Title:   Khovanov homology and exotic planes Speaker:   Yikai Teng, Rutgers University-Newark Abstract:   Since the 1980s, mathematicians have discovered uncountably many "exotic" embeddings of R^2 in R^4, i.e., embeddings that are topologically but not smoothly isotopic to the standard xy-plane. However, until today, there have been no direct, computable invariants that could detect such exotic behavior (with prior results relying on indirect arguments). In this talk, we define the end Khovanov homology, which is the first known combinatorial invariant of properly embedded surfaces in R^4 up to ambient diffeomorphism. Moreover, we apply this invariant to detect new exotic planes, including the first known example of an exotic plane that is a Lagrangian submanifold of the standard symplectic R^4.

Speaker:   Gao Chen
Title:   Gravitational Instantons and Gauge Theory
Abstract:   In this talk, we survey three problems concerning hyper-Kähler 4‑manifolds with finite energy:

1. 
   The asymptotic structures at infinity.
2. 
   The Torelli-type theorems.
3. 
   Their relationship with the moduli space of the dimension reductions of anti-self-dual instantons.”

Speaker:   Mingyang Li
Title:   Poincare-Einstein 4-manifolds with complex geometry
Abstract:   Poincare-Einstein manifolds are important objects in geometric analysis and mathematical physics, while constructing them beyond the perturbative method remains challenging. In this talk, I will present a large-scale, non-perturbative construction that yields infinite-dimensional families of such manifolds in the presence of complex geometric structures. The approach reduces the Einstein equation to a Toda-type system, essentially due to ansatz by LeBrun and Tod. Joint work with Hongyi Liu. 

Title:

The linear algebra of the decomposition theorem

Speaker:   Matt Larson, Princeton U and IAS Abstract:   The decomposition theorem is one of the deepest known facts about the topology of complex projective varieties. Given a map X -> Y of complex projective varieties, with X smooth, it implies strong restrictions on the structure of the cohomology H*(X) as a module over H*(Y). We show that many of these restrictions are linear-algebraic consequences of classically-known properties of H*(X). This enables us to deduce these restrictions in situations where one cannot apply the decomposition theorem, such as in combinatorial Hodge theory and for Chow rings modulo numerical equivalence. Joint work with Omid Amini and June Huh.

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Speaker:   Lan-Hsuan Huang
Title:   Monotonicity of causal killing vectors and geometry of ADM mass minimizers
Abstract:   We address two problems concerning ADM mass minimizing initial data sets: the equality case of the positive mass theorem and the resolution of Bartnik's 1989 stationary vacuum conjecture. A key new ingredient is a monotonicity formula for the Lorentzian length of a causal Killing vector field. This is joint work with Sven Hirsch based on the paper https://arxiv.org/abs/2510.10306

Title:

Counting Curves and Surfaces in Calabi–Yau Threefolds and Modular Forms

Speaker:

Artan Sheshmani, Beijing Institute for Mathematical Sciences and Applications (BIMSA)

Abstract:   I will discuss a famous 40-year-old conjecture from string theory known as the S-duality modularity conjecture. It predicts that a certain partition function—encoding the “count” of stable solutions to the partial differential equations describing D-brane interactions, supported on complex surfaces deforming inside a Calabi–Yau threefold—is given by a modular form. Depending on how these surfaces deform in the ambient Calabi–Yau threefold, one obtains different counting problems and correspondingly different versions of the S-duality conjecture. I will explain an algebro-geometric reformulation of this problem and survey a series of results obtained with collaborators over the past 15 years toward proving the conjecture in various geometric settings. Finally, I will describe ongoing work on the most difficult version of the conjecture, which involves tools such as Tyurin degeneration, derived intersection theory, and the categorification of Donaldson–Thomas invariants.

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Speaker:   Qi Yao
Title:   Holomorphic disc foliation and the local higher regularity of the HCMA equation
Abstract:   The Homogeneous Complex Monge-Ampère (HCMA) equation plays a central role in Kähler geometry, effectively describing geodesics in the space of Kähler metrics. A major open question concerns the regularity of weak solutions to this equation.

In this talk, I will present a new local higher regularity result for the HCMA equation on complete Kähler manifolds. The proof relies on constructing a local foliation of the space by holomorphic discs and redeveloping the global pluripotential theory. I will point out a subtle regularity issue in the parameter dependence of these foliations and show how to resolve it using a Nash-Moser technique. By constructing a global plurisubharmonic subsolution, I will show that the local solution determined by the foliation agrees exactly with the global $C^{1,1}$ solution. As an application, I will discuss the consequences of this regularity on the ALE end.

Roland Roeder, IUPUI
TBA

Speaker:

Georgios Dimitroglou Rizell, Uppsala University

Abstract:

TBA