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In mathematical physics, a wealth of information comes from the exact, non-perturbative, solution of quantum models in one-dimension and classical models in two-dimensions. This subject is an introduction to this beautiful and deep subject. Yang-Baxter equations, Bethe ansatz and matrix product techniques are developed in the context of the critical two-dimensional Ising model, dimers, free fermions, the 6-vertex model, percolation, quantum spin chains and the stochastic asymmetric simple exclusion model. The algebraic setting incorporates the quantum groups, and the Temperley-Lieb and braid-monoid algebras.
Intended learning outcomes
After completing this subject students should:
- have learned how exactly solvable models apply to a variety of problems in applied mathematics and mathematical physics;
- appreciate the role of exact solutions and universality in mathematical physics and be able to use concepts of real and complex analysis to determine asymptotic behaviour;
- be able to compute correlation functions using matrix product techniques or random matrix theory;
- be familiar with the basic mathematical techniques of exactly solvable models including Yang-Baxter equation, Bethe Ansatz, commuting transfer matrices and matrix product states;
- understand the basic concepts of random matrix theory and appreciate their applicability;
- have the ability to pursue further studies in these and related areas.
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team;
- time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 2 December 2019