Old abstracts
Fall 2023
Date: Sep. 12.
Speaker: Kelly Jabbusch (University of Michigan - Dearborn)
Title: The minimal projective bundle dimension and toric 2-Fano manifolds
Abstract: In this talk we will discuss higher Fano manifolds, which are Fano manifolds with positive higher Chern characters. In particular we will focus on toric 2-Fano manifolds. Motivated by the problem of classifying toric 2-Fano manifolds, we will introduce a new invariant for smooth projective toric varieties, the minimal projective bundle dimension, m(X). This invariant m(X) captures the minimal degree of a dominating family of rational curves on X or, equivalently, the minimal length of a centrally symmetric primitive relation for the fan of X. We'll present a classification of smooth projective toric varieties with m(X) ≥ dim(X)-2, and show that projective spaces are the only 2-Fano manifolds among smooth projective toric varieties with m(X) equal to 1, 2, dim(X)-2, dim(X)-1, or dim(X). This is joint work with Carolina Araujo, Roya Beheshti, Ana-Maria Castravet, Svetlana Makarova, Enrica Mazzon, and Nivedita Viswanathan.
Date: Sep. 26.
Speaker: Jaiung Jun (SUNY New Paltz)
Title: Schemes over the natural numbers
Abstract: In 2006, Baker and Norine proved a Riemann-Roch theorem for finite graphs. To generalize this result to higher dimension, one is naturally led to study the scheme theory over the tropical semifield (or more generally schemes over natural numbers). I will introduce basic notions and examples for schemes over the natural numbers. Then, I will discuss several properties of line bundles and vector bundles in this setting. This is joint work with James Borger, Kalina Mincheva, and Jeffrey Tolliver.
Date: Oct. 10.
Speaker: Jason Lo (California State University, Northridge)
Title: Stability of line bundles and the deformed Hermitian-Yang-Mills equation on elliptic surfaces
Abstract: In the 1980s, Donaldson and Uhlenbeck-Yau established a correspondence between the existence of solutions to the Hermitian-Yang-Mills equation associated to a vector bundle on a compact Kahler manifold, and the Mumford-Takemoto stability (i.e. slope stability) of the vector bundle. Motivated by the recent developments of deformed Hermitian-Yang-Mills (dHYM) equations and Bridgeland stability conditions, Collins-Yau asked if a similar relation might hold between the dHYM equation and the Bridgeland stability of line bundles. In this talk, we will present a partial result on elliptic surfaces. This talk is based on joint work with Tristan Collins, Yun Shi, and Shing-Tung Yau.
Date: Oct. 24.
Speaker: Han-Bom Moon (Fordham University)
Title: Algebraic geometry over finite fields
Abstract: I will share my experience on an REU project during last summer. We explored two related projects, which can be described in an elementary way but are about the number of rational points on schemes defined over a finite field. The results are joint work with six students, Alana Campbell, Flora Dedvukaj, Ritik Jain, Donald McCormick III, Joshua Morales, and Peter Wu.
Date: Nov. 7.
Speaker: Mee Seong Im (United States Naval Academy)
Title: Correspondence between automata and one-dimensional Boolean topological theories and TQFTs
Abstract: Automata are important objects in theoretical computer science. I will describe how automata emerge from topological theories and TQFTs in dimension one and carrying defects. Conversely, given an automaton, there is a canonical Boolean TQFT associated with it. In those topological theories, one encounters pairs of a regular language and a circular regular language that describe the theory.
Date: Dec. 5.
Speaker: Nicola Tarasca (Virginia Commonwealth University)
Title: Coinvariants of metaplectic representations and abelian varieties
Abstract: Spaces of coinvariants have classically been constructed by assigning representations of affine Lie algebras, and more generally, vertex operator algebras, to pointed algebraic curves. Removing curves out of the picture, I will construct spaces of coinvariants at abelian varieties with respect to the action of an infinite-dimensional Lie algebra. I will show how these spaces globalize to twisted D-modules on moduli of abelian varieties, extending the classical picture from moduli of curves. This is based on the preprint arXiv:2301.13227.