
Combinatorics, Algebra, Number Theory Seminar Spring 2006 Archive:
 April 24
 Alexander Burstein: Statistics on permutation tableaux
Abstract: We define permutation tableaux and
describe a bijection from permutation tableaux to permutations and its
properties, then prove that a certain bistatistic on permutation
tableaux has the same distribution as the bistatistic (number of weak
excedances, number of cycles) on permutations that refines both the
Eulerian numbers and signless Stirling numbers of the first kind. We
also give some bijections from certain restricted sets of permutation
tableaux to other (wellknown) combinatorial objects such as lattice
paths and noncrossing partitions.
 April 17
 Key One Chung: Representations of Symmetric Groups
Abstract: Since every finite group is isomorohic to
a subgroup of a symmetric group by Cayley's theorem, the study of the
representation of symmetric groups is likely help the study for the
representation theory of finite groups. For any finite group, the
number of irreducible representations is the same as the number of
conjugacy classes of group elements. But in general, there is no
natural way to match up irreducible representations with conjugacy
classes. In this talk, I will introduce the method for decomposing a
representation of the symmetric group into its irreducible factors
known as Young symmetrizers which has a natural relation with
conjugacy classes.
 April 10
 Gargi Bhattacharyya: The Terwilliger Algebra of a hypercube
Abstract: In this presentation I will focus on the
Terwilliger algebra of a distanceregular graph with focus on the
Terwilliger Algebra of a hypercube. This is a bipartite distance
regular graph. Let X denote the vertex set of the hypercube. We fix a
vertex x and let T=T(x) denote the associated Terwilliger Algebra. A
is the adjacency matrix and we define A* to be the diagonal matrix
A*=A*(x). A and A* satisfy two fundamental relations that enable us to
show that for each irreducible Tmodule there are two orthogonal
bases, called the standard and the dual standard basis. Finally I will
discuss the center of T and show that it is generated by the function
phi which consists of terms involving only A and A*.
 April 3
 Jayce Getz (University of Wisconsin): Hilbert modular
forms, intersection homology, and base change (joint work with Mark
Goresky at IAS)
 March 27
 Fan Chung (UCSD): Random
graphs and Internet graphs (special colloquium)
Abstract:
We will discuss some recent developments on random graphs with given expected degree distributions.Such ramdom graphs can be used to model various very large graphs arising in Internet and telecommunications. In turn, these "massive graphs" shed insights and lead to new directions for random graph theory. For example, it can be shown that the sizes of connected components depend primarily on the average degree and the secondorder average degree under certain mild conditions. Furthermore, the spectra of the adjacency matrices of some random power law graphs obey the power law while the spectra of the Laplacian follow the semicircle law. We will mention a number of related results and problems that a re suggested by various applications of massive graphs.
 March 20
 Bruce C.
Berndt (UIUC): THE FIVE STRANGEST, MOST FASCINATING, MOST
INTERESTING RESULTS IN RAMANUJAN'S LOST NOTEBOOK (IN THE SPEAKER'S MOST
HUMBLE OPINION)
Abstract: Many of Ramanujan's results, especially
from his lost notebook, are so strange and surprising that it would
seem that no one else, either in the present or the future, would have
had the foresight to discover them. Five entries from Ramanujan's lost
notebook have been chosen for presentation and detailed discussion.
Each of them is surprising. All have been proved, except for one (as
of this writing). At the conclusion of the lecture, members of the
audience will be asked to rank on supplied paper ballots their choices
from 1 to 5 as to which are the strangest, most fascinating, and most
interesting.
 March 13
 Spring Break
 March 8, 2:103p.m, Carver 290
 Ye Tian (McGill University): Rational Points on Twisted Fermat
Curves
Abstract: This talk is based on my joint works with
Diaconu and with Zhang. We will discuss rational points on twisted
Fermat curves X^{ p}+Y^{ p}=nZ^{
p}.
 March 6
 LuenChau Li (Penn State University): Compatible Poisson structures
on Poisson algebras and Virasoro symmetries
Abstract: Two Poisson brackets on the same manifold
are said to be compatible if their sum is also a Poisson bracket. In
this talk, our focus is on the algebraic aspects of a general
construction of compatible Poisson structures on Poisson algebras.
More precisely, given a classical rmatrix on a Poisson algebra, I
will show how to construct a natural family of compatible Poisson
structures for the Hamiltonian formulation of Lax equations. In
particular, a family of vector fields satisfying the Virasoro
relations will be shown to play the key role in this investigation.
Examples for which our formalism applies includes the Benny hierarchy,
the dispersionless KP hierarchy etc.
 Feburary 27
 SungYell Song: On Terwilliger Algebras and Modules.
 Feburary 20
 Gargi Bhattacharyya: An Introduction to Terwilliger Algebra
Abstract: I will introduce Terwilliger Algebra and
discuss some examples .
 Feburary 13
 Wensheng
Zhang (Iowa State University): A Random PerturbationBased Scheme
for Highly Secure Pairwise Key Establishment in Sensor Networks
Abstract: Establishing pairwise keys between nodes
in a wireless sensor network is a prerequisite for the nodes to
securely communicate with each other. Since deploying the public key
cryptosystem or maintaining an online key distribution center has
high overhead, many pairwise key establishment schemes are based on
key predistribution. Although these solutions have low overhead, they
are subject to selective node compromises, and cannot deal with
colluding attacks when a large number of nodes are compromised. In
some schemes, two nodes have to rely on other nodes to help establish
the key, which may expose the established keys to the helper nodes. To
address these limitations, we propose a novel random perturbation
based scheme, which guarantees that any two nodes can agree on a
pairwise key without involving other nodes; at the same time, even
after a large number of nodes have been compromised, selectively or
not, the pairwise keys shared by the noncompromised nodes are still
highly secure.
 Feburary 6
 Ling Long : qHypergeometric Series and qIdentities
Abstract: Let
(a;q)_{n}=(1a)(1aq)…(1aq^{n}1)
be the qseries. The basic qhypergeometric series is
defined as
F(a,b;t:q) =å _{n =0}
((a;q)_{n}/(b;q)_{n})
t^{n}. In this talk, we will first discuss some
fundamental properties of basic qhypergeometric series. Then
method of iteration will be introduced and used to obtain several
wellknown qidentities, including the famous Jacobi triple product
identity. Finally, we define general qhypergeometric series
and briefly mention their applications.
 January 30
 Christian Roettger: Primitive prime divisors of Mersenne numbers,
cyclotomic polynomials and uniform distribution
Abstract: This talk is about the same weighted
average of primitive prime divisors of Mersenne numbers as in my last
talk. Based on an idea of Pomerance, the sum is shown to be close to a
sum of values of cyclotomic polynomials, which in turn can be
approximated using uniform distribution. The tools used are entirely
different from my last talk, but again fairly elementary. They allow
to improve the error term to O(T) as opposed to
O(T log T) before.
 January 23
 Chad Brewbaker: Lonesum (0,1)matrices and the polyBernoulli
numbers
Abstract: We will show that the number of
(0,1)matrices with n rows and k columns uniquely reconstructible from
their row and column sums are enumerated by the polyBernoulli numbers
of negative index, B_{n}(k). Two proofs of this
main theorem are presented giving a combinatorial bijection between
two polyBernoulli formula found in the literature. Next, we will show
some connections to Fermat showing that for a positive integer n and
prime number p
B_{n}(p) = 2^{n}
(mod p),
and that for all positive integers {x, y, z, n} greater than
two there exist no solutions to the equation:
B_{x}(n) + B_{y}(n) =
B_{z}(n).
In addition we might discuss directed graphs with sum
reconstructible adjacency matrices, and enumerations of similar
(0,1)matrix sets.
Archive
of other semesters
Back
to the Mathematics Institute
Back to
Main Street

