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Mathematics & Statistics

# Discrete Mathematics (Leader: Prof. Greg McColm)

## Monday, April 23, 2007

 Title The dynamics of icosahedral viruses: what does Viral Tiling Theory teach us? Speaker Anna Taormina Durham University United Kingdom Time 3:00-4:00 p.m. Place PHY 120

Abstract

I shall give a brief review of key aspects of Viral Tiling Theory and exploit the symmetry properties of viruses whose protein shell is invariant under the icosahedral group $$H_3$$, with a view to provide biologists with predictions on the normal modes of vibration of such viruses and assist them, for instance, in their study of mechanisms of genetic material release during viral replication.

## Monday, April 16, 2007

 Title Assembly Graphs Speaker Angela Angeleska Time 3:00-4:00 p.m. Place PHY 120

Abstract

Motivated by problems in gene recombination observed in ciliates, we define assembly graphs as a finite graphs with rigid vertices of degree four and even number of vertices of degree one. We investigate certain properties and applications of assembly graphs, such as: the assembly number, smoothing, Hamiltonian and polygonal paths. Also some results for words associated with assembly graphs are presented.

## Monday, April 9, 2007

 Title Graphs With Few Triangles Speaker Brendan Nagle Time 3:00-4:00 p.m. Place PHY 120

Abstract

In 1978, Ruzsa and Szemerédi proved that if a graph $$G$$ has few triangles, then it is possible to remove relatively few edges from $$G$$ to destroy all of its triangles. This result, dubbed the ‘triangle removal lemma’, quickly implies some deep results from combinatorial number theory and geometry, including Roth's Theorem on arithmetic progressions of length $$3$$. In this talk, we shall consider some of these applications and will discuss the proof of the triangle removal lemma. We attempt to motivate recent work of the speaker and others concerning extending the triangle removal lemma to hypergraphs, and hope to introduce some of the technicalities that arise in such an extension.

## Monday, April 2, 2007

 Title Kauffman-Harary Conjecture for Virtual Knots Speaker Matt Williamson Time 3:00-4:00 p.m. Place PHY 120 Note Defense of Master's Thesis

Abstract

In this talk, I will present some basics of virtual knot theory, leading into a conjecture called the Kauffman-Harary Conjecture. This conjecture was proved to be true for many classical knots, and I present a type of virtual knot diagram where it is true as well. Furthermore, I use a move called a $$k$$-swap throughout the talk and I show that it does not change the involutory quandle.

## Thursday, March 29, 2007

 Title Skew Hadamard difference sets from commutative semifields and symplectic spreads Speaker Qing Xiang University of Delaware Time 2:00-3:00 p.m. Place LIF 261 Note Special Seminar

Abstract

Let $$G$$ be a finite group of order $$v$$ (written multiplicatively). A $$k$$-element subset $$D$$ of $$G$$ is called a $$(v,k,\lambda)$$ difference set if the list of “differences” $$xy^{-1}$$, $$x,y\in D$$, $$x\ne y$$, represents each nonidentity element of $$G$$ exactly $$\lambda$$ times. Let $$q$$ be a prime power congruent to $$3$$ modulo $$4$$. The set of nonzero squares of $$\mathrm{GF}(q)$$ is a $$(q,\frac{q–1}{2},\frac{q–3}{4})$$ difference set in $$(\mathrm{GF}(q),+)$$. This construction dates back to 1933, and it is due to Paley. The difference sets coming from this construction are usually called Paley difference sets.

A difference set $$D$$ in a finite group $$G$$ is called skew Hadamard if $$G$$ is the disjoint union of $$D$$, $$D^{(-1)}$$, and $$\{1\}$$, where $$D^{(-1)}=\{d^{-1}:d\in D\}$$. The Paley difference sets provide a family of examples of skew Hadamard difference sets. For more than 70 years, these were the only known examples in abelian groups. It was conjectured that no further examples in abelian groups can be found. This conjecture was disproved by Ding and Yuan in 2005. Subsequently, we found another construction using certain permutation polynomials from the Ree-Tits slice symplectic spreads in $$\mathrm{PG}(3,3^{2h+1})$$. In this talk, we will discuss these developments and raise several questions about skew Hadamard difference sets.

## Monday, March 26, 2007

 Title Area of the smallest triangle among $$n$$ points in the unit square Speaker Niluk John Time 3:00-4:00 p.m. Place PHY 120

Abstract

The Heilbronn problem is defined as follows.

Given a configuration $$F$$ of $$n$$ points in the unit square $$[0,1]^2$$, let $$T(F)$$ be the area of the smallest triangle. And Heilbronn's problem is about determining $$T(n)=\max T(F)$$, where the maximum is taken over all possible configuration $$F$$ of $$n$$ points in $$[0,1]^2$$. We shall give a review of this problem and the existing upper and lower bounds for $$T(n)$$.

We shall also discuss the average case, where the $$n$$ points are independent and uniformly distributed on $$[0,1]^2$$.

## Monday, February 28, 2007

 Title The chromatic polynomial of graphs as the Euler characteristic Speaker Masahiko Saito Time 3:00-4:00 p.m. Place PHY 120

Abstract

This talk is a brief overview of the work by Helme-Guison and Rong on a categorification of the chromatic polynomial of graphs. The chromatic polynomial counts the number of colorings of vertices such that colors are distinct when two vertices are connected by edges. The categorification, in this case, means that the chromatic polynomial can be regarded as the Euler characteristic, the famous topological formula for polyhedrons: $$[(\text{# of vertices})-(\text{# of edges})+\dotsb]$$.

My motivation of this review is for possible applications to DNA recombinations and other situations.

## Monday, February 12, 2007

 Title An analogue of conjugation of groups for Hopf algebras and representations of braid groups Speaker Masahiko Saito Time 3:00-4:00 p.m. Place PHY 120

Abstract

The conjugation $$(x,y)\to y^{1}xy$$ of groups are related to braid groups by assigning a mapping $$(x,y)\to (y,y^{1}xy)$$ to a braiding.

In this talk, an analog of conjugation is defined for Hopf algebras that are vector spaces with multiplication and some other structures, which I will explain by pictures. Such an analog of conjugation gives braid group representations on vector spaces, that have been of interest for physicists by the name “Yang-Baxter equation”.

The talk is an exposition of facts known to algebraists and mathematical physicists since the 1990’s, from a topological and combinatorial point-of-view of knots and braids. In particular, linear maps are represented by graphs, and matrix equations are proved by pictures.

## Monday, February 5, 2007

 Title Does the Number Seven “Exist”? Speaker Greg McColm Time 3:00-4:00 p.m. Place PHY 120

Abstract

Ever since Pythagoras, the philosophy of mathematics has been a spectrum from Realism — the contention that mathematical objects like The Number Seven actually “exist” — and various anti-realists, who contend that the Number Seven is a social convention. This will be an outline of the debate from Pythagoras to the present, and its implications for the practice of mathematics … and of science in general.

## Monday, January 22, 2007

 Title Rigid Structures Speaker Greg McColm Time 3:00-4:00 p.m. Place PHY 120

Abstract

A class of complex structures constructible from simple building blocks can be described as follows. Start with a diagram giving the articulation possibilities of the various types of blocks. This diagram can be used to devise a semigroup of all possible (relative) positions of different blocks in a complex. If we want to embed the complex in a geometric space (such as Euclidean space), we can do this block by block. By imposing restrictions the (use of) the semigroup, we can capture the notion of the complex being rigid, and of the blocks not being able to occupy the same space. This is joint work with Nataša Jonoska.

## Monday, January 8, 2007

 Title From Potts to Tutte and Back Again… A Graph Theoretic View of Statistical Mechanics Speaker Jo Ellis-Monaghan St. Michaels College Time 3:00-4:00 p.m. Place PHY 120

Abstract

This talk will provide basic principles of the Ising and $$q$$-state Potts model partition functions of statistical mechanics. These models play important roles in the theory of phase transitions and critical phenomena in physics and have applications as widely varied as muscle cells, foam behaviors, and social demographics. The Potts model is constructed on various lattices, and when these lattices are viewed as graphs (i.e., networks of nodes and edges) then, remarkably, the Potts model is also equivalent to one of the most renown graph invariants, the Tutte polynomial. Thus, the talk will also give a general introduction to the Tutte polynomial. The Tutte polynomial is the universal object of its type, in that any invariant that obeys a certain deletion/contraction relation (or, equivalently, a particular state-model formulation) must be an evaluation of it. The talk includes a (very brief) history and some interesting properties of the Potts model partition function and Tutte polynomial, but the emphasis will be on how the Potts model and Tutte polynomial are related and how research into one has informed the theory of the other, and vice-versa. The talk includes computational complexity results and a brief excursion into Monte Carlo simulations.