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Spring Semester 2013

Date / Time Speaker Title Location
* 29 January 2013
13:30-14:30 		Dr. Vivek Shende
MIT
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Algebraic Geometry and Moduli Seminar

Title Hilbert schemes and knot invariants
Speaker, Affiliation Dr. Vivek Shende, MIT
Date, Time 29 January 2013, 13:30-14:30
Location HG G 43
Hilbert schemes and knot invariants
HG G 43
22 February 2013
16:00-17:00 		Dr. Artan Sheshmani
MPI Bonn
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Algebraic Geometry and Moduli Seminar

Title Donaldson-Thomas invariants of 2-dimensional torsion sheaves and modular forms
Speaker, Affiliation Dr. Artan Sheshmani, MPI Bonn
Date, Time 22 February 2013, 16:00-17:00
Location 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.
Donaldson-Thomas invariants of 2-dimensional torsion sheaves and modular formsread_more
HG G 43
1 March 2013
16:00-17:00 		Jorgen Rennemo
Imperial College, London
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Algebraic Geometry and Moduli Seminar

Title The homology of Hilbert schemes of a locally planar curve
Speaker, Affiliation Jorgen Rennemo, Imperial College, London
Date, Time 1 March 2013, 16:00-17:00
Location 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.
The homology of Hilbert schemes of a locally planar curveread_more
HG G 43
15 March 2013
16:00-17:00 		Prof. Dr. Diane Maclagan
University of Warwick
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Algebraic Geometry and Moduli Seminar

Title A tropical approach to effective cones
Speaker, Affiliation Prof. Dr. Diane Maclagan, University of Warwick
Date, Time 15 March 2013, 16:00-17:00
Location 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.
A tropical approach to effective conesread_more
HG G 43
22 March 2013
16:00-17:00 		Qizheng Yin
University of Amsterdam
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Algebraic Geometry and Moduli Seminar

Title Interesting cycles on symmetric powers of curves
Speaker, Affiliation Qizheng Yin, University of Amsterdam
Date, Time 22 March 2013, 16:00-17:00
Location 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.
Interesting cycles on symmetric powers of curvesread_more
HG G 43
29 March 2013
16:00-17:00 		Dr. Flavia Poma
SISSA
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Algebraic Geometry and Moduli Seminar

Title Virtual classes of Artin stacks
Speaker, Affiliation Dr. Flavia Poma, SISSA
Date, Time 29 March 2013, 16:00-17:00
Location 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.
Virtual classes of Artin stacksread_more
HG G 43
19 April 2013
16:00-17:00 		Prof. Dr. Gavril Farkas
Humboldt Universität Berlin
Details

Algebraic Geometry and Moduli Seminar

Title Singularities of theta divisors and the geometry of A_5
Speaker, Affiliation Prof. Dr. Gavril Farkas, Humboldt Universität Berlin
Date, Time 19 April 2013, 16:00-17:00
Location 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.
Singularities of theta divisors and the geometry of A_5read_more
HG G 43
26 April 2013
16:00-17:00 		Prof. Dr. Alessandro Chiodo
Jussieu
Details

Algebraic Geometry and Moduli Seminar

Title LG/CY correspondence, global mirror symmetry and Orlov equivalence
Speaker, Affiliation Prof. Dr. Alessandro Chiodo, Jussieu
Date, Time 26 April 2013, 16:00-17:00
Location 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.
LG/CY correspondence, global mirror symmetry and Orlov equivalenceread_more
HG G 43
3 May 2013
16:00-17:00 		Prof. Dr. Harry Tamvakis
Uniersity of Maryland
Details

Algebraic Geometry and Moduli Seminar

Title Schubert polynomials and degeneracy loci for the classical Lie groups
Speaker, Affiliation Prof. Dr. Harry Tamvakis, Uniersity of Maryland
Date, Time 3 May 2013, 16:00-17:00
Location 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.
Schubert polynomials and degeneracy loci for the classical Lie groupsread_more
HG G 43
10 May 2013
16:00-17:00 		Prof. Dr. Gabriele Vezzosi
Jussieu
Details

Algebraic Geometry and Moduli Seminar

Title Derived symplectic geometry
Speaker, Affiliation Prof. Dr. Gabriele Vezzosi, Jussieu
Date, Time 10 May 2013, 16:00-17:00
Location 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.
Derived symplectic geometryread_more
HG G 43
24 May 2013
16:00-17:00 		Prof. Dr. Daniel Greb
Universität Bochum
Details

Algebraic Geometry and Moduli Seminar

Title Compact moduli spaces for slope semistable sheaves
Speaker, Affiliation Prof. Dr. Daniel Greb, Universität Bochum
Date, Time 24 May 2013, 16:00-17:00
Location 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.
Compact moduli spaces for slope semistable sheavesread_more
HG G 43
* 13 June 2013
16:15-17:15 		Prof. Dr. Eleny Ionel
Stanford
Details

Algebraic Geometry and Moduli Seminar

Title Gopakumar-Vafa formula for symplectic manifolds
Speaker, Affiliation Prof. Dr. Eleny Ionel, Stanford
Date, Time 13 June 2013, 16:15-17:15
Location 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.
Gopakumar-Vafa formula for symplectic manifoldsread_more
HG G 43
* 5 July 2013
14:30-15:30 		Felix Janda
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Title Relations in the tautological ring from stable quotients
Speaker, Affiliation Felix Janda, ETH Zürich
Date, Time 5 July 2013, 14:30-15:30
Location 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.
Relations in the tautological ring from stable quotientsread_more
HG G 43
5 July 2013
16:00-17:00 		Noah Giansiracusa
UC Berkeley
Details

Algebraic Geometry and Moduli Seminar

Title Equations of tropical varieties
Speaker, Affiliation Noah Giansiracusa, UC Berkeley
Date, Time 5 July 2013, 16:00-17:00
Location 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.
Equations of tropical varietiesread_more
HG G 43

Notes: events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

Organisers: Rahul Pandharipande

Archive: SS 25 AS 24 SS 24 AS 23 SS 23 AS 22 SS 22 AS 21 SS 21 AS 20 SS 20 AS 19 SS 19 AS 18 SS 18 AS 17 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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