Abstract. Delta Conjecture of Haglund, Remmel and Wilson is the identity describing the action of Macdonald operators on elementary symmetric functions. The conjecture was proved independently by Blasiak-Haiman-Morse-Pun-Seelinger, and D'Adderio-Mellit. In this talk, I will give a geometric model for Delta conjecture using affine Springer fibers. This is a joint work with Sean Griffin and Maria Gillespie.

Abstract. The amoeba and coamoeba of a subvariety $Z \subset (\bC^\times)^n$ are the images under its projections to $\bR^n$ and $T^n$, respectively. In this talk we discuss joint work with Chris Kuo studying the coamoebae of Lagrangian submanifolds of $(\bC^\times)^n$, specifically how the combinatorics of their degenerations encodes the homological algebra of mirror coherent sheaves. Concretely, we associate to a free resolution $F^\bullet$ of a coherent sheaf on $(\bC^\times)^n$ a tropical Lagrangian coamoeba $T(F^\bullet)$, a certain simplicial complex in $T^n$. We show that the discrete information in $F^\bullet$ can be recovered from $T(F^\bullet)$ in common situations, and that in general there is a constructible sheaf supported on $T(F^\bullet)$ which is mirror to the coherent sheaf in the relevant sense. This sheaf can be interpreted as a singular Lagrangian brane supported on the stratified conormal bundle of $T(F^\bullet)$, and in some cases can be expressed as a degeneration of smooth Lagrangian branes. The resulting interplay between coherent sheaves on~$(\bC^\times)^n$ and simplicial complexes in $T^n$ provides a higher-dimensional generalization of the spectral theory of dimer models in~$T^2$, as well as a symplectic counterpart to the theory of brane brick models.

Abstract. We introduce a mod $p$ analogue of the Mumford—Tate conjecture, which governs the $p$-adic monodromy of families of mod $p$ abelian varieties. It turns out that the conjecture is closely related to a notion of formal linearity on the moduli space of abelian varieties. Surprisingly, the conjecture can be reduced to an unlikely intersection problem of Ax—Schanuel type, a phenomenon that is unique to positive characteristic. This gives rise to new perspectives for attacking the Mumford—Tate problem, say, algebraization and p-adic O-minimal theory...

Abstract. The talk is based on joint work with L. Rozansky. In our previous work we explained how one can associate to a braid on $n$ strands complex of coherent $\mathbb{C}^* \times \mathbb{C}^*$ -equariant sheaves on $\mathrm{Hilb}_n(\mathbb{C}^2)$. In my talk, I explain how the categorical Nakajima operators fit into the picture. Roughly, the Nakajima operators correspond to adding extra strands to the braid. I will also show how this observation allows us to compute the homology of torus link (and more).

Abstract. In this talk, we will first explain a dimensional reduction of pseudoholomorphic curves to contact manifolds, which we call `contact instantons'. Then we explain how we can apply the analytical machinery combining it with contact Hamiltonian geometry, to prove Shelukhin's conjecture on any closed contact manifold. The conjecture concerns some quantitatve aspect of contact topology and contact dynamics, and the proof also utilizes construction of contact product and its anti-contact involutive symmetry.

Abstract. Compared to varieties, vector bundles and derived categories on stacks are less understood. (Bridgeland) stability conditions are effective to study these objects, however in general they are difficult to construct. In this talk we construct tilt stability conditions on root stacks over smooth surfaces, and show their deformation dimension is at least the even rational Chen-Ruan cohomology. I will briefly recall the structure of derived categories of root stacks and the notion of Chen-Ruan cohomology. Then I will show the ideas of proof via 2 key examples. We will also discuss some potential further questions and applications.

Abstract. Non-commutative Hodge structures were introduced by Katzarkov, Kontsevich, and Pantev as a generalization of classical Hodge structures for non-commutative spaces. These structures naturally arise from the study of mirror symmetry, enumerative geometry, and singularity theory. The nc-Hodge structures consist of de Rham data and Betti data, and those from geometry are expected to satisfy a property called “exponential type.” In this talk, we will discuss how the Betti data changes under the Fourier-Laplace transform. In particular, we will explain how to construct a perverse sheaf with vanishing cohomologies on the complex plane from a Stokes structure of exponential type. This will give rise to two equivalent descriptions of B model nc-Hodge structures associated to Landau-Ginzburg models. We will also relate the spectral decomposition of nc-Hodge structures to the vanishing cycle decomposition after Fourier transform via certain choices of Gabrielov paths, motivated by the study on A model. This talk is based on joint work with Tony Yue Yu.

Abstract. Quillen proves that the universal ring for formal group laws is isomorphic to the homotopy of complex cobordism MU. I will discuss equivariant complex oriented cohomology theories and discuss an equivariant version of Quillen's theorem for abelian compact Lie groups.

Abstract. A natural class of dynamical systems is obtained by iterating polynomial maps, which can be viewed as maps from projective space to itself. One can ask which other projective varieties admit endomorphisms of degree greater than 1. This seems to be an extremely restrictive property, with all known examples coming from toric varieties (such as projective space) or abelian varieties. We describe what is known in this direction, with the new ingredient being the "Bott vanishing" property. Joint work with Tatsuro Kawakami.

Abstract. Quantum K-theory studies enumerative invariants of varieties through counting nodal curves. It naturally arises in the framework of 3D mirror symmetry, a duality between mirror holomorphic symplectic varieties, of which (type-A) flag varieties are among the first examples. In this talk, we study the genus-zero quantum K-theory of flag varieties from the perspective of Givental cone. We prove a reconstrction theorem of the cone, and discuss several implications. The proof involves torus fixed point localization and abelian/non-abelian correspondence.

Abstract. In this talk, I will present the notion of F-bundle, an abstraction of the quantum D-module motivated by non-commutative Hodge structures. I will present two theorems: the spectral decomposition theorem, decomposing an F-bundle based on the eigenvalues of the action of the Euler vector field, and the extension of framing theorem, which allows to reconstruct isomorphisms between F-bundles. I will present applications of the theory of framing to the decomposition of the quantum D-module of a projective bundle and a blow-up. Based on arXiv:2411.02266, joint with Tony Yue Yu, Shaowu Zhang and Chi Zhang.

Abstract. In this talk, I will present homological mirror symmetry with the complex side being a theta divisor in a principally polarized abelian variety and the symplectic side being a locally toric noncompact Calabi-Yau manifold, together with a complex valued function on it that is a symplectic fibration away from a singular fiber.  For example, a theta divisor in 1 complex dimension is a genus two curve.  We will also discuss the identifications between the complex and Kahler moduli spaces under mirror symmetry.  In addition, for genus two curves, we will talk about the Kahler cones in the mirror Kahler space and see that it in fact corresponds to the cones in the Voronoi decomposition for symmetric positive definite matrices.  This is a joint work in preparation with Haniya Azam, Catherine Cannizzo, and Chiu-Chu Melissa Liu.