mathematical finance diagram

Undergraduate Research

follow your interests & work closely with faculty experts

Undergraduate? It's time to become a researcher. 

Independent work can be an important part of the undergraduate educational experience. The mathematics department offers opportunities to work in specialized areas with a faculty mentor during the academic year, over the summer, or through the selective Freiwald Scholars program. Students most often become involved in independent study projects or research during their junior or senior years, after taking several advanced courses. 


Independent Study

Undergraduates can work with a faculty member over the course of a semester, conducting independent research for a maximum of three units of course credits. To learn more, talk with your advisor or a reach out to a faculty member whose work you admire. 

Honors Thesis

One of the purposes of an honors thesis is to work together with a WashU faculty member during the final few semesters of study, exploring new material and tying together skills acquired during the major.  

Freiwald Scholars Program

The Freiwald Scholars program is directed to undergraduates with a proven record of strong mathematical skills who are motivated to carry out research in areas with substantial mathematical content. The program engages students in a reading and research project that lasts for at least two semesters and a funded 8 weeks in summer.

      The Freiwald Scholars Program

      About the Freiwald Scholars Program

      In 2012 the Advanced Research Training For Undergraduates (ARTU) program was founded and was originally funded by the National Science Foundation. Its goal is to introduce and train bright undergraduate students to do research in mathematics and closely related disciplines. The program allows students to work over the course of multiple semesters with a faculty mentor, and includes a funded summer residency. By 2017 the ARTU program had evolved from its original founding and received a name change to honor our Professor Ronald Freiwald, who retired in 2017.

      The Freiwald Scholars are supported by a donation from a department alumnus in  honor of Professor Freiwald, and also by the Office of Undergraduate Research.


      Juniors, sophomores, and exceptionally strong first-year students from Washington University in St. Louis will be considered. It is a small and selective program.


      Student work takes place under the supervision of mathematics faculty who are leading researchers in their respective fields.

      Timeline & Commitment 

      Minimum participation time is 1 semester and 1 summer. The expectation is that most students will want to continue significantly longer. Students will write progress reports approximately every 4 months, and after their first summer of funding, they are also expected to do oral or poster presentations every four months. These students are also expected to fulfill requirements outlined by the Office of Undergraduate Research, including presenting their work at the Undergraduate Research Symposium.


      Students and faculty must be in residence in St. Louis during the summer research months. Long-distance supervision is not usually allowed.


      Summer and Spring 2022 Program

      The primary faculty mentor/advisor for the Summer 2022 program will be Professor Greg Knese.  More details are below.  Students with other interests may still apply and may still be accepted into the program with a different advisor.

      The Freiwald Scholars led by Professor Knese in the Spring-Summer 2022 will consist of weekly seminars during the spring semester and two months of intense research work during the summer. Students who wish to begin a research project or who are already engaged in undergraduate research under the guidance of a professor in the math/statistics department are welcome to apply. All activities will be coordinated by Prof. Knese.

      The weekly seminars during the spring semester will be the equivalent in time commitment to a 3 credit course. They will be dedicated to discussing potential topics students may adopt for their summer research, acquiring the necessary technical background, as well as presentations by the students. Summer activities will revolve around the development and writing of results, and presentations among the participants. Freiwald Scholars will also give poster presentations of their results at the Washington University Undergraduate Research Symposium in the Fall of 2021.

      How to Apply

      Applications for Spring-Summer 2022 are due on November 8, 2021. Applications are reviewed and finalists are notified and invited to an informal interview to assess the applicants' motivations and consider matches for faculty membership. An application is made up of the following:

      • Updated resume (PDF format)
      • Statement of interest (PDF format; please keep to one page or less)
      • Two letters of recommendation from faculty members familiar with your mathematical knowledge and skills.

      Strong academic records are the primary qualification for admission into the Freiwald Scholar program.

      For more information or to apply contact Blake Thornton


        Recent Honors Theses & Projects

        Predicting Outcomes for Batted Balls in MLB
        (Advisor: Professor Victor Wickerhauser)

        An Essay on Independent Component Analysis
        (Advisor: Professor Jimin Ding)

        Representations of the Symmetric Group and Nonstandard Quantum Statistics
        (Advisor: Professor Renato Feres)

        Applications of Mixed Effects Modeling in Observational Studies and Clinical Trials for Alzheimer's Disease
        (Advisor: Professor Edward Spitznagel)

        Hecke Algebra Characters Evaluated at Kazhdan-Lusztig Basis Elements Give the Betti Numbers of Hessenberg Varieties
        (Advisor: Professor John Shareshian)

        Predicting Pitching Injuries in Major League Baseball
        (Advisor: Professor Nan Lin)

        Quantum Techniques for Classical Systems
        (Advisor: Professor Renato Feres)

        A Walk Through Einstein's General Theory of Relativity
        (Advisor: Professor Quo-Shin Chi)

        Nonabelian Group Based Cryptography
        (Advisor: Professor John Shareshian)

        Two Partitioned Numerical Methods for Solving the Hodgkin-Huxley Equations
        (Advisor: Professor Ari Stern)

        On Numbers with Equal Cototients
        (Advisors: Professors Matt Kerr and John Shareshian)

        Manifesting Polynomials with Small Galois Groups as Rational Points
        (Advisor: Professor John Shareshian)

        Discrete Morse Theory and Some Applications
        (Advisor: Professor John Shreshian)

        Generating Functions for Enumerating Chains of Partitions with Distinct Parts
        (Advisor: Professor John Shareshian)

        A Multivariate Extension of the von Neumann Inequality
        (Advisor: Professor Greg Knese)

        Analysis of Peer-Led Team Learning (PLTL) Data for Calculus Using MIxed Linear Effects Models
        (Advisor: Professor Jimin Ding)

        Algorithms for Reducing the Complexity of High-Dimensional Data Sets
        (Advisor: Prof. Victor Wickerhauser)

        Quantal Response: Models of Irrationality and Correlation in Games
        (Advisor: Professor John Shareshian)

        Group Decision-Making with Negative Influence Actors
        (Advisor: Professor John Shareshian)

        Hodge Decomposition and the Shapley Value of a Cooperative Game
        (Advisor: Professor Ari Stern)

        Spectral Gap of a Specific Linear Operator
        (Advisor: Professor Xiang Tang)

        Semiparametric Modeling of Train Accident Costs with Implications for the Importance of Positive Train Control
        (Advisor:Professor Todd Kuffner)

        Modeling the Spread of Amyloid in Patients with Alzheimer's Disease
        (Advisor:Professor John McCarthy)

        Elements of the Mathematical Formulation of Quantum States
        (Advisors: Professors Xiang Tang and Renato Feres)

        Exponential Random Graph Models Under Measurement Error
        (Advisor:Professor Nan Lin)

        Risk Factor Analysis of Thrombocytopenia
        (Advisor: Professor Edward Spitznagel)

        Stochastic Thermodynamics of Markov Chain from Billiards Model
        (Advisor: Professor Renato Feres)

        The Hermitian Symplectic Group, Neumann Extension Theory, and Scattering on Quantum Graphs
        (Advisor: Professor Renato Feres)

        A Comparison of the Lasso and Dantzig Selector in Linear Regression Models
        (Advisor: Professor Todd Kuffner)

        N-Complexes and Sn Representations
        (Advisor: Professor John Shareshian)

        Schatten-class Truncated Toeplitz Operators
        (Advisor: Professor Richard Rochberg)

        Computing the Dispersion Relation of Periodic Quantum Graphs
        (Advisor: Professor Renato Feres)

        Modeling Air Pollution with GIS
        (Advisor: Professor Edward Spitznagel)

        Polynomial Automorphisms
        (Advisor: Professor David Wright)

        An Alternative Graph Theoretic Proof of the Amitsur-Levitzki Identity  
        (Advisor: Professor John Shareshian)

        Nash-Moser Inverse Function Theorem and its Application to Ricci Flow 
        (Advisor: Professor Quo-Shin Chi)

        Automatic Classification of Formal and Informal Text
        (Advisor: Professor Edward Spitznagel)

        The Relativistic Heat Equation 
        (Advisor: Professor Ari Stern)

        Modifications to Nowak et al.'s Mathematical Population Genetic Model of The Evolution of Eusociality 
        (Advisors: Professor Renato Feres and Prof. David Queller (Biology))

        Categories, Groupoids, and Groupoidifications 
        (Advisor: Professor Xiang Tang)

        Aliquot Cycles for Elliptic Curves with Complex Multiplication 
        (Advisor: Professor Matt Kerr)

        On a Small Non-Shellable Lie Algebra
        (Advisor: Professor John Shareshian)

        Applications of Mixed Effects Models: An Analysis of fMRI data in Anesthesiology
        (Advisor: Professor Nan Lin)

        Lie Groups and Lie Algebras
        (Advisor: Professor Xiang Tang)

        Nonholonomic Diffusion
        (Advisor: Professor Renato Feres)

        Life-Cycle Investment and Consumption Retirement Strategy via Markov Chain Monte Carlo with R
        (Advisor: Professor Renato Feres)

        Questions to Remember: An Analysis of Cognitive Data for Alzheimer’s Disease Detection
        (Advisor: Professor Jimin Ding)

        Proctor or Doctor: An Analysis of Survey Data from Patients with Alzheimer's Disease
        (Advisor: Professor Jimin Ding)

        Applications of Linear Mixed Effect Models: An Analysis of Missouri School Data
        (Advisor: Professor Jimin Ding)

        I started at WashU knowing I was interested in math, but I wasn't sure how I could make that into a career. The department not only helped me find the area of math I'm most passionate about, but also guided me in finding opportunities to pursue it further. Along the way I've made a lot of great relationships with professors, advisors, and other majors in the department, which has been one of the best parts of my undergrad experience!

        ―Ryan SchneiderMathematics Major

        Have questions?

        To learn more about undergraduate research opportunities, contact the associate director of undergraduate studies. 

        Contact Blake Thornton