University of Minnesota

Algebraic Geometry Seminar 2014-2015

Spring Semester Meetings: 2:30-3:30pm on Wednesday at Vincent Hall 364

Wednesday, March 25 at 3:35pm in VinH 209

Kawamata-Viehweg type vanishing theorems with coefficients

Junecue Suh (University of California, Santa Cruz)
Abstract: After reviewing various vanishing theorems, we present new Kawamata-Viehweg type vanishing theorems, that allow general coefficients (polarisable variations of Hodge structures and mixed Hodge modules). Some applications will be discussed, and also perhaps some ideas in the proof.

Wednesday, March 4

Extending the Landau-Ginzburg/Calabi-Yau correspondence to non-Calabi-Yau hypersurfaces in weighted projective space

Pedro Acosta (University of Michigan)
Abstract: In the early days of mirror symmetry, physicists noticed a remarkable relation between the Calabi-Yau geometry of a hypersurface in projective space defined by a homogenous polynomial W and the singularity theory of the Landau-Ginzburg model with superpotential W. This relation came to be known as the Landau-Ginzburg/Calabi-Yau correspondence. In this talk, I will explain how this correspondence can be extended to non-Calabi-Yau hypersurfaces in weighted projective space using the recently introduced Fan-Jarvis-Ruan-Witten theory as the mathematical formalism behind Landau-Ginzburg models.

Wednesday, February 18

Birational geometry of moduli spaces of stable rational curves

Ana-Maria Castravet (The Ohio State University)
Abstract: I will talk about joint work with Jenia Tevelev on the birational geometry of the Grothendieck-Knudsen moduli space of stable rational curves with n markings. The main result is that for n large this space is not a Mori Dream Space, thus answering a question of Hu and Keel.

Fall Semester Meetings: 2:30-3:30pm on Wednesday at Vincent Hall 313

Wednesday, November 19

Motivic Gottsche's curve-counting invariants

Yu-jong Tzeng (University of Minnesota)
Abstract: On smooth algebraic surfaces, the number of nodal curves in a fixed linear system is universal polynomial of Chern numbers (conjectured by Gottsche, now proven). Recently Gottsche and Shende defined a "refined" invariant which can count real and complex nodal curves and is an invariant in tropical geometry. In this talk I will define two "motivic" invariants which generalize the universal polynomials and refined invariant to the algebraic cobordism group and to the Grothendieck ring of varieties. The properties and the possible meaning of those motivic invariants will be discussed.

Wednesday, November 5

Homology of stabilized moduli of Lefschetz fibrations

Craig Westerland (University of Minnesota)
Abstract: This talk is about the space of relatively minimal Lefschetz fibrations over surfaces X with at most one node in each fibre. We study the homology of these spaces as the number of nodal fibres tends to infinity, and relate the stable homology to the homology of the function space of maps from X to a variant of the Deligne-Mumford compactification of M_g. Restricting our answer to H_0 yields a form of Auroux's stable classification of Lefschetz fibrations.

Wednesday, October 29

Virtual Invariants on Quot Schemes over del Pezzo Surfaces

Daniel Schultheis (University of Minnesota)
Abstract: The Quot scheme over a smooth projective curve C offers a compactification of the space of maps from C to a Grassmannian. Numerous mathematicians have studied the intersection theory of these Quot schemes, culminating in proof that the virtual count of these maps satisfies the Vafa-Intriligator formula from physics. In this talk we will explore the history of this problem, as well as recent generalizations when C is replaced by a del Pezzo surface.

Wednesday, October 22

Rank 2 Donaldson-Thomas theory vertex

Amin Gholampour (University of Maryland)
 Abstract: We study the geometry of moduli space of rank 2 torsion free sheaves on toric 3-folds. As an outcome we develop a vertex theory for rank 2 Donaldson-Thomas invariants. These invariants are expressed combinatorially in terms of a new labelled box configuration. We compare this to the vertex theories for rank 1 Donaldson-Thomas theory of Maulik-Nekrasov-Okounkov-Pandharipande as well as the stable pair theory of Pandharipande-Thomas.

 Wednesday, October 8 and October 15

Quasimaps,wall-crossings, and Mirror Symmetry I and II

Ionut Ciocan-Fontanine (University of Minnesota)
Abstract: Quasimaps provide compactifications, depending on a stability parameter epsilon, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter epsilon varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas in genus zero, and will explain why they can be viewed as generalizations (in several directions) of Givental's toric mirror theorems. I will also describe extensions of wall-crossing to higher genus, and (time permitting) to orbifold GIT targets as well. The talk is based on joint works with Bumsig Kim, and partly also with Daewoong Cheong and with Davesh Maulik.

Wednesday, September 10 and September 17

Preparation for Kisin's lectures: p-curvature and the Grothendieck-Katz Conjecture I
and II

William Messing (University of Minnesota)

Last update: March 15, 2015