The Graduate Colloquium

 The Graduate Colloquium is a colloquium-style series for mathematics graduate students to share their current ideas with the rest of their colleagues. Interspersed within are talks and panels focused on career development.



Fall 2017 Schedule

Date Speaker Title
Wednesday
Aug 23
Panel Q & A with Current Graduate Students
locationLeConte 412 @ 11am-1pm
Tuesday
Sept 5
Alex Duncan Hilbert's 13th Problem
locationLeConte 412 @ 4:30pm-5:30pm
Tuesday
Sept 12
Hays Whitlatch Phylogeny, Pressing Sequences and Bill Gates' Burnt Pancakes: What do they have in common?
locationLeConte 412 @ 4:30pm-5:30pm
Tuesday
Sept 19
Panel The Academic Job Search
locationLeConte 412 @ 4:30pm-5:30pm
Tuesday
Sept 26
Duncan Wright TBA
locationLeConte 412 @ 4:30pm-5:30pm
Tuesday
Oct 3
Zhiyu Wang Erdős-Szekeres Theorem for Cyclic Permutations
locationLeConte 412 @ 4:30pm-5:30pm
Tuesday
Oct 10
Josiah Reiswig Aphid Sequences: Turning Fibonacci Numbers Inside Out
Tuesday
Oct 17
Robert Vandermolen TBA
Tuesday
Oct 24
Frank Thorne The Distribution of the Primes
Tuesday
Oct 31
Keller Vandebogert Classification of Elementary Particles via Symplectic Induction
Tuesday
Nov 7
Andy Kustin Syzygies
Tuesday
Nov 14
Alicia Lamarche Toric Varieties
Tuesday
Nov 21
TBA TBA
Tuesday
Nov 28
TBA TBA
Tuesday
Dec 5
TBA TBA


back to top



Abstracts


Panel -- Q & A with Current Graduate Students

An open and wide-ranging discussion of the mathematics graduate program at USC led by experienced graduate students and young faculty members.

Alex Duncan -- Hilbert's 13th Problem

The highlight of the standard algebra sequence is the Abel-Ruffini Theorem, which states that there is no solution in radicals to a general polynomial equation of degree 5 or greater. This is not the end of the story, though. There are many other types of functions besides radicals. When can equations be solved using more general classes of functions?

David Hilbert, in his 1900 Paris lecture, famously outlined 23 problems as a challenge to mathematicians for the twentieth century. His 13th problem asked if there was a solution to a degree 7 polynomial using only functions of two variables. Unfortunately, Hilbert was vague about exactly what kind of functions were allowed --- the answer depends on what kind of mathematician you are!

In this talk I will explore whether and how we can solve polynomials of various degrees using assorted kinds of functions.

Hays Whitlatch -- Phylogeny, Pressing Sequences and Bill Gates' Burnt Pancakes: What do they have in common?

In the field of phylogenetics one studies the evolution of genomes through mutations such as inversions, transpositions, translocations, and fusions and fissions. One problem of interests in this field is to provide an evolutionary distance between species by studying the number of (likely) mutations needed to transform one genome into another. In computer science and mathematics one studies related problems such as permutation editing and sorting, pressing sequences in graphs and other combinatorial objects. In this talk we discuss some of these mathematical problems as well as some interesting associated work.

Panel -- The Academic Job Search

Join us to get all of your academic job search questions answered and to hear advice from professors with experience as recent job seekers and/or advising job seekers.

Zhiyu Wang -- Erdős-Szekeres Theorem for Cyclic Permutations

We show a cyclic permutation analogue of Erdős-Szekeres Theorem. In particular, we show that every cyclic permutation of length \((k-2)(m-2)+2\) has either an increasing cyclic sub-permutation of length \(k\) or a decreasing cyclic sub-permutation of length \(m\). We also show that the result is tight.

Josiah Reiswig -- Aphid Sequences: Turning Fibonacci Numbers Inside Out

Fibonacci numbers are well known to enumerate pairs of rabbits assuming liberally interpreted reproductive patterns and immortality. We turn these numbers inside out by considering the reproductive behavior of aphids, whose offspring can begin to reproduce before they are even born. We study the sequences that arise both from allowing aphids to be immortal and from assuming a given lifespan, tying together many different generalizations of Fibonacci numbers. We determine recurrence relations for them and generating function. We then use the generating functions to determine bounds on the growth rates of these sequences.

Frank Thorne -- The Distribution of the Primes

A famous recent result -- due to Zhang, Maynard, Tao, and the Polymath collaboration -- is that there are infinitely many pairs of primes which are at most 246 apart. Another recent result, due to Helfgott: every odd integer at least 7 can be written as the sum of three primes.

I will explain at least two things in my talk: (1) why such results were firmly expected, long before they were proved, and (2) the first step in any such proof, a "level of distribution" result for the primes: what this means, and why it opens doors for clever people like those listed above.

This talk will be an advertisement for a graduate course on the subject which I plan to teach here, and an accompanying book which I will be writing with Robert Lemke Oliver.

Keller Vandebogert -- Classification of Elementary Particles via Symplectic Induction

The Poincare group G has a linear representation which can be understood as the composition of a Lorentz transform and a spacetime "boost" for points sitting in Minkowski space. As it turns out, the (abelian) normal subgroup of spacetime boosts H induces a particularly simple orbit structure of our Poincare group, but only for elements of H. Luckily, there is an elaborate construction known as Symplectic Induction for which you can induce a symplectic structure on the entirety of a space based off of the symplectic structure of H-orbits. These induced orbits allow us to classify the standard elementary particles: photons, tachyons, and particles with mass, and characterize these systems as Hamiltonian G-spaces. (This talk will be made as accessible as possible, but some familiarity with Lie groups/algebras and coadjoint representations would certainly help!)

back to top



Previous Colloquia