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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 13:3014:30 
Dr. Vivek Shende MIT 
Hilbert schemes and knot invariants  HG G 43  
22 February 2013 16:0017:00 
Dr. Artan Sheshmani MPI Bonn 
DonaldsonThomas invariants of 2dimensional torsion sheaves and modular forms  HG G 43  
Abstract: We study the DonaldsonThomas invariants of the 2dimensional 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 NoetherLefschetz 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 DTtheoretic 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 Bmodel side.  
1 March 2013 16:0017:00 
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 nth 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 MaulikYun and MiglioriniShende 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 16:0017:00 
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 16:0017:00 
Qizheng Yin University of Amsterdam 
Interesting cycles on symmetric powers of curves  HG G 43  
Abstract: We discuss a simple way of detecting nontrivial 'interesting' cycles on symmetric powers of curves. This in particular gives an elementary proof of a result of Green and Griffiths.  
29 March 2013 16:0017:00 
Dr. Flavia Poma SISSA 
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 (GromovWitten, DonaldsonThomas). We suggest a construction of virtual fundamental classes of Artin stacks over a Dedekind domain endowed with a perfect obstruction theory.  
19 April 2013 16:0017:00 
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, SalvatiManni and Verra we give a complete classification of all 5dimensional 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 AndreottiMayer 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 16:0017:00 
Prof. Dr. Alessandro Chiodo Jussieu 
LG/CY correspondence, global mirror symmetry and Orlov equivalence  HG G 43  
Abstract: I will explain the LandauGinzburg/CalabiYau 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 16:0017:00 
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 16:0017:00 
Prof. Dr. Gabriele Vezzosi Jussieu 
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 DonaldsonThomas 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 16:0017:00 
Prof. Dr. Daniel Greb Universität Bochum 
Compact moduli spaces for slope semistable sheaves  HG G 43  
Abstract: While the variation of moduli spaces of Hslope/Giesekersemistable sheaves on surfaces under change of the ample polarisation H is wellunderstood, research on the corresponding question in the case of higherdimensional 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 higherdimensional analogues of the DonaldsonUhlenbeck compactification exists, and I will discuss our construction of such a compactification.  
*
13 June 2013 16:1517:15 
Prof. Dr. Eleny Ionel Stanford 
GopakumarVafa formula for symplectic manifolds  HG G 43  
Abstract: The GopakumarVafa conjecture predicts that the GromovWitten invariants of a CalabiYau 3fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic GromovWitten theory, we prove that the GopakumarVafa formula holds for any symplectic CalabiYau 6manifold, and hence for CalabiYau 3folds. The results extend to all symplectic 6manifolds and to the genus zero GW invariants of semipositive manifolds.  
*
5 July 2013 14:3015:30 
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 FaberZagier relations. Last year Pixton has conjectured an extension of the FZrelations 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 FaberZagier relations.  
5 July 2013 16:0017:00 
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 schemetheoretic 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 Tvalued points this recovers KajiwaraPayne'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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