### 2010 Semester 2

Talks are Tuesday 3pm in room 67-442 unless otherwise noted.
 July 6 Ryan Budney, University of Victoria, BC, CanadaKnots so far Abstract: A knot is a possibly tangled loop of string in 3-dimensional space. A mathematical knot is a formalization of this idea. A typical question you could ask about knots is "can we write an efficient computer algorithm that decides if a knot can be untangled without cutting it?" I will describe how our understanding of such problems has evolved. This story will start with the work of Gauss and Kelvin, and ends with the recent work of Perelman. July 12-14 Combinatorics and Mathematical Physics conference, UQ July 15, 2pmAndrew Rechnitzer, UBC, CanadaGAS sampling of polygons and knot probability ratios Abstract: Polygons in the cubic lattice are simple closed curves in three space and have well-deﬁned knot types. Denote the number of lattice polygons of length n and knot type K by p_n(K). The exact computation of p_n(K) for arbitrary n and knot type K is an extremely difficult numerical problem. Instead of exact computation we have used an approximate enumeration method - the GAS algorithm. This can be thought of as a generalisation of the famous Rosenbluth method for sampling and approximate enumeration of self-avoiding walks.  Using this algorithm we have estimated p_n(K) for a range of different length and knots on three different three-dimensional lattices. The results of these simulations provide strong direct evidence that the entropic exponent of polygon of a fixed knot type K is a simple function of the entropic exponent of unknotted polygons and the number of prime components in K. I will also discuss the relative frequencies of different knot types and show evidence that these ratios are in fact universal. Aug 3Stephan Tillmann, Maths UQVolume of representations Abstract: I describe a volume function defined on the set of representations of the fundamental group of an n-dimensional manifold into the group of isometries of hyperbolic n-space. This function has humour: its definition is somewhat technical and messy, but many of its properties and applications are elegant and easy to state. This talk explores some of its properties and applications and is largely based on joint work with Steve Boyer (UQAM). Sept 7Tim Trudgian, University of Lethbridge, CanadaOn the zeroes of the Riemann zeta-functionAbstract: This talk, which will be accessible to an undergraduate audience will provide an overview to study of the distribution of the zeroes of the zeta-function. That the zeroes should all have real part 1/2 is the Riemann Hypothesis, and the margin for this abstract is so narrow that I cannot write down a proof here. Some reasons for its possible validity will be given, along with examples of the gulfs between what we know and what is conjectured. Sept 14 Abdollah Khodkar, University of West Georgia, USASuper edge-graceful labelings of complete bipartite graphsAbstract: Let [n]* denote the set of integers {-(n-1)/2, ..., (n-1)/2} if n is odd, and  {-(n-1)/2, ..., -1,1, ..., (n-1)/2} if n is even.  A super edge-graceful labeling f of a graph G of order p and size q is a bijection f : E(G) -> [q]*, such that the induced vertex labeling f* given by f*(u) = \sumuv in E(G)}{f(uv)}\$ is a bijection f* : V(G) -> [p]*. A graph is super edge-graceful if it has a super edge-graceful labeling. We show by construction that all complete bipartite graphs are super edge-graceful except for K2,2, K2,3, and K1,n if n is odd.Joint work with Sam Nolen and James Perconti. Sept 27-30AustMS Annual Conference Oct 19Alexander Hulpke, Colorado State University, USAComputing Conjugacy Classes in Matrix GroupsAbstract: Recent progress in the matrix group recognition project has reached the stage at which the resulting composition tree can be used for concrete calculations. I will describe how to extend existing algorithms for the determination of Conjugacy classes to the case of matrix groups, using this setup. This is Joint work with Max Neunhöffer (St Andrews) Oct 26Henry Segerman, University of Melbourne A generalisation of the deformation varietyAbstract: The deformation variety is similar to the representation variety in that it describes (generally incomplete) hyperbolic structures on 3-manifolds with torus boundary components. However, the deformation variety depends crucially on a triangulation of the manifold: there may be entire components of the representation variety which can be obtained from the deformation variety with one triangulation but not another, and it is unclear how to choose a "good" triangulation that avoids these problems. I will describe the "extended deformation variety", which deals with many situations that the deformation variety cannot. In particular, given a manifold which admits some ideal triangulation we can construct a triangulation such that we can recover any irreducible representation (with some trivial exceptions) from the associated extended deformation variety.