Lecture 1 (13:30 - 15:00): Equivariant operations in symplectic topology

Operations on Floer-theoretic invariants are defined by using counts of J-holomorphic curves as structure constants. When the domain curve has extra symmetry, one can exploit the symmetry to define equivariant analogues of the ordinary operations. I will survey the basic construction of such equivariant operations, and introduce quantum Steenrod operations which deform the classical Steenrod operations on mod p singular cohomology.

Lecture 2 (15:15 - 16:45): Quantum Steenrod operations and equivariant mirror symmetry

Quantum Steenrod operations are Z/p-equivariant operations defined on mod p quantum cohomology. We review their basic properties, in particular their flatness with respect to the (small) quantum connection. The flatness property determines the operations in low degree in terms of classical Steenrod operations and ordinary Gromov—Witten invariants. We will explain the first nontrivial computation that goes beyond the scope of flatness property, and explain its relationship to enumerative mirror symmetry in positive characteristic.

Let $(X,D)$ be a log-smooth and log-canonical (lc) pair. Via the use of the residue functions, a sequence of analytic adjoint ideal sheaves, which fit into the short exact sequences induced from the residue maps to lc centres of $(X,D)$, can be defined. Such ideal sheaves and exact sequences facilitate a proof of Fujino's conjecture, a generalisation of Kollár's injectivity theorem to lc pairs $(X,D)$ with compact Kähler $X$, by an induction on the codimension of the lc centres. The inductive argument as well as its potential use in obtaining some results in the extension problems will be discussed in this talk. The content is based on the joint work with Young-Jun Choi and the work in progress with Young-Jun Choi and Shin-ichi Matsumura.

In the positive characteristics algebraic geometry, the theory of F-singularities arises from the theorem of Kunz. For a local domain R of finite type over an algebraically closed field k, the theorem states that the variety X=Spec R is non-singular if and only if the Frobenius push-forward F^e_*R (resp. F^e_*O_X ) is a free R (resp. O_X)-module for all e>0. Hence, if R is singular, counting the number of free R-copies of F^e_*R measures the singularities of X. The F-signature plays a crucial role when measuring singularities of varieties in the above sense. For instance, s(R) = 1 implies R is regular, and 0 < s(R) < 1 implies R is strongly F-regular, which is a char p analog of klt singularities. For a globally F-regular variety X, the F-signature of an ample invertible sheaf is defined as the F-signature of the section along the invertible sheaf over X. In this talk, we will discuss the F-signature is well-defined and is (locally Lipschitz) continuous on the ample cone and how the signature extends to the boundary of the cone. This is joint work with Swaraj Pande (University of Michigan). 

Lecture 1 (Hyunbin Kim) Introduction to Tropical Geometry

We briefly review some basic definitions and theorems in classical tropical geometry. We also take a look at how tropical geometry plays a role in other fields of geometry.

Lecture 2 (Eric Dolores) Integral RELU Neural Networks and Tropical Geometry

In this talk we will give a friendly introduction to neural networks. In particular the audience is not required to know how to code. Our goal is to understand the connection between tropical geometry and neural networks discovered in 2018 by L. Zhang, G. Naitzat and L. Lim.

In enumerative geometry, Virasoro constraints were first conjectured for the moduli of stable curves (the Witten conjecture) and stable maps. Recently, the analogous constraints were conjectured in some sheaf theoretic contexts. In joint work with A. Bojko and M. Moreira, we generalize and reinterpret Virasoro conjecture in sheaf theory using Joyce’s vertex algebra. A new interpretation makes use of a conformal element and primary states of vertex operator algebras which are classical subjects in representation theory. As an application, we prove the Virasoro constraints for any moduli of torsion-free sheaves on curves and surfaces via Joyce's wall-crossing formulas.