Faculty in the mathematics department at WCU are engaged in research in several areas. Below, faculty members write about their research interests.
My research interests are focused on the development, theoretical analysis and implementation of high-order numerical schemes that accurately and efficiently solve linear and nonlinear Partial Differential Equations (PDEs).
My specialty is Finite Elements Methods specifically the Discontinuous Galerkin Finite Element (DG-FE) method, but I am comfortable in utilizing other spatial discretizations when appropriate, e.g. finite differences and spectral methods.
One aspect of my work is the numerical solution of Cahn-Hilliard (CH) type equations, arising in models describing phase separation and biological growth, e.g. cancerous tumors, biofilms. Adaptive mesh refinement, error estimation, and stable time-stepping techniques are addressed in my work. For the solution of the resulting algebraic linear or nonlinear systems, efficient solvers based on the multi-grid method are designed. These equations are naturally combined with other famous equations of fluid dynamics, like the Navier Stokes equation, in the context of multiphase flows which their numerical solution and various applications I am also currently studying.
For this portion of my research I have been awarded in 2017 a prestigious 3-year long NSF grant that is expected to total $100,115.00. For more information see the NSF webpage for the grant.
Another equation and its numerical solution that recently I became interested in, is the focusing Shrödinger equation in the semiclassical limit where the Planck constant ε is small.
The development of hybrid discrete cellular automata systems (HD-CA) by coupling linear and nonlinear reaction- diffusion partial differential equations with discrete (stochastic) cellular automata systems is also a part of my work. I utilize HD-CA in tissue engineering, cancer modeling, evolutionary game theory and biofilm modeling.
Multidisciplinary research has been a natural application of my computational background. In a collaborative project with colleagues from Temple University and the Center for Biofilm Engineering (CBE), I am studying the various aspects of biofilm formation. The study of these and other systems in a microgravity environment is also one of my recent interests. I have also been involved in an ongoing project in collaboration with colleagues from Duke University, using agent-based modeling techniques to understand the various mechanisms governing the dorsal closure in drosophila embryogenesis as a model for wound healing.
I always strive to advance my work by expanding to new related fields and the formation of collaborations with researchers from Biology, Medicine, Engineering and Physics to name a few. Multidisciplinary projects naturally would give rise to various individual sub-projects that students (undergraduate and graduate) of various levels of expertise could work on, as a part of their degree requirements.
I’ve always believed statistics is a form of communication. My research has been focused on ways to properly tell the story of the respective data I’m analyzing. My dissertation focused on Semi-Markov models which focused on explaining disease progression and recovery within epidemics. The main application was in the progression of HIV/AIDS. Can we describe how patients transition to worse or better stages?
I was fortunate to work in a research group at UPENN as I completed my Ph.D. work. This gave me a jump right into analytical approaches to real life data analysis. My work focused on the analysis of Randomized Clinical Trials (RCT). Communication is still a key component of RCTs but where the focus is properly describing and interpreting on-average change over time. These change over time models are referred to as Longitudinal Data Analysis (LDA). My research has focused on finding advanced techniques for the analysis of LDA consisting of: Mixed Effects Models, Generalized Mixed Effects Models, Partially Nested Mixed Effects Models, Mixed Effects Markov Models (which combined my dissertation work with LDA models), and Functional Mixed Effects Models.
Most RCTs focus on which intervention perform best versus a comparative group, recent research has focused on whether intervention effects occurs through or around a third intermediate mediating variable corresponding to indirect and direct effects, respectively. These analyses are referred to a Mediation models. Standard mediation analyses assume sequential ignorability, i.e., conditional on covariates the intermediate or mediating factor is randomly assigned, as is the treatment in a randomized clinical trial. My current research has focused on the applications without the standard sequential ignorability assumption, using Causal Models techniques.
My field of research is mathematical biology. I use a combination of mathematical modeling, simulation, and statistics to analyze data and answer theoretical questions of interest in environmental biology. I am currently co-PI with J. Auld (biology, WCU) on a $213,858 RUI grant from the National Science Foundation from July 1, 2014-June 30, 2017 titled “A theoretical and experimental investigation of optimal mating strategies in a hermaphrodite”; for more details see National Science Foundation Award Abstract #1406231. The grant supports our interdisciplinary research program in evolutionary ecology with WCU undergraduate and graduate mathematics and biology majors. See the West Chester UniversityRESEARCH, SCHOLARLY, AND CREATIVE ACTIVITIES ANNUAL REPORT 2014-2015 for a profile of our NSF funded research (pages 5 and 16). Our latest publication appears in the Journal of Evolutionary Ecology in 2016; see the abstract.
I am currently preparing a paper for publication on the kinematics of swimming of the manta ray (Manta birostris), with F. Fish (biology, WCU) and A. Crossett (statistics, WCU). I also have a graduate student working on developing a java applet to simulate a swarm for research and educational (Women in Science workshops) purposes. This stems from my dissertation research modeling the self-organized collective motion of swarms of animals such as schooling fish and flocking birds. My students and I regularly present our research on-campus and at SIAM meetings.
I was honored for my research at the first annual “Spotlight on Research” reception at the WCUPA Foundation; see the Daily Local Newsarticle.
For more details on my research and teaching, please go to my website
One of my current research interests is to develop a matched Alternating Direction Implicit (ADI) method for solving a large class of problems, called the interface problems, arising in Biology, Materials, and Engineering. The mathematical model of interface problems consists of a parabolic partial differential equation (PDE) and imposed jump conditions on the interface to relate the solutions across the interface. Without appropriately addressing such jump conditions in the mathematical formulation, the standard numerical methods are known to be inaccurate, or even fail, when solving the interface problems. This motivates the development of this matched ADI method in which the central difference is locally corrected to essentially 1D jump conditions near the interface. By incorporating the 1D jump conditions into the ADI framework, the matched ADI method is unconditionally stable and able to restore the second order of accuracy in both time and space for complex 2D and 3D interfaces. This work is a joint research with Dr. Shan Zhao from the University of Alabama.
In present, my second research interest focuses on developing efficient parallel computing techniques to accelerate the computations of generating the 3D molecular surfaces, solving the Poisson-Boltzmann Equation (PBE) for electrostatic potential, and calculating corresponding electrostatic energies. In nowadays, calculating electrostatic potential and corresponding energies has become a standard computational approach for studying biomolecules and nano-objects immersed in water and salt phase. One most recognized math model in molecular biology is the PBE, which can only be solved numerically for irregular-shaped molecules and proteins. Moreover, no existing sequential PBE solver is ready to be utilized to solve the PBE for large macromolecules and complexes due to high computational time and memory requirements. It calls for developing efficient parallel computing techniques to significantly speedup the calculations and make it feasible for large macromolecules and complexes. This work is a joint research with Dr. Emil Alexov from Clemson University and supported by NIH grant 5R01GM093937-07.
My original research interests, as well as my dissertation research, focused on Mathematical Finance. More specifically, I studied the relationship between arbitrage and state price deflators. Over the years my research interests have substantially broadened. I am extremely interested in Statistical Education and have published articles on my pedagogical practices in the peer-reviewed journals Mathematics Teacher and PRIMUS. I also enjoy more traditional, collaborative statistical work and have co-authored multiple papers using statistical methods to compare various approaches to treating trauma. Currently, I have been working with one of our graduate students on a meta-analysis of trauma-treatment papers. I also enjoy doing more methodological work and most recently have co-authored with a former student a paper in the Annals of Epidemiology on the Misuse of the Odds Ratio.
More recently, I have returned to my academic roots. I have begun work with one of our graduate students on the study of virtual economies. The recently gathered data should lend itself to many different analyses and, as such, should provide research opportunities for any other interested students. If you are one of those students then please don’t hesitate to send me an email!
Finally, I dream of someday working with students to design and implement a study to confirm (or refute) Snapple Fact #33 that termites eat wood faster while listening to rock and roll music (https://www.snapple.com/real-facts).
I am classically trained as an applied functional analyst with a specialization in abstract nonlinear evolution equations. Over the past 15 years, I have augmented this training to include stochastic evolution equations and stochastic control theory. My work has applications spanning fields of study, including engineering, epidemiology, population dynamics, pharmacokinetics, and non-Newtonian fluids, to name a few. My most recent research is concerned with developing a theory for different classes of abstract functional stochastic evolution equations driven by a Levy process in a Hilbert space: this work is applicable particularly in electrical engineering and the theory of neural networks.
I have published two graduate-level textbooks on deterministic and stochastic evolution equations and one book on advanced differential equations for senior-level undergraduate students, all published by CRC Press. I wrote all these employing the pedagogical approach of guiding the readers on a journey of discovery of the subject with the intention of instilling intuition in the reader.
Over the past 20 years, I have published more than 30 peer-reviewed journal articles on nonlinear analysis, evolution equations, control theory, and stochastic analysis in national and international journals. A complete list of these can be found on my personal site (http://www.wcupa.edu/sciences-mathematics/mathematics/mmckibben/). I am also a member of the editorial board of three mathematics journals devoted to the study of function spaces, abstract analysis, and non-autonomous evolution equations.
My passion for teaching has prompted me also to be heavily involved in textbook-writing for freshmen and sophomore-level mathematics courses for the past 15 years. Of the various projects in which I am involved, I am the supplement author for a nationally-successful suite of textbooks for Pre-calculus, authored by C. Y. Young and published by J. Wiley and Sons, and am a contributing author, along with B. Boyce and J. Brannan, for one of J. Wiley and Sons’ two main differential equations textbooks.
Presently, my research interests are mostly in the area of basic hypergeometric series and related topics, including integer partitions, mock theta functions, and q-continued fractions.
I just recently (January 2017) finished a book on basic hypergeometric, entitled "Methods and Topics in q-series", which will be published by World Scientific.
Starting in 2004 and roughly every two years since that, myself and Nancy Wyshinski organized a special session on continued fractions at the Joint Mathematics Meetings (at the most recent such special session in January 2017, we were assisted by Geremías Polanco). Some information about these special sessions may be obtained by following the links:
More on my research may be found on my personal page. Here are quick links to not-too-technical Wikipedia pages on some of my favourite research topics: basic hypergeometric series, integer partitions,continued fractions, the Rogers-Ramanujan continued fractions (the most famous example of a q-continued fraction), and the mock-theta functions.
I also enjoy working with students on research. Ricky Sparks completed a Masters thesis under my supervision on a topic in the area of basic hypergeometric series. I am currently (Spring 2017) supervising a Masters thesis on a topic in the area of Galois theory.
My current research focuses on two main areas: (1) gender disparities in mathematics participation and achievement, and (2) the knowledge, skills, practices, and beliefs of elementary teachers as they transition from teacher preparation programs into teaching.
For my dissertation, I investigated factors contributing to the retention of female mathematics doctoral students. To continue this line of research, I am currently exploring factors that may contribute to differences in publication rates upon graduation between male and female doctoral students in various disciplines. I have also conducted research using The Early Childhood Longitudinal Study, Kindergarten Class of 2010-2011 (ECLS-K:2011), a large-scale, nationally representative data set. This data set was used to investigate early indicators of gender disparities in mathematics achievement for elementary school students (Link to article: http://journals.sagepub.com/doi/full/10.1177/2332858416673617). Finally, I am involved in a project investigating qualitative differences in the problem solving strategies of high-achieving male and female middle school students.
My second line of research focuses on the preparation of future elementary mathematics teachers. I have been involved with a project aimed at empirically examining relationships between the experiences of pre-service teachers while enrolled in their teacher preparation program and their teaching capabilities post-graduation. Specifically, teaching capabilities were analyzed in terms of mathematical and pedagogical content knowledge, skills for analyzing teaching, lesson planning, and professional vision.