Mathematics

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

Class: 

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
CRG: 
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

Class: 

$A^1$ 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

Class: 

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
CRG: 
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.

Class: 

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

Class: 

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.

Class: 

Public Goods, from Biofilms to Societies.

Speaker: 
Simon Levin
Date: 
Sat, Jun 15, 2019
Location: 
PIMS, University of Victoria
Conference: 
Levin Fest
CRG: 
Abstract: 

Ecological and economic systems are alike in that individual agents compete for limited resources, evolve
their behaviors in response to interactions with others, and form exploitative as well as cooperative interactions as a result. In these complex adaptive systems, macroscopic properties like the flow patterns of resources like nutrients and capital emerge from large numbers of microscopic interactions, and feed back to affect individual behaviors. In this talk, I will explore some common features of these systems, especially as they involve the evolution of cooperation in dealing with public goods, common pool resources and collective movement. I will describe examples from bacteria and slime molds to vertebrate groups to insurance arrangements in human societies and international agreements on environmental issues.

Class: 

Mathematical ecology: A century of progress, and challenges for the next century

Speaker: 
Simon Levin
Date: 
Sat, Jun 15, 2019
Location: 
PIMS, University of Victoria
Conference: 
Levin Fest
CRG: 
Abstract: 

The subject of mathematical ecology is one of the oldest and most exciting
in mathematical biology, and has helped in the management of natural
systems and infectious diseases. Though many problems remain in those
areas, we face new challenges today in finding ways to cooperate in
managing our Global Commons. From behavioral and evolutionary
perspectives, our societies display conflict of purpose or fitness across
levels, leading to game-theoretic problems in understanding how
cooperation emerges in Nature, and how it might be realized in dealing with
problems of the Global Commons. This lecture will attempt to weave these
topics together, tracing the evolution from earlier work to challenges for the
future.


 



Simon Levin is the J.S. McDonnell distinguished university professor in Ecology and Evolutionary Biology at Princeton University. He is a recipient of the National Medal of Science, the Kyoto Prize and a Robert H. MacArthur Award.

Class: 

Modularity of Calabi-Yau Varieties

Speaker: 
Noriko Yui
Date: 
Thu, Apr 25, 2019
Location: 
PIMS, University of Saskachewan
Conference: 
Hugh C. Morris Lecture
CRG: 
Abstract: 

Let $X$ be a Calabi-Yau variety of dimension d. We will confine ourselves to Calabi-Yauvarieties of small dimensions, e.g., $d < 3$. Dimension one Calabi–Yaus are elliptic curves, those of dimension two are $K3$ surfaces, and dimension three ones are Calabi-Yau threefolds. Geometry and physics are both very much in evidence on Calabi-Yau varieties over the field of complex numbers.

Today I will focus on Calabi-Yau varieties defined over the field $Q$ of rational numbers (or number fields), and will discuss the modularity/automorphy of Calabi-Yau varieties in the framework of the Langlands Philosophy.

In the last twenty-five years, we have witnessed tremendous advances on the modularity question for Calabi-Yau varieties. All these results rest on the modularity of the two-dimensional Galois representations associated to them. In this lecture, I will present these fascinating results. If time permits, I will discuss a future direction for the realization of the Langlands Philosophy, in particular, for Calabi-Yau threefolds.

Biography

Noriko Yui is a professor of mathematics at Queen’s University in Kingston, Ontario. A native of Japan, Yui obtained her B.S. from Tsuda College, and her Ph.D. in Mathematics from Rutgers University in 1974 under the supervision of Richard Bumby. Known internationally, Yui has been a visiting researcher at the Max-Planck-Institute in Bonn a number of times and a Bye-Fellow at Newnham College, University of Cambridge.

Her research is based in arithmetic geometry with applications to mathematical physics and notably mirror symmetry. Currently, much of her work is focused upon the modularity of CalabiYau threefolds. Professor Yui has been the managing editor for the journal Communications in Number Theory and Physics since its inception in 2007. She has edited a number of monographs,and she has co-authored two books.

Subject: 
Class: 

An Introduction to Randomized Algorithms for Matrix Computations

Speaker: 
Ilse C.F. Ipsen
Date: 
Thu, Mar 14, 2019
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
CRG: 
Abstract: 

The emergence of massive data sets, over the past twenty or so years, has led to the development of Randomized Numerical Linear Algebra. Fast and accurate randomized matrix algorithms are being designed for applications like machine learning, population genomics, astronomy, nuclear engineering, and optimal experimental design.

We give a flavour of randomized algorithms for the solution of least squares/regression problems. Along the way, we illustrate important concepts from numerical analysis (conditioning and pre-conditioning), probability (concentration inequalities), and statistics (sampling and leverage scores).

Subject: 
Class: 

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