warning: Creating default object from empty value in /www/www.mathtube.org/modules/taxonomy/taxonomy.pages.inc on line 33.

Mathematics

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.

The Life and Numbers of Richard Guy

Speaker: 
Hugh Williams
Date: 
Tue, Jun 21, 2016
Location: 
PIMS, University of Calgary
Conference: 
Canadian Number Theory Association 2016
Abstract: 
Fifty years ago Richard Kenneth Guy joined the then Department of Mathematics, Statistics and Computer Science at the nascent University of Calgary. Although he retired from the University in 1982, he has continued, even in his 100th year, to come in every day and work on the mathematics that he loves. Hugh Williams, Richard’s long-time friend and colleague, will share with us a glimpse into the life and research of this most remarkable man. You will not want to miss this inspirational talk!

Nonlocal equations from various perspectives - lecture 3

Speaker: 
Enrico Valdinoci
Date: 
Wed, Jun 15, 2016
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Workshop on Nonlocal Variational Problems and PDEs
Abstract: 
We would like to give a detailed presentation of some equations which exhibit some nonlocal phenomena. Often, the nonlocal effect is modelled by a diffusive operator which is (in some sense) elliptic and fractional. Natural example arise from probability, geometry, quantum physics, phase transition theory and crystal dislocation dynamics. We will try to discuss some of the mathematical tools that are useful to deal with these problems, explain in detail some of the main motivations, describe some recent results on these topics and list some open problems.

Nonlocal equations from various perspectives - lecture 2

Speaker: 
Enrico Valdinoci
Date: 
Tue, Jun 14, 2016
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Workshop on Nonlocal Variational Problems and PDEs
Abstract: 
We would like to give a detailed presentation of some equations which exhibit some nonlocal phenomena. Often, the nonlocal effect is modelled by a diffusive operator which is (in some sense) elliptic and fractional. Natural example arise from probability, geometry, quantum physics, phase transition theory and crystal dislocation dynamics. We will try to discuss some of the mathematical tools that are useful to deal with these problems, explain in detail some of the main motivations, describe some recent results on these topics and list some open problems.

Nonlocal equations from various perspectives - lecture 1

Speaker: 
Enrico Valdinoci
Date: 
Mon, Jun 13, 2016
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Workshop on Nonlocal Variational Problems and PDEs
Abstract: 
We would like to give a detailed presentation of some equations which exhibit some nonlocal phenomena. Often, the nonlocal effect is modelled by a diffusive operator which is (in some sense) elliptic and fractional. Natural example arise from probability, geometry, quantum physics, phase transition theory and crystal dislocation dynamics. We will try to discuss some of the mathematical tools that are useful to deal with these problems, explain in detail some of the main motivations, describe some recent results on these topics and list some open problems.
Syndicate content