Scientific

The connection between textiles and mathematics is intimate but not often explored, possibly because textiles and fiber arts have traditionally been the domain of women while mathematics was viewed as a male endeavour. How times have changed! Today, textiles and mathematics, like art and science, are recognized for their interwoven, complimentary attributes. In this presentation, mathematics professor Gerda de Vries will examine the connection between textiles and mathematics, in the context of both traditional and contemporary quilts. In a sense, every quilt is a mathematical object, by virtue of the fact that it has shape and dimension. But some quilts are more mathematical than others, and in very different ways. She will show how mathematical concepts such as symmetry, fractals, and algorithmic design show up in the world of quilting through serendipitous and intentional design.

This lecture is for a general audience. A background in mathematics is not needed, nor the ability to sew!

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Abstract

The Langlands program, initiated in the 1960s, is a set of conjectures predicting a unification of number theory and the representation theory of groups. More precisely, the Langlands correspondence provides a way to interpret results in number theory in terms of group theory, and vice versa.

In this talk we sketch a few aspects of the local Langlands correspondence using elementary examples. We then comment on some questions raised by the emerging "mod p" Langlands program.

Biography

Professor Ollivier works in the Langlands Programme, a central theme in pure mathematics which predicts deep connections between number theory and representation theory. She has made profound contributions in the new branches of the "p-adic" and "mod-p" Langlands correspondence that emerged from Fontaine's work on studying the p-adic Galois representation, and is one of the pioneers shaping this new field. The first results on the mod-p Langlands correspondence were limited to the group GL2(Qp); but Dr. Ollivier has proved that this is the only group for which this holds, a surprising result which has motivated much subsequent research.

She has also made important and technically challenging contributions in the area of representation theory of p-adic groups, in particular, in the study of the Iwahori-Hecke algebra. In joint work with P. Schneider, Professor Ollivier used methods of Bruhat-Tits theory to make substantial progress in understanding these algebras. She has obtained deep results of algebraic nature, recently defining a new invariant that may shed light on the special properties of the group GL2(Qp).

Rachel Ollivier received her PhD from University Paris Diderot (Paris 7), and then held a research position at ENS Paris. She subsequently was an assistant professor at the University of Versailles and then Columbia University, before joining the UBC Department of Mathematics in 2013.

Rachel is the recepient of the 2015 UBC Mathematics and Pacific Institute for the Mathematical Sciences Faculty Award.

More information on this event is available on the event webpage

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Let F be a number field. The Arakelov class group Pic_0 F of F is an analog to the Picard group of a curve. This group provides us the information of F such as the class number h_F, the class group Cl(F) and the regulator R_F. In this talk, we will first introduce Pic_0 F then show that how this group gives us the information of F. Next, we will discuss a tool to compute this group-reduced Arakelov divisors- and their properties. Finally, some open problems relating to this topic will be presented.

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For elliptic curves, the Mordell-Weil Theorem allows to relate bisections (pre-images of the multiplication by 2) in the group of points of a curve defined over F_q and the quadratic reciprocity of some elements in the field F_q, which can be used to obtain an algorithm to bisect points in E(F_q). For reduced divisors D=[u(x),v(x)] (in Mumford representation) in the Jacobian of imaginary hyperelliptic curves y^2=f(x) (with f(x) squarefree and of odd degree), we show a relation between the existence of F_q-rational bisections and the quadratic character of u(x) when it is evaluated at the roots of the polynomial f(x) (i.e. at the x-coordinates of the Weierstrass points). This characterization allows us to compute all the bisections of a reduced divisor computing a few square roots (2g square roots if f(x) has 2g+1 roots in F_q) and solving a small system of linear equations.For hyperelliptic curves of genus 2, we obtain an equivalent characterization for curves with a real model (with f(x) squarefree of degree 6) when working with balanced divisors.