Department of Mathematical Sciences

Faculty Research Interests

Laura Anderson
My research focuses on interactions between combinatorics and topology, particularly those involving oriented matroids, convex polytopes, and other concepts from discrete geometry. Much of my work involves combinatorial models for topological structures such as differential manifolds and vector bundles. The aims of such models include both combinatorial answers to topological questions (e.g., combinatorial formulas for characteristic classes), and topological methods for combinatorics (e.g. on topology of posets).

Robert Bieri
My original interest in homological methods for infinite groups (cohomological dimension and Poincare type duality) shifted towards geometric and -- more recently -- asymptotic methods. I find it interesting to relate geometric properties at infinity of groups and G-spaces with algebraic properties of these groups, their group rings and their modules. The focus is on familar groups like metabelian, soluble, free and linear ones, or fundamental groups of 3-manifolds, but I also met Thompson's group F and other PL-homeomorphism groups on the way, and had an encounter with tropical geometry.

Benjamin Brewster
Algebra, group theory. Topics of particular interest:
  1. Sylow subgroups,how the group acts on them via conjugation,and how they intersect.
  2. Solvable groups-their conjugacy classes of subgroups.
  3. Subgroup lattices-intervals in the lattice and the influence of permutable subgroups on this lattice.
  4. Characterizing subgroups with embedding properties in direct products.

Matthew G. Brin
I am currently interested in the mathematical interactions of a collection of groups that arose first in logic and universal algebra. The groups are generalizations of three groups first discovered by Richard Thompson. The groups show up in a strong way in logic, homotopy and shape theory, dynamical systems, categorical algebra and its relation to physics, and the combinatorial group theory of the word problem and of infinite simple groups. They are a source of examples or potential examples in geometric group theory, cohomology of groups, string rewriting systems and abstract measure theory.

Alex Feingold
Finite dimensional semisimple Lie algebras, tensor product decomposition of irreducible modules, representation theory of the infinite dimensional Kac-Moody Lie algebras, bosonic and fermionic creation and annihilation operators, affine and hyperbolic Kac-Moody algebras, topics in combinatorics, power series identities, modular forms and functions, Siegel modular forms, conformal field theory, string theory, and statistical mechanical models, vertex operator algebras, their modules and intertwining operators, the theory of fusion rules.

Ross Geoghegan
I am interested in the interplay between group theory and geometry/topology. In particular: geometric and homological group theory, fixed point theory, and certain parts of dynamical systems. Some of the questions motivating this work are algebraic, involving the algebraic K-theory of rings associated with the fundamental group; this is how I got interested in Nielsen fixed point theory, particularly parametrized versions of that theory. Other questions are about how an action by a discrete group on a non-positively curved space can lead to group theoretic information. I'm also interested in understanding the asymptotic topological invariants of a group. I have recently finished a book on that subject called "Topological Methods in Group Theory".

Andrey Gogolev
My general interest is in smooth dynamics and related areas of geometry and topology. More specifically I am interested in various classification problems in hyperbolic dynamics.

Fernando Guzman
My mathematical interests are algebraic in a broad sense. From universal algebra, lattice theory and ordered structures, through more classical algebraic topics like group theory and homology to interactions of algebra with computer science and logic.

David Hanson
Research interests: The Wiener process. Asymptotic properties of weighted sums of random variables. Certain problems in non-parametric statistics. Utility theory.

Luise-Charlotte Kappe
Group theory ( properties of groups, Engel conditions, commutator calculus, supplementation, p-groups, nonabelian tensor products); ring theory (derivations, commutativity conditions in rings); number theory (transcendental numbers, diophantine approximations).

Dikran Karagueuzian
My research for the past few years has been primarily in the representations and cohomology of finite groups. For the past few years I have been studying problems in algebra that arise from techniques of algebraic topology. Sometimes there is a theorem hidden behind the feasibility of a well-known method. An example of this phenomenon is my most recent preprint, written in collaboration with Peter Symonds of the University of Manchester Institute of Science and Technology. In this case the theorem was uncovered through exploration with the computer algebra package Magma, which is well worth checking out. Often such software lets us investigate mathematical phenomena which would be very difficult to understand otherwise.

Paul Loya
The underlying theme of my research is the investigation of topological, geometric, and spectral invariants of (singular) Riemannian manifolds using techniques from partial differential equations. For example, the Euler characteristic of a surface is a topological invariant based its usual definition in terms of a triangulation of the surface. However, it may also be considered geometric in view of the Gauss-Bonnet theorem or spectral in view of the Hodge theorem. I am interested in such relationships on general singular Riemannian manifolds.

Marcin Mazur
My research interests concentrate around areas where number theory and group theory intersect. Topics of particular interest are group rings, group schemes over rings of algebraic integers, Galois module structures and Galois representations.

Tam Nguyen-Phan
My interests lie in Differential Geometry and Geometric Topology. Recently I have been focusing on understanding the topology of noncompact, complete, finite volume, nonpostively curved Riemannian manifolds. For example, I study the question which closed manifolds occur as a cross section of a cusp of such manifolds. I am also interested in constructing new examples of this class of manifolds, and more generally, aspherical manifolds.

Pedro Ontaneda
My general interest is the geometry and topology of aspherical spaces. I have done some work in the study of the relationship between exotic structures and (negative, non-positive) curvature, and its applications to the limitations of PDE methods in geometry. Other interests: geometric group theory, K-theory, mechanics.

Aleksey Polunchenko
Mathematical statistics and specifically the problem of sequential (quickest) change-point detection, currently focusing on the case of composite hypotheses.

Xingye Qiao
Statistical machine learning and data mining, especially classification problems, and high-dimensional inference.

Anton Schick
Uses of large sample theory in statistics, the characterization and construction of efficient estimators and tests for semiparametric and nonparametric models, statistical inference for Markov chains and stochastic processes, estimation and comparison of curves, the behavior of plug-in estimators, optimal inference for bivariate distributions with constraints on the marginal, modelling with incomplete data, and the theory and application of finite and infinite order U-statistics.

Vaidyanathan Sivaraman
My research interests lie in combinatorics, especially graph theory and matroid theory. Particular topics of interest include well-quasi-ordering, graph invariants (particularly chromatic number, Hadwiger number, girth), minor-closed classes of graphs, induced subgraphs, signed graphs, matroids coming from graphs, signed-graphic matroids, and statistical properties of matroids.

Adrian Vasiu
My area of research is Arithmetic Algebraic Geometry, which is the common part of Number Theory, Algebra, and Geometry. I am very much interested in moduli spaces, group schemes, Lie algebras, formal group schemes, representation theory, cohomology theories, Galois theory, and the classification of projective, smooth, connected varieties. My research is focused on:
  1. Shimura varieties of Hodge type (which are moduli spaces of polarized abelian varieties endowed with Hodge cycles),
  2. arithmetic properties of abelian schemes,
  3. classification of $p$-divisible groups,
  4. representations of Lie algebras and reductive group schemes,
  5. crystalline cohomology of large classes of polarized varieties, and
  6. Galois representations associated to abelian varieties.

Xiangjin Xu
  1. Harmonic Analysis on Manifolds: eigenfunction estimates and multiplier problems on Riemannian manifolds, Gibbs' phenomenon and Pinsky's phenomenon for Fourier inversion and eigenfunction expansion.
  2. Nonlinear differential equations: Well-posedness problems for nonlinear hyperbolic differential equations on manifolds; Boundary stabilization, controllability problems for (linear and nonlinear) parabolic and hyperbolic PDE's on manifolds; Periodic solutions, subharmonics and homoclinic orbits

Qiqing Yu
My research interests are mainly in three fields.
  1. Survival analysis. Since 1987, I have been working in this field, in particular on modeling the interval censored data, studying consistency and asymptotic normality of the generalized maximum likelihood estimator (MLE) of survival function or the semi-parametric estimator under linear regression model.
  2. Statistical decision theory. My thesis was on admissibility and minimaxity of the best invariant estimator of a distribution function.
  3. Probability model and computing methods for pattern recognition in the Genome project.

Thomas Zaslavsky
My research is in combinatorics, especially matroids and their connections with combinatorial geometry and graph theory. The main topic of my work is signed, gain, and biased graphs. These are graphs with additional structure that leads to new graphical matroids and other new kinds of graph theory, such as colorings and geometrical representations, of which ordinary graphical matroids, colorings, etc., are special cases. In combinatorial geometry I work on arrangements of hyperplanes and lattice-point counting. Other research interests are in graph theory and in generalizing Sperner's theorem.

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