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## 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.

### Sanjeevi Krishnan, krishnan.118@osu.edu

There are two main projects:

1. The first is to develop fast tools for matrix calculations subject to positivity constraints. Some goals are a fast algorithm and implementation thereof for efficiently describing the intersection of the null space of a matrix with the set of all vectors with non-negative components; here "efficiently describing" means find the set of all extremal vectors of length 1 bounding this intersection. These sort of calculations tend to blend techniques from optimization and linear algebra. Such methods are part of a larger program to extend algebraic topology so that positive cones usefully describe features on a directed space (e.g spacetime or a directed graph) in the same way that vector spaces describe features on a space. Ideally, the student can program (or learn to program) in a computational environment (e.g. Julia, Python/Numpy/Cython) and has a basic understanding of finite-dimensional linear algebra.

2. The second is to experiment with certain extensions of a constructive logic used in automated reasoning. Specifically, this project will evaluate various proposed extensions of a recent dependent type theory useful for proving certain assertions in topology. These extensions purport to make HOTT useful for reasoning about real-world dynamical systems, and in particular establish foundations for secure protocols in cybersecurity. Ideally, the student has some basic experience programming and has seen either some logic or topology.

### 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 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 Opportunities in Mathematics for Underrepresented Students (ROMUS)

The Department of Mathematics at the Ohio State University invites undergraduate students, primarily from traditionally underrepresented groups, to pursue research under the tutelage of experienced faculty members. Students work with a faculty member on a project of mutual interest for 8-10 weeks during the summer. In addition to the research projects, all accepted students will participate in various cohort activities. These experiences will virtual/remote for the summer of 2022.

The projects will be in the areas of topological and geometric data analysis; probability; infectious disease modeling; dynamics; applied math; algebraic geometry; operator algebras; quantum theory; computational number theory; computational math; combinatorics and model theory.

Applications are welcome from all students from all schools. REU participants are paid a stipend during the summer. Please note, REUs are intended primarily for US citizens and Permanent Residents. Because of funding limitations, international students will be considered on an ad-hoc basis.

Applications will begin being evaluated in early March, and accepted until all slots are filled. All applications will be submitted to mathprograms.org through this link: https://www.mathprograms.org/db/programs/1259

## Mathematical Biosciences Institute (MBI) Summer REU Program

The Mathematical Biosciences Institute (MBI) hosts a multi-institution REU program in the mathematical biosciences, facilitated by the MBI located on the campus of The Ohio State University. The objectives of the program are: (1) to introduce a diverse cohort of undergraduate students to the mathematical biosciences, broadly interpreted to include areas such as biostatistics, bioinformatics, and computational biology, in addition to biologically-inspired mathematical modeling; (2) to encourage students to pursue graduate study in the mathematical biosciences; and (3) to increase the number of students who enter the workforce with training in this field.

REU participants work on projects in areas such as molecular evolution, neuronal oscillatory patterning, cancer genetics, epidemics and vaccination strategies, and animal movement. Participants work individually or in pairs under the guidance of expert mentors to make specific research contributions in these areas, often leading to a peer-reviewed publication and conference presentations. The REU program incorporates various professional and research-skills development activities throughout the summer to help ensure the participants’ success in completing their summer project and to prepare them for graduate study or entering the workforce.

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.