KPPY 74

During biological processes, enzyme (protein) interaction with DNA can change the topology of DNA which results in knotted or linked DNA. The topology of DNA can sometimes be determined by using biological methods. However, it is a difficult andlaborious process which often doesn't work. Thus tangle analysis was introduced tostudy various enzyme(protein) actions mathematically. A tangle is a 3-dimensionalball with strings properly embedded in it. In a tangle model, we assume the proteincomplex as a 3-dimensional ball and the DNA segments bound by protein as stringsembedded inside the ball. When a protein binds to DNA at n-sites, then the DNA-protein complex can be modeled by an n-string tangle. In theearly 1990's, C. Ernst and D. Sumners used 2-string tangle to analyze the conformation of DNAsegments within the Tn3 and Phase lamda proteins based on N. Cozzarelli's experiments. In late 2000's, a 3-string tangle analysis of DNA topology within Mu-protein is introducedby I. Darcy et. al. Their work is motivated by S. Pathaniaet. al.'s difference experiment of Mu-transpososome. Recently, the author and I. Darcy developed 4-string tangle analysis of DNA-protein complexes and this theory is generalized to n-string tangle analysis.

Vector colouring was introduced by Karger, Motwani and Sudan (1998) as a way to approximate chromatic number using semidefinite programming. A vector colouring is a map from the vertices of a graph to unit vectors in some Euclidean space. The goal is to map adjacent vertices to vectors with small inner product. Representations of a graph on its least eigenspace are examples of vector colourings. For distance-regular graphs, these colourings are optimal, and in a sense canonical. We will look at vector colourings of distance-regular graphs, a condition for determining when these colourings are unique, and consequences for the existence of homomorphisms between graphs. In particular, this machinery can be used to give new proofs that the Kneser and q-Kneser graphs are cores.

A sequence is called uniformly distributed mod 1 if the fractional part of the sequence is equidistributed in the unit interval. In this talk we will summarize some classical results in the theory of uniform distribution and present new results of equidistribution of sequences involving primes. Also we will discuss applications of this result to combinatorial number theory.

Dominic Welsh conjectured in 70's that almost all matroids have a simple structure called sparse paving. The Johnson graphs, from coding theory, is a useful tool in analyzing the sparse paving matroids, and a recent result by Pendavingh and van der Pol uses the Johnson graph to estimate the proportion of the sparse paving matroids. We present several theorems relating the independence number of the Johnson graph with the convexity of the Tutte polynomial. We also suggest a possible approach to a weaker version of the Welsh's conjecture, that most of the matroids are highly connected.

We prove that the spectra of the weighted Laplacians of matroid complexes are polynomials in the weights of ground set elements, generalizing the result by Kook, Reiner, and Stanton that the spectra of the combinatorial Laplacians of matroid complexes are integers. This generalization provides a simple proof for the orthogonal decomposition of the chain groups of matroid complexes of which is finer than the eigenspace decomposition for the combinatorial Laplacians. Using the result for the weighted Laplacians, we give a new formula for the weighted high-dimensional tree-numbers of matroid complexes in terms of matroid invariants: the $\alpha$-invariant (the unsigned reduced Euler characteristic) and Crapo's $\beta$-invariant. To derive the formula, we develop a weighted analogue of high-dimensional Temperley's formula which gives a relationship among the determinants of the weighted Laplacians, the weighted tree-numbers, and the weights of ground set elements. Applying the formula, we obtain several interesting examples for weighted high-dimensional tree-numbers. This is a joint work with Woong Kook.