1. Grant Barkley

Title: Coxeter Groups and the Lattice of Total Orders

Abstract: A Coxeter group is a group generated by elements of order 2, called reflections, along with certain relations between the elements. The group of permutations on n elements (generated by swapping 1 and 2, 2 and 3, etc.) is an important example of a Coxeter group. A permutation is said to invert the pair (i,j) if i < j and the permutation puts j before i. Each permutation has a unique set of inversion pairs; we may define a partial order, called weak order, on permutations by ordering these sets by containment. This order has many important properties, such as being a semidistributive lattice. We may define inversions and weak order similarly for arbitrary Coxeter groups, but when the group has infinite size, the weak order is no longer a lattice. It has been conjectured by Dyer and others that, by considering instead of inversion sets a generalization called biclosed sets, we re-obtain a lattice with similar properties to the weak order on permutations. We prove this conjecture for classical affine Coxeter groups by interpreting permutations as total orders. (Faculty mentor: Dr. D. Speyer, REU)

2. Ryan Brandt

Title: Optimal Control: Getting the Most Out of Controllable Systems

Abstract: The methods of linear algebra and differential equations describe and predict the behavior of a staggering number of real-world systems. Quite often, the behavior of these systems is dependent on a driving force or input function, called a control. These controls can be used to change the system’s behavior. The question then arises: how do you pick a control that causes the system to behave in an optimal manner? This talk will begin to explore the answer to this question. From heat transfer and vibrational analysis to fluid flow and electromagnetism, the field of optimal control has diverse applications across many disciplines of science, mathematics, and engineering. We assume that the audience is familiar with basic calculus and linear algebra. We will develop further concepts as necessary during the talk. (Faculty mentor: Dr. L. Bociu)

3. Jonathan Dunay

Title: Counting the Number of Matrices that are Squares and Higher Powers

Abstract: We seek to find combinatorial results concerning the problem “How many $n \times n$ matrices with entries in a finite field are squares?” That is, let $\mathbb{F}_q$ be the finite field of order $q$, and let $Mat(n,q)$ be the set of $n \times n$ matrices with entries in $\mathbb{F}_q$. Then for how many $A\in Mat(n,q)$ does there exist some $B\in Mat(n,q)$ such that $B^2=A$? The method that we seek to use is to analyze the invariant factors of the matrices. The invariant factors of a matrix are sufficient to characterize a matrix up to similarity (where two matrices $A$ and $B$ are similar if and only if there is another invertible matrix $P$ such that $A=PBP^{-1}$). The idea is to take a matrix $B$ and determine the relationship between the invariant factors of $B$ and the invariant factors of $B^2$. This allows us to determine what the possible sets of invariant factors are for squares of matrices. To generalize this problem, we can ask how many matrices $A \in Mat(n,q)$ have the form $B^l$ where $l$ is a positive integer, $B \in Mat(n,q)$. For this case, we can seek to relate the invariant factors of a matrix $B$ to those of the matrix $B^l$. (Faculty mentor: Dr. M. Kang)

4. Nick Randolph

Title: Stability Analysis of Nonhomogeneous Delay Differential Equation System of the Valsalva Maneuver

Abstract: Mathematical modeling is a crucial and powerful tool used to describe physical and biological systems. Commonly involving differential equations, modeling characterize, describe, predict, and analyze system dynamics. In many systems, oscillatory dynamics may arise due to parameter variations and interactions or due to the inclusion of a forcing function. In physiological systems, this instability may signify (i) an attempt to return to homeostasis or (ii) system dysfunction. In this brief talk, we analyze a nonlinear, nonautonomous, nonhomogeneous open-loop neurological control model describing the autonomic nervous system response to the Valsalva maneuver. This model is a system of ordinary differential equations and one delay differential equation. Unstable modes have been identified as a result of parameter interactions between the sympathetic delay and time-scale. In a two-parameter bifurcation analysis, we examine both the homogeneous and nonhomogeneous systems, examining the stabilizing effect of a forcing function. We use analytical methods involving the Lambert W function for the homogeneous system, identifying transcendental relationships between the parameters, and we use computational methods for the nonhomogeneous system to determine stability regions. The presence of a Hopf bifurcation within the system is discussed and solution types from the sink and stable focus regions are compared to two control patients and a patient with postural orthostatic tachycardia syndrome (POTS). The model and its analysis support the current clinical hypotheses that patients suffering from POTS experience altered nervous system activity. (Faculty mentor: Dr. M. Olufsen)

5. Claire Steffen

Title: Signature Schemes for the Additive RSA

Abstract: Since its original publication in 1976, the RSA scheme has become a popular cryptosystem and two main digital signatures were developed for additional security when utilizing this system. The goal of this paper is to investigate the lesser known additive RSA and digital signatures to accompany the cryptosystem. In order to test these signature schemes, I wrote a program in Python to run sufficiently large integer examples. The results revealed that the two digital signatures used with the traditional RSA, not only hold for the additive RSA, but function in a similar fashion as well. The additive RSA cryptosystem provides a more simple scheme that can be implemented in secondary schools as a way to introduce younger students to cryptography. (Faculty mentor: Dr. E. Stitzinger)

6. Sreeram Venkat

Title: Sobolev Orthogonal Polynomials on the Sierpinski Gasket

Abstract: Building on the theory of Legendre orthogonal polynomials on the Sierpinski Gasket (SG), we develop a theory of Sobolev orthogonal polynomials on SG. Initially, we define several notions of a Sobolev inner product on SG using powers of the canonical Laplacian. We use these inner products to find general recurrence relations connecting the Sobolev polynomials to the Legendre polynomials on SG. We then analyze the finer properties of the Sobelev inner product by presenting estimates for the $L^2$, $L^\infty$ and $H^m$ norms of the polynomials and studying their convergence properties with respect to the parameters in the $H^m$ inner product. We also highlight the major differences and similarities between the polynomials on SG and those on $\mathbb{R}$ resulting from the properties of the self-similar measure and the Laplacian. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation on SG. (Faculty mentor: Dr. R. Strichartz and Dr. K. Okoudjou, REU)