Research

Main (broadly understood) research groups:

• Algebra and number theory (Dubravka Ban, Kwangho Choiy, Andrew Earnest, Don Redmond)
• Combinatorics and graph theory (John McSorley)
• Geometry and topology (Jerzy Kocik, Mike Sullivan)
• Logic and complexity (Wesley Calvert)
• Probability theory and stochastic systems (Randy Hughes, Henri Schurz)
• Statistics and Data Science (David Olive, S. Yaser Samadi)
• Algebra Group Theory and Combinatorics ( Lindsey-Kay Lauderdale)
• Numerical Analysis (Jianhong Xu, Mingqing Xiao, Henri Schurz)
• Differential Equations and Mathematical Biology (Dashun Xu, Henri Schurz)
• Partial Differential Equations (Mathew Gluck, Mingqing Xiao)

Alan Schoen discovered a minimal surface that he named the gyroid. The gyroid is becoming increasingly popular as more and more new occurrences of it in nature are being discovered (google it up!). It has been discussed at Cornell University and John Baez's site at Univ. of California-Riverside and on wired.com's wired science blog. You may purchase a sculpture of it from Bathsheba. Currently Alan is developing an amazing site Geometry Garret. We hope that some day a big gyroidal sculpture will embellish our backyard.

---

In SIU School of Mathematics and Statistical Sciences, we encourage undergraduate students to participate in mathematical research and supervised readings of special topics under guidance of our faculty. Annually, we organize a research conference in undergraduate mathematics research jointly with the Southeast Missouri State University.

Below are names of students and their projects in recent past.

---

Title:  Use reinforcement learning to optimize a game agent to compute with scripted agents and novice human players

• Details: (1) iterative algorithms for solving Riccati equations
  (2) characterizing Lyapunov types of matrix stability
• Term: Spring 2018
  Student: Preston R. Yun
  Professor: Xu, Jianhong

---

Title: A Solution to Pell's Equation

• Details: A classical problem in the history of mathematics was calculating all the integral solutions (m, n) for a fixed D > 0 of m^2 – D·n^2 = +-1. In fact, as long as D is not a perfect square, there are infinitely many distinct pairs (m, n) satisfying the equation.  This project aims to provide a solution to Pell's Equation by introducing some elementary tools from Algebra and Number Theory, and giving similar solutions to other special higher degree cases like that of Pell's equations.
• Term: Spring 2017
• Student: Zeid Ghalyoun
• Professor: Calvert

---

Title: Improving Numerical Integration Estimates with Inverse Functions

• Details: Estimating the integral for power functions with powers 0<p<1 takes much more computational power via standard numerical methods than functions with powers p>1. This is, at least partially due to the fact that such power functions are not Lipschitz continuous. However, as was first shown by C.A. Laisant in 1905, the integrals of certain functions can be expressed as integrals of their inverses. This paper hopes to show that by expressing hard to approximate integrals as integrals of their inverses, methods such as Newton-Cotes formulas and Monte Carlo methods become much more useful and effective than by approximating the integral of the original function alone.
• Term: Spring 2017
  Student: Thomas Campbell
  Professor: Schurz

---

Title: Natural Computation in Gene Regulatory Networks

• Details: The purpose of this research is to implement computability theory in gene regulatory networks. A gene regulatory network is a collection of molecular regulators that interact with each other and other elements in the cell to output gene expression levels. These networks exhibit natural computation. The class of partial recursive functions is the smallest class satisfying the following five axioms: the successor function, the constant function, projection functions, composition, and recursion. This seeks to realize these axioms in gene regulatory networks.
• Term: Fall 2016--Spring 2017
  Student: Brianna Martin
  Professor: Calvert

---

Title: Sports Data Analysis Project

• Details: The group analyzed the Missouri Valley conference Men and Women's basketball data using R software, clustering, and discriminant analysis.
• Term:  Fall 2015
• Students:  Nicole Staples, Philip Kains
• Professors:  Budzban, Olive 

---

Title: Research in Visualization in Mathematics 

• Details: Research in geometry, computing, and visualization.  The work started in Fall 2014 and continued through Fall 2015. Our weekly meetings (Thursday 2pm) were frequented also by a number of other interested students.
• Term:  Fall 2014
  Students:  Bradley Dragun, Thomas Finkenkelle, Aaron Zolotor
  Professor:  Kocik
• Term:  Spring 2015
  Students:  Jeffrey Lawrence, Aaron Zolotor
  Professor:  Kocik
• Term:  Fall 2015
  Student: Bradley Dragun
  Professor:  Kocik