Cluster algebras are commutative rings with a combinatorial structure encoding an algorithmic construction of a generating set from a seed, which consists of a distinguished set of elements, called a cluster, and a mutation rule, encoded in an integer matrix.
Their algorithmic nature is a powerful tool, but it is also what renders defining a structure preserving map between cluster algebras a challenge. This project is concerned with expanding on new advances in a categorical approach to cluster algebras and combinatorially related objects. It entails a selection of concrete computational examples, parallel to the potential to apply and develop abstract categorical machinery to the study of cluster algebras and their friends.
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Project description: Symmetries in geometry and mathematical physics are often encoded as group actions on parameter spaces. Geometric Invariant Theory of Mumford is a machinery to construct quotients of varieties by reductive group actions. Recently developed extension of GIT to non-reductive symmetry groups provides new powerful machinery to attack long-standing open questions in complex geometry and enumerative geometry. A high point is the proof of two long-standing conjectures in complex geometry: the Green-Griffiths-Lang and Kobayashi hyperbolicity conjectures for generic projective hypersurfaces with polynomial degree.
Prospective PhD topics include:
Administrative information:
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We seek a PhD student interested in building further on recent advances in the analytic theory of automorphic forms and L-functions, a central part of modern number theory. The project would likely concern problems of estimation (moment asymptotics for families of L-functions, subconvex bounds, ...). The field is characterized by problems rather than methods, but recent progress has involved techniques coming from representation theory and microlocal analysis, among other sources, and the candidate should have some interest in learning such techniques.
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