Andrew Rechnitzer


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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2021)
Enumeration Problems in Directed Walk Models (2015)

Self-avoiding walks appear ubiquitously in the study of linear polymers as it naturally captures their volume exclusion property. However, self-avoiding walks are very difficult to analyse with few rigourous results available. In 2008, Alvarez et al. determined numerical results for the forces induced by a self-avoiding walk in an interactive slit. These results resembled the exact results for a directed model in the same setting by Brak et al., suggesting the physical consistency of directed walks as polymer models. In the directed walk model, three phases were identified in the infinite slit limit as well as the regions of attractive and repulsive forces induced by the polymer on the walls. Via the kernel method, we extend the model to include two directed walks as a way to find exact enumerative results for studying the behaviour of ring polymers near an interactive wall, or walls. We first consider a ring polymer near an interactive surface via two friendly walks that begin and end together along a single wall. We find an exact solution and provide a full analysis of the phase diagram, which admits three phase transitions. The model is extended to include a second wall so that two friendly walks are confined in an interactive slit. We find and analyse the exact solution of two friendly walks tethered to different walls where single interactions are permitted. That is, each walk interacts with the wall it is tethered to. This model exhibits repulsive force only in the parameter space. While these results differ from the single polymer models, they are consistent with Alvarez et al.Finally, we consider the model with double interactions, where each walk interacts with both walls. We are unable to find exact solutions via the kernel method. Instead, we use transfer matrices to obtain numerical results to identify regions of attractive and repulsive forces. The results we obtain are qualitatively similar to those presented in Alvarez et al. Furthermore, we provide evidence that the zero force curve does not satisfy any simple polynomial equation.

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Master's Student Supervision (2010 - 2020)
Multidimensional Lattice Walk Enumeration through Coefficient Extraction Operators (2016)

In this thesis, we investigate the enumeration of lattice walk models, with or without interactions, in multiple dimensions, through the use of linear operators comprised of coefficient or term extractions. This is done with the goal of furthering our abilities to automate the derivation and solutions of the functional equations for the generating functions for the models. In particular, for a fairly large class of d-dimensional lattice walk models with interactions and arbitrary step sets, the generating function Q satisfies the functional equation (1 - tΓS)Q = q, where Γ is an operator, and S and q are Laurent polynomials. We can automatically expand this equation to obtain an explicit functional equation satisfied by Q. For example, we derive an equation for d-dimensional lattice walks with interactions and small steps living in an orthant. We also use this operator approach to unify and extend the algebraic and obstinate kernel methods through the use of a weighted orbit summation operator and substitutions. Other topics include: a partial classification of two-dimensional models with interactions and small steps on the quarter plane, explicit relations for Q and partially-interacting versions of itself for some models, and an analysis of some of the more abstract properties of the operators involved.

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Enumeration Problems in Baumslag-Solitar Groups (2010)

Geometric group theory refers to the study of finitely generated groups and their properties by exploring the algebraic and topological structure. This thesis will look at various enumeration problems that arises in Baumslag-Solitar groups. Initially, this thesis aims to reproduce and validate some of the work that has been done on the questions of growth, cogrowth and geodesic elements via an enumeration approach. This approach will then be used to explore specific examples of Baumslag-Solitar groups where these questions have not been fully answered.The first part of this thesis will look at the growth of a horocyclic subgroup in Baumslag-Solitar groups. It will then build upon this to develop an algorithm to count the elements of the group in general out to a fixed radius with the intention of further understanding the unresolved cases. The second part of this thesis will use Baumslag-Solitar groups as a basis to develop numerical tests to estimate cogrowth of groups. Since the cogrowth of a group is directly related to the amenability of the group, these numerical tests for cogrowth can be applied to groups such that Thompson's group F, where the question of amenability is still highly debated.

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