**The Graduate Colloquium**

Department of Mathematics

University of South Carolina

*Fall 2015 Lectures and Events*

Abstracts

**Introductory panel** - This will be an open and wide-ranging discussion of graduate program led by some experienced graduate students and young faculty.

**Daniel Kamenetsky** - Binary forms are homogeneous polynomials in two variables. It is natural to consider the effect of invertible linear changes of variables on binary forms and to investigate any properties that are unaffected by such transformations. I will provide an overview of the basic invariant theory in the simplest cases of complex and real binary quadratic, cubic, and quartic forms. I will use the language of group actions and discuss the role of invariants in the problem of distinguishing orbits. Additionally, I will point out some connections to number theory, where the study of integral binary forms, from the work of Gauss to the recent work of Bhargava, has led to many significant results.

**Joshua Cooper** - I will discuss a few problems in combinatorics and discrete math that have been the subject of my research in recent years. We'll visit the partition-regularity of Pythagorean triples, a problem at the intersection of Ramsey Theory, hypergraphs, and number theory; unsolved problems arising from pressing sequences of bicolored graphs, a constellation of questions that started in bioinformatics but now connects graphs, permutations, computational complexity, probability, and binary matrix theory; and spectral hypergraph theory, a topic that involves hypergraphs, multilinear algebra, probability, algebraic geometry, and optimization theory.

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

**Tyler Lewis** - Singularity theory is a broad field that has fascinating relations to algebraic geometry, commutative algebra, complex analysis, representation theory, Lie groups, topology, and many other fields. Fundamentally, singularity theory is the study of finitely many differentiable, analytic, or algebraic functions in the neighborhood of a point where the Jacobian matrix of these functions is not of locally constant rank. In this talk, we will discuss how to think about these singularities algebraically, given that different equations can give rise to the same singularity. We will look at some interesting examples and define the main invariants and ideas that go into studying singularities.

**Paula Vasquez** - What do shampoo, paint, milk, cell membranes, and mucus have in common? They are all complex fluids.
Complex fluids are soft materials with the ability to behave both as viscous liquids and elastic solids. In addition, their behavior at the macroscopic scale is dictated by complicated dynamics at the microscopic level. This multi-scale, multi-physics coupling leads to unique and often unforeseen behaviors. From the mathematical point of view, these fluids offer a wealth of research opportunities for both experienced and novice mathematicians from fields such a functional analysis, PDEs, numerical analysis, and graph theory, just to mention a few.
In this talk, I will discuss two of my established research projects: the study of flow and diffusive transport processes in pulmonary mucus and the organization and dynamics of chromosomes in yeast cells. I will also discuss new opportunities in the math biology group that include the implementation of models derived from network theory to study tomato plants and pituitary cells.

**Erik Palmer** - Mucus inside the lungs serves a fundamental purpose to our well-being. From the health of prematurely born babies to cystic fibrosis, much depends on the viscoelastic properties of lung mucus. However, due to the complexities which arise from the ability of mucus to behave as both a viscous liquid and an elastic solid, few mathematical models have produced significant insight into the rheology of this system. In response to these challenges, I will present an elastic dumbbell chain model that leverages the parallel processing power of GPUs to create a unique micro-macro scale driven design that describes the breaking and reforming of entanglement points in the mucin molecule network. The added fidelity which this model provides will allow a full reconstruction of the microstructure-flow coupling giving it the ability to demonstrate the influence of well-known microscopic characteristics on the fluid flow properties of lung mucus.

**Alexander Duncan** Consider the field C(x,y) consisting of all rational functions in the variables x and y with complex coefficients. A complex automorphism g of the field C(x,y) is obtained by replacing x and y with new rational functions in x and y (with some constraints). The set of all automorphisms form the plane Cremona group. We'll discuss how you can study the structure of this group using the geometry of algebraic surfaces.

**Jesse Kass** - A celebrated result of Eisenbud--Kimshaishvili--Levine shows how to use a quadratic form to compute the multiplicity of the zero of a polynomial with real coefficients. I will explain a parallel result when the polynomial has more arithmetically interesting coefficients and then pose some related problems in computational and commutative algebra.

**Blake Farman** - Named for Nobuo Yoneda (1930 - 1996), a Japanese mathematician and computer scientist, the Yoneda Lemma is the starting point for the view that objects in a category can readily be determined by the morphisms into that object. In particular, it is the cornerstone of Grothendieck's method of studying schemes by way of the so called Functor of Points. In this talk we’ll introduce the notion of functor and natural transformation, which we’ll use to construct the category of all presheaves of sets on a locally small category, and discuss how the Yoneda Lemma allows us to embed a locally small category into its category of presheaves.

**George McNulty** - We all know a handful of five or six equations that define the notion of a group. Suppose someone where to hand you a finite set of equations and ask whether they also define the notion of a group. What would you do about answering? A finite list of equations could be typed in at a compute keyboard. Could you write a computer program, which upon input of such a list, would eventually be able to respond with the correct answer? How hard could it be? Or again, suppose someone handed you a finite graph (a bunch of nodes with edges joining some of them), and asked you whether you could color the nodes with three colors so that any nodes joined by an edge got distinct colors. How could you answer? Could you write a computer program for this? How be a strain would such a program put on your computational resources? How hard could it be? There are many open problems like these at the confluence of algebra, mathematical logic, discrete mathematics, and computer science. Some are significant for the development of mathematics and some have important practical applications.

**Frank Thorne** - Analytic number theory is the part of number theory that deals with quantitative questions arising in number theory. For example, how many primes are there less than X, as a function of X? The subject also studies certain natural functions that arise when investigating these questions -- in particular, the zeta and L-functions, which are notable for their elegant complex and Fourier analytic properties.
In this lecture I'll give an overview of some of my favorite questions and theorems in analytic number theory. The works discussed will range from the 19th century to last year.

For more information about The Graduate Colloquium, contact ballard@math.sc.edu.