Interplay between ribbon graphs and CohFT.
We will review an axiomatic formulation of a 2D TQFT whose formalism is based on the edge-contraction operations on graphs drawn on a Riemann surface (cellular graphs). We will describe a new result, that ribbon graphs provide both cohomological field theory and a visual explanation of Frobenius-Hopf duality. This is based on work in progress with Motohico Mulase.
The log canonical threshold revisited.
In recent years the minimal model program has extended its reach into other areas. One such area is the study of algebraic foliations. We propose one possible way to define the log canonical threshold in this context and we conjecture that it satisfies the ascending chain condition.
Coregularity of Fano varieties.
In this talk, we will introduce the coregularity of Fano varieties. The coregularity measures the singularities of the anti-pluricanonical sections of the Fano variety. Philosophically, most Fano varieties have coregularity 0. In the talk, we will explain some theorems that support this philosophy. We will show that a Fano variety of coregularity 0 admits a non-trivial section in |-2K_X|, independently of the dimension of X. This is joint work with Fernando Figueroa, Stefano Filipazzo, and Junyao Peng.
Semi-orthogonal decompositions of moduli spaces.
Let C be a smooth projective curve of genus at least 2 and let N be the moduli space of stable rank 2 vector bundles on C of odd degree. We construct a semi-orthogonal decomposition of the bounded derived category of N conjectured by Narasimhan and by Belmans, Galkin and Mukhopadhyay. It has two blocks for each i-th symmetric power of C for i = 0,...,g−2 and one block for the (g − 1)-st symmetric power. Our proof is based on an analysis of wall-crossing between moduli spaces of stable pairs, combining classical vector bundles techniques with the method of windows. Joint work with Sebastian Torres.
Calabi-Yau varieties of large index.
A projective variety X is called Calabi-Yau if its canonical divisor is Q-linearly equivalent to zero. The smallest positive integer m with mK_X linearly equivalent to zero is called the index of X. Using ideas from mirror symmetry, we construct Calabi-Yau varieties with index growing doubly exponentially with dimension. We conjecture they are the largest index in each dimension based on evidence in low dimensions. We also give Calabi-Yau varieties with large orbifold Betti numbers or small minimal log discrepancy. Joint work with Louis Esser and Burt Totaro.
Computing invariants of GIT quotients.
Let G be a connected reductive group with maximal torus T, and let V and E be two representations of G. Then E defines a vector bundle on the orbifold V//G; let X//G be the zero locus of a regular section. The quasimap I-function of X//G encodes the geometry of maps from P^1 to X//G and is related to Gromov-Witten invariants of X//G. Especially when X//G is Fano, the I-function is useful for classification problems or computing quantum cohomology. By directly analyzing the maps from P^1, we explain how to relate the I-function of X//G to that of V//T. In joint work with Nawaz Sultani, we use our formulas to validify a mirror symmetry computation of Oneto-Petracci that relates the quantum period of X//G to a certain Laurent polynomial defined by a Fano polytope.
Mirror structure constants via non-archimedean analytic disks.
For any smooth affine log Calabi-Yau variety U, we construct the structure constants of the mirror algebra to U via counts of non-archimedean analytic disks in the skeleton of the Berkovich analytification of U. This generalizes our previous construction with extra toric assumptions. The technique is based on an analytic modification of the target space as well as the theory of skeletal curves. Consequently, we deduce the positivity and integrality of the mirror structure constants. If time permits, I will discuss further generalizations and virtual fundamental classes. Joint work with S. Keel.