Algebra and Topology Seminar 

08/01/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building
Luke Wolcott, University of Western Ontario

Bousfield lattices, quotients, ring maps, and non-Noetherian rings

Given an object $X$ in a compactly generated tensor triangulated category $C$ (such as the derived category of a ring, or the stable homotopy category), the Bousfield class of $X$ is the collection of objects that tensor with $X$ to zero. The set of Bousfield classes forms a lattice, called the Bousfield lattice $\mathrm{BL}(C)$. First, we will look at examples of when a functor $F:C\to D$ induces a lattice map $\mathrm{BL}(C)\to \mathrm{BL}(D)$, and will describe several lattice quotients and lattice isomorphisms. Second, we will focus on homological algebra; a ring map $f:R\to S$ induces, via extension of scalars, a functor $D(R)\to D(S)$, and this induces a map on Bousfield lattices. Third, we specialize to a specific map between some interesting non-Noetherian rings.

See also

https://www.math.tecnico.ulisboa.pt/~ggranja/Wolcott-IST.pdf

15/11/2012, 15:30 — 16:30 — Room P3.10, Mathematics Building
Enxin Wu, University of Western Ontario

Doing differential geometry on the irrational torus, a rush introduction to diffeology.

The quotient group of the $2$-torus modulo a line of an irrational slope is an interesting geometric object. However, as a quotient topological space it is indiscrete. In noncommutative geometry, the study of this geometric object (called the irrational torus) is based on the study of a $C^{*}$-algebra related to it. In this talk, we will introduce a geometric way (called diffeology) to study the irrational torus. If time permits, I will talk about the usual geometric information of the irrational torus from the diffeological point of view: the tangent bundle, de Rham cohomology, smooth homotopy groups, etc.

17/07/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
Marco Robalo, Université Montpellier 2

Noncommutative Motives: A universal characterization of the motivic stable homotopy theory of schemes.

In this talk I will explain a new approach to the theory of noncommutative motives based in the construction of a motivic stable homotopy theory for the noncommutative schemes of Kontsevich.

In the first part of the talk, we review the original motivic homotopy theory of schemes as constructed by Voevodsky-Morel-Jardine and explain its recent universal characterization. The fundamental step in this characterization is to understand the nature of the construction of symmetric spectrum objects in a model category M. I will try to sketch the idea in detail.

In the second part we explain the construction of a new motivic theory for the noncommutative schemes which mimics the classical one for schemes. Every scheme gives rise to a noncommutative one and because of the universal property described in the first part, this assignment gives birth to a canonical monoidal comparison map between the commutative and the new noncommutative motivic theories. This work is ongoing and it is part of my PhD thesis under the direction of B. Toën in the Université de Montpellier.

The talk will require (at all times) the language of $\mathrm{\infty }$-categories, for which I will provide a small introduction in the beginning.

References

• M. Kontsevich, Noncommutative motives
• J. Lurie, Higher Algebra
• M. Robalo, Non-commutative motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes
• G. Tabuada, A guided tour through the Garden of Noncommutative Motives

12/07/2012, 16:30 — 17:30 — Room P4.35, Mathematics Building
Tibor Beke, University of Massachusetts

The sign pattern theorem and Brouwer's fixed point theorem

This work grew out of my attempt at concocting a proof of the Brouwer fixed point theorem that is suitable for a first course in topology. It should not involve algebraic topology and special tricks like the no-retraction theorem, and should make the statement itself plausible. Already in dimension two, Brouwer's fixed point theorem is quite surprising and (visually) not very compelling --- a contrast to the one-dimensional case where the statement is equivalent to the intermediate value theorem that is visually "obvious". We present a proof the Brouwer fixed point theorem as a higher-dimensional generalization of the intermediate value theorem. The proof itself is purely combinatorial and reduces to the "sign pattern theorem" about (higher dimensional) matrices containing two types of symbols, + and -. This talk should be suitable (and hopefully, ideal) for undergraduate students.

03/07/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
Maria Vaz Pinto, Instituto Superior Técnico

The regularity and the vanishing ideal of parameterized codes associated to even cycles

Let $X$ be an algebraic toric set in a projective space over a finite field $K$, parametrized by the $s$ edges of a graph $G$. We give an explicit combinatorial description of a set of generators of the vanishing ideal of $X$, $I(X)$, when $G$ is an even cycle or a connected bipartite graph with pairwise vertex disjoint even cycles. We also show a formula for the regularity of $S/I(X)$, where $S$ is a polynomial ring in $s$ variables over $K$.

15/12/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Mike Paluch, Instituto Superior Técnico

A simplicial guide to Voevodsky's motives

Let $V$ be a smooth complex projective variety. Since $V$, when endowed with the classical topology carries the homotopy type of a finite CW complex, it follows that the normalized chain complex $NZ\mathrm{Sin}(V)$ of the free simplicial abelian group of the singular complex of V can be viewed as an object in the derived category $D(\mathrm{Ab})$ of bounded complexes of finitely generated abelian groups. As stated by Beilinson and Vologodsky a basic objective of the theory of motives is to find a triangulated category $\mathrm{DM}$, whose formulation does not depend on the metric topology of the field of complex numbers, and functors from the category algebraic varieties to $\mathrm{DM}$ and from $\mathrm{DM}$ to $D(\mathrm{Ab})$ such that the composite functor carries $V$ to $NZ\mathrm{Sin}(V)$. In this talk I shall present a simplicial perspective of Voevodsky's theory of motives in terms of additive and simplicially additive Grothendieck topologies and a simplicial candidate for $\mathrm{DM}$.

17/11/2011, 15:00 — 16:00 — Room P4.35, Mathematics Building
Stavros Papadakis, CAMGSD, IST

Betti bounds for the Stanley-Reisner ring of a stellar subdivision

The talk will be about recent joint work with Janko Boehm (Kaiserslautern) which uses unprojection theory to give bounds for the betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex.

22/09/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Pedro Brito, University of Aberdeen

Enriched manifold calculus and operads

Goodwillie-Weiss' manifold calculus is, in some sense, an extension of Gromov's $h$-principle. Given a topological presheaf $F$, it produces a tower of (homotopy) sheafifications of $F$, where the first sheafification corresponds to Segal's scanning map. In this talk I will present a continuous (as in enriched over spaces) formulation of manifold calculus and discuss an interesting connection to operads along the way. This is joint work with Michael Weiss.

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.