Thursday, February 17th
Mike Wilson, University of Vermont
Abstract: After reminding the audience of a few basic astronomical facts, we will sketch the development of some ancient theories of the solar system. Via Mathematica, we will explore the concentric-sphere model of Eudoxus of Cnidus (4th Century BC), go on to the deferent-epicycle model of Apollonius of Perga (3rd Century BC) and its tweakings by Hipparchus of Nicea (2nd Century BC), and finish with the fully developed deferent-epicycle-equant model of Ptolemy of Alexandria (2nd Century AD). In a brief aside we will describe Hipparchus' very successful eccentric-circle model of solar motion. We will use no math beyond trigonometry.
Thursday, February 3rd
Iterative Numerical Methods for Solving Nonlinear Equations, and Techniques for Their Acceleration
Taras Lakoba, University of Vermont
Abstract: This talk will consist of two parts. First, I will review the concept behind iterative methods for obtaining solutions of linear and nonlinear (systems of) equations. One prominent application is finding stationary solitary wave solutions of nonlinear wave equations.
In the second part of the talk, I will consider three techniques to accelerate these methods. The first of these techniques eliminates the numerical stiffness of the system of equations caused by the need to include high Fourier harmonics. The second technique eliminates the slowest-decaying mode in the error of the iterative solution. The third technique is known as the Momentum method in Machine learning and consists of modifying the iterated matrix so as to greatly decrease its condition number.
I will try to make most of the talk accessible to the MS-level graduate and upper-level
Thursday, January 20th
Analytic, Number-Theoretic, and Combinatorial Aspects of Finite Point Configurations, and Applications to Data Science
Alex Iosevich, University of Rochester
Abstract: The basic question we are going to address is, does a sufficiently "large" subset of Euclidean space contain a point configuration of a given type, such as an equilateral triangle, a long chain, or an arbitrary angle? The notion of large can be measured in terms of Hausdorff dimension, Lebesgue density, or even the number of points in the discrete case. This simple-sounding question has attracted the attention of many mathematicians in the past 75 years or so, with the flagship problems being the Erdos distance conjecture in geometric combinatorics and the Falconer distance conjecture in geometric measure theory. We are going to discuss these problems, the variety of techniques and ideas that are used to approach them, and, towards the end of the talk, describe some ongoing efforts to apply the resulting technology to the study of dimensionality of large data sets.