Math 196: Student Colloquium

Fall 2018

Coordinator: Alex Cloninger

Email: acloninger (at) ucsd (dot) edu

Office: AP&M 5747

Colloquium Location: AP&M B402A

Colloquium Time: Tuesdays, 1-1:50pm

Description: This is a weekly colloquium meant to give students a fun and informative introduction to ideas in research mathematics.

Grading: There will be a sign-in sheet at each colloquium. You may miss at most one colloquium to receive credit.

Schedule

10/2/2018: Alex Cloninger
Title: Laplacian Eigenvectors and Applications
Abstract: The graph Laplacian is a simple way to summarize which nodes are connected on a graph, but it contains a lot of information in its eigendecomposition. It also can be used to synthesize information about data points that lie in a high-dimensional space, and has connections to everything from differential geometry to image processing. We will discuss the intuition, theory, and applications of these eigenvectors, and discuss some (as yet) unanswered questions.

10/9/2018: Brendon Rhoades
Title: Shoving boxes into corners
Abstract: A partition is a way to shove a finite collection of square boxes into a corner. We will explain some combinatorial, algebraic, and geometric applications of box shoving.

10/16/2018: Wenxin Zhou
Title: Huber Regression
Abstract: Big data can easily be contaminated by outliers or contain variables with heavy-tailed distributions, which makes many conventional methods inadequate. To address this challenge, we propose the adaptive Huber regression for robust estimation and inference.

10/23/2018: Melvin Leok
Title: Computational Geometric Mechanics: A Synthesis of Differential Geometry, Mechanics, and Numerical Analysis
Abstract: Geometric mechanics involves the use of differential geometry and symmetry techniques to study mechanical systems. In particular, it deals with global invariants of the motion, and how they can be used to describe and understand the qualitative properties of complicated dynamical systems, without necessarily explicitly solving the equations of motion. This approach parallels the development of geometric numerical methods in numerical analysis, wherein numerical algorithms for the solution of differential equations are constructed so as to exactly conserve the invariants of motion of the continuous dynamical system. This talk will provide a gentle introduction to the role of geometric methods in understanding nonlinear dynamical systems, and why it is important to develop numerical methods that have good global properties, as opposed to just good local behavior.

10/30/2018: Martin Licht
Title: Numerical methods for partial differential equations
Abstract: We take on a wild tour through the jungle of numerical methods for partial differential equations. In this lecture you will learn about what brings together computer simulations, partial differential equations, linear algebra, and modeling in physics and biology.

11/6/2018: Andrew Suk
Title: Sum versus product: number theory, graph theory, and geometry
Abstract: In this talk, I will sketch a surprising proof due to Gy├Ârgy Elekes on a non-trivial lower bound for the sums-versus-product problem in combinatorial number theory.

11/13/2018: Todd Kemp
Title: Calculus and the Heat Equation on Matrix Lie Groups Abstract: In Math 20, we learned how to differentiate and integrate functions defined on Euclidean spaces. There is a much wider world of smooth spaces (manifolds) where a generalization of calculus is possible, but it requires a steep learning curve and a lot of new language to understand. There is a class of manifolds, however, that is both large and interesting, and also retains enough Euclidean-like structure to do calculus almost the same way as in Math 20. These are called Lie groups.

11/20/2018: TBA

11/27/2018: Rayan Saab
Title: The cocktail party problem
Abstract: I will talk about the problem of separating multiple signals from each other when we only have access to a few linear (or non-linear) combinations of them. An example of this type of problem is at a cocktail party when you are trying to have a conversation with a friend but there are several converations happening around you. Your ears provide you with a superposition of all the voices, and your brain does remarkably well at focusing on your friend's voice and drowning out all the others. We will talk about one computer algorithm (or time permitting, more) that does such a task (reasonably) successfully. Along the way, we will talk about important tools in mathematical signal processing, including the Fourier transform and sparsity.

12/4/2018: Adrian Ioana
Title: Orbit equivalence of group actions
Abstract: I will give a gentle introduction to the theory of orbit equivalence. This aims to study the structure of orbits arising from group actions on measure spaces. I will first explain the basic notions in the theory, and then present (both classic and recent) results in the particular case of actions by rotations on the n-dimensional sphere.