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 (well-known) 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 distance-regular 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 T-module 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 second-order 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 semi-circle law. We will mention a number of related results and problems that a re suggested by various applications of massive graphs.
March 20
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, 2:10-3p.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
Luen-Chau 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 r-matrix 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
Sung-Yell 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 Perturbation-Based 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 on-line 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 non-compromised nodes are still highly secure.
Feburary 6
Ling Long : q-Hypergeometric Series and q-Identities
Abstract: Let (a;q)n=(1-a)(1-aq)…(1-aqn-1) be the q-series. The basic q-hypergeometric series is defined as

F(a,b;t:q) =å n =0 ((a;q)n/(b;q)n) tn.

In this talk, we will first discuss some fundamental properties of basic q-hypergeometric series. Then method of iteration will be introduced and used to obtain several well-known q-identities, including the famous Jacobi triple product identity. Finally, we define general q-hypergeometric 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 poly-Bernoulli 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 poly-Bernoulli numbers of negative index, Bn(-k). Two proofs of this main theorem are presented giving a combinatorial bijection between two poly-Bernoulli formula found in the literature. Next, we will show some connections to Fermat showing that for a positive integer n and prime number p

Bn(-p) = 2n (mod p),

and that for all positive integers {x, y, z, n} greater than two there exist no solutions to the equation: Bx(-n) + By(-n) = Bz(-n).

In addition we might discuss directed graphs with sum reconstructible adjacency matrices, and enumerations of similar (0,1)-matrix sets.

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