Algebraic geometry and moduli seminar

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Spring Semester 2013

Note: The highlighted event marks the next occurring event and events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

Date / Time Speaker Title Location
* 29 January 2013
Dr. Vivek Shende
Hilbert schemes and knot invariants HG G 43 
22 February 2013
Dr. Artan Sheshmani
MPI Bonn
Donaldson-Thomas invariants of 2-dimensional torsion sheaves and modular forms  HG G 43 
Abstract: We study the Donaldson-Thomas invariants of the 2-dimensional stable sheaves in a smooth projective threefold. The DT invariants are defined via integrating over the virtual fundamental class when it exists. When the threefold is a K3 surface fibration we express the DT invariants of sheaves supported on the fibers in terms of the the Euler characteristics of the Hilbert scheme of points on the K3 surface and the Noether-Lefschetz numbers of the fibration. Using this we prove the modularity of the DT invariants for threefolds given as K3 fibrations as well as local P^2 which was predicted in string theory. We develop a DT-theoretic conifold transition formula through which we compute the generating series for the invariants of Hilbert scheme of points for singular surfaces. We also use our geometric techniques to compute the generating series for DT invariants of threefolds given as complete intersections such as the quintic threefold. Finally if time permits I explain further application of this study such as deep connections between our torsion DT invariants and higher dimensional Knot theory as well as proof of Crepant resolution conjecture on the B-model side.
1 March 2013
Jorgen Rennemo
Imperial College, London
The homology of Hilbert schemes of a locally planar curve  HG G 43 
Abstract: Associated with an algebraic curve C are the Hilbert schemes C^[n] parametrising length n subschemes of C. If C is smooth, C^[n] is the n-th symmetric product of C, hence topologically simple, but if C is singular, the topology of C^[n] depends on the singularities of C. When C has locally planar singularities, recent work of Maulik-Yun and Migliorini-Shende shows that the homology groups of the C^[n] are explicitly determined by a certain filtration on the homology of the compactified Jacobian of C. I will explain this result and a new proof.
15 March 2013
Prof. Dr. Diane Maclagan
University of Warwick
A tropical approach to effective cones  HG G 43 
Abstract: Tropical geometry associates a polyhedral complex, called the tropical variety, to a subvariety X of a torus. The tropical variety is a "combinatorial shadow" of X which preserves many invariants of X and of a good compactification Y of X. In this talk I will explain how tropical techniques can give bounds on the nef and effective cones of a projective variety Y, and more generally on the cones of effective cycles of arbitrary dimension.
22 March 2013
Qizheng Yin
University of Amsterdam
Interesting cycles on symmetric powers of curves  HG G 43 
Abstract: We discuss a simple way of detecting non-trivial 'interesting' cycles on symmetric powers of curves. This in particular gives an elementary proof of a result of Green and Griffiths.
29 March 2013
Dr. Flavia Poma
Virtual classes of Artin stacks  HG G 43 
Abstract: Virtual classes of moduli stacks play a central role in enumerative geometry as they represent a major ingredient in the definition of deformation invariants (Gromov-Witten, Donaldson-Thomas). We suggest a construction of virtual fundamental classes of Artin stacks over a Dedekind domain endowed with a perfect obstruction theory.
19 April 2013
Prof. Dr. Gavril Farkas
Humboldt Universität Berlin
Singularities of theta divisors and the geometry of A_5  HG G 43 
Abstract: In joint work with Grushevsky, Salvati-Manni and Verra we give a complete classification of all 5-dimensional principally polarized abelian varieties whose theta divisor has a quadratic singularity not of maximal rank. We then determine the slope of the effective cone of A_5 and show that the component N_0' of the Andreotti-Mayer divisor has minimal slope 54/7. Furthermore, the Iitaka dimension of the linear system corresponding to N_0' is equal to zero.
26 April 2013
Prof. Dr. Alessandro Chiodo
LG/CY correspondence, global mirror symmetry and Orlov equivalence  HG G 43 
Abstract: I will explain the Landau-Ginzburg/Calabi-Yau correspondence in simple terms starting from a variation of stability condition in geometric invariant theory. Then I will present the quantum candidate counterpart to this transition involving a theory of singularities based on spin curves formulated first by Fan, Jarvis and Ruan following ideas of Witten (FJRW theory). In collaboration with Iritani and Ruan we prove this quantum correspondence and we relate it to Orlov's equivalence.
3 May 2013
Prof. Dr. Harry Tamvakis
Uniersity of Maryland
Schubert polynomials and degeneracy loci for the classical Lie groups  HG G 43 
Abstract: Let G be a classical complex Lie group, P any parabolic subgroup of G, and X=G/P the corresponding homogeneous space, which parametrizes (isotropic) partial flags of subspaces of a vector space. In the mid 1990s, Fulton and Pragacz asked for global formulas which express the cohomology classes of the universal Schubert varieties in flag bundles -- when the space X varies in an algebraic family -- in terms of the Chern classes of the vector bundles involved in their definition. We will explain our recent combinatorially explicit solution to this question, in terms of data coming from the Weyl group.
10 May 2013
Prof. Dr. Gabriele Vezzosi
Derived symplectic geometry  HG G 43 
Abstract: Motivated by a general program of quantization of moduli paces, we will discuss a generalization of symplectic geometry into the world of derived algebraic geometry, show existence theorems for derived symplectic structures and draw consequences for underived moduli spaces. In particular we will show that (-1)-shifted derived symplectic forms are a good replacement for symmetric obstruction theory, and briefly discuss applications to Donaldson-Thomas theory. Time permitting, we will also discuss derived coisotropic and derived Poisson structures. This is joint work with T. Pantev, B. Toen and M. Vaquié.
24 May 2013
Prof. Dr. Daniel Greb
Universität Bochum
Compact moduli spaces for slope semistable sheaves  HG G 43 
Abstract: While the variation of moduli spaces of H-slope/Gieseker-semistable sheaves on surfaces under change of the ample polarisation H is well-understood, research on the corresponding question in the case of higher-dimensional base manifolds revealed a number of pathologies. After presenting these, I will discuss recent joint work with Matei Toma (Nancy), which resolves some of these pathologies by looking at curves instead of divisors. This naturally leads to the question whether higher-dimensional analogues of the Donaldson-Uhlenbeck compactification exists, and I will discuss our construction of such a compactification.
* 13 June 2013
Prof. Dr. Eleny Ionel
Gopakumar-Vafa formula for symplectic manifolds  HG G 43 
Abstract: The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa formula holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.
* 5 July 2013
Felix Janda
ETH Zürich
Relations in the tautological ring from stable quotients  HG G 43 
Abstract: In 2000 Faber and Zagier conjectured a set of relations between tautological classes in the Chow ring of the moduli space M_g of smooth curves of genus g. For at least g<24 it has been verified that these give all relations between tautological classes. In 2010 Pandharipande and Pixton used the geometry of the moduli space of stable quotients of P^1 in order to produce a set of relations which as they show imply the Faber-Zagier relations. Last year Pixton has conjectured an extension of the FZ-relations to the moduli space Mbar_{g, n} of stable, nodal marked curves. I want to explain how the stable quotient methods can be extended to produce relations in Mbar_{g, n} which imply Pixton's generalized Faber-Zagier relations.
5 July 2013
Noah Giansiracusa
UC Berkeley
Equations of tropical varieties  HG G 43 
Abstract: I'll discuss joint work with J.H. Giansiracusa (U. Swansea) in which we describe a framework for producing/studying equations cutting out tropical varieties. This entails working with an extension of scheme theory based on semirings rather than rings, developed by various authors in the context of the "field with one element". We construct a scheme-theoretic tropicalization functor sending closed subschemes of a toric variety over a valued field to closed subschemes of the corresponding tropical toric variety. Upon restricting to T-valued points this recovers Kajiwara-Payne's tropicalization functor. We show that for projective subschemes the Hilbert function is preserved under tropicalization, thereby revealing a hidden flatness in the degeneration sending a variety to its polyhedral skeleton.

Organizers: Rahul Pandharipande 

Archive: SS 17  AS 16  SS 16  AS 15  SS 15  AS 14  SS 14  AS 13  SS 13  AS 12  SS 12  AS 11 

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