The Department of Mathematics is staffed with faculty members who are more than just impassioned mentors and educators; they are well respected leaders in their field. Many of the faculty engage in research. Below is a sampling of some of the varied areas in which Monmouth mathematics professors apply their theoretical knowledge in the practical world.
Statistical Analysis of Veterinary or Medical Procedures
Veterinary and medical procedures often balance therapeutic results against risk of side effects; moreover, practitioners often tweak procedures to minimize side effects. Statistical study of the efficacy of these changes helps assess the costs and benefits and facilitates publication of studies. Several projects involving mathematics for statistical analysis of efficacy of veterinary or medical procedures are being conducted. In addition to research involving statistical analysis and interpretation of data, students in my lab learn the basics of client communication and other essential entrepreneurship skills necessary for developing their own consulting practice. Potential projects in consultations with a veterinary practice include “Urinary Incontinence in Spayed Dogs” and “Regenerative Stem Cell Therapy in Dogs”. Consulting projects in conjunction with a urologist include “Serum Prostate Specific Antigen Levels as Predictors of Prostate Cancer”
Knot Theory and Virtual Knot Theory
My current research interest is in knot theory and virtual knot theory. A classical knot is an embedding of the circle into a three dimensional space (e.g., take a rope, tangle it up, and glue the ends together). The main question in knot theory is: given two knots J and K, is it possible to manipulate J so that you get K? Surprisingly, this turns out to be a deep question about the geometry and topology of three manifolds. An alternate approach to the main question is via virtual knots. Virtual knots are an extension of classical knots where one has overcrossings, undercrossings (as in the classical case), and virtual crossings. Because of the additional crossing type, virtual knots lend themselves naturally to combinatorial and algebraic analysis. My recent research has focused on trying to understand how these structures play a role in finite-type invariants of virtual knots. The Goussarov Theorem states that every finite-type invariant of classical knots can be represented by a combinatorial formula. This is important for the main question in knot theory because all finite-type invariants can be computed quickly. Indeed, they can be computed in polynomial time. Thus, studying knots helps us understand three manifolds, while studying finite-type invariants helps us to determine properties of those knots quickly.
The primary mathematical interest in knot theory lies in its exceptional beauty. However, knot theory is also studied for its interrelations with other branches of mathematics like representation theory, category theory, and graph theory.
My research is in scattering theory or, more specifically, the scattering of electromagnetic or acoustic waves. Scattering theory can be divided into two categories: forward and inverse. In the forward problem, the main goal is to compute the scattered wave given an incident field and the medium in which the wave travels. In the inverse problem, one tries to determine the object that actually scattered the wave given the incident and scattered field. I tend to concentrate my efforts in the area of numerical analysis, where I mainly focus on the computational aspects of scattering theory. More specifically, I work on finite element methods for the forward problem and regularization/sampling techniques for the inverse problem.
Partial Differential Equations
My primary research area is numerical solutions of partial differential equations. My research interests are in the fields of numerical analysis, partial differential equations, scientific computation, mathematical modeling, and computational fluid dynamics. I have recently been concentrating on the following two fields of research: numerical analysis and computer simulation. For numerical analysis, I develop numerical methods to approximate the solutions of partial differential equations, prove the unique solvability of the numerical algorithms, and carry out the error analysis of the numerical solutions. For computer simulation, I develop 3D mathematical models to simulate the blood flows in human atherosclerotic arteries, to study the blood flow pattern in curved arteries with or without stenosis, and to investigate the effect of the stenosis on the wall shear stress, the pressure drop, and the flow disturbance. Numerical computations are carried out to allow for simulations of different geometries and flow parameters under the physiological conditions, and the numerical results are analyzed.
Algebra and Number Theory
My research is in pure mathematics and has included overlapping areas of algebra, number theory, and geometry. More specifically, I have done work applying Iwasawa theory techniques in algebraic number theory; I have applied arithmetical algebraic geometry techniques to the study of genus 1 curves; and have written pedagogical materials and a textbook for elementary number theory courses. Recently I have been working on projects related to bilinear and quadratic forms over rings and fields of even characteristic, which is an interesting and often overlooked case that is usually omitted from classical treatments on the subject. I’m always interested in hearing about and working on new problems whose solutions are potentially susceptible to the methods of modern algebra, especially those coming from number theory and geometry.
Inverse Problems on Hilbert Spaces
My main research interests are in neighborhood hypothesis and inverse problems on Hilbert spaces. Neighborhood hypothesis is an improved testing method that is both mathematically convenient and practically relevant to replace the usual hypothesis. The advantage is that the asymptotes remain essentially the same with the neighborhood null hypothesis and corresponding alternative reversed. For inverse problems on Hilbert spaces, I focus particularly on improving estimators of inverse function and regression models in Hilbert Spaces. For the past two years, I have also been including student researchers in the above topics with applications.