Department of Mathematical Sciences
Faculty Research Interests
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
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.
Algebra, group theory. Topics of particular interest:
- Sylow subgroups,how the group acts on them via conjugation,and
how they intersect.
- Solvable groups-their conjugacy classes of subgroups.
- Subgroup lattices-intervals in the lattice and the influence
of permutable subgroups on this lattice.
- Characterizing subgroups with embedding properties in direct
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.
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.
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".
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.
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.
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.
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.
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
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
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.
statistics and specifically the problem of sequential (quickest)
change-point detection, currently focusing on the case of composite
Statistical machine learning and data mining, especially
classification problems, and high-dimensional inference.
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.
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.
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
- representations of Lie algebras and reductive
- crystalline cohomology of large classes of polarized
- Galois representations associated to abelian
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: 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
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
semi-parametric estimator under linear regression model.
- Statistical decision theory. My thesis was on admissibility
and minimaxity of the best invariant estimator of a distribution
- Probability model and computing methods for pattern
recognition in the Genome project.
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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matt at math.binghamton.edu