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 H3, 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, λ) difference
set if the list of “differences” xy
--1,x,y 0 D, x ≠ y,
represents each nonidentity element of G exactly λ times.
Let q be a prime power congruent to 3 modulo 4. The set of nonzero
squares of GF(q) is a (q, (q – 1)/2,
(q – 3)/4) difference set in (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 0 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 PG(3, 32h+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: [(# of vertices) - (# of edges) + …]
.
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) → y1
xy of groups are related to braid groups by assigning a mapping
(x, y) → (y, y1xy)
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.