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 Gspaces
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 3manifolds, but I also met Thompson's
group F and other PLhomeomorphism groups on the way, and had an encounter
with tropical geometry.

Benjamin Brewster

Algebra, group theory. Topics of particular interest:
 Sylow subgroups,how the group acts on them via conjugation,and
how they intersect.
 Solvable groupstheir conjugacy classes of subgroups.
 Subgroup latticesintervals in the lattice and the influence
of permutable subgroups on this lattice.
 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 KacMoody Lie algebras, bosonic and
fermionic creation and annihilation operators, affine and
hyperbolic KacMoody 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
Ktheory 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 nonpositively 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 nonparametric statistics.
Utility theory.

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 wellknown 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 GaussBonnet 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 NguyenPhan

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, nonpositive) curvature, and its applications
to the limitations of PDE methods in geometry. Other interests: geometric
group theory, Ktheory, mechanics.

Aleksey Polunchenko

Mathematical
statistics and specifically the problem of sequential (quickest)
changepoint detection, currently focusing on the case of composite
hypotheses.

Xingye Qiao

Statistical machine learning and data mining, especially
classification problems, and highdimensional 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 plugin 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 Ustatistics.

Vaidyanathan Sivaraman

My research interests lie in combinatorics, especially graph theory and
matroid theory. Particular topics of interest include
wellquasiordering, graph invariants (particularly chromatic number,
Hadwiger number, girth), minorclosed classes of graphs, induced
subgraphs, signed graphs, matroids coming from graphs, signedgraphic
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:
 Shimura varieties of Hodge type (which are
moduli spaces of polarized abelian varieties endowed with Hodge cycles),
 arithmetic properties of abelian schemes,
 classification of
$p$divisible groups,
 representations of Lie algebras and reductive
group schemes,
 crystalline cohomology of large classes of polarized
varieties, and
 Galois representations associated to abelian
varieties.

Xiangjin Xu


Harmonic Analysis on Manifolds: eigenfunction estimates and multiplier
problems on Riemannian manifolds, Gibbs' phenomenon and Pinsky's phenomenon
for Fourier inversion and eigenfunction expansion.

Nonlinear differential equations: Wellposedness 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.
 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
semiparametric estimator under linear regression model.
 Statistical decision theory. My thesis was on admissibility
and minimaxity of the best invariant estimator of a distribution
function.
 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
latticepoint counting. Other research interests are in graph theory and
in generalizing Sperner's theorem.
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