|Avery St. Dizier|
Avery St. Dizier is an undergraduate mathematics student at LSU. His interests lie in pure mathematics, particularly in Probability Theory and Combinatorics. Avery has worked at the University of Nebraska - Lincoln studying discrete calculus under Dr. Allan Peterson and at Cornell University studying Jacobian groups of regular matroids under Dr. Farbod Shokrieh. At LSU, he has done work on the matrix exponential and the Laplace transform under Dr. Frank Neubrander. He currently works as a tutor for Dr. Neubrander’s calculus classes, and also as a private tutor. He has taken multiple graduate level mathematics courses as an undergraduate and has presented at several research conferences at LSU and elsewhere.
My long-term goal is to become a research mathematician and to spend my life creating new mathematics and inspiring the next generation of students to embrace mathematics. In the course of discovering my own research interests, I transitioned from computational work, implementing mathematical algorithms, to more pure work, including matroid theory and combinatorics. My diverse research experiences have broadened my mathematical toolkit while focusing my interests.
In my first semester at Louisiana State University (LSU), I dove into research with Dr. Frank Neubrander. When I started college, I only knew basic calculus. However, Dr. Neubrander relentlessly pushed my peers and me each week, opening our minds to the vibrant world of higher math. After a couple of months of weekly, late-night lectures, he introduced us to our research topic. Motivated by sharp error estimates for approximating strongly continuous operator semigroups using higher order Pade Approximations, he tasked us with implementing and testing new algorithms for approximating the matrix exponential using Mathematica and Matlab. We then carried the computation of the matrix exponential into the numerical inversion of the Laplace Transform. Despite what was suggested by the sharp error bounds, we discovered that the scaling and squaring used by the mainstream algorithms remained more computationally efficient than higher order approximations. I was then tasked with summarizing the algorithms and results in a technical report. We continued on to produce a poster which we presented at the Louisiana State University Undergraduate Research Conference. Dr. Neubrander showed me how research is actually done, not by following a predetermined path from the problem to the solution, but by slowly breaking down a seemingly impenetrable question into manageable pieces that can yield insight into the bigger mathematical picture.
After a year of working with Dr. Neubrander, I attended the Summer Research Program in applied mathematics at the University of Nebraska-Lincoln (UNL). I was mentored by Dr. Allan Peterson in the area of discrete calculus. Working with a larger group of students, I applied the knowledge of the Laplace Transform and the research philosophy that I had gained working with Dr. Neubrander to develop an analogue of the Laplace Transform in a radically different setting. We considered a set called the scaled number line, consisting of the orbit of a fixed based point under an affine map and its inverse. On this set, we first examined the expected analogues of the elementary functions and verified that they possessed properties similar to those of their continuous counterparts. Then, we focused on the discrete integral and the basic theory of difference equations. Guided by our intuitions from calculus, we formulated a definition for the discrete Laplace Transform, derived its properties, proved convergence and injectivity results, and demonstrated its effectiveness in solving discrete difference equations on the scaled number line. We collected our results into a paper that we submitted to the Pan-American Mathematical Journal. I also presented a poster on the research at the LSU Discover Research Conference and again at the 2014 Joint Mathematics Meeting. My research at UNL was my first experience contributing to mathematical theory. I greatly enjoyed the project, and it drew me to studying more abstract subjects when I returned to LSU. As I worked throughout the next semester to complete and submit the paper, I also began delving into my newfound interests by taking the first-year graduate courses in algebra, complex analysis, and topology.
Once I had completed a year of graduate-level coursework, I was able to pursue the new interests in pure mathematics that I had been exploring since the previous summer by attending Cornell University's REU program. I worked with another undergraduate, Esco He (Cornell University), and was mentored by Professor Farbod Shokrieh in the area of chip-firing games on graphs. Given a finite, connected graph with no loops, a chip-firing game is played by first assigning integer values, called “chips,” to each vertex. The objective of the game is to get each vertex out of “debt,” that is to make each vertex have a nonnegative value through a sequence of moves, where a move consists of allowing a vertex to fire by sending one chip to each adjacent vertex. The analysis of this game motivates the study of the Jacobian group, a quotient of the group of initial chip configurations under addition.
One interesting property of the Jacobian group is that the order of the group is exactly the number of spanning trees of the graph. There are many different, known explicit bijections between the set of spanning trees and the Jacobian group. However, all known bijections depend heavily on the graph-theoretic notion of vertices. Our goal was to construct a combinatorial bijection that could be extended to regular matroids, structures that capture the dependence relations between the columns of real matrices in which all subdeterminants are 0, 1, or -1. We began by studying an equivalent definition of the Jacobian compatible with regular matroids. Then, we analyzed a geometric construction that encoded the set of spanning trees as a tiling of space by higher-dimensional parallelepipeds and encoded Jacobian elements as corners of the parallelepipeds modulo an equivalence relation derived from the cycles in the matroid. After repeatedly varying the parameters and working out dozens of low-dimensional examples, I established a method for recreating the geometric bijection using only the combinatorics of the underlying matroid with no reference to the geometry. I then proved the equivalence of the geometric and combinatorial methods, establishing that the combinatorial method was indeed a bijection. However, unlike the geometric bijection, which has a natural inverse, the combinatorial bijection is difficult to invert. The project is ongoing, and we are in the process of drafting a paper with our results.
Since last summer, I have been focusing on probability theory. During the Cornell REU, I attended several fascinating seminars, including talks on mixing times of stochastic processes and connectivity of randomly growing networks. I have taken graduate courses in probability theory and real analysis in previous semesters, and I am currently taking stochastic analysis to prepare to undertake graduate studies in probability theory. I have applied to a number of graduate schools with strong programs and active research in probability theory, and I am eagerly awaiting replies.