I am available to supervise Master's and PhD theses. I can suggest the following research topics, but I am also eager to hear your suggestions.
Deep learning, quivers and deformed W-algebras
Free field realizations provide a way to construct complicated algebras from simpler oscillator-like operators. Recent work shows that these constructions can be encoded by decorated quivers, with transformations of the graph producing new realizations. As the number of possible transformations grows rapidly, it becomes difficult to explore them all by hand. The student will generate examples in low rank and train a graph neural network to recognize quivers that are likely to define valid free field realizations. Its predictions will then be checked using exact algebraic relations. The goal is not to replace mathematical proofs, but to identify patterns and suggest new constructions.
Suggested background: linear algebra, Python and introductory abstract algebra; previous experience with deep learning would be useful.
Special functions from coloured partitions
Affine Laumon functions are special functions constructed by summing over integer partitions, which can be represented as arrangements of boxes. They appear both in supersymmetric gauge theories and as wave functions of exactly solvable quantum systems. The student will compute the first terms of the simplest two-colour examples, study how they simplify for special values of their parameters, and compare them with familiar families of symmetric polynomials. Symbolic calculations can then be used to test predicted symmetries and equations.
Suggested background: linear algebra and basic complex analysis; some combinatorics or programming would be useful.
Polynomial solutions of deformed integrable equations
The KP and Toda hierarchies are families of nonlinear equations possessing many exact solutions. Their polynomial solutions are described by symmetric functions called Schur functions. Recent calculations suggest that generalized Macdonald functions play a similar role for deformed versions of these equations. The student will construct low-degree examples, verify the first equations of the deformed hierarchy, and study parameter limits in which known symmetric functions are recovered.
Suggested background: linear algebra, calculus and differential equations; symbolic programming would be useful.
Small-system models of the non-Abelian quantum Hall effect
In the fractional quantum Hall effect, interacting electrons form collective states with unusual excitations. A matrix model for certain non-Abelian states can be rewritten as a system of interacting particles with internal degrees of freedom. The student will study this Hamiltonian for a small number of particles, calculate its spectrum, and compare its eigenstates with the predicted wave functions. If time permits, the model can be extended by adding a simple excitation sector described by a two-node quiver.
Suggested background: undergraduate quantum mechanics, linear algebra.