Research Interests
Georgia Benkart.
Georgia Benkart's work involves groups such as the group
of invertible complex matrices or the group of matrices whose
determinant is 1. Each such group is what is called a
Lie group, and it has a Lie algebra associated with it.
The Lie algebra for the group of matrices of determinant 1
is just space of all matrices whose trace is 0, together with
the product xy-yx.
Lie algebras have interesting actions on vector spaces
and often a beautiful combinatorics connected with
them. They arise in many different contexts in physics.
For example, the Lie algebra of 3 x 3 matrices of trace 0
acts by multiplication on a 3-dimensional space. Physicists
identify the basis elements of that space with 3 quarks.
The way quarks behave can be described using some facts about
modules for Lie algebras.
Every Lie algebra has attached to it an infinite-dimensional
associative algebra, and the two have the same modules.
These associative algebras are examples of noncommutative
domains - so they are of interest to ring theorists.
Some of my latest work studies associative algebras that
are like these algebras - they arose in a very combinatorial
way by considering the operators up and down on a set with
a partial order. Another part of my current research
is devoted to studying the structure of certain
(possibly infinite-dimensional) Lie algebras.
She has recently completed a long-time project
which solves a missing piece in the classification of the finite-dimensional
simple Lie algebras of prime characteristic.
Lev Borisov's research is in the general area of algebraic
geometry. His thesis was on the quotients of Siegel upper
half space by subgroups of Sp(4,Z), but most of the
later research is connected to the mirror symmetry,
and more generally conformal field theory. Lev Borisov is
also interested in the theory of modular forms, in particular
the geometry of the modular curves X_1(n).
Lev Borisov has recently worked on elliptic genera, vertex algebras,
toric varieties and their hypersurfaces, modular forms. He has
also engaged in miscellaneous projects related to stacks (aka
orbifolds) and the flag variety of G2. Lev Borisov is currently
interested in derived categories. He is currently accepting
new students who are interested in algebraic geometry or
number theory.
Richard A. Brualdi's research interests are in combinatorics and
matrix theory, especially the interplay between the two. He also has
worked on problems in graph theory, coding theory, partially ordered
sets, and matroid theory. His interests in matrix theory are enhanced
and influenced by by his co-editorship of the journal "Linear
Algebar and its Applications." Teaching currently a course in
cryptography, he has also become interested in cryptography.
Combinatorial matrix theory is concerned with the effect of
combinatorial structure on linear-algebraic properties of matrices,
the investigation of combinatorial parameters on a combinatorially
defined class of matrices, and the use of matrix theory in the
solution of combinatorial problems. In many combinatorial problems,
the only known techniques come from linear algebra.
Andrei Caldararu's research is concentrated in Algebraic Geometry, with influences from String
Theory and Homological Algebra. I am mostly interested in the study of derived
categories, with applications in Algebraic Geometry. I have devoted most of my
time to two special areas: twisted sheaves and their geometric applications
(moduli problems, etc.), and Hochschild structures (homology, cohomology) of
algebraic varieties and
orbifolds.
Jordan Ellenberg's research is in arithmetic algebraic geometry: his
specific interests include rational points on varieties over number fields,
enumeration of number fields and other arithmetic objects, Galois
representations attached to varieties and their etale fundamental groups,
non-abelian Iwasawa theory, automorphic forms, Hilbert-Blumenthal abelian
varieties, Q-curves, geometry of moduli spaces, curves of low genus, Serre's
conjecture, the ABC conjecture, and Diophantine problems related to all of the
above.
I. Martin Isaacs.
I. Martin Isaacs specializes in finite group
theory and related topics. His principal interest
is in the character theory of finite groups, and
especially of those groups that have lots of normal
subgroups.
A character of a group is a certain type of function
from the group into the complex numbers, Historically,
characters have been used to prove theorems about
groups. (The first major application, for example, was
W. Burnside's proof that the order of a simple group
cannot have just two prime divisors.) Isaacs is
primarily interested in studying the subtle interplay
between the characters of a group and the subgroup
structure of the group.
Arnold Johnson.
Arnold Johnson
specializes in quadratic forms and
classical groups.
Let V denote a finite dimensional vector space over a
field F. Given a suitable inner product on V, one
can define a mapping from V to F called a classical
quadratic form. The space V, together with a classical
quadratic form on V, is called a quadratic space over
F. For some fields F, the quadratic spaces over F have
been classified. Some examples are the finite fields, the
algebraically closed fields, and the algebraic number fields.
Some more examples are the local fields, which arise in
number theory.
Associated with a quadratic form on V is a certain classical
group, called the orthogonal group and denoted O(V). This
group consists of all invertible linear transformations
on V which preserve the quadratic form. There are
some more classical groups which appear as subgroups
of O(V); an example is the commutator subgroup. There are
a number of open questions concerning the structure of
these classical groups. For instance, we would like to
understand their automorphisms.
A quadratic form on V yields a certain geometric structure.
Johnson is interested in the interplay between the quadratic
form, the associated geometric structure, and the associated
classical groups.
Ken Ono's research interest is in number theory. He has
worked actively on questions related to Diophantine equations,
partitions, L-functions, elliptic curves and modular forms.
These objects are the main players in Wiles' proof of Fermat's
Last Theorem, and they are the subject of intense investigation.
Recently, Ono has been thinking about deceptively simple questions
such as:
1) When does the typical elliptic curve have a rational point?
2) Where do values of L-functions exist in nature?
3) What are ranks and cranks about?
4) What objects are modular?
Donald S. Passman works in group theory, ring theory
and the interplay between the two subjects. Specifically,
he is interested in group algebras, which are rings
constructed from groups. Group algebras have important
applications to both finite and infinite group theory.
Furthermore, certain generalizations called skew group
rings and crossed products have important ring-theoretic
consequences. Passman is also interested in related
rings such as Hopf algebras and enveloping algebras
of Lie algebras.
In the case of group algebras of infinite groups, one
knows, for example, when they have nilpotent ideals,
when they are prime rings, when they are Artinian, and
when they satisfy a polynomial identity. On the other
hand, one does not know (precisely) when they have zero
divisors, when they are Noetherian, or when they are
Jacobson semisimple. Thus, we understand a good deal
about group algebras, but there is much more to learn.
Arun Ram's research centers around the connections
between combinatorics, symmetry groups, geometry,
and harmonic analysis. The main goal is
to understand how symmetry groups can act on vector spaces
and whether these actions can be understood combinatorially.
Combinatorics is the art of counting, and some of
the counting problems that arise in this type of work
are questions like:
(a) How many different ways can a symmetry group act
on a vector space?
(b) What is the size of the vector space that the group is
acting on?
(c) How many pieces can you divide the vector space into and
still manage to get an action of the group on each piece?
Almost by definition, symmetry groups arise from geometry,
since, after all, they are defined as the groups of transformations
that preserve some geometrical symmetry. Thus, understanding
this geometry is an inherent part of understanding how these
groups act on vector spaces. Sometimes the geometry that enters
into play is very discrete (like the geometry of cubes and
tetrahedra) and sometimes it is very continuous
(like the geometry of rotating spheres).
It all comes together in the kind of work that we are
trying to do. One day we might be working with
cohomology and sheaf theory, the next day with lots of integrals,
and the third day by counting the number of graphs that satisfy
certain conditions--All with the goal of understanding symmetry
groups.
Paul Terwilliger does combinatorics, algebra and special
functions. However, for the past several years his heart
has been stolen by a certain algebraic object which he
calls a "Leonard pair". The inspiration comes from
combinatorics and special functions. The idea is so simple
it can easily be described right here:
Let K denote a field, and let V denote a vector space over
K with finite positive dimension. Consider a pair of linear
transformations A: V -> V and B: V -> V that satisfy (i), (ii)
below:
(i) there exists a basis for V with respect to which the matrix
representing A is diagonal and the matrix representing B
is irreducible tridiagonal,
(ii) there exists a basis for V with respect to which
the matrix representing B is diagonal and the matrix
representing A is irreducible tridiagonal.
(A tridiagonal matrix is said to be irreducible whenever the
entries immediately above and below the main diagonal are nonzero).
We call such a pair A,B a "Leonard pair". (The name comes
from a connection to a theorem of Leonard involving
the q-Racah polynomials).
Leonard pairs are interesting on several levels.
First of all, they are elegant linear algebraic objects
and there is a very nice classification. Secondly,
Leonard pairs are associated with a family of orthogonal
polynomials, consisting of the q-Racah polynomials
and their relatives. The q-Racah polynomials describe the
Racah coefficients for representations of the quantum
algebra U_q(sl_2), and also play a role in the quantum
theory of angular momentum. Roughly speaking, it is
appropriate to think of Leonard pairs as "q-Racah polynomials
in disguise". Thirdly, Leonard pairs all "come from"
irreducible representations of a certain algebra, called
the Tridiagonal algebra. This algebra is a kind of
generalization of U_q(sl_2). Fourthly, Leonard pairs
arise naturally in combinatorics, in the context
of distance-regular graphs. A distance-regular graph is
a finite, undirected, connected graph that is "highly regular"
in a certain way. These distance-regular graphs support
representations of the Tridiagonal algebra, and these
representations give Leonard pairs.
Algebra:
I am interested in group generalizations: What do you get if you relax the defining
properties for a group? (If you don't know what a group is, look here.)
There are many different sets of axioms people use to define groups, and you have to be
a bit careful as to how you choose when you decide to make changes. E.g., a loop is
sometimes described as "a group without associativity". But if you start with
group axioms which do not specifically state that the identity works on both sides,
relying on associativity to imply that, then dropping associativity requires you to add
the requirement for the identity to be two-sided.
It is probably better to start from the other end and build up. A quasigroup is an
algebraic object consisting of a set and a binary operation which meets a very simple
requirement: You want to be able to "divide", and the usual way to say this is
that if x denotes the operation, the combination a x b = c has a unique solution for any
one of a, b, and c if the other two are specified in the set. A loop is a quasigroup with
an additional requirement, that it have a (two-sided) identity element.
One of the things which interests me is the interplay between the algebraic description
of an object and its combinatorial properties. I am usually interested in the case when
the set is finite, which simplifies some of these descriptions. A quasigroup, for example
is just the case where the operation table is a Latin Square, i.e. has each entry
appearing exactly once in each row and in each column of the table. Some surprising
results come from combining these views.
In algebra we usually study properties which are preserved by isomorphisms. I am
interested in a generalization not only of the objects but also this characterizing
mapping. An isomorphism is a function which "takes" one algebraic object to
another which, among other things, satisfies that f(a) * f(b) = f(a x b) where f is the
function and * and x are the operations on the two objects. An interesting variation is to
allow different functions in those three places, and the resulting triple of functions is
called an isotopism. Viewed in terms of the picture suggested above, the one that gave a
relation between a quasigroup and a Latin Square, an isomorphism requires a permutation
which works simultaneously on the rows, the columns, and the entries in the table, while
an isotopism allows three different permutations.
Mathematics Education:
I am also interested in mathematics education, both in the immediately practical sense
of "how can we teach our classes better" and in more basic work on how people
learn mathematics and what forces shape their mathematical education.
One of my primary interests at the present is the extent to which our cultural context
affects how students succeed in mathematics. Data from international studies show that US
kids at grades K-12 do not do very well in comparison with students from some other
countries. (You can see some results at http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2001028.)
It is easy to blame schools and teachers. But this is also a country where many (highly
educated in some cases) people are actually proud that they can't do math, where a
"nerd" is something kids don't want to be, and where toys and the media
broadcast an anti-math, anti-science, anti-thinking point of view far more than in many of
the countries that did well on those tests. Can we expect K-12 students to do well at
something they are constantly being told they should avoid if they are to be socially
acceptable? If that is not possible, what can we do to change society's image and message?
If it is possible, how? What could/should be done in teaching to give the societal message
less impact? (It is interesting to note that US student's abilities in math and science
fall away as they progress from 4th to 8th to 12th grade, exactly as they become more
aware of and sensitive to social pressures!)