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 current work is in the theory of approximation methods for inverse and ill-posed problems framed within the context of Hilbert space operator theory and in numerical solution of integral equations. Specific projects include investigation of an iterative method for semi-linear bivariate operator equations of the first kind and classical approximation methods applied to stabilized numerical differentiation.
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.
Logic Regression is an alternative regression methodology that is applied to binary data and uses Boolean logic to find combinations of the predictors for classification. It is especially efficient in finding interactions of binary factors that lead to increased disease risk and, thus, is often used with SNP data. Many complex diseases, however, manifest as a result of combinations of both continuous and binary factors. This project attempts to adjust the logic regression framework to allow for the inclusion of continuous variables. It introduces an additional move to the logic regression algorithm that will dichotomize a variable at the “optimal” value so that the variable can be included in the logical combination of predictors that discriminate disease outcome.
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.
I am willing to work with students, but I do not have an active research program. My work with a student would probably originate with a problem in Mathematical Monthly or a similar publication.
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).
More to come soon....