My research areas are matrix theory and numerical methods in linear algebra and optimization. I am currently working on mathematical modeling with large data sets using regression methods. Approximation errors are analyzed to improve the models. The developed methods have been implemented in MatLab for some real-world applications in COVID-19 and climate change. I am also working on mathematical modeling using systems of linear and nonlinear differential equations for ecosystem changes and pandemic.
The past decade has witnessed significant developments in high-dimensional data analysis, driven primarily by a wide range of applications in many fields such as genomics and signal processing. My current research focuses on feature selection. It aims to rank features that are dependent and select the covariates appropriately which benefit our model the most under certain conditions. This area would be a revolution in the sense of replacing the general overuse of ‘Testing the Hypothesis of Equality’ with new decision-theoretic models for ordering populations with prescribed confidence in the resulting decision. In general, ranking and selecting important covariates/coefficients can be more fundamental and desirable than covering the true coefficient or testing its significance. Students who are interested in this topic are welcome to stop by my office and get more information/ideas about it.
I have been primarily interested in the resampling schemes in nonparametric and semiparametric models, by their “almost” minimal to moderate distributional assumptions in data analysis. In particular, the topics are rank statistics, multiple comparisons, quasi-likelihood estimation method, generalized linear models, microarray gene expression data analysis, and RNA sequencing gene expression data analysis. In addition, I have been interested in computational statistics, high-dimensional data analysis, and student research in real-world applications.
My research is in the theory of differential equations, difference equations, and dynamic equations on time scales and their fractional derivative counterparts. My main focus is in two areas:
The first is studying the smoothness of solutions with respect to boundary data and parameters. Although this research employs the use of advanced mathematical theorems, the work itself is accessible to undergraduate students with calculus II and differential equation knowledge. I have published several papers with undergraduate students and worked with them to present in various forums.
The second is seeking sufficient conditions that guarantee the existence or nonexistence of a solution(s) to all types of differential (or similar) equations using fixed point theorems. This is useful so that one does not utilize time and resources searching for a solution that does not exist.
I like almost every type of mathematics. The beauty of mathematics does not have bounds. However, my research is focused in combinatorics; especially graphs theory, matroids, and lattice paths. Another area of my interest is elementary number theory. I like to work research with students (undergraduate and graduate).
My research area is mainly geometric group theory and number theory. In the past, my collaborators Dr. Flórez, Mr. Higuita, and I have explored several properties present in the Hosoya polynomial triangle, GCD properties of generalized Fibonacci polynomials, etc. In addition, we have worked on geometries in the Hosoya triangle and provided proofs of Fibonacci identities using geometries present in the triangle.
I am currently starting to explore jump sums in the Hosoya triangle. This project will involve at least two undergraduate students. Dr. Flórez and I have been working on research projects with undergraduate students since 2014. These projects involve solving open problems from the journal Fibonacci Quarterly. The problems mainly involve Fibonacci and Lucas number sequences.
In the past I have also worked on geometric group theory topics like isoperimetric inequalities and hyperbolic groups.
I am working on several projects related to augmented happy functions with collaborators from High Point University, Denison University, Loras College, and the American Mathematical Society. A number is considered happy if the sum of the digits squared equals 1 after some number of repetitions. For example, 86 is happy since 64+36=100 and 1+0+0=1. In several of our projects, we are looking at adding an augmenting constant to the sum of the squares of the digits in arbitrary bases and describe several properties of these functions. We have a project on 1.5-happy numbers which takes the sum of the floor of the digits to the 1.5 power. In the past, I have mentored Marcus Harbol ‘17 on a project looking at augmented happy functions of higher powers. He also looked at these functions applied to complex integers. There are many questions left to be answered about these functions, and I’m happy to have interested students working with me on any of these projects.
I have been working with Shankar Banik and Michael Verdicchio on a project with James Andrus ’18 dealing with multicast routing using delay intervals for collaborative and competitive applications. This project is interdisciplinary between computer science and mathematics, specifically graph theory.
My PhD research focused on graph theory which is an area I still have interest. Any students wanting to work on a graph theory project are welcome to stop by my office and discuss options.
Todd Wittman, Ph.D. – Applied Mathematics
My research focuses on applying optimization techniques and differential equations to image processing and other problems in mathematical modeling. I am very happy to work with any student interested in applied mathematics. Some past student projects I have supervised include:
- Removing the noise from MRI brain images
- Determining connections between industries by looking for correlations between stock prices
- Tracking crime hotspots based on home burglary reports
- Speeding up the rendering time of computer animations (sponsored by PIXAR Studios)
- Enhancing the resolution of satellite images (sponsored by the National Geospatial-Intelligence Agency)
- Detecting blood vessels in medical images (sponsored by UCLA Medical School)
I am doing research with Dr. Sarvate at CofC on "3-GDDs with Block Size 4 and Different Group Sizes". I might be looking for student(s) working with me in the Fall 2018 or in 2019. I'd like the student to have a GPA 3.2 or higher and to be interested in applied math or interdisciplinary studies (students from outside of the department are welcome).