### Cluster Algebras, Tim 2013-03-03

### Part 1

Review of Lie algebras; weights and roots.

### Part 2

(tentatively) Root systems, etc ... leading up to cluster algebras.

Review of Lie algebras; weights and roots.

(tentatively) Root systems, etc ... leading up to cluster algebras.

An introduction to the $ XXX_{1/2} $ Heisenberg spin chain and magnons. Using the Bethe ansatz to solve for 2-particle states.

- Gleb Arutyunov,
*Student seminar on integrable systems*, http://www.staff.science.uu.nl/~aruty101/StudentSeminar.pdf - L. D. Fadeev,
*How Algebraic Bethe Ansatz works for integrable model*, http://arxiv.org/abs/hep-th/9605187

An introduction to the Bethe ansatz and the Yang-Baxter equation, based on a simple physics example.

- C. N. Yang,
*From the Bethe-Hulthen Hypothesis to the Yang-Baxter equation*, Oskar Klein memorial lecture - The Yang-Baxter equation http://math.ucr.edu/home/baez/braids/node4.html

A glance at Yang-Mills scattering amplitudes and some of the driving philosophy behind recent developments. An introduction to twistors, from a practical viewpoint. Electromagnetism formulated using spinors.

- Twistor Primer http://users.ox.ac.uk/~tweb/00004/
- Witten's lectures at PiTP 2004 http://www.sns.ias.edu/pitp/2004/schedule.html, Witten's paper on Twistor Strings http://arxiv.org/abs/hep-th/0312171

Given the vastness of the topics under consideration, it is imperative to understand what the final purpose of the talk is. We shall start with looking at the familiar elliptic functions and integrals, and see how they lead to elliptic curves. The connection between elliptic curves, modular invariance and modular groups is then studied. The next section will pick up from here, and introduce the theory of modular forms briefly. This will involve the definition of modular forms, and a summary of some of their properties. This is connected with the Eisenstein series, and the discriminant cusp form, and we conclude with an outline of theta functions.

If time permits (or more likely in a second talk) we will connect these up with physics, in particular CFT. Connections would be made between these two areas (modular forms & elliptic curves vs CFT) by examining scenarios such as the free massless scalar field, Weyl and Maxwell field, etc. The importance of modular forms in the Bianchi classification, self-dual Yang-Mills, 3D quantum gravity and statistical physics (the Yang-Baxter equations) will be highlighted.

TBA at the talk

- For a simple and concise introduction to elliptic curves TWF-13

Killing vector fields are generators of symmetry in differential geometry and provide an invariant way to characterize solutions of the Einstein's Field Equations. We will also briefly survey the related results known as the Bianchi classification and Frobenius theorem.

A discussion on the hydrogen atom and harmonic oscillator problems in 3 (space) dimensions and special symmetries which renders these systems completely solvable. Eg: the $\frac{1}{R}$ and ${R^2}$ potentials in central force problems viz. the hydrogen atom and the harmonic oscillator.

An introduction to Lie superalgebras and a discussion on the orthosymplectic $osp(m|n)$ and special linear $sl(m|n)$ cases.

- Intro to Supersymmetry (Cambridge Monographs on Mathematical Physics), Peter Freund
- A sketch of Lie superalgebra theory (Communications in Mathematical Physics 1977), Victor Kac