Math 196: Student Colloquium
Fall 2016

Coordinator: Brendon Rhoades
Email: bprhoades (at) math.ucsd.edu
Office: 7250 APM
Colloquium Location: B402A APM
Colloquium Time: Tuesdays, 12:00-12:50 pm

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

9/27/2016: Brendon Rhoades
Title: Shoving boxes into corners
Abstract: A *partition* is a way to shove a collection of square boxes into a corner. We will describe some of the ways in which partitions arise in mathematics.

10/4/2016: Brendon Rhoades
Title: The Schensted correspondence
Abstract: A *standard Young tableau* is a way to put the numbers 1, 2, ... , n in a corner in such a way that we have increase going down columns and across rows. We will describe a relationship between standard Young tableaux and permutations in S_n.

10/11/2016: David Quarfoot
Title: Curiosity Reborn
Abstract: Research mathematics is about asking questions. When's the last time you did? Bring your cell phone or computer, and be prepared to change the way you approach everything in mathematics, and life.

10/18/2016: Jon Novak
Title: Polya's random walk theorem
Abstract: This lecture will be about a remarkable law of nature discovered by George Polya. Consider a particle initially situated at a given point of the d-dimensional integer lattice. Suppose that, at each tick of the clock, the particle jumps to a neighboring lattice site, with equal probability of jumping in any direction. Polya's law states that the particle returns to its initial position with probability one in dimensions d = 1,2, but with probability strictly less than one in all higher dimensions. Thus, a drunk person wandering a city grid will always return to their starting point, but if the drunkard can fly s/he might never come back.

10/25/2016: 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.

I will discuss (with two or three guiding examples) how to do calculus on Lie groups, which can usually be realized as groups of square matrices. I will then discuss the most important differential equation in the world -- the heat equation -- in the context of matrix Lie groups, and the beautiful interplay between geometry and heat flow. Finally, I'll talk about my research into the heat flow of eigenvalues in matrix Lie groups -- and there'll be a lot of cool pictureses.

11/1/2016: Ioan Bejenaru
Title: Interesting problems in Harmonic Analysis
Abstract: I will introduce some problems in Harmonic Analysis and explain their relations with other fields of Mathematics.

11/8/2016: 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.

11/15/2016: Eric Gelphman
Title: Algorithm for numerically computing Green's Theorem with applications to physics and engineering
Abstract: The Hyades Algorithm is an algorithm which I have invented to numerically compute the area of any complicated shape in R^2 using Green's Theorem by means of parametrizing the boundary of the shape using a traversal algorithm. This has many applications to thermodynamics involving the power and efficiency of internal combustion engines.

11/22/2016: TBA
Title: TBA
Abstract: TBA

11/29/2016: Jeff Rabin
Title: The unreasonable effectiveness of mathematics in physics: Differential geometry and general reletivity
Abstract: 2015 was the Centinnial year of Einstein's General Theory of Relativity, and fittingly concluded with the discovery of gravitational waves, which he had predicted. Despite knowing the key physical principles, Einstein was only able to formulate his theory after learning differential geometry from mathematician Marcel Grossmann in 1912. In a sense, General Relativity simply *is* applied differential geometry. This talk will sketch the key ideas of differential geometry and how they apply to Einstein's theory of gravity. The presentation will emphasize ideas and pictures, rather than equations.