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.

Duncan Wright -- An Introduction to Quantum Mechanics through Random Walks

We give a brief overview of the major differences of classical and quantum physics, focusing on statistical mechanics. The main goal of this talk is to introduce the simple (classical) random walk and its quantum generalization and to explore the similarities and differences between them.

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.

Robert Vandermolen -- I Assure you it is Geometry

This talk will try and introduce algebraic geometry to a more general audience. We will discuss the "geometry" behind algebraic geometry with fun examples like the affine plane \(\mathbb{A}_k^n\) and spaces like the zero locus of \(x^2+1\), while giving beautiful motivation from the world of quantum physics, with fun parables like Schrodinger's cat and Heisenberg's uncertainty principle.

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!)

Andy Kustin -- Syzygies -- (slides)

Hilbert introduced the notion of ``syzygies'' in his 1890 paper ``Ueber die Theorie der algebraischen Formen''. This paper contains the proof of the Hilbert basis theorem, the proof of the Hilbert syzygy theorem, the proof of Hilbert's Nullstellensatz, and the first version of the Hilbert-Burch theorem. Hilbert's motivation was combinatorial. He wanted to count how many homogeneous forms of each degree live in a given ring of invariants.

In this talk, syzygies will introduced slowly and gently. Various applications of syzygies to combinatorics, algebraic topology, and algebraic geometry will be described. For example, information about the singularities of a parameterized curve can be read from the syzygies of the parameterizing functions. Also, the syzygies of the homogeneous coordinate ring of a variety encode geometric information about the variety. Finally, there will be a discussion about how to find syzygies. We will discuss the Nike method, techniques from combinatorics and algebraic topology, computational techniques (that is, Groebner bases), and the geometric method. I am offering a course on the geometric method for finding syzygies in the Spring of 2018. The course and the textbook are both called ``The cohomology of vector bundles and syzygies''.

Alicia Lamarche -- Toric Varieties

Toric varieties form a special class of algebraic varieties whose structure can be described via combinatorial methods. We’ll focus on one particular example of a toric variety (the projective plane over \(\mathbb{C}\)!) in order to demonstrate how we can extract geometric information from the combinatorial description.

Garner Cochran -- Quick Trips: On the oriented diameter of graphs

I will begin with an introduction to diameter problems on graphs, showing some techniques that have been used previously on problems in the area. Then I will introduce the concept of oriented diameter and show the outline of a proof strengthening a previously known result for the largest oriented diameter of a graph from \(7\frac{n}{\delta+1}\) to \(5\frac{n}{\delta-1}\) and describe an algorithm that builds such an orientation.

Blake Farman -- Kernels for Noncommutative Projective Schemes

In their 1994 paper, Noncommutative Projective Schemes, Michael Artin and J.J. Zhang introduce a generalization of usual projective schemes to the setting of not necessarily commutative algebras over a commutative ring. Gonçalo Tabuada in 2005 endows the category of differential graded categories with the structure of a model category and in 2007 Toën shows that its homotopy category is symmetric monoidal closed. In this talk, we’ll give a brief overview of these results, adapting Artin and Zhang’s noncommutative projective schemes for the language of DG categories, and discuss a “geometric” description of this internal Hom for two noncommutative projective schemes. As an immediate application, we give a noncommutative projective derived Morita statement along the lines of Rickard and Orlov.

Erik Palmer -- Mathematical Modeling for High Performance Computing

In this talk we will examine two instances of mathematical modeling in a high performance computing setting. The first, will be the construction of a polymer mixing model that leverages the parallel processing power of graphics processing units to create a dynamic multi-level multi-physics simulation. This model not only reproduces the nonlinear response of hydrogel polymers in isolation but also provides a platform to capture the unique characteristics produced when two or more hydrogels are mixed. In the second, we will examine the particle collision models and tracking techniques used in the multiphase fluid flow simulation software, MFiX. Making efficient use of future exascale computing machines presents several challenges to the coupling of fluid dynamics with discrete element methods for particle tracking used in this software. Several strategies are developed to address these issues, such as adaptive mesh refinement, multicore and multithread programming and neighbor lists, but each strategy effects and is effected by the mathematical models employed.