## Archive for the ‘2008’ Category

### Young Soo Kwon (권영수), Maps, regular maps and Belyi functions

Saturday, August 23rd, 2008
Maps, regular maps and Belyi functions
Young Soo Kwon (권영수)
Dept. of Mathematics, Yeungnam University, Kyeongsan, Korea.
2008/09/26 Fri, 3PM-4PM 5PM-6PM

A map is a 2-cell embedding of a graph into a closed surface and a regular map is a highly symmetric map like five Platonic solids. A map is not merely a topological object. It is also a sequence of permutations, which provides a relation to group theory, and a ramified covering of the Riemann sphere, which gives a relation to Riemann surface. Furthermore, it can be realized by a complex algebraic curve called Belyi function.

In this talk, classifications of regular maps are considered from several aspect. And, some connections between maps and permutation group, Riemann surface and Belyi functions and their applications are given.

### Thomas Lam, Symmetric functions and variations on tableaux

Friday, August 22nd, 2008
Symmetric functions and variations on tableaux
Thomas Lam
Dept. of Mathematics, Harvard University, Cambridge, USA.
2008/09/09 Tue, 5:30PM-6:30PM
It is well known that the the central objects in the theory of symmetric functions, Schur functions, are generating functions of semistandard Young tableaux. We present a collection of generalizations of semistandard tableaux. The generating functions of these tableaux are symmetric functions occurring in the study of the (co)homology of the affine Grassmannian, and the K-(co)homology of the Grassmanian.

### Heesung Shin (신희성), Counting labelled trees with given indegree sequence

Friday, August 22nd, 2008
Counting labelled trees with given indegree sequence
Heesung Shin (신희성)
Institut Camille Jordan, Université Claude Bernard Lyon 1, France.
2008/08/21 Thu, 4PM-5PM

For a labeled tree on the vertex set [n]:={1,2,…,n}, define the direction of each edge ij as i to j if i<j. The indegree sequence λ=1e12e2 … is then a partition of n-1. Let aλ be the number of trees on [n] with indegree sequence λ. In a recent paper (arXiv:0706.2049v2) Cotterill stumbled across the following remarkable formulas $a_\lambda = \dfrac{(n-1)!^2}{(n-k)! e_1! (1!)^{e_1} e_2! (2!)^{e_2} \ldots}$ where k = Σi ei. In this talk, we first construct a bijection from (unrooted) trees to rooted trees which preserves the indegree sequence. As a consequence, we obtain a bijective proof of the formula. This is a joint work with Jiang Zeng.

### Chung-Kil Hur (허충길), Term Equational Systems and Logics

Friday, August 22nd, 2008
Term Equational Systems and Logics
Chung-Kil Hur (허충길)
Computer Laboratory, University of Cambridge, Cambridge, UK.
2008/08/11 Mon, 4PM-5PM

Equational reasoning is fundamental in automated theorem proving (that is, the proving of mathematical theorems by a computer program), and rewriting is a powerful method for equational reasoning. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.

Using category theory, we have developed a framework for equational reasoning. A Term Equational System (TES) is given by a semantic universe and an abstract notion of syntax; and given this, we automatically derive a sound logical deduction system, called Term Equational Logic (TEL). Furthermore, we provide an algebraic free construction for the system, which may be used to synthesize a sound and complete rewriting system for it.

Existing systems arising in this framework include:

• first-order equational logic and rewriting system;
• combinatory reduction system of Klop;
• binding equational logic and rewriting system of Hamana; and
• nominal equational logics independently developed by Gabbay and Matheijssen, and Clouston and Pitts.

Especially, following the above scenario in our framework, we have newly developed a sound and complete rewriting system for nominal equational logic.

In this talk, rather than going into the technical details, I will focus on explaining basic ideas of category theory and how it can be used in practice.

This is joint work with Marcelo Fiore.

### Ae Ja Yee (이애자), Combinatorics of generalized q-Euler numbers

Friday, August 22nd, 2008
Combinatorics of generalized q-Euler numbers
Ae Ja Yee (이애자)
Dept. of Mathematics, Pennsylvania State University, University Park, USA.
2008/07/03 Thu, 4PM-5PM

The Euler number En counts the number of alternating permutations on the set [n]. It is well known that its exponential generating function equals Tan z + Sec z. For this reason, E2n and E2n+1 are called secant numbers and tangent numbers, respectively. Certain polynomials arising in series expansions for zeros of generalized Rogers-Ramanujan functions provide a q-analog of the tangent numbers, which is part of a wider class of polynomials with similar combinatorial interpretations. In this talk, we will discuss various q-Euler numbers. This is a joint work with Tim Huber from Iowa State University.