Speaker: Thomas Hangelbroek, UH-Manoa

Title: Kernel approximation and PDEs (Part 2)

Abstract: Fundamental solutions to elliptic partial differential equations can serve as a useful tool for solving a variety of computational problems (e.g., data fitting, denoising, quadrature, numerical solution of differential equations). In these talks, I’ll develop some key results about meshless approximation with kernels arising as solutions to elliptic PDE — focusing primarily on analytic properties which derive directly from the differential equation, such as their approximation power and localized structure. I’ll include a number of examples on spheres, the rotation group, compact Riemannian manifolds without boundary, and Euclidean regions with boundaries.

TITLE: On Diophantine equations

ABSTRACT:

A Diophantine equation is an equation of the form F(X_1, X_2, … , X_m) = c (with a fixed c in Z) for which we look for the solutions (x_1, x_2, … , x_m) in Z^m verifying F(x_1, x_2, … , x_m) = c. The most famous result is probably the solution of Fermat’s last theorem X^n + Y^n = Z^n found by Andrew Wiles using so-called elliptic curves. A small survey of a few results will be given and the notion of elliptic curve will be introduced. The lecture is accessible to anyone, most particularly to undergraduates.

Speaker: Pat Collins, UH-Manoa

Title: Approximating with Symmetric Positive Definite Reproducing Kernels

Abstract: Symmetric positive definite kernels arise naturally as the reproducing kernels of Hilbert spaces, including certain Sobolev spaces of the form $ W_{2}^{m}(Omega) $. We explore the connection between reproducing kernels and symmetric positive definite kernels, and show that the corresponding Hilbert spaces contain unique norm-minimizing solutions to interpolation problems.

Speaker: David Jeffrey (U. Western Ontario)

Title: Twenty years of Lambert W

Abstract: The year 2016 marks 20 years since the publication of the paper

“On the Lambert W function”. As will be pointed out, the function was

studied before 1996, but this publication has proved to be the most cited reference. The talk will review some of the decisions made in defining

the function, particularly in defining its branches.

The talk will present some of the interesting and beautiful properties

of the function, including recent work that improves on the results in

the original papers. Specifically, expressions for derivatives and

series expansions, the role of differences between branches,

the convergence of series expansions.

See organizers Rufus, Alan or Robin for details.

Speaker: Asaf Hadari (UH Mānoa)

Title: Hilbert’s third problem – how to cut and paste using linear algebra

Abstract: In the year 1900 the mathematician David Hilbert famously gave a list of 21 problems that he felt were the most important challenges facing the mathematical community of the day.

The third problem, though stated differently, essentially asked whether it was necessary to use calculus to do basic geometry in three dimensions. For instance, is there a geometric way to calculate the volume of a pyramid?

This was the first of his problems that was answered, using a neat idea from linear algebra. I’ll show you how, and discuss some of the neat mathematics surrounding this problem.

Speaker: Amy Feaver (The King’s University)

Title: The Structure of Multiquadratic Number Fields

Abstract: In this talk, we will discuss the basic structure of multiquadratic number fields: that is, fields of the form $K:=\mathbb{Q}\left(\sqrt{a_1},\sqrt{a_2},…,\sqrt{a_n}\right)$ for $a_1,…,a_n\in\mathbb{Z}$ squarefree. We will begin by discussing subfields of $K$ and what we are able to know about $K$ based on its subfields. We will then define the ring of integers for these fields, and discuss several properties of these rings. These properties will include prime factorizations and the existence of a Euclidean algorithm (or lack thereof).