SURF Mentoring

Potential projects/topics: The theory of total positivity, introduced by G. Lusztig, has become an essential tool for investigating quantum groups and cluster algebras. Broadly speaking, for certain topological spaces over the complex numbers, the subspace of positive real numbers admits a decomposition that structures it as a regular CW-complex. Remarkably, this subspace is often homeomorphic to a closed ball, a property that leads to significant existence results in quantum groups. Motivated by the mentor's current research, this project extends total positivity—traditionally applied to a single topological space—to the study of a specific family of complex projective lines. These lines exhibit an interesting degeneration upon taking the closure, a phenomenon that arises naturally in the study of local models of Shimura varieties. While the background is deep, the project focuses on a concrete goal: computing the totally positive part of this family in detail to demonstrate that it is homeomorphic to a closed ball. This research combines conceptual theory with concrete calculations, utilizing techniques from advanced linear algebra and algebraic topology. The problem offers natural avenues for follow-up research for interested participants.

Potential skills gained: Good understanding of advanced linear algebra (exterior products, Jordan canonical forms), basics of projective geometry, elementary algebraic topology (CW-complexes), familiarity with the theory of complex algebraic curves

Required qualifications:

• Preferred course completions: A good knowledge linear algebra and calculus I.

Direct mentor: Post-doctorate