# Previous Undergraduate Research Projects

**Name:** Caroline Henson

**Class:** Undergraduate, Computer Engineering

**Faculty:** Alexandria Volkening, Assistant Professor of Mathematics

**Project:** Spring and Summer 2022 semesters

Her research focused on developing a program that would transform mathematical model data into realizstic zebrafish images. Caroline's software succesffuly produced convincing images of zebrafish from simulated data that has potential to be used in futute zebrafish studies.

Caroline presented her work at the Purdue Undergraduate Research Symposium (Spring 2022), the Indiana Undergraduate Math Conference (Summer 2022) and the Purdue Summer Undergraduate Research Fellowship Symposium (Summer 2022).

**Name:** Mahimna Vyas

**Class:** Undergraduate, Physics Honors/Mathematics/Philosophy

**Faculty:** Yuan Gao, Assistant Professor of Mathematics

**Project:** Spring 2022 and Fall 2022

Mahimna has been working on non-equilibrium biochemical reactions from a Hamiltonian viewpoint. She is working on a 2-D reaction system to find it's detailed balance case then solving for a steady-state solution and energy landscape in which the equations would give different behaviors.

Mahimna presented her work at the Purdue Undergraduate Research Symposium (Spring 2022)

**Name**: Matthew Griffin, and Aneesh Khilnani

**Class: **Math and Physics Honors, Aerospace Engineering

**Faculty: **Thomas Sinclair, Assistant Professor of Mathematics

**Project: **Summer 2022

Using techniques from semidefinite programming, Matthew and Aneesh studied the problem of finding a closest quantum channel to the projection mapping onto a matricial subsystem. They defined a Hilbert space on the set of linear operators on square matrices and used this to derive two invariants of matricial subsystems that are related to the quantum Lovász theta function of Duan, Severini, and Winter.

Matthew and Aneesh Presensted their work at the Purdue Undergraduate Research Symposium (Spring 2022) and has submitted their paper to the Journal of Linear Algebra.

**Name:** Meenakshi McNamara

**Class:** Undergraduate, Math and Physics Honors

**Faculty:** Rolando de Santiago, Assistant Professor of Mathematics

**Project:** Spring and Fall 2022 semesters

Meenakshi is investigating quantum chromatic numbers of quantum graphs. These are defined in terms of quantum games and the generalization of graphs using operator algebras.

Her research defined the lexicographic product of quantum graphs and is deriving bounds upon the quantum chromatic number of the resultant quantum graph.

Meenakshi presented her work at the Purdue Undergraduate Research Symposium (Spring 2022) and Purdue Mathematics Society Undergraduate Talks (Spring 2022).

**Names**: Ryan Branstetter, Manas Paranjape, and Mengqi Liu

**Class: **Mathematics, Computer Engineering, Computer Engineering

**Faculty: **Alexandria Volkening, Assistant Professor of Mathematics

**Project: **Spring, Summer and Fall 2022

Ryan, Manas and Mengqi's work focuses on forecasting U.S. elections. They use an adapted Susceptible-Infected-Susceptible (SIS) model developed by Alexandria Volkening. The work they have done can be found on a responsive design website they created where they will post their forecasts throughout the Fall 2022 semester. Their website can be viewed here.

Ryan, Manas, and Mengqi presented their research at the Spring 2022 Purdue Undergraduate Research Conference, the Summer 2022 Purdue Undergratuate Research Fellowship Symposium and teh Summer 2022 Indana Undergraduate Research Conference.

**Name:** Zijin Liu

**Class:** Undergraduate, Math Honors

**Faculty: **Jing Wang, Assistant Professor of Mathematics

**Project:** Spring and Fall 2021 semesters, and Spring 2022

Zijin worked on Brownian Motion on sub-Riemannian models and visualizations. He brought probabiity into some interesting geometric structures. The models he used are the Heisenberg group, the 3-sphere and the anti-desitter sapce.

Zijin used the 3-sphere and the anti-de sitter for sub-Riemannian models with positive and negative curvature. The Heisenburg group was used as the flat model.

With some fibrations of manifold, Zijin generated a Brownian Motion path on each model by solving stochastic differential equations and change of coordinates.

Zijin presented his work at the Young Mathemeatics Conference (Summer 2021).

## More Research Projects

Name:** ****Chiara Travesset and Joseph Veltri**

- Project:Number Theory and Analysis
- Published in Quarterly Journal in Mathematics: reprint found here
- 2021

Name: **Sydney Kohn**

- Project: Intersection of geometry and children's thinking in mathematics
- Class: Undergraduate, Math Honors
- Fall 2020

Name: **Zijie (Jerry) Zhou**

- Project:RIPS REU
- Submitted to the journal SIAM Undergraduate Research Online (SIURO): Arxiv link
- Jun 2019 to Aug 2019

- Project: Polymath REU
- Polymath Presentation (PDF)
- Jun 2020 to Aug 2020

Name: **Juliet Aygun**

- Project: Riemannian and symplectic geometry
- Presented at three virtual conferences
- Summer 2020

Name: **Victor Hughes**

- Project: Hamiltonian systems and stochastic differential equations
- VERUM Summer Research Program
- Completed VERUM Projects
- 2019

**REU Projects under Professor Jonathon Peterson:**

Mike Cinkoske (Purdue), Joe Jackson (Swarthmore College), Claire Plunkett (Case Western Reserve)

- 8 week REU at Purdue University in the summer of 2017. Part of an NSF funded REU grant: Purdue Research in Mathematics Experience (PRiME)
- "On the speed of an excited asymmetric random walk"
- Presented at 2017 Indiana Undergraduate Math Research Conference, 2017 Mathfest, and 2018 Joint Mathematics Meetings
- Published in Rose Hulman Undergraduate Mathematics Journal. https://scholar.rose-hulman.edu/rhumj/vol19/iss1/6/
- Poster presentation.pdf

**Jacob Menix (Western Kentucky), Paige Schoonover (Tennessee)**

- 8 week REU at Purdue University in the summer of 2017. Part of an NSF funded REU grant: Purdue Research in Mathematics Experience (PRiME)
- "Statistical tests for convergence of some random walks to perturbed Brownian motion"
- Presented at 2017 Indiana Undergraduate Math Research Conference and 2017 Mathfest.
- Poster presentation.pdf (PDF)

**REU Projects under Professor Birgit Kaufmann:**

Name: **Justin Copenhaver and Raunaq Kumaran**

- DURI project about quantum computing in quantum chemistry
- Jan 20--Sep 20
- Funded by a DURI grant and Kaufmann’s CAREER grant.
- Publication: J.Copenhaver, A.Wasserman and B. Wehefritz-Kaufmann, Using quantum annealers to calculate ground state properties of molecules, J. Chem. Phys. 154 (2021) 034105

Name: **Kyler Overton**

- Class: Undergraduate Student and Ascarelli Fellow
- Project: Worked on graph Hamiltonians as an REU
- Jan 19--May 19

Name: **Tibor D’ome**

- Class: Undergraduate visiting from ETH Zurich
- Project: Wrote term paper "The Periodic Zero-Field Six-Vertex-Model"
- Jan 18--Apr 18

Name: **Brant Coburn**

- Class: Undergraduate
- Project: Worked on Hecke algebras and quantum spin chains as an REU; presented a poster at the Purdue Undergraduate Poster Symposium
- Aug 14--May 15

Name: **Nolan Teasdale**

- Class: Undergraduate REU student
- Project: Worked on Monte{Carlo simulations of reaction-difussion models
- Jan 10--May 10

Name: **Tong Ding**

- Faculty: Professor Jingwei Hu
- Project: Particle method for the Landau equation
- This project was presented by the student at the 2019 Purdue Undergraduate Research Conference.
- Summer 2018 to Spring 2019

Name: **Logan J. Cross**

- Faculty: Professor Xiangxiong Zhang
- Project: On the monotonicity of high order discrete Laplacian. A very challenging project, resulted in a high-quality paper, currently under review in a top numerical analysis journal.
- Publication: https://arxiv.org/abs/2010.07282
- 2019-2020

Name: **Fernando Davis, Grant Bowman, Truman Bennet**

- Faculty: Professor Greg Buzzard
- Project: Hyperparameter optimization for Deep Learning
- Support: NSF CCF-1763896, Purdue SURF Program
- June-August 2019

Name: **Trevor Crupi**

- Faculty: Professor Greg Buzzard
- Project: Foundations of Deep Learning
- January-May 2020

Name: **Yuzhang Wu**

- Faculty: Distinguished Professor Laszlo Lempert
- Project: Hyperbolic geometry in the Poincare model, that is, in the unit disc in the complex plane, where the motions are the Moebius maps of the disc. The purpose was to rediscover what had already been discovered, and accordingly no publication resulted.
- September 2019- December 2019

Working with Professor Jim McClure, sophomore **Alec McGail** read David Bressoud’s book, *A Radical Introduction to Real Analysis*. Alec also worked on a project to reconstruct all of the standard properties of sin and cos by using only facts about their power series. Since Alec had a special interest in philosophy, Professor McClure asked him to write a philosophical reflection on whether the sin and cos he creates this way are “the same” as the ones he knew in high school.

In independent discussions with Professor Uli Walther, senior **Kathryn Marsh** investigated various topics in mathematics, touching on subjects in group theory, topology, knot theory, and prime numbers. Weekly meetings supplemented and guided individual reading. Between students, communication was in the form of editing a class Rhea page on topics they were individually researching.

In a project with Professor Jon Peterson, junior **Lirong Yuan** and senior **Jingdan Liu** focused on studying the probabilistic properties of eigenvalues of random matrices and roots of random polynomials. While the statements of the problems they were interested in were very basic, the research brought up a variety of topics not often encountered by undergraduates: Rouche’s theorem, Cardano’s formula, weak convergence of probability distributions, and the change of variables formula for *n*-dimensional integrals. In addition to learning some more advanced mathematics, Lirong and Jingdan were able to experience how research in mathematics works. In particular, they were able to see that hard problems are attacked by first solving smaller simplifications of the problem, and that sometimes one needs to try several different approaches to a problem before finding one that works. Lirong likened mathematical research to climbing a mountain: when standing at the base of the mountain there are many possible routes to the top, but it might take several attempts to find the best route to the summit; moreover, the joy when reaching the summit makes all the previous failed attempts worthwhile.

Other summer math REUs were seniors **Brett Ernst**, **Sean Fancher** and junior **Abi Komanduru**, who worked with former Golomb Assistant Professors. Seniors **Ronald Archer**, **Benito Martinez**, junior **Han Liu**, and sophomore **Steve Mussmann** engaged in projects under the guidance of former Professor Edray Goins.