeen

Manifold functor calculus and h-principles

In the 90's, and inspired by Goodwillie's homotopy functor calculus, M. Weiss developed a variant of functor calculus (called at the time embedding calculus) in order to study the homotopy type of certain spaces of maps between manifolds such as spaces of immersions and spaces of embeddings. In this talk I will give a brief overview of the basic definitions and properties of this manifold functor calculus and describe how it can be used to prove Vassiliev's h-principle.

05/05/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Ayhan Gunaydin, CMAF

Small multiplicative subgroups of fields

We are aiming to give an overview of a model theorist's take on certain number theoretic topics of diophantine nature. Our approach is based on the study of 'small' subgroups of the multiplicative group of a field of characteristic zero. Here 'small' is quite a technical term which happens to be the model theoretic counter-part of 'finite rank'. Model theory needed for this talk is at the minimum and will be summarized in the first few minutes.

17/03/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Pedro Resende, IST, CAMGSD

Facets of topos theory

Grothendieck toposes can be regarded in several ways, and in this talk I will give a bird's eye view of three such ways: (i) toposes as categories of sheaves (on sites, groupoids, quantales, etc.); (ii) as generalized universes of sets in which mathematics can be carried out; and (iii), last but not least, as generalized spaces in their own right, for instance orbit spaces of groupoids, with (noncommutative) quantales playing the role of "coordinate rings". I will begin with a crash course on locale theory, intended to provide the appropriate "commutative algebra".

23/11/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
Abdó Roig-Maranges, Universitat Politècnica de Catalunya

Morphic cohomology of toric varieties

In this talk I will describe morphic cohomology, a cohomology theory for complex algebraic varieties introduced by Friedlander and Lawson in the 90's. This cohomology is in between motivic cohomology and singular cohomology, and captures geometric information of the variety such as cycles modulo algebraic equivalence. Unlike its motivic cousin, this theory has a very explicit and concrete definition in terms of homotopy groups of spaces of cycles. Then I will talk about some problems it presents, and will describe a spectral sequence that computes it rather explicitly in the case of toric varieties.

23/09/2010, 15:00 — 16:00 — Room P4.35, Mathematics Building
James Lewis, University of Alberta

An Archimedean height pairing on the equivalence relation defining Bloch's higher Chow groups

The existence of a height pairing on the equivalence relation defining Bloch's higher Chow groups is a surprising consequence of some recent joint work by myself and Xi Chen on a nontrivial $K_1$-class on a self-product of a general $K3$ surface. I will explain how this pairing comes about.

21/09/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
James Lewis, University of Alberta

New invariants on algebraic cycles

I will explain the intertwining role of Hodge theory and algebraic cycles, beginning from the classical constructions in the 1960's to the more recent developments using arithmetical normal functions.

14/07/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
Jorge Vitória, University of Warwick

Equivalences for non-commutative projective spaces

Even though noncommutative projective spaces (in the sense of Artin and Zhang) are not very well understood in high dimensions, by considering a restricted class of such spaces one can hope for interesting classification theorems. In this talk we recall the key concepts of this approach to noncommutative algebraic geometry and discuss the case of noncommutative projective spaces associated to multiparameter skew-polynomial rings. We prove a classification theorem and we make some observations on birational equivalences and point varieties.

12/07/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
Bjorn Poonen, Massachussetts Institute of Technology

Existence of rational points on smooth projective varieties

Let $k$ be a finite extension of the field $\mathbb{Q}$. We prove results including:

1. If there is an algorithm to decide whether a smooth projective $k$-variety has a $k$-point, then there is an algorithm to decide whether an arbitrary $k$-variety has a $k$-point.
2. If there is an algorithm to decide whether a smooth projective 3-fold has a $k$-point, then there is an algorithm to compute $X(k)$ for any curve $X$ over $k$.

See also

http://math.mit.edu/~poonen/papers/chatelet.pdf

08/07/2010, 15:00 — 16:00 — Room P4.35, Mathematics Building
Javier Elizondo, Universidad Nacional Autónoma de México

The motivic Euler-Chow series

Consider the formal power series $\sum [C_{p,\alpha (X)}]t^{\alpha }$ (called Motivic Chow Series), where $C_p(X)=\coprod C_{p,\alpha }(X)$ is the Chow variety of $X$ parametrizing the $p$- dimensional effective cycles on $X$ with $C_{p,\alpha }(X)$ its connected components, and $[C_{p,\alpha }(X)]$ its class in $K(ChM{)}_{A^1}$, the $K$-ring of Chow motives modulo $A^1$-homotopy. Using Picard product formula and Torus action, we will show that the Motivic Chow Series is rational in many cases.

09/06/2010, 15:00 — 16:00 — Room P4.35, Mathematics Building
Pedro Martins Rodrigues, IST, CAMGSD

A Profinite Module associated to a hyperbolic toral automorphism

In this talk we describe and discuss some of the properties of a profinite module associated to a hyperbolic toral automorphism, defined by its action on periodic orbits. We are particularly interested in understanding to what extent the algebraic/topological classification of those automorphisms is determined by that action.

02/06/2010, 15:00 — 16:00 — Room P4.35, Mathematics Building
Carlo Rossi, CAMGSD

Formality for two branes and Lie algebras-II

We discuss some results of work in progress, where we apply the recent machinery of formality for two branes in Lie algebra theory, aiming to prove more general Duflo-like theorems; if possible, Lie algebroids and global issues will also be discussed.

26/05/2010, 15:00 — 16:00 — Room P4.35, Mathematics Building
Carlo Rossi, CAMGSD

2-brane formality and Lie algebras

We discuss some results of work in progress, where we apply the recent machinery of 2-brane formality in Lie algebra theory, aiming to prove more general Duflo-like theorems; if possible, Lie algebroids and global issues will also be discussed.

15/04/2010, 10:00 — 11:00 — Room P3.10, Mathematics Building
Marcin Szamutolski, Instituto Superior Técnico

Galois theory of Hopf-Galois extensions

For a not necessarily commutative comodule algebra over a Hopf algebra, we construct a Galois correspondence between the complete lattices of of subalgebras and the complete lattice of generalised quotients of the structure Hopf algebra. The construction involves techniques of lattice theory and of Galois connections. Such a 'Galois Theory' generalises the classical Galois Theory for field extensions, and some important results of S.Chase and M.Sweedler, F. van Oystaeyen, P.Zhang and P.Schauenburg. Using the developed theory we positively answer the question raised by S. Montgomery: is there a bijective correspondence between generalised subobjects of a Hopf algebra and its generalised quotients? If time permits, I will present a proof, based on our results, for finite dimensional Hopf algebras. This is joint work with Dorota Marciniak (UAB Barcelona).

17/03/2010, 15:00 — 16:00 — Room P4.35, Mathematics Building
Joana Ventura, IST, CAMGSD

Reduced and tame fusion systems

The talk will have two parts. In the first, we'll define saturated fusion systems and present some properties. Then, we'll introduce two new classes of saturated fusions systems: reduced and tame, and explain how they are related in our search for exotic fusion system.

03/03/2010, 15:00 — 16:00 — Room P4.35, Mathematics Building
João Boavida, Instituto Superior Técnico

$O(n)$ periods of Eisenstein series on $O(n\mathrm{,1})$

We will walk over the calculation of the $O(n)$ period on an Eisenstein series on $O(n\mathrm{,1})$. We will see how this period unwinds into an Euler product, and how the factors can be computed (especially how high values of $n$ can be reduced to lower values). We will also see how this period fits into the bigger picture.

18/02/2010, 15:00 — 16:00 — Room P4.35, Mathematics Building
João Boavida, Instituto Superior Técnico

Quadratic forms and the structure of orthogonal groups

This talk will be a leisurely review of some classical material from the first two thirds of the last century, as background for the next talk (on how to actually calculate a period over $O(n\mathrm{,1})$). Specifically, today we will discuss the following topics: non-degenerate quadratic forms $Q$ over a field $k$ (for example, over $Q^3$, $Q(x)=x^2+y^2+z^2$), the parabolic subgroups of orthogonal groups $O(Q,k)$ (for the same example, $O(3)$), and the role they play on the structure of $O(Q)$. We will also see how to construct Eisenstein series over $O(Q)$.

26/01/2010, 14:00 — 15:00 — Room P4.35, Mathematics Building
Rachid el Harti, Hassan I University, Morocco

Projective limits of C*-algebras

The underlying C*-subalgebra $A_b$ of a projective limit algebra $A$ is regarded as the non-commutative version of the Stone-Cech compactification, and the functor $A\to A_b$ from topological *-algebras to C*-algebras as a coreflector. The C*-algebra topology can be described by two projective structures: the "trivial" and "maximal" ones. Besides the necessary background, I aim to describe another non-trivial structure for the group algebra C*($F_n$) associated to the free group $F_n$ and three for C*($F_{\mathrm{\infty }}$). For these we need a new characterisation of projective limits of C*-algebras. Joint work with Paulo Pinto.