I strongly support undergraduate research opportunities in which students work closely with a faculty mentor to produce new knowledge within a field of mathematics.Â
Project: Mirror maps, local Yukawa couplings, and the twist construction
Student Researchers:
C. Crane, Sep. 2026 - Present
Objectives: Determine precise relationships between the q-series of mirror maps and holomorphic periods of Calabi-Yau pencils related by the iterative fibration structure of Doran & Malmendier's pure twist construction. Apply these results to local Yukawa couplings a la the method developed here.
Project: Mirror symmetry for lattice polarized K3 surfaces and holomorphic conformal geometry
Student Researchers:
L. Smith, May 2026 - Present
M. Lucas, June 2024 - May 2025 (Currently CEO at The Math Tree)
Objectives: Understand the integrable holomorphic conformal structure associated to higher dimensional moduli spaces of families of lattice polarized K3 surfaces with surjective period map established here from the perspective of Cartan connections. Apply to the projective normal form of associated multivariate Picard-Fuchs systems and determine precise relationship between the differential geometric structure and the coincidence for such lattice polarized K3 surfaces of the associated quantum cohomology rings with the ordinary complex cohomology ring.
Project: Projective differential geometry of surfaces associated to Appell's bivariate hypergeometric functions
Student Researchers:
M. Ryan, Oct. 2023 - Jan. 2026 (Selected 2026 VT Math Outstanding Senior in Undergraduate Research)
Objectives: Learn and effectively apply the GKZ method for A-hypergeometric functions to generalized euler integrals. Learn about differential geometric content of associated holonomic systems of PDEs and the projective differential geometry of surfaces in projective 3-space. Apply results to understand both differential geometric content of and Laplace invariants of surfaces determined by Appell's bivariate hypergeometric functions of rank-4.
Outcome: Together wrote journal article "Line congruences associated to Appell's hypergeometric functions of rank-4", accepted with revisions to Involve, a Journal of Mathematics, Sept. 2026.