Fields Undergraduate Network

Abstract from CAIMS-MPrime Industrial Mathematics Prize Lecture given at CAIMS2012 Meeting:
http://www.fields.utoronto.ca/programs/scientific/11-12/CAIMS_SCMAI/abstracts/lectures.html
High accuracy solutions to industrial problems

Industrial partners often have very high expectations concerning numerical modeling: accuracy, efficiency, robustness and, whenever possible, low computation costs and this, for very complex problems. In the last few years, we have concentrated our efforts at GIREF on the development of quadratic discretizations (in both space and time) for a large variety of applications. This is a good compromise between linear solutions (wrongly believed as low cost solutions) and higher order finite element approximations requiring higher solution regularity. To further enhance the accuracy of our solutions, we have also developed an adaptive remeshing strategy that can be applied to high order discretizations and leads to optimal meshes in a sense that will be explained. This also led to the development of iterative methods that maintain their convergence properties on very anisotropic meshes. In this presentation, we will briefly describe some of these methods and present a few industrial applications.

As part of the CMS Student Committee’s initiative to coordinate with FUN, we are hosting an event as part of the CMS Studc Fields Trip. We are planning a few morning activities while the CMS Studc is putting together the afternoon activities.

The morning session includes a short meet and greet with Professor Fraser and some of the other student attendants of the larger CMS Winter Meeting. Following this will be a short recap on all the events up to date held by FUN as well as some announcements of future directions.

Professor Fraser readily agreed to give a talk and answer questions about his talk on the History of Complex Analysis. Please review the abstract and consider the following references,

Helena Pycior, Symbols, Impossible Numbers, and Geometric Entanglements British Algebra
Through the Commentaries on Newton’s Universal Arithmetik (1997)

Ivo Schneider, “Der Mathematiker Abraham de Moivre (1667–1754)”, Archive for History of
Exact Sciences 5 (1968), pp. 177-317.

Hope to see you there!

Brigitte Stepanov, a mathematics student of Queen’s University, has put together a great event for the Fields Undergraduate Network.

She has invited three of Queen’s algebraic geometers, Mike Roth, Anthony Geramita, and Gregory Smith to give this set of specialized talks for undegraduates.

We hope to see you there and also that you would join us for dinner during the evening of the conference. If you choose to attend the dinner, please send an email to 8bns@queensu.ca

Please see below for the abstracts of the talks.

Algebraic Geometry as provider of insight
Mike Roth, Queen’s University

Abstract: One of the most appealing features of algebraic geometry is the way in which translating an algebraic problem to a geometric one can illuminate it, revealing aspects invisible from the point of view of equations. As a sample we will consider the problem of trying to find polynomial solutions to a single equation and see how the underlying geometry of the complex solutions completely resolves this algebraic question.

Sums of Squares:  Evolution of an Idea

Anthony V. Geramita, Queen’s University and the University of Genoa

Abstract: Questions about sums of squares of integers were considered in Number Theory by Gauss, Lagrange, Fermat and others. I will show, in this talk, how these considerations in Number Theory evolved into a wonderful question in Geometry, particularly in Algebraic Geometry.  Furthermore, that question still has aspects of it that are open problems which can be considered by undergraduates.

Polynomial Equations and Convex Polytopes

Gregory G. Smith, Queen’s University

Abstract: How many complex solutions should a system of n polynomial equations in n variables have?  When n = 1, the Fundamental Theorem of Algebra bounds the number of solutions by the degree of the polynomial.  In this talk, we will discuss generalizations for larger n.  We will focus on some of especially attractive bounds which depend only on the combinatorial structure (i.e. the associated Newton polytopes) of the polynomials.