Let G be a reductive algebraic group over ℂ, which is the complexification of a real Lie group K. Consider a holomorphic principal G-bundle together with a K-invariant Lagrangian submanifold in the total space and its quotient Lagrangian submanifold in the base. This setup appears in the context of GIT quotients and symplectic reduction. We develop a method to lift holomorphic disks from the base to the total space. Using the developed method, we explain how the Gross-Hacking-Keel-Kontsevich superpotential of the basic affine space can be obtained via lifted holomorphic disks from the flag variety.

In this talk, we establish correspondences between polygon spaces in Euclidean spaces of dimensions 2, 3, 5 and 9 and the quotient spaces of 2-Stiefel manifolds. By introducing the Hopf map over the normed division algebras 𝔽, we derive the induced SO-action on the Euclidean space ℝ⊕𝔽 ≃ℝ ^(1+dim_{ℝ}𝔽). This alternative perspective enables us to extend the foundational results the foundational results of Hausmann and Knutson regarding complex 2-Grassmannians and spatial polygon spaces.

A coadjoint orbit admits a Hamiltonian maximal torus action with moment map 𝜇: 𝛰(𝜆) → ℝⁿ. We completely characterize the circle subgroup actions of this maximal torus for which freeness and local freeness coincide on every level set of the corresponding moment map. Using the Duistermaat--Heckman theorem, we describe the chamber structure on ℝⁿ such that the reduced spaces at values in the same chamber are diffeomorphic. We also show that, for these circle actions, each chamber consists precisely of values whose level sets admit free actions. This is joint work with Yoosik Kim.

Geometric quantization requires understanding how the metaplectic structure interacts with particular choices of polarization. Depending on the polarization type, the stabilizer of a linear polarization in the metaplectic group may have one, two, or four connected components. In this talk, we will classify Lie group double covers over a possibly disconnected, split base Lie group using group cohomology. We will then construct Hess characters on subgroups of the metaplectic group fixing each polarization type.

Dimer models are well-known combinatorial models that provide a powerful dictionary between symplectic geometric structures on the A-side and algebraic structures on the B-side. In this talk, we focus on their role in closed-string mirror symmetry. In particular, combinatorics of dimer models provide natural Lagrangians in associated punctured Riemann surfaces, and the Jacobi algebra of the dual dimer gives an explicit description of the corresponding Lagrangian Floer complex on the A-side. This allows us to establish equivalences between Hochschild invariants of wrapped Fukaya categories and matrix factorization categories.

This talk presents cylindrical varieties through their connection to birational geometry. We discuss basic examples, polar cylinders on projective varieties, and their relation to additive group actions. Fano varieties, particularly del Pezzo surfaces, serve as guiding examples throughout the talk. The talk serves as an introduction to the subject, with emphasis on natural questions arising from classification problems.

I would like to report my recent observation that the normalized Kahler-Ricci flow on compact hyperbolic Riemann surface stabilizes the first eigenvalue of an initial metric monotonically to that of the constant Gaussian curvature metric. If time permits, I will discuss the possibility of generalizing the result to the case of canonically polarized compact Kahler manifolds.

This talk is an introductory overview of how equivariant vector bundles can be described by combinatorial data. I will start with the complexity-zero case, namely toric varieties, where Klyachko described equivariant vector bundles using compatible filtrations attached to the rays of a fan. I will then explain how these filtrations can also be viewed as piecewise linear maps to a building. In the last part, I will introduce a recent extension to complexity-one (T)-varieties, where divisorial fans and Bruhat–Tits building-valued support maps replace the usual fan and Klyachko filtrations.