Program of SWAG7

Abstracts:

Grothendieck’s question on Brauer groups (Chang-Yeon Chough)

Grothendieck posed a question of whether the natural map from the Brauer group of a scheme to its cohomological one is an isomorphism of abelian groups. It’s not true in general, but we have some positive results from Grothendieck and Gabber (and de Jong), among many others. After a brief review of Brauer groups in algebraic geometry, I’ll talk about some recent progress in the setting of derived and spectral algebraic geometry.

K-stability of log del Pezzo surfaces with small α-invariants (In-Kyun Kim)

In this talk, we estimate beta-invariants and delta-invariant of some singular log del Pezzo surfaces with quotient singularities. As a result we prove their K-stability and the existence of Kähler-Einstein metrics.

Real Lagrangian submanifolds in toric symplectic del Pezzo surfaces (Joontae Kim)

We explore the topology of real Lagrangian submanifolds in toric symplectic del Pezzo surfaces using symmetries of polytopes. In particular, we give the complete classification of connected real Lagrangians in monotone toric symplectic del Pezzo surfaces up to diffeomorphism. This is joint work with Joe Brendel and Jiyeon Moon.

Mini-talks (morning session):

Introduction to Divisorial Stability and Okounkov Body (Sungwook Jang)

Kento Fujita suggested the divisorial stability that is the notion weaker than K-stability. To determine divisorial stability of given Fano variety, we should calculate an invariant about prime divisors on the variety. K.Fujita observes that this invariant is related to the first coordinate of the barycenter of Okounkov body. I will briefly explain this relation.

Polygon spaces in the Euclidean plane and connectedness (Eunjeong Lee)

The polygon spaces are the collections of polygons in Euclidean spaces. In particular, the polygon space is related to a moduli space in algebraic geometry and toric degenerations in the theory of mirror symmetry. The foundations of geometry and topology of the polygon spaces were established by Kapovich and Millson. By applying the theory of rectified simplex, we introduce a new approach to geometry and topology of the polygon spaces. In this talk, we discuss the connectedness of the polygon space of fixed side-lengths. This is a joint work with Jae-Hyouk Lee.

Quantum Teichmüller Space and Bonahon-Wong Quantum Trace Map (Miri Son)

In this talk, I introduce the quantum Teichmüller space and Bonahon-Wong’s quantum trace map, which is a map between two non-commutative algebras, the skein algebra and Chekhov-Fock algebra. For this work, it is necessary to construct square-root quantum Teichmüller space and redefine quantum coordinate change isomorphism relevant to Chekhov-Fock square-root algebra.