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Abstracts:

 

Entangled polymers/clusters and pattern theorem

Mahshid Atapour, University of Saskatchewan

Klarner proved in 1967 that the number of n-edge animals in the simple cubic lattice (up to translation) grows exponentially. Let e_n be the number (up to translation) of all n-edge linked clusters in which the connected components (animals) are (topologically) linked. It was a conjecture/open problem that e_n also has a finite exponential growth rate until we found a proof a few years ago. In this talk I will briefly review the methodology used for solving this open problem. I will also talk about pattern theorems in general and their surprising use in other combinatorial questions related to pattern-avoiding permutations. This is a joint work with Neal Madras.  

 

Models of pulled and compressed polymers

Nicholas Beaton, University of Melbourne

I will discuss several models of polymers, located at the boundary between a solvent and an impenetrable surface, under the influence of forces which either pull the polymer away from or compress it against the surface. These include the classic models of self-avoiding walks and Dyck paths. Results that I’ll mention include a new, simple proof of the critical pulling force for self-avoiding walks, and an investigation of the behaviour of Dyck paths under compression.   Collaborators on these works include Tony Guttmann, Stu Whittington and Buks van Rensburg, with additional help from Brendan McKay and Robin Pemantle.  

 

The Thermodynamics and Dynamics of Glasses: Insights From Confined Particle Packings.

Richard Bowles, University of Saskatchewan

Amorphous, glassy materials have been used for thousands of years to make everything from decorative art to functional window coverings. Modern day applications of amorphous materials include the preservation of food, the storage of nuclear materials and the production of high performance optical fibres used in communications and computing. However, despite our considerable ability to manipulate these materials, our understanding of the fundamental chemistry and physics of glasses is limited. This talk will outline the key challenges glasses present to our understanding of the thermodynamics and dynamics of materials. I will then highlight how the study of particle packings in confined geometries can give us insights into these problems.  

 

A Random Walk Proof of the Matrix Tree Theorem and Applications to Markov Chains

Michael Kozdron, University of Regina

The matrix tree theorem, also called Kirchhoff's theorem after the 19th century German physicist Gustav Kirchhoff, relates the number of spanning trees in a graph to the determinant of a matrix derived from the graph. There are a number of proofs of Kirchhoff's theorem known, most of which are combinatorial in nature. In this talk we will present a relatively elementary random walk-based proof of Kirchhoff's theorem which follows from Greg Lawler's proof of David Wilson's 1996 algorithm for generating spanning trees uniformly at random. Since Wilson's algorithm is interesting in its own right and easily understood, we will spend some time discussing his technique for generating uniform spanning trees. As a curious side note, most other algorithms for generating spanning trees do not yield the matrix tree theorem as a consequence; Wilson's algorithm does! Moreover, these same ideas can be applied to other computations related to general Markov chains and processes on a finite state space. Based on joint work with Larissa Richards (Toronto) and Dan Stroock (MIT).  

 

Natural computing and bioinformatics using formal language theory

Ian McQuillan, University of Saskatchewan

 

 

An overview of polymer sampling methods

Thomas Prellberg, Queen Mary University of London

I shall present a brief review of general polymer sampling methods. In particular, I will focus on sampling algorithms that are based on stochastic growth methods, and review modern extensions of these algorithms that capable of uniform sampling across a whole range of temperatures in one simulation.  

 

Knotted Open Chains in Polymer Models

Eric Rawdon, University of St. Thomas

Some proteins recently have been classified as being knotted. However, proteins have free ends and knotting, traditionally, has only been defined formally for closed curves. How should we define the existence of knotting within open chains? Once we settle on a definition, we can see simpler knots within complex knot. We discuss recent results for classifying the knotting within nice knots, random knots, and proteins. This is joint work with Ken Millett, Andrzej Stasiak, and Joanna Sulkowska.  

 

Approximate counting and the link form(al/er)ly known as Hopf

Andrew Rechnitzer , University of British Columbia

 

 

Minimum step number of knots in the simple cubic lattice

Koya Shimokawa, Saitama University

 

 

An Overview of Polymer Knots

De Witt Sumners, Florida State University

This will be an introduction to the subject of knots in polymers, discussing the theory of random knotting and some applications in biology and chemistry.  

 

Tree factorials and hook length formulas for trees: the same story in two different languages

Karen Yeats, Simon Fraser University

The enumerative combinatorics community and the community taking a Hopf algebraic approach to renormalization both, independently, came to the same result regarding weighted generating functions of classes of trees. I will describe these two perspectives and comment on some ways in which the insights of one community can be carried over to the other. Joint work with Bradley Jones.