Areas – Mathematics

Our faculty conduct research across a broad spectrum of mathematical disciplines. Below you can explore the primary research areas and the faculty members engaged in each.

Algebra and Representation Theory

Groups, algebras, and their representations.

Faculty: Robert Boltje (modular representation theory, finite groups), Chongying Dong (infinite-dimensional Lie algebras, vertex operator algebras), Laura Escobar (combinatorics, polytopes, combinatorial commutative algebra), Kasia Jankiewicz (geometric group theory), Geoff Mason (vertex algebras; quasiHopf algebras and modular tensor categories, finite groups). Beren Sanders (triangulated categories, homological algebra), Martin Weissman (p-adic groups, algebraic groups)

Algebraic Geometry

Geometry of solution sets to polynomial equations.

Faculty: Laura Escobar (toric and Schubert geometry), Jesse Kass (curves, abelian varieties, moduli spaces), Beren Sanders (derived categories, tensor triangular geometry), Junecue Suh (Hodge theory, abelian varieties)

Analysis and Spectral Theory

Differential equations, functional analysis, operators.

Faculty: Torsten Ehrhardt (operator theory; Toeplitz, Hankel, and Wiener-Hopf systems; spectral theory), Roozbeh Gharakhloo (complex analysis, structured determinants, systems of orthogonal polynomials), Longzhi Lin (geometric analysis), François Monard (applied functional analysis, microlocal analysis, inverse problems), Pedro Morales-Almazán (spectral geometry, spectral zeta functions), Jie Qing (harmonic analysis, nonlinear analysis)

Applications and History of Mathematics

Connections to other disciplines.

Faculty: Torsten Ehrhardt (random matrix theory, statistical physics), Jesse Kass (African-Americans in mathematics), Debra Lewis (learning analytics), François Monard (mathematical models of imaging sciences, numerical implementations), Richard Montgomery (emeritus) (control theory), Martin Weissman (computational biology, analysis of ‘omics data)

Differential Geometry

Geometry of smooth manifolds.

Faculty: Viktor Ginzburg (symplectic geometry), Longzhi Lin (geometric analysis), Richard Montgomery (emeritus) (sub-Riemannian geometry), Jiayin Pan (Riemannian geometry, Ricci curvature and topology), Jie Qing (Riemannian geometry, conformal geometry)

Dynamical Systems

Geometric mechanics, Hamiltonian dynamics, symplectic geometry.

Faculty: Viktor Ginzburg (Hamiltonian dynamics, periodic orbits, Conley Conjecture), Debra Lewis (geometric mechanics, optimal control, relative equilibria and their stability), Richard Montgomery (emeritus) (N-body problems, celestial mechanics, sub-Riemannian geometry)

Foundations of Mathematics

Logic, complexity theory, category theory.

Faculty: Frank Bäuerle (complexity theory, recursion theory), Beren Sanders (higher category theory)

Mathematics Education

Pedagogy, new modalities, equity and inclusion.

Faculty: Frank Bäuerle (online calculus), Debra Lewis (learning analytics), Pedro Morales-Almazán (inclusive pedagogy), Anthony Tromba (online instruction)

Mathematical Physics

Mechanics/dynamics, conformal field theory, mathematical relativity.

Faculty: Chongying Dong (conformal field theory), Roozbeh Gharakhloo (random matrix theory, Riemann-Hilbert problems), Debra Lewis (geometric mechanics), Geoff Mason (Archimedean and non-archimedean conformal field theory, the modular bootstrap program), Richard Montgomery (emeritus) (celestial mechanics, nonlinear control theory), Pedro Morales-Almazán (heat kernel, Casimir effect), Jie Qing (mathematical relativity, mathematics of AdS/CFT correspondence), Hirotaka Tamanoi (topological quantum field theory)

Number Theory

Whole numbers and prime numbers, arithmetic, Diophantine equations.

Faculty: Robert Boltje (algebraic number theory), Jesse Kass (arithmetic enumeration), Geoff Mason (vector-valued modular forms, applications of automorphic forms in algebra and theoretical physics), Junecue Suh (Shimura varieties, arithmetic geometry), Martin Weissman (Langlands program, classical number theory)

Partial Differential Equations

Systems of multivariate quantities changing over time.

Faculty: Longzhi Lin (calculus of variations, geometric PDEs), François Monard (inverse problems for elliptic and transport PDEs, integral geometry), Jie Qing (calculus of variations, geometric PDEs; fluid mechanics), Anthony Tromba (calculus of variations, minimal surfaces)

Topology

Geometry which remains after continuous deformation.

Faculty: Viktor Ginzburg (symplectic topology, Floer homology), Kasia Jankiewicz (geometric group theory, low-dimensional topology), Jiayin Pan (fundamental groups of open manifolds, Milnor conjecture), Beren Sanders (stable homotopy theory, higher category theory), Hirotaka Tamanoi (homotopy theory, algebraic topology, loop spaces)