Abstracts

David Anderson: Old and new formulas for degeneracy loci

Abstract : Lines in flag varieties have been extensively investigated. In particular, for a semi-simple group G and a maximal parabolic subgroup P, it is known that the lines in G/P passing through the base point form a smooth projective variety on which P acts with one or two orbits. By contrast, little seems to be known about lines in Schubert varieties. The talk will discuss these and the related notion of minimal rational curves on natural desingularizations of Schubert varieties, based on joint work with S. Senthamarai Kannan.

Anders Buch: Positivity determines quantum cohomology

Abstract: I will show that the small quantum cohomology ring of a Grassmannian is, up to rescaling the deformation parameter q, the only graded q-deformation of the singular cohomology ring with non-negative Schubert structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants are uniquely determined by Witten's presentation of the quantum ring and the fact that they are non-negative. A similar statement appears to be true for any flag variety of simply laced Lie type. For the variety of complete flags, this statement is equivalent to Fomin, Gelfand, and Postnikov's conjecture that the quantum Schubert polynomials are uniquely determined by positivity properties. The proof for Grassmannians answers a question of Fulton. This is joint work with Chengxi Wang.

Linda Chen: Quantum K-theory of homogenous spaces

Abstract: The quantum K-theory ring of homogeneous spaces, defined by Givental and Lee, is a generalization of the quantum cohomology ring. I will describe recent progress on the quantum K-theory of homogeneous spaces, including joint work with Dave Anderson and Hsian-Hua Tseng showing that the quantum product, which a priori produces infinitely many terms, is finite for G/P.

Gabi Farkas: Green's Conjecture via Koszul modules.

Abstract: Using ideas from geometric group theory we provide a novel approach to Green's Conjecture on syzygies of canonical curves. Via a strong vanishing result for Koszul modules we deduce that a general canonical curve of genus g satisfies Green's Conjecture when the characteristic is zero or at least (g+2)/2. Our results are new in positive characteristic (and answer positively a conjecture of Eisenbud and Schreyer), whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Joint work with Aprodu, Papadima, Raicu and Weyman.

Angela Gibney: Vector bundles of conformal blocks defined by modules over vertex algebras of CohFT-type

Abstract: Finitely generated admissible modules over certain conformal vertex algebras, together with stable pointed curves of arbitrary genus g can be used to construct dual vector spaces of coinvariants and conformal blocks. I'll describe vertex algebras V and conditions on V that guarantee these spaces have a number of good properties, including that they give rise to vector bundles on the moduli space of stable n-pointed curves of genus g. This talk is about joint work with C. Damiolini, and N. Tarasca.

June Huh: Lorentzian polynomials

Abstract: Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions. Although no specific background beyond linear algebra and multivariable calculus is needed to enjoy the talk, I advertise the talk to people with interest in at least one of the following topics: graphs, convex bodies, stable polynomials, projective varieties, Potts model partition functions, tropicalizations, Schur polynomials, highest weight representations. Based on joint works with Petter Brändén, Christopher Eur, Jacob Matherne, Karola Mészáros, and Avery St. Dizier.

Ravi Vakil: Stabilization of Hurwitz spaces in algebraic geometry

Abstract: A recurring theme in geometry is that moduli spaces become better and better behaved "in the limit". Stabilization of the Grothendieck ring or Chow ring is an algebro-geometric analogue of stabilization in topology. After briefly introducing stabilization in the Grothendieck ring (joint with Wood), I will describe how it applies to low-degree Hurwitz spaces (with Landesman and Wood) in analogy with Bhargavology and Ellenberg-Venkatesh-Westerland. Hurwitz spaces stabilize in other ways (joint with Patel, and with Deopurkar-Landesman-Patel).

Sjuvon Chung: A primer on Schubert calculus (New Facets talk)

Abstract: Schubert calculus today bustles with activity from various areas of math—from the geometry of flag varieties, to the combinatorics of symmetric functions, combinatorics which govern the many cohomology rings of these varieties. How did it get this way, this calculus from nineteenth century enumerative geometry? In this talk, we will present a glimpse of Schubert calculus and its modern landscape; we will do this by returning to its origins and discussing some of the developments that have shaped the field today.

Isabel Vogt: An introduction to the moduli of curves (New Facets talk)

Abstract: We'll begin the talk with the definition of stable n-pointed curves of genus g, and introduce their moduli space \bar{M}_g,n. Then for most of the remaining time, I'll try to convince you, mostly through examples, that this is a good and interesting definition, both for studying the geometry of curves, and in order to obtain a nice moduli space.