Scientific

Coincidences between homological densities, predicted by arithmetic - 2 of 2

Speaker: 
Benson Farb
Date: 
Thu, Jun 13, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
In this talk I'll describe some remarkable coincidences in topology that were found only by applying Weil's (number field)/(f unction field) analogy to some classical density theorems in analytic number theory, and then computing directly. Unlike the finite field case, here we have only analogy; the mechanism behind the coincidences remains a mystery. As a teaser: it seems that under this analogy the (inverse of the) Riemann zeta function at (n+1) corresponds to the 2-fold loop space of P^n. This is joint work with Jesse Wolfson and Melanie Wood.

 


This is the second lecture in a two part series: part 1

Representation stability and asymptotic stability of factorization statistics

Speaker: 
Rita Jimenez-Rolland
Date: 
Wed, Jun 12, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
In this talk we will consider some families of varieties with actions of certain finite reflection groups – such as hyperplane complements or complex flag manifolds associated to these groups. The cohomology groups of these families stabilize in a precise representation theoretic sense. Our goal is to explain how these stability patterns manifest, and can be recovered from, as asymptotic stability of factorization statistics of related varieties defined over finite fields.

Quillen's Devissage in Geometry

Speaker: 
Inna Zakharevich,
Date: 
Tue, Jun 11, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 

In this talk we discuss a new perspective on Quillen's devissage theorem. Originally, Quillen proved devissage for algebraic $ K $-theory of abelian categories. The theorem showed that given a full abelian subcategory $ \mathcal{A} $ of an abelian category $ \mathcal{B} $, $ K(\mathcal{A})\simeq K(\mathcal{B}) $ if every object of $ \mathcal{B} $ has a finite filtration with quotients lying in $ \mathcal{A} $. This allows us, for example, to relate the $ K $-theory of torsion $ \mathbf{Z} $-modules to the $ K $-theories of $ \mathbf{F}_p $-modules for all $ p $. Generalizations of this theorem to more general contexts for $ K $-theory, such as Walhdausen categories, have been notoriously difficult; although some such theorems exist they are generally much more complicated to state and prove than Quillen's original. In this talk we show how to translate Quillen's algebraic approach to a geometric context. This translation allows us to construct a devissage theorem in geometry, and prove it using Quillen's original insights.

A^1 enumerative geometry: counts of rational curves in P^2 - 2 of 2

Speaker: 
Kirsten Wikelgren
Date: 
Wed, Jun 12, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
We will introduce A^1 homotopy theory, focusing on the A^1 degree of Morel. We then use this theory to extend classical counts of algebraic-geometric objects defined over the complex numbers to other fields. The resulting counts are valued in the Grothendieck--Witt group of bilinear forms, and weight objects using certain arithmetic and geometric properties. We will focus on an enrichment of the count of degree d rational plane curves, which is joint work with Jesse Kass, Marc Levine, and Jake Solomon.

 


This is the second lecture in a two part series: part 1

Conjectures, heuristics, and theorems in arithmetic statistics - 1 of 2

Speaker: 
Wei Ho
Date: 
Wed, Jun 12, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
We will begin by surveying some conjectures and heuristics in arithmetic statistics, most relating to asymptotic questions for number fields and elliptic curves. We will then focus on one method that has been successful, especially in recent years, in studying some of these problems: a combination of explicit constructions of moduli spaces, geometry-of-numbers techniques, and analytic number theory.

E_2 algebras and homology - 1 of 2

Speaker: 
Soren Galatius
Date: 
Wed, Jun 12, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
Block sum of matrices define a group homomorphism GL_n(R) \times GL_m(R) \to GL_{n+m}(R), which can be used to make the direct sum of H_s(BGL_t(R);k) over all s, t into a bigraded-commutative ring. A similar product may be defined on homology of mapping class groups of surfaces with one boundary component, as well as in many other examples of interest. These products have manifestations on various levels, for example there is a product on the level of spaces making the disjoint union of BGL_n(R) into a homotopy commutative topological monoid. I will discuss how it, and other concrete examples, may be built by iterated cell attachments in the category of topological monoids, or better yet E_2 algebras, and what may be learned by this viewpoint. This is all joint work with Alexander Kupers and Oscar Randal-Williams.

 


This is the first lecture in a two part series: part 2

The Grothendieck ring of varieties, and stabilization in the algebro-geometric setting - part 1of 2

Speaker: 
Ravi Vakil
Date: 
Tue, Jun 11, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
A central theme of this workshop is the fact that arithmetic and topological structures become best behaved “in the limit”. The Grothendieck ring of varieties (or stacks) gives an algebro-geometric means of discovering, proving, or suggesting such phenomena

 


In the first lecture of this minicourse, Ravi Vakil will introduce the ring, and describe how it can be used to prove or suggest such stabilization in several settings.

 


This is the first lecture in a two part series: part 2

 


In the second lecture of the minicourse, Aaron Landesman will use these ideas to describe a stability of the space of low degree covers (up to degree 5) of the projective line (joint work with Vakil and Wood). The results are cognate to Bhargava’s number field counts, the philosophy of Ellenberg-Venkatesh-Westerland, and Anand Patel’s fever dream.

Point counting and topology - 1 of 2

Speaker: 
Benson Farb
Date: 
Tue, Jun 11, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
In this first talk I will explain how the machinery of the Weil Conjectures can be used to transfer information back and forth between the topology of a complex algebraic variety and its F_q points. A sample question: How many F_q-points does a random smooth cubic surface have? This was recently answered by Ronno Das using his (purely topological) computation of the cohomology of the universal smooth, complex cubic surface. This is part of a much larger circle of fascinating problems, most completely open.

 


This is the first lecture in a two part series: part 2

A^1 enumerative geometry: counts of rational curves in P^2 - 1 of 2

Speaker: 
Kirsten Wikelgren
Date: 
Tue, Jun 11, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
We will introduce A^1 homotopy theory, focusing on the A^1 degree of Morel. We then use this theory to extend classical counts of algebraic-geometric objects defined over the complex numbers to other fields. The resulting counts are valued in the Grothendieck--Witt group of bilinear forms, and weight objects using certain arithmetic and geometric properties. We will focus on an enrichment of the count of degree d rational plane curves, which is joint work with Jesse Kass, Marc Levine, and Jake Solomon.

 


This is the first lecture in a two part series: part 2
Syndicate content