Positivity in cluster algebras and their generalizations & Tilted Richardson varieties

Time: 3:00 PM – 4:00 PM

Speaker: Amanda Burcroff (Harvard)

Title: Positivity in cluster algebras and their generalizations

Abstract:

This dissertation connects several perspectives on positivity in cluster algebras and scattering diagrams, which are powerful combinatorial tools deeply related to geometry, representation theory, and physics. We prove the Laurent positivity conjecture for generalized cluster algebras via a manifestly-positive combinatorial formula for rank 2 scattering diagrams.  We also prove bijective results relating scattering diagrams and Dyck path combinatorics for quantum cluster algebras. Portions of this work are joint with Kyungyong Lee and Lang Mou.

Time: 4:15 PM – 5:15 PM

Speaker: Jiyang Gao (Harvard)

Title: Tilted Richardson varieties

Abstract:

We introduce and develop the theory of tilted Richardson varieties, a new family of subvarieties of the flag variety that generalize classical Richardson varieties. We establish their fundamental geometric properties, including irreducibility, explicit dimension formulas, and a stratification indexed by tilted Bruhat intervals. We construct a tilted analog of the Deodhar decomposition, leading to a combinatorial formula for tilted Kazhdan–Lusztig R-polynomials. We also develop a theory of total positivity for tilted Richardson varieties, showing that their totally nonnegative parts form a CW complex, thus addressing a question of Björner. Finally, we relate tilted Richardson varieties to quantum Schubert calculus, proving they coincide with minimal-degree two-point curve neighborhoods, and use this to compute their cohomology classes and derive new relations among Gromov–Witten invariants. This dissertation is based on joint work with Shiliang Gao and Yibo Gao.

For information about the Richard P. Stanley Seminar in Combinatorics, visit https://math.mit.edu/combin/