Students

Student research

I supervise undergraduate projects in analysis and mathematical physics, often centered on models that combine rigorous mathematics with computation and physical interpretation.

Many projects begin with a one-dimensional or exactly solvable model, where students can explore examples numerically, formulate conjectures, and then prove results using tools from differential equations, linear algebra, Fourier analysis, and spectral theory.

Research themes and topics

• Quantum optics in real space: exact or effective equations of motion arising from quantum-field theoretic models of light interacting with two-level atoms.
• Dispersive equations: decay estimates for discrete, fractional, Schrödinger, and Dirac-type systems.
• Time- and space-periodic dynamics: Floquet theory and the effect of periodic potentials or forcing on transport and decay.
• Random media: radiative transport, localization, and effective equations for waves in disordered systems.
• Exactly solvable spectral problems: point interactions, piecewise constant coefficients, and symmetry-reduced periodic operators.

Background and getting started

Each project requires a different background and can be chosen to a student's experience. A first course in real analysis (Math 350) is very helpful, but many projects can begin with linear algebra (Math 250), and differential equations (Math 309). Programming experience and some physics coursework are useful for some projects but are not always required.

Williams students interested in a thesis, summer project, or independent study can contact me at jk34@williams.edu. A brief note describing your mathematical background and the topics that interest you is helpful.

Letters of recommendation

I am happy to consider requests for letters of recommendation from students I know well enough to write a detailed letter. When asking, it is useful to ask whether I would be able to write a strong letter for the particular opportunity.

Please contact me well before the first deadline—ideally at least three or four weeks in advance, and earlier when you are applying to many programs. Along with your request, please provide:

• a list of the programs, scholarships, jobs, or other opportunities, including deadlines and submission instructions;
• a brief explanation of why you are applying and what you hope to do in the program;
• your current CV or resume and an unofficial transcript;
• drafts of any personal, research, or application statements you have been asked to write;
• a reminder of the courses, projects, or other work you completed with me, together with any accomplishments or experiences you would especially like the letter to address.

Please keep me informed if deadlines or application plans change. A short reminder about one week before the first deadline is also helpful.