Abstracts - Spring 2023

Speaker:  Louis Esser

Title: Minimal log discrepancies and mirror symmetry

Abstract:  For certain quasismooth Calabi-Yau hypersurfaces in weighted projective space, the Berglund-Hübsch-Krawitz (BHK) mirror symmetry construction gives a concrete description of the mirror.  We prove that the minimal log discrepancy of the quotient of such a hypersurface by its toric automorphism group has a simple description in terms of the weights and degree of the BHK mirror.  This illuminates a new connection between mirror symmetry and invariants from birational geometry.  We use this result to construct klt Calabi-Yau pairs with very small minimal log discrepancies.

Speaker:  Svetlana Makarova

Title:  Moduli problems in abelian categories

Abstract: I will explain why viewing moduli problems as functors allows one to recover the structure of a variety on the set of isomorphism classes of objects, and then I will talk about modern methods of studying moduli problems. The modern theory ​"Beyond GIT" provides a "coordinate-free" way of thinking about classification problems. Among giving a uniform philosophy, this allows to treat problems that can't necessarily be described as global quotients, like objects in an abstract abelian category. I will show how the methods of BGIT can be applied to prove existence and projectivity of moduli spaces of objects in a class of abelian categories. This is based on a joint work with Andres Fernandez Herrero, Emma Lennen. 

Further, applying these methods to moduli of quiver representations allows us to focus on geometry and obtain new results. I will define a determinantal line bundle which descends to a semiample line bundle on the moduli space and provide effective bounds for its global generation. For an acyclic quiver, we can observe that this line bundle is ample and thus the adequate moduli space is projective over an arbitrary noetherian base. This part is based on a preprint with Belmans, Damiolini, Franzen, Hoskins, Tajakka (https://arxiv.org/abs/2210.00033).

Speaker:  Joaquin Moraga

Title:  Fundamental groups of algebraic singularities

Abstract:   In this talk we will review the history of fundamental groups of algebraic singularities and explain some new developments. We start by reviewing classic results regarding normal, quotient, rational, and Cohen-Macaulay singularities. Then, we will discuss more modern classes of singularities: log terminal and log canonical.

Speaker:  Isabel Vogt 

Title: Interpolation for Brill--Noether curves

Abstract: The interpolation problem is one of the oldest in mathematics.  In its most broad form it asks: when can a curve of a given type be passed through a given number of general points?  I'll discuss my recent joint work with Eric Larson that completely solves this problem for curves of general moduli.