Speaker: Gideon Zamba (U. Iowa)

Title: Data-Driven Sciences: Another Way to Bring Math to the World and the World to Math

Abstract: Applied mathematics is a field of constant adaptability to the world’s contingencies. Such adaptability requires a solid training and understanding of theoretical and pure mathematical thinking—as the activity of applied thinking is vitally connected with research in pure mathematics. One such applied mathematical field is the field of statistics. As the world continues to rely more on data for decision making, statistics and associated data-driven fields have gained increased recognition. The purpose of this talk is to educate the audience about the field of statistics, about statistical involvements, and further provide examples of settings where statistical theory finds an application and where real world application calls for a new statistical development. The presentation further elaborates on Biostatistics and provides some general advice about mathematical and computational skills needed for a successful graduate degree in Biostatistics or Statistics.

The presentation is semi-technical.

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.