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Scientific

On the local Langlands conjectures

Speaker: 
Rachel Ollivier
Date: 
Fri, Sep 30, 2016
Location: 
PIMS, University of British Columbia
Conference: 
UBC-PIMS Mathematical Sciences Faculty Award
Abstract: 

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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PDE Aspects of Fluid Flows

Speaker: 
Vladimir Sverak
Date: 
Fri, Sep 23, 2016
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 
We explain some of the recent results in concerning PDEs describing fluid flows, as well as some of the difficulties. Model equations will also be discussed. For more information, see the event webpage for this event.

The Arakelov Class Group

Speaker: 
Ha Tran
Date: 
Thu, Sep 22, 2016
Location: 
PIMS, University of Calgary
Conference: 
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract: 
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. See the event webpage for more information.

Sparsity, Complexity and Practicality in Symbolic Computation

Speaker: 
Mark Giesbrecht
Date: 
Thu, Mar 17, 2016
Location: 
PIMS, University of Manitoba
Conference: 
PIMS-UManitoba Distinguished Lecture
Abstract: 

Modern symbolic computation systems provide an expressive language for describing mathematical objects. For example, we can easily enter equations such as

$$f=x^{2^{100}}y^2 + 2x^{2^{99}+1}y^{2^{99}+1}+2x^{2^{99}}y+y^{2^{100}}x^{2}+2y^{2^{99}}x+1$$

into a computer algebra system. However, to determine the factorization

$$f -> (x^{2^{99}}y+y^{2^{99}}x+1)^2$$

with traditional methods would incur huge expression swell and high complexity. Indeed, many problems related to this one are provably intractable under various reasonable assumptions, or are suspected to be so. Nonetheless, recent work has yielded exciting new algorithms for computing with sparse mathematical expressions. In this talk, we will attempt to navigate this hazardous computational terrain of sparse algebraic computation. We will discuss new algorithms for sparse polynomial root finding and functional decomposition. We will also look at the "inverse" problem of interpolating or reconstructing sparse mathematical functions from a small number of sample points. Computations over both traditional" exact and symbolic domains, such as the integers and finite fields, as well as approximate (floating point) data, will be considered.

Bisections and Squares in Hyperelliptic Curves

Speaker: 
Nicolas Theriault
Date: 
Fri, Jun 10, 2016
Location: 
PIMS, University of Calgary
Conference: 
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract: 
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.
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