Faculty Research

Robert Boltje and his students work in the representation theory of finite groups. They are primarily involved with the conjectures of Alperin, Broué and Dade. Current research interests include canonical induction formulas, biset functors, fusion systems, block theory of group algebras, and equivalences between such blocks. Professor Boltje has also worked in the area of algebraic number theory, where he has developed functorial methods to understand Galois actions on rings of algebraic integers, and other structures associated to number fields.

Frank Bäuerle has been in the trenches of undergraduate mathematics education at UC Santa Cruz since his arrival in Fall of 1994. He just concluded his latest stint as undergraduate vice chair, a position that he has held on and off for the last decade. He has worked on statewide projects to improve Mathematics Instructions across all three systems of higher education in California. 

Most importantly, over the last decade Frank, together with his colleague and friend Tony Tromba, has developed the online versions of the Calculus sequence for Science and Engineering. These courses thus far have been taken by about 35,000 students and are open for enrollment and articulated for major’s credit to students from all nine undergraduate-serving UC campuses (the first and only such courses in the 157 year history of UC).

Chongying Dong works in the area of vertex operator algebras. This area has its origins in two-dimensional conformal field theory, monstrous moonshine and vertex operator representations of affine Kac-Moody algebras. Together with Geoffrey Mason, he is interested in vertex operator algebras, infinite dimensional Lie algebras, Hopf algebras, category theory and mathematical physics. His current research focuses on the structure and representation theory of vertex operator algebras.

Torsten Ehrhardt uses Operator Theory to study the spectral and asymptotic properties of various types of operators such as Toeplitz, Hankel, or Wiener-Hopf operators.

Part of his work is motivated by applications to Random Matrix Theory and Statistical Physics. His research interests also include Wiener-Hopf factorization theory, Riemann-Hilbert problems as well as aspects of the theory of Banach algebras.

Laura Escobar‘s research focuses on the interplay between combinatorics and algebraic geometry. She has developed combinatorial models as part of her program to understand the singularities of Schubert varieties and the action of tori on these varieties. This work has led her to construct and study polytopes using tools from symplectic and algebraic geometry. Lately, Escobar has been working on a generalization of the polytope-variety dictionary of toric varieties.

Roozbeh Gharakhloo studies the asymptotic behavior of principal objects in mathematical physics and random matrix theory which have a characterization in terms of structured determinants and/or a system of orthogonal polynomials. A primary objective in his work has been to expand the application of the Riemann-Hilbert approach to studying the asymptotic properties of various structured determinants, including Toeplitz and Hankel determinants, as well as their altered forms like Toeplitz+Hankel, bordered Toeplitz, framed Toeplitz, and more general forms such as pj‑qk or slant Toeplitz determinants.

Viktor Ginzburg’s work from the last two decades has mainly been in symplectic dynamics, an area at the interface of symplectic topology and dynamical systems. He has worked on various aspects of the existence problem for periodic orbits: the Hamiltonian Seifert conjecture, the Conley conjecture, multiplicity of periodic orbits of Reeb flows in higher dimensions, periodic orbits of a charge in a magnetic field and several others. His more recent work focuses on connections between Floer theory and dynamics beyond periodic orbits, including dynamics of Hamiltonian pseudo-rotations in all dimensions, invariant sets, entropy, and Floer homology persistence modules. 

In the 90s he worked in Poisson geometry and Hamiltonian actions of compact groups.

Kasia Jankiewicz works in geometric group theory. She is interested in nonpositively curved groups and spaces, Artin groups, Coxeter groups, small cancellation theories, separability properties, and coherence.

Jesse Kass studies algebraic geometry and related topics in commutative algebra, number theory, and algebraic topology. He has major projects on moduli spaces of sheaves on singular curves and on counting algebraic curves arithmetically using motivic homotopy theory. He is also interested in the history of mathematics, particularly the representation of African American mathematicians.

Debra Lewis is an expert in relative equilibria for Hamiltonian systems and their stability, as well as bifurcations of relative equilibria. A relative equilibrium for a dynamical system is a solution that is an equilibrium modulo a symmetry group action. Her work is an interplay between group theory and symplectic geometry, and uses a good deal of symbolic manipulation. Her research focuses on geometric mechanics and optimal control theory, including models influenced by psychological and social pressures. She is presently involved with math placement and preparation, where she uses learning analytics to assess and increase equity in math access and outcomes

Longzhi Lin works on geometric partial differential equations, focusing on geometric flows (such as mean curvature flow and harmonic map heat flow), harmonic and biharmonic maps, minimal surfaces, prescribed mean curvature surfaces (H-surfaces), and min-max theory. His recent research explores the convexity properties of various conformally invariant energy functionals to establish quantitative uniqueness of their critical points — such as conformal-harmonic maps and solutions to the H-surface system — and to develop corresponding min-max constructions.

Francois Monard’s work focuses on parameter-reconstruction (“inverse”) problems in PDEs and integral geometry, with applications to imaging sciences. His most recent work is concerned with the inversion of X‑ray/Radon transforms (mappings which integrate a geometric object along a given family of curves) and their generalizations to non‑Euclidean geometries and to the non‑abelian integration of matrix fields.

Richard Montgomery is best known for  work in two areas: the classical  N‑body problem and sub‑Riemannian geometry.  In the first area, he and Alain Chenciner (IMCEE Paris) rediscovered the figure eight orbit of Cris Moore and  rigorously prove of this solution to the three-body problem, a solution in which which three equal massed stars or planets chase each other around a figure-eight shaped curve. Their paper is mentioned in the best-selling science fiction book “The Three Body Problem” by Cixin Lin.  The figure eight work led to the discovery of a  huge variety of “choreography orbits” N-body solutions in which a number of  equal masses chase each other around various aesthetically pleasing curves.  Their result  and the techniques  introduced by the paper drove much work in mathematical celestial mechanics from 2000 to 2015.  In the second area he is best known for his counterexample and his book “A Tour of SubRiemannian Geometry”.  The counterexample, a certain three-dimensional geometry with a geodesic which does not satisfy the previously accepted geodesic equation, changed the face of subRiemannian geometry.   One of his most cited papers is  on coin flipping  and was written with  Persi Diaconis and Susan Holmes (Stanford).  This team showed  that a flipped coin is more likely to land heads up if starts heads up, and tails up if it starts tails up.   This paper and  the two research areas just described have their roots in his early work of his on how a cat changes shape in mid-air in order to land on its feet. 

Pedro Morales-Almazán is interested in regularization problems arising in quantum field theory and asymptotic methods in number theory. His focus has been the use of spectral zeta functions arising from pseudo‑differential operators in Riemannian manifolds to compute Casimir energy and vacuum energy of quantum fields. He is also interested in number theory problems involving regularization and resummation methods, specifically using zeta function regularizations.

Jiayin Pan works in Riemannian geometry – more specifically, manifolds with Ricci curvature bounded below, and their limit spaces under Gromov-Hausdorff convergence. Their research concerns the fundamental groups of open manifolds with nonnegative Ricci curvature, as well as the structure of Ricci limit spaces. He is also interested in sub-Riemannian geometry and Lorentzian geometry.

Jie Qing works on conformal geometry, the AdS‑CFT correspondence, and most recently in general relativity. In conformal geometry, only the angle between two vectors can be measured but not the vector’s lengths. The AdS‑CFT correspondence relates the Riemannian or pseudo-Riemannian (general relativistic) geometry on one manifold to the conformal geometry of a manifold that bounds it. AdS‑CFT has roots going back about 30 years but became very popular in the last decade due to its relevance to string theory. Qing has some of the strongest uniqueness results available in AdS/CFT for conformal spheres.

Beren Sanders works in algebra and topology. His research centers on triangulated categories and their applications, especially tensor triangular geometry and examples arising in stable homotopy theory, modular representation theory, and algebraic geometry. Other interests include equivariant homotopy theory, motivic homotopy theory, higher category theory, and the representation theory of groups and associative algebras.

Junecue Suh is interested in arithmetic or number‑theoretic topics in algebraic geometry, such as the zeta function of varieties over finite fields, the cohomology of Shimura varieties, and mixed Hodge modules.

Hirotaka Tamanoi has worked in homotopy theory, in particular, generalized cohomology theories known as cobordism theories. He is fascinated by mathematical ideas and methods inspired by string theory and conformal field theory in mathematical physics. His work has been centered around algebraic topological aspects of loop spaces such as elliptic genus (which was given a quantum field theoretical interpretation by Witten’s work on string theory), and Sullivan’s string topology and its relation to symplectic topology. He has written a book on vertex operator algebras and elliptic genera. More recently, he is interested in topological quantum field theories and their applications.

Tony Tromba‘s main body of research is on minimal surfaces, a subject about which he has written several Grundlehren volumes. These are surfaces that minimize area among all surfaces bounding a given curve in space (think of soap films bounding wires in space). He developed a Morse Theory and Degree Theory for such surfaces paralleling Morse’s theory for geodesics, results Morse failed to achieve, theories which were consequently honored by an invitation to deliver an address to The International Congress of Mathematicians. He also developed a new approach to Teichmüller Theory and applied this to solve one of the main open questions in the field, namely to compute the sectional curvature of Teichmüller Space with respect to its Weil Petersson Metric. Most recently he has found new positive results for Hilbert’s famous 19th problem in the Calculus of Variations. 

Martin Weissman‘s research involves the interaction between representation theory, geometry, and number theory. Specifically, he works on automorphic forms and representations— the network of theorems and conjectures known as the Langlands program. Within the Langlands program, he is interested in exceptional and metaplectic groups, and broad questions in the representations of p‑adic groups. He has also studied connections between arithmetic and Coxeter groups, and the visualization of algebra and number theory.