School of Mathematics and Statistics Vacation Scholarships

Mathematical Physics & Statistical Mechanics

Random juggling
(posted for 2017/2018)

If we take a snapshot of a juggler juggling some balls, we can record how far into the future each ball is due to land. This information can be encoded as a binary string known as a ``juggling state''. For a fixed number of balls, we can draw a graph of all juggling states, and the throws that connect them. A walk on this graph then corresponds to a juggling pattern.

For this project, we wish to investigate random walks on these juggling state graphs, which will give us random juggling patterns. We intend to model lazy jugglers and energetic jugglers by introducing a temperature-like parameter to the edge probability distribution, which will allow us to adjust the energy required for various patterns. The project will involve writing some code to calculate the statistics of the juggling states.

Anthony Mays and Thomas Wong

New models of lattice polygons and polyominoes
(posted for 2017/2018 - also listed in Discrete Mathematics & Algebraic Combinatorics)

Self-avoiding walks were originally conceived in the 1940s as a model of long polymers like DNA, but have since been the subject of many results in combinatorics, mathematical physics and probability theory, including some exciting developments in just the last few years. One useful tool is the concept of irreducible self-avoiding walks, which act a bit like the primes in number theory, and are connected to renewal processes in probability theory.

The goal of this project is to study the related concepts of irreducible polygons and polyominoes. This may involve looking at the corresponding renewal processes, doing some numerical series analysis, or using analytic combinatorics to search for solvable models.

Contact Nicholas Beaton

Self-interacting polymers in confinement
(posted for 2017/2018 - also listed in Discrete Mathematics & Algebraic Combinatorics)

Self-avoiding walks on a lattice are a classical model of long polymer chains, and when equipped with a nearest-neighbour interaction energy they form a simple yet rich model of collapsing polymers. However, very little has been proven rigorously about the model in two or more dimensions.

The goal of this project is to study a simplified version of the model, namely interacting walks and/or loops in a two-dimensional strip or three-dimensional tube. These can be analysed using a transfer-matrix approach. One question is the zero-temperature limit in both the positive and negative energy regimes — that is, when very few or very many interactions are preferred. Another is the possibility of using self-interacting walks as a (highly simplified) model of strand-exchange operations in polymers like DNA, with potential applications to Monte Carlo methods for dense polymers.

Contact Nicholas Beaton

Time-reversal in quantum theory and group representations beyond unitary operators
(posted for 2016-2017)

According to a famous theorem by E. Wigner (1931), symmetries in quantum mechanics need to be implemented by unitary or anti-unitary operators acting on Hilbert space. While known for a long time and responsible for some of the astonishing properties of certain physical models, the implications for some more modern areas of mathematics and mathematical physics seemingly have not been analysed so far.

In this project, the summer vacation scholar will first of all reproduce the statement and the proof of Wigner's Theorem before trying to understand its consequences in various physical situations and making an attempt to incorporate the newly gained freedom in order to generalise established higher algebraic structures such as Hopf algebras and quantum groups.

Contact: Thomas Quella

Topological invariants in quantum systems
(posted for 2016-2017)

The physical properties of a quantum system generally depend on parameters which determine the strength of various interactions, e.g. the coupling to a magnetic field. Upon variation of these parameters the system exhibits different physical phases with qualitatively different features. Some of these phases can be distinguished by a discrete invariant that takes one value in one phase and another one in a second. This observation provides a link to the mathematical field of topology which studies the properties of geometric objects, such as knots, up to continuous deformations. In view of this connection, one frequently speaks about topological phases of matter. There are various prominent examples which have only been discovered in the last couple of years - first theoretically, then also experimentally.

Building on the example of Kitaev's so-called Majorana chain, a simple free fermion model of a 1D superconductor, the Vacation Scholar will develop some intuition about the associated topological invariant which, essentially, counts the number of Majorana edge modes. She or he will then apply these insights to a closely related system of so-called parafermions and try to derive a topological invariant for these. While the project has a strong analytical/mathematical component, there will also be the possibility to analyse different parafermion systems using computer algebra in case of interest.

Contact: Thomas Quella

Number systems
(Also listed in Discrete Mathematics & Algebraic Combinatorics and Algebra, Number Theory & Representations, posted for 2016-2017)

What distinguishes the real and complex numbers? Beginning with this question, the project aims to develop properties of the octonion number system.

Contact: Peter Forrester

Partition function zeros and the Ising model in a field
(Also listed in Complex Systems posted for 2016-2017)

The Ising model is one of the fundamental models of statistical physics. The sites of a lattice are occupied by spins pointing either up or down. Spins on nearest-neighbor sites interact so as to favour alignment and in addition an external magnetic field may by applied which favours say up-spins. Much is yet unknown about this model in a magnetic field. The aim is to study the properties of this models by studying the distribution of the partition function zeros. How does the distribution of zeros change with field, how does the zeros move and interact in the complex plane and is there some generic (perhaps universal) aspects to the dynamics of this movement?

Contact: Iwan Jensen

Plane partitions, generating series and free fermions
(Also listed in Discrete Mathematics & Algebraic Combinatorics, posted for 2016-2017)

Plane partitions are important combinatorial objects, with a rich theory that has influenced many different disciplines of contemporary mathematics and physics. Plane partitions generalize the concept of integer partitions to two dimensions. Given an integer \(n \geq 1\), a plane partition \(\pi\) of weight n is an array of non-negative integers \(\pi(i,j)\) such that for all \(i,j \geq 1\),

\pi(i,j) \geq \pi(i+1,j),
\pi(i,j) \geq \pi(i,j+1),

It is often convenient to think of plane partitions as columns of cubes of height \(\pi(i,j)\) stacked over each coordinate square \((i,j)\), giving rise to a three dimensional representation.

Calculating the number of plane partitions of a given weight \(n\) is a classical enumerative problem that was solved by Percy MacMahon in 1912. The solution of this problem is phrased in terms of a generating series, a standard device in discrete mathematics which acts as a 'clothes-line' for the numbers that interest us. Today many different generating functions of plane partitions exist, corresponding to the various ways of refining their counting, for example by allowing different coloured cubes (as in the figure above).

The aim of this project is to calculate the generating series of some classes of plane partitions which have not yet been considered in the literature, using a powerful framework from mathematical physics, the fermionic Fock space. This technique has already led to great success in studying the statistics of plane partitions, as seen most notably in the work of Fields Medallist Andrei Okounkov and collaborators.

Contact: Michael Wheeler

Benchmarking a Novel Electromagnetic Scattering Method
(Also listed in Apllied Mathematics posted for 2015-2016)

Our group has developed a novel, robust and efficient method of studying electromagnetic scattering phenomenon. There is an opportunity for a student with strong background and interest in differential equations, vector calculus, special functions and methods in mathematical physics as well a knowledge of electrodynamics to help implement the efficient evaluation of analytical benchmark results. Experience with using Mathematica or Matlab or interest in learning to use such computational tools will be an advantage.

This is a well structured project with the opportunity to work in a team in Mathematics and Engineering to help validate computation results. The work program can extend to undertaking more complex calculations using boundary integral methods.

Interested candidates are encouraged and welcome to seek further information prior to lodging an application. See also

Contact: Derek Chan

Integrable systems, supersymmetric gauge theories, topological strings, and related topics
(re-posted for 2015-2016)

My research interest is in integrable systems (systems of differential equations that can be solved
exactly), and their applications to supersymmetric gauge theories (quantum field theories with
equal numbers of integral-spin and half-integral spin particles), topological string theories (very
simple, but mathematically important versions of the full-fledged string theory), etc.

I am happy to supervise projects that will take the form of reading introductory material on the
above topics.

Contact: Omar Foda

Exact solutions for polygon and walk models
(Also listed in Discrete Mathematics & Algebraic Combinatorics posted for 2014-2015)

Lattice paths and polygons are studied extensively in combinatorics and statistical physics where they serve as simple toy models for polymers and vesicles. Mostly these have been studied on the square lattice. One project is to find exact solutions to such models on other two-dimensional, say Archimedean, lattices. In another project we aim to extend the exact solution for punctured staircase polygons to more general cases, see I Jensen and A Rechnitzer, Exact perimeter generating function for a model of punctured staircase polygons, J. Phys. A: Math. Theor. 41 215002 (2008).

Contact: Iwan Jensen

Densities and Moments of the \(\beta \) Ensembles in Random Matrix Theory
(posted for 2014-2015, related report)

Random matrix theory has provided theoretical paradigms in many applications including the distribution of mutation fitness effects across species, principal component analysis, wireless communication and antenna networks, queuing models and quantum transport in mesoscopic systems just to name a few examples. However some of the most exciting developments in this area are happening in the purely mathematical arena.

There are a number of tools available to study for example the density of eigenvalues of a random matrix for general \( \beta \) (the inverse temperature) in various ensembles each with their own advantages and drawbacks. One of these is the loop equation formalism which is well suited to the task of computing the large rank or \( N \) expansion, and has been used to compute the topological expansions of certain matrix integrals.

We have recently extended the loop equation formalism to the circular ensembles, where the objects computed are generalisations of matrix integrals over \( U(N) \) with Haar measure. The project will involve using some novel tools developed within our group to evaluate these and study their properties.

Contact: N.S. Witte and P.J. Forrester

For more information on this research group see:
Mathematical Physics & Statistical Mechanics

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