## Research Opportunities for Undergraduate Math Majors

Here is a list of Mathematics faculty, who are interested in working with undergraduate mathematics majors on research projects. If you are interested in working with one of these faculty members, please feel free to contact them directly.

### David Anderson, anderson.2804@osu.edu

Algebraic Geometry, Combinatorics

I have several projects related to interactions between linear algebra and combinatorics. Here is the flavor of one of them. You can place constraints on the space of all k by n matrices by requiring certain submatrices have bounded ranks. What rank conditions are feasible? What are more efficient ways of specifying such conditions? And how does the problem change if instead of all matrices, you work with only symmetric matrices?

These problems are related to Coxeter groups in combinatorics, and Schubert varieties in algebraic geometry, and many natural questions remain open. No knowledge of Coxeter groups or algebraic geometry is needed to begin working on them, but a good understanding of linear algebra is essential. Some experience with a programming language (C, Python, Maple, Mathematica, or Sage) will be helpful for running experiments to test hypotheses.

### Ghaith Hiary, hiary.1@math.osu.edu

Implement a new method to compute the Riemann zeta function, which is of comparable complexity to the Riemann-Siegel formula, but is much easier to derive, and with a remainder term that is easier to control. The method is elementary (requiring no more than basic algebra and employing the geometric series), and it does not use the functional equation, nor the standard analytic continuation, but had been missed by researchers in the field thus far.

The method can be generalized to Dirichlet L-functions, which will constitute an additional contribution (filling a gap in the literature) as we do not have a complete analogue of the Riemann-Siegel formula in that case. However, this generalization is technical, and so might make for a separate project.

Ideally, the student will have basic experience with C/C++ or FORTRAN programming (but nothing advanced), or a willingness to learn as needed. I estimate that the project can finish during a semester of 1 or 2 meetings per week.

### Barbara Keyfitz, keyfitz.2@osu.edu **On Leave 2016-2017**

Partial Differential Equations; Hyperbolic Conservation Laws

Although Conservation Laws (in fact PDE in general) is not usually considered a topic for undergraduate research, several topics related to my research are accessible to undergraduate mathematics majors.

- Parameter exploration in models that change type.

There are some simple and appealing systems of equations, including a model for two-way traffic flow, some chromatography models, and a model for thin films, where solutions have not been found even for some elementary problems. Students can use a combination of numerical simulation and analysis. For the simulations, there is available software, such as the Matlab Toolboxes, and they can also program simple finite-difference methods. The analysis involves investigations of equilibria in planar dynamical systems. - Multidimensional equations.

Another interesting direction is the study of scalar multidimensional problems. The theory of scalar problems is complete. However, the qualitative behavior of solutions of these equations -- formation and decay of shocks, interactions of shocks and rarefactions, and so on -- is made complicated by the fact that no fluxes can be genuinely nonlinear. Hence, looking at self-similar initial data provides insight into the nature of genuine nonlinearity and bifurcation.

### Hoi Nguyen, nguyen.1261@math.osu.edu

I am interested in any kind of Combinatorics, Discrete Mathematics, and Probability (in particular Random Matrix Theory).

### Crichton Ogle, ogle@math.ohio-state.edu

Topology, Analysis, Algebra

Bart Snapp, snapp.14@osu.edu

I am interested in working with undergraduates in on various projects including (but not limited to) elementary number theory, geometry, and algebra.

### David Terman, terman.1@osu.edu

Mathematical Neuroscience

I am interested in working with undergraduate math majors on projects related to using mathematical methods to understand models for neuronal activity in the brain. These models arise in the study of neurological diseases such as Parkinson’s disease and stroke.

Research Experience for Undergraduates (REU)

On occasion, individual faculty attach a Research Experience For Undergraduates (REU) onto their current research. Currently there are no REU's offered in this department. However, there are many mathematics-specific REU’s available across the country supported by the National Science Foundation.

## Young Mathematicians Conference (YMC)

The YMC is a annual conference for undergraduate student researchers in mathematics. Talks and poster presentations are given by the students of their results and discoveries. They also discuss research ideas and experiences with their peers at the conference.

Students involved in REU's and similar research programs from all over the United States apply each summer to actively participate in YMC. Accepted students (typically around 70) are invited with full support to the conference. It is held during a weekend in August at the Department of Mathematics of The Ohio State University. See YMC for more information.

## Undergraduate Research Office (URO)

The Undergraduate Research Office helps students pursue research opportunities at The Ohio State University. Research can be conducted independently, as part of a team, in collaboration with faculty, here at the university or elsewhere. The URO website includes information about getting started with research, how to find research opportunities, what's involved in presenting your work, and a variety of other information sources for undergraduates interested in making research a part of their college experience.