Mathematics

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

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

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.

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.

Given a matrix-valued function F that depend nonlinearly on a single
parameter z, the basic nonlinear eigenvalue problem consists of finding complex scalars z for which F(z) is singular. Such problems arise in many areas of computational science and engineering, including acoustics, control theory, fluid mechanics and structural engineering.

In this talk we will discuss some interesting mathematical properties of
nonlinear eigenvalue problems and then present recently developed
algorithms for their numerical solution. Emphasis will be given to the linear algebra problems to be solved.