10:00 - 10:45
Stuart W. Margolis : Monoids and Their Algebras
Associated to Coxeter Arrangements and Bruhat Order
Abstract
Whereas the representation theory of finite groups has played a central role in
group theory and its applications for more than a century, the same cannot be said
for the representation theory of finite semigroups. While the basic parts of the
representation theory of finite semigroups were developed in the 1950s by Clifford,
Munn and Ponizovksy, there were no ready-made applications of the theory either
internal to semigroup theory or in applications of that theory to other parts of
mathematics and science.
Over the past few years, this situation has changed completely arising from at
least three sources. The first is the theory of monoids of Lie type developed by
Putcha, Renner and others. The second is the theory of quasihereditary and stratified
algebras; finite regular semigroups have quasihereditary algebras whereas an
arbitrary finite semigroup has a stratified algebra. The third is applications to algebraic
combinatorics and Markov chain theory via monoid structures on objects such
as real and complex hyperplane arrangements, ordered matroids, interval greedoids
and analyzing random walks on them via the representation theory of the associated
monoid.
The purpose of this talk is to survey these latter combinatorial applications where
finite semigroups and their linear representations play a central role. We concentrate
on the monoid structures associated to the Coxeter hyperplane arrangement
and Bruhat order associated to a finite Coxeter group.
Time permitting, we give a new non-commutative interpretation of Leray numbers,
by showing that the Leray number of a simplicial complex can be identified
with the global dimension of a related monoid algebra.
This is joint work with Franco Saliola, Département de Mathématiques LaCIM,
Université du Québec à Montréal, and Benjamin Steinberg, Department of Mathematics,
CCNY, CUNY.
Coffee break
11:15 - 12:00
Miriam Cohen : Are we counting or measuring something
Abstract
Let H be a semisimple Hopf algebra over an algebraically closed field k of characteristic
0. We define Hopf algebraic analogues of commutators and their generalizations
and show how they are related to H?, the Hopf algebraic analogue of the
commutator subgroup. We introduce a family of central elements of H', which on
one hand generate H' and on the other hand give rise to a family of functionals on
H. When H = kG, G a finite group, these functionals are counting functions on G.
It is not clear yet to what extent they measure any specific invariant of the Hopf
algebra. However, when H is quasitriangular, they are at least characters on H.
Joint work with Sara Westreich.
12:15 - 13:00
C.S. Aravinda : Twisted doubles and nonpositive curvature
Abstract
In this talk we address the question of whether it is possible to always prescribe
a nonpositively curved metric on a closed manifold which is homeomorphic to a
closed nonpositively curved manifold. In particular, we discuss this question on
certain examples which arise as twisted doubles of finite volume real hyperbolic
manifolds.
Lunch
15:00 - 15:45
Tali Kaufman : Locally testable codes and expanders
Abstract
Expanders and error correcting codes are two celebrated notions, coupled to-
gether. E.g. it is known that expander graphs lead to some of the best error
correcting codes.
In recent years there is a growing interest in codes admitting algorithms that run
in *constant* time. E.g. codes for which it is possible to distinguish in constant
time between codewords and vectors far from the code. Such codes are known as
locally testable codes (LTCs) and they are intimately related to the PCP Theorem.
In this talk I'll show that codes based on random expanders are not LTCs, so
expansion is strongly related to codes but not to LTCs. I'll then show that high
dimensional expansion is the correct notion related to testability. In particular,
high dimensional expansion implies local testability.
Based on Joint work with Alex Lubotzky.
Coffee break
16:15 - 17:00
Jack Sonn : Groups as Galois groups with few ramified primes
Abstract
The classical inverse Galois problem asks if every finite group is
realizable over the field Q of rational numbers.Although this problem is still
open, many groups are known to be realizable over Q, and their realizations raise
further questions.One is the following: every nontrivial extension field of Q has
only finitely many ramified primes, but at least one ramified prime (Minkowski),
so one may ask, given a group G which is known to be realizable over Q, what is
the minimal number of ramified primes in a realization of G over Q?
There is an evident lower bound (the minimal number of conjugacy classes in G whose union
generates G), and ”the minimal ramification problem” is, can this lower bound be
realized? Failing this, can one find a good upper bound for the minimal number of
ramified primes in a realization of G?
This will mainly be a survey talk.
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