Here you can find a list of some possible themes for student projects (bachelor or master) in Mathematics or Mathematical Physics. Contact me if you are interested in working with me on these or similar projects.
Keywords: Integrable spin chains, Lie algebras, moduli spaces of curves of genus zero
Outline: Using the Gaudin integrable models of statistical mechanics as a motivating physical example, we would like to prove that the Gaudin subalgebras are in a one to one correspondence with points in the Knudsen compactification of the moduli space of genus zero curves.
Reference: L. Aguirre, G. Felder, A. P. Veselov. Gaudin subalgebras and stable rational curves. Compos. Math. 147 (2011) 1463-1478. (Mainly Section 2 and Appendix) arXiv/1004.3253
Keywords: WDVV equations, Frobenius manifolds, reflection groups, Hurwitz spaces, integrable hierarchies.
Outline: The first part of this project would be a review of the origin of the Witten-Dijkgraaf-Verlinde-Verlinde associativity equations and their geometric formulation as Frobenius manifolds. In the second part we will address the construction of a specific class of Frobenius manifolds, e.g., orbit spaces of Coxeter groups, Hurwitz spaces, etc. or some other related topic, like bi-Hamiltonian structures of hydrodynamic type.
Reference: B. Dubrovin. Geometry of 2D topological field theories. hep-th/9407018