Mathematics

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

A1-homotopy of the general linear group and a conjecture of Suslin

Speaker: 
Ben Williams
Date: 
Mon, Jun 10, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
Following work of Röndigs-Spitzweck-Østvær and others on the stable A1-homotopy groups of the sphere spectrum, it has become possible to carry out calculations of the n-th A1-homotopy group of BGLn for small values of n. This group is notable, because it lies just outside the range where the homotopy groups of BGLn recover algebraic K-theory of fields. This group captures some information about rank-n vector bundles on schemes that is lost upon passage to algebraic K-theory. Furthermore, this group relates to a conjecture of Suslin from 1984 about the image of a map from algebraic K-theory to Milnor K-theory in degree n. This conjecture says that the image of the map consists of multiples of (n-1)!. The conjecture was previously known for the cases n=1, n=2 (Matsumoto's theorem) and n=3, where it follows from Milnor's conjecture on quadratic forms. I will establish the conjecture in the case n=4 (up to a problem with 2-torsion) and n=5 (in full). This is joint work with Aravind Asok and Jean Fasel.

Geometric Aspects of Arithmetic Statistics - 2 of 2

Speaker: 
Jordan Ellenberg
Date: 
Mon, Jun 10, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
Arithmetic statistics asks about the distribution of objects arising “randomly” in number-theoretic context. How many A_5-extensions of Q are there whose discriminant is a random squarefree integer? How big is the Selmer group fo a random quadratic twist of an elliptic curve over Q? How likely is the the maximal extension of Q unramified away from a random set of 3 primes to be an extension of infinite degree? Traditionally we ask these questions over Q or other number fields, but they make sense, and are expected to have broadly similar answers, when we replace Q with the function field of a curve over a finite field F_q. Having done this, one suddenly finds oneself studying the geometric properties of relevant moduli spaces over F_q, and under ideal circumstances, the topology of the corresponding moduli spaces over the complex numbers. Under even more ideal circumstances, results in topology can be used to derive theorems in arithmetic statistics over function fields. I’ll give an overview of this story, including some recent results in which the arrow goes the other way and analytic number theory can be used to prove results about the geometry of moduli spaces.

 


This is the second of a two part series: part 1

Geometric aspects of arithmetic statistics - 1 of 2

Speaker: 
Jordan Ellenberg
Date: 
Mon, Jun 10, 2019
Location: 
PIMS, University of British Columbia
Conference: 
Workshop on Arithmetic Topology
Abstract: 
Arithmetic statistics asks about the distribution of objects arising “randomly” in number-theoretic context. How many A_5-extensions of Q are there whose discriminant is a random squarefree integer? How big is the Selmer group fo a random quadratic twist of an elliptic curve over Q? How likely is the the maximal extension of Q unramified away from a random set of 3 primes to be an extension of infinite degree? Traditionally we ask these questions over Q or other number fields, but they make sense, and are expected to have broadly similar answers, when we replace Q with the function field of a curve over a finite field F_q. Having done this, one suddenly finds oneself studying the geometric properties of relevant moduli spaces over F_q, and under ideal circumstances, the topology of the corresponding moduli spaces over the complex numbers. Under even more ideal circumstances, results in topology can be used to derive theorems in arithmetic statistics over function fields. I’ll give an overview of this story, including some recent results in which the arrow goes the other way and analytic number theory can be used to prove results about the geometry of moduli spaces.

 


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