Fall 2012

**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**

** 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.