Moduli spaces of pseudoholomorphic curves and discs with Lagrangian boundary conditions are fundamental objects in symplectic geometry, and especially in Lagrangian Floer theory. These spaces are in general highly singular and must be regularized in order to be useful. One method for doing so is to use Kuranishi structures. A particularly useful kind of a Kuranishi structure is a global Kuranishi chart, first developed by Abouzaid-McLean-Smith in 2021. The earlier constructions, first developed by Fukaya-Oh-Ohta-Ono in the 1990s, require multiple charts with complicated relationships between charts and are quite difficult to work with. A global Kuranishi chart is a Kuranishi structure with only one chart, which is significantly more practical.

We give a brief introduction to global Kuranishi charts and outline a new construction. The construction is highly analytic, much more like the original Kuranishi structure construction of FOOO than the algebro-geometric construction of AMS.

The quantum connection is a flat connection arising from genus 0 Gromov--Witten theory. They can be defined integrally for sufficiently positive symplectic manifolds, allowing one to consider their characteristic p or p-adic versions which bear similarity to Gauss--Manin connections in arithmetic geometry. I would like to survey aspects of this theory, focusing on the case of Calabi--Yau threefolds and the role of quantum power operations. This talk is mostly based on joint work with Shaoyun Bai and Daniel Pomerleano.

Cohomological Hall algebras of quivers with potentials give a geometric way to realize quantum groups. In this talk, I will define a certain vertex coproduct called a Joyce vertex coalgebra on the Cohomological Hall algebras of a quiver with potential. This structure is compatible with the multiplication and upgrades the cohomological Hall algebra into a vertex bialgebra. I will then explain how to compare this structure to Deformed drinfeld coproducts on ADE Yangians.This is joint work with Shivang Jindal and Alexei Latyntsev.

Coisotropic branes were introduced by Kapustin-Orlov in order to enlarge the Fukaya category of a symplectic manifold so that it matches the homological mirror symmetry prediction. However, how such branes can be defined as an object in the Fukaya category is wildly unknown. In this talk, I would like to present a mathematical rigorous definition for the self-hom of the so-called canonical coisotropic branes, which is a space-filling brane. We begin with a semi-flat SYZ fibration $X\to B$ that carries a semi-affine canonical coisotropic brane. Such brane carries a natural complex structure $I$. We define the mirror B-brane by taking fiberwise geometric quantization, and by using family Toeplitz construction with a gauge transformation, we obtain an chain map from the Dolbeault complex of certain I-holomorphic deformation quantization of $(X,I)$ to that of the endomorphism algebra of the mirror B-brane. Our main result states that this map is a chain isomorphism of algebras. This provides an intrinsic definition of the self-hom and also the first mirror theorem of such coisotropic branes. As an application of our construction, we provide a rigorous mathematical definition of brane quantization in the semi-flat SYZ setting. This is a joint work with Kwokwai Chan, Nai-Chung Conan Leung, Qin Li, and Yu-Tung Tony Yau.

It has been observed in various settings—such as dripping faucets, insect populations, thermal convection in fluids, and electronic circuits—that the transition from regular dynamics to chaos occurs through a characteristic sequence of period-doubling bifurcations. Remarkably, the quantitative characteristics of this phenomenon appear to be universal, meaning they do not depend on the specific details of the systems under consideration. These observations have been rigorously established for 1D unimodal maps by Sullivan, McMullen, and Lyubich. In their work, they pioneered a new technique, known as renormalization, which has since become fundamental to the field. Together with S. Crovisier, M. Lyubich, and E. Pujals, we have recently extended this theory to the far more intricate 2D setting of Hénon-like maps.

In my talk, I will provide an intuitive explanation of the renormalization method, both in the classical 1D setting and in the new 2D context. I will then discuss how the tools we developed for the renormalization of Hénon-like maps contribute to the broader study of 2D dynamical systems.

TBA

Let X be a normal hypersurface of degree d with with prescribed singularities. The set of prescribed singularities is also called the basket of singularities. This talk asks which properties of X depend only on the degree and the basket not on the positions of the singular points. If time permits, we introduce a topological formula for Q-factoriality defect which measures the failure of the variety to be Q-factorial. This talk is partially based on the joint work with Prof Morihiko Saito. 

In this talk, we will explore the 130 families of Fano 3-fold weighted hypersurfaces, with a particular focus on their non-rationality and K-stability. This is an ongoing collaboration with T. Okada.

TBA