Abstract: Differentials on Riemann surfaces correspond to translation surfaces with cone points, where the zero orders of differentials determine the cone angles of translation surfaces. Affine transformations act on translation surfaces and preserve the zero type. These perspectives make differentials and their moduli spaces extensively studied in many areas. In this talk, I’ll first introduce the background of differentials and their moduli spaces. Then I’ll explain several recent AI-assisted works relevant to moduli spaces of differentials, which include elementary number theory, automorphisms of Riemann surfaces, topology of hyperplane arrangements, and special generating functions. These AI works were not prompted by me, and they are not as fancy as the million-dollar problems. However, the related questions naturally arose in my research field, and I did think about them before the era of AI. Both parts of the talk should be accessible to a general audience.
Abstract: To some extent, a perfect obstruction theory in the sense of Behrend--Fantechi is the shadow of a quasi-smooth derived Deligne--Mumford stack, while a cosection of a perfect obstruction theory in the sense of Kiem--Li is the shadow of a (-1)-shifted function on a quasi-smooth derived Deligne--Mumford stack. In this talk, we will introduce shifted logarithmic functions on derived stacks equipped with log structures of Deligne--Faltings type, and logarithmic cosections of perfect obstruction theories. As an application, we will present a decomposition theorem for virtual cycles, after Log GLSM. If time permits, we will briefly mention the iterative decomposition algorithm and discuss another facet of shifted logarithmic functions: their derived critical loci carry shifted logarithmic symplectic structures, building on work of Calaque-Safronov.
Oct 13: Distinguished Lecture Series by Laura DeMarco (Harvard), tentatively scheduled for 4 pm, room TBD
We will present recent progress in understanding the moduli stack of all reduced algebraic curves by means of moduli spaces of subalgebras called territories. We will discuss a stratification of arbitrarily singular curves generalizing the stratification of stable curves by dual graph, present results on the existence of non-smoothable curve singularities, and answer the question of whether the stack of all reduced curves is connected.
Any smooth complex projective variety admits a Lefschetz pencils, and such pencils are a useful tool for studying the topology of complex varieties. Topological versions of Lefschetz pencils similarly play an important role in 4-dimensional topology, as Donaldson and Gompf showed that any closed, oriented 4-manifold admits a topological Lefschetz pencil if and only if it admits a symplectic form. However, the question of how many topological Lefschetz pencils that any symplectic 4-manifold admits remains poorly understood. In this talk we will discuss a construction of infinitely many topological Lefschetz pencils of the same fiber genus on the underlying smooth 4-manifolds of ruled surfaces of genus g ≥ 2. This is joint work with Carlos A. Serván.