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.
The History of Mathematics
My research treats the history of mathematics in the late 19th and early 20th centuries. In my work, I focus on not only the origin, evolution, and reception of mathematical ideas, but also on how mathematics is communicated (and to whom, and why), and the different ways in which mathematics is impacted by broader cultural, social, and political issues. I am particularly interested in the history of complex analysis in the advent of set theory; the histories of mathematical journals and congresses; the history of women in mathematics; and the internationalization and nationalization of mathematics around the time of the First World War.
More recently, I have been working on a project involving the origins of our axioms for various mathematical structures, and a related project concerning the ways in which axiom systems are connected to broader educational concerns in the early 1900s. If you’re interested in learning more about the history of mathematics, don’t hesitate to stop by my office!
Information Dynamics of Natural and Artificial Systems
My research focuses on understanding how dynamical systems communicate information from their past through their present to their future. I draw on tools from machine learning, statistics, dynamical systems, network science, and information theory. The research is both theoretical, deriving relevant measures of communication and methods for their estimation from data, and computational, implementing those methods via efficient algorithms. Previous research projects include predicting user behavior on Twitter, characterizing the time-varying exchange of information in the brain, quantifying the change in market dynamics before and after the introduction of automated trading, and modeling the information dynamics of nanoscale electronics.
Read more about Professor Darmon’s research at David Darmon, PhD
Potential Student Collaborations
confcurve: Development of an R Package for Statistical Inference with Confidence Distributions (this is an SRP 2020 project; applications due 3/1/20)
A reproducibility crisis is brewing in the application of statistics to medicine and the social sciences. The reproducibility crisis has many causes, including P-hacking, the garden of forking paths, and the file drawer effect. Much of the blame for the reproducibility crisis has been heaped on P-values and their abuses. Confidence intervals have been proposed as a supplement or alternative to P-values. However, even confidence intervals suffer from the pre-assignment of a default coverage level, typically 95%, and thus fall into the same trap as using the standard 0.05 cutoff for statistically significant versus non-significant estimates.
Confidence distributions provide a straightforward and readily graspable way to extract all of statistical information that can be inferred from a given data set under a given model, including point estimates, confidence intervals, and P-values. The theory of confidence distributions is well-developed, and the practical aspects of their implementation with many commonly used models has reached maturity. However, without readily available and user-friendly software for generating confidence distributions from data, confidence distributions will likely remain outside the toolkit of practicing scientists who lack extensive training in mathematical statistics.
During this project, students will design statistical software in R to construct confidence distributions for a diverse set of statistical procedures. This software will be validated using real world data sets. Over the course of the summer, students will gain skills in statistical inference, numerical analysis, and software development that will serve them well in any future endeavor in the applied sciences.
Susan Fiske, former president of the Association for Psychological Science, has called advocating for methodological reform in her own field of social psychology “methodological terrorism.” One person’s terrorist is another person’s freedom fighter. Join the resistance this summer!
Applications of Moving Frames in Geometry
My research interests lie at the interface of applied mathematics, differential geometry, and discrete geometry. Specifically, I am interested in the theory of transformation groups and the method of (equivariant) moving frames, and their applications to differential equations and their finite difference approximations. In the recent years, have been using the moving frame techniques to solve differential and difference equations that admit a group of Lie point symmetries. I am also interested in symmetry-preserving numerical methods, the study of geometric submanifold flows, and the method of equivalence. If you’d like to learn more about what I do, feel free to stop by my office!
Read more about Professor Valiquette’s research at Francis Valiquette, PhD
My research falls into the broad category of numerical analysis. In particular, I have an interest in numerical approximations of wave phenomena or scattering theory. 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. Recently I have concentrated my efforts on the forward problem looking at discontinuous Galerikn methods and preconditioning techniques.
Potential Student Collaborations
Impact of Compression of Similar Signals Using Wavelets (this is an SRP 2020 project; applications due 3/1/20)
The process of compressing an audio signal is, in part, the result of an effort to reduce the file size. This is necessary in applications related to both storage and transmission of the file. In a general sense, compression is either lossy or lossless where the former employs techniques or algorithms that approximate or discard information making it impossible to fully reverse the compression process. In other words, there is no free lunch; efficient compression comes at a cost. Of course, the consequences of the loss of information in lossy compression varies.
It is reasonable to consider that a signal may have different variations. Between studio and live versions of a song could be thought of as variations of the same signal. One may think of the studio version of a song as the (original) signal and consider live versions as the variations. In this situation, the differences exist before compression begins.
Basic wavelet compression employs a signal representation in terms of a basis. Consequent levels of decomposition based on nested subspaces and leads to a version of the signal that largely conserves the energy of the signal among other characteristics. Portions of these characteristics may be approximated or even dismissed to reduce file size and result in a lossy compression. The aim of this project is to investigate the impact of wavelet compression on different versions of the same signal. We will attempt to understand what similarities or differences exist initially and what the impact of compression has. In particular we will explore the ratio and related threshold parameters and how they relate to different versions.
Fixed Point Theory
My research is in an area called fixed point theory. The big idea is to determine which broad classes of functions have fixed points; that is, which functions that have inputs that aren’t changed by applying the function. Somehow, knowing which functions have (or don’t have) fixed points can tell you quite a lot of useful geometric and topological information, and this is what I find most intriguing. For more detailed information, feel free to visit my research page where you can find information about my personal research as well as undergraduate research projects, or stop by my office to chat!
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”
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.
Potential Student Collaborations
Maximal Length Divisor Cycles (this is an SRP 2020 project; applications due 3/1/20)
A finite sequence of distinct positive integers is called a divisor chain if each term is either a divisor or a multiple of the preceding term. A divisor chain is called a divisor cycle (or loop) if the first term is also either a divisor or multiple of the last term. The chain (or cycle) is bounded by the natural number n if the terms of the sequence are all less than or equal to n. It’s natural to ask: what is the longest chain or cycle that can be constructed with a given bound n?
Erdös and Pomerance initiated the study of maximal bounded divisor chains in the 1980’s by looking at their asymptotic behavior. Our project will focus on smaller, computational problems concerning maximal bounded divisor cycles and other similar structures, as well as the structure of their graph-theoretic models. Students working on this project will have the opportunity to increase their knowledge in the areas of number theory, abstract algebra, and graph theory.
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.