Math 196/296: Student Colloquium
Fall 2012

Coordinator: Brendon Rhoades
Email: bprhoades (at)
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.


10/2/2012: Brendon Rhoades
Title: Shoving boxes into corners.
Abstract: If n is a nonnegative integer, a partition of n is a sequence of weakly decreasing positive integers which sum to n. Partitions arise in the study of the symmetric group of permutations of the set {1, 2, ..., n}, the geometry of the Grassmannian of k-dimensional subspaces of an n-dimensional vector space, and in numerous combinatorial areas such as combinatorial asymptotics and finite field theory. I will show how a visualization of partitions obtained by shoving boxes into a corner can be used to define a polynomial refinement of the binomial coefficients called the Gaussian polynomials and will discuss various properties of this polynomial refinement (and present at least one open problem).

10/9/2012: Alireza Salehi Golsefidy
Title: Random walk on graphs and expanders.
Abstract: I will explain what a random walk is and we will see how it can help us to see some of the properties of the underlying graph. Then I define what expanders are and mention some of their applications.

10/15/2012: Jelena Bradic
Title: Why do my friends have more friends then I do (Friendship Paradox in Statistics).

10/23/2012: Mia Minnes
Title: Orders.
Abstract: Ordered-sets show up in many examples: the natural numbers, the dictionary order of words in the language, the cardinals indexing ``sizes of infinity". We will talk about special properties of orders on countable sets and see how the rational numbers give a universal ordering. Then, we will talk about a characterization of all linear orders and mention research directions in this area.

10/30/2012: Jim Lin
Title: Careers in Mathematics.
Abstract: Many undergraduates think that math majors either become professors or high school teachers. There are actually many other career options. In this talk, we will review data from a survey of employers and other mathematicians to try to answer what employers value about mathematicians and what courses help you to develop the skills that make you employable.

11/6/2012: Adrian Ioana
Title: The Banach Tarski paradox.
Abstract: I will explain the Banach Tarski paradox: suppose that we are given a sphere. One can slice the sphere into a finite number of pieces such that these pieces can be re-assembled into two spheres of the exact size as the original.

11/13/2012: Daniel Rogalski
Title: The quaternions.
Abstract: William Hamilton discovered the quaternions in 1843. This was the first example of a system of numbers in which every nonzero element has a multiplicative inverse, but for which multiplication is not commutative. Besides still being a fundamental example in noncommutative algebra, the quaternions have many connections to the analysis of real 3-space and are still important today in applications such as computer graphics. We will define the quaternions and investigate some of their interesting properties and applications.

11/20/2012: Alina Bucur
Title: Size doesn't matter: heights in number theory.
Abstract: How complicated is a rational number? Its size is not a very good indicator for this. For instance, 1987985792837/1987985792836 is approximately 1, but so much more complicated than 1. We'll explain how to measure the complexity of a rational number using various notions of height. We'll then see how heights are used to prove some basic finiteness theorems in number theory. One example will be the Mordell-Weil theorem: that on any rational elliptic curve, the group of rational points is finitely generated.

11/27/2012: Ron Evans
Title: The mathematics of Futurama and Stargate SG-1.
Abstract: Episodes of the acclaimed sci-fi television series Futurama and Stargate SG-1 feature a two-body mind-swapping machine which will not work more than once on the same pair of bodies. Crises are created when participants in the swapping are unable to figure out how to return everyone back to normal. We show how group theory saves the day, by giving the best possible algorithms for undoing the swapping.

12/4/2012: Dragos Oprea
Title: Enumerative geometry.
Abstract: The basic question of enumerative geometry can be simply stated as: How many geometric objects of a given type satisfy given geometric conditions?
For instance, one may ask for:
- the number of lines intersecting 4 fixed general lines in space (the answer is 2);
- the number of lines on a general cubic surface (the answer is 27);
- the number of lines on a quintic threefold in P^4 (the answer is 2,875)...
In this talk, I will give an introduction to some aspects of enumerative geometry.