Algebraic geometry and moduli seminar

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Spring Semester 2012

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
* 22 February 2012
Prof. Dr. Richard Thomas
Imperial College, London, UK
Counting curves on surfaces  HG G 43 
Abstract: I will discuss the counting of holomorphic curves on complex algebraic surfaces in two ways: a modified (reduced) Gromov-Witten theory, and a modified theory of stable pairs. The latter can be computed completely in terms of various topological numbers. It is related to the former by the MNOP conjecture, but unfortunately with some correction terms we cannot yet compute.
2 March 2012
Aaron Pixton
Divisors on the moduli space of stable n-pointed curves of genus 0  HG G 43 
Abstract: I'll discuss various conjectures about divisors on \bar{M}_{0,n} and describe a counterexample to one of them: it is not true that every nef divisor is numerically equivalent to an effective sum of boundary divisors. The counterexample is combinatorial in nature and is closely related to the (11, 5, 2) biplane.
9 March 2012
Felix Janda
ETH Zürich
The housing theorem  HG G 43 
Abstract: The moduli space of curves of compact type is a partial compactification of the moduli space of smooth curves. Its ring of kappa classes has been studied by Pandharipande leaving open some questions in the unpointed case. I want to discuss the housing theorem, whose mainly combinatorical proof in particular answers one of the questions and which has been extended to give a connection to the smooth case. This is joint work with Aaron Pixton.
16 March 2012
Alex Massarenti
Automorphisms group of the moduli space of stable pointed curves  HG G 43 
Abstract: he moduli stack \overline{\mathcal{M}}_{g,n} parametrizing Deligne-Mumford stable n-pointed genus g curves and its coarse moduli space \overline{M}_{g,n}: the Deligne-Mumford compactification of the moduli space of n-pointed genus g smooth curves from several decades are among the most studied objects in algebraic geometry, despite this many natural questions about their biregular and birational geometry remain unanswered. In particular we are interested in their automorphisms groups. The symmetric group on n elements S_{n} acts on \overline{\mathcal{M}}_{g,n} and \overline{M}_{g,n} by permuting the marked points. We will prove that the automorphisms groups of \overline{\mathcal{M}}_{g,n} and \overline{M}_{g,n} are isomorphic to the symmetric group S_{n} for any g,n such that 2g-2+n\geq 3, and compute the remaining cases. In doing this we will give an explicit description of \overline{M}_{1,2} as a weighted blow-up of the weighted projective plane \mathbb{P}(1,2,3).
23 March 2012
Jacopo Stoppa
University of Cambridge, UK
A degeneration formula for quiver moduli and its Gromov-Witten equivalent  HG G 19.2 
Abstract: Motivated by string-theoretic arguments Manschot, Pioline and Sen discovered a new remarkable formula for the Poincare polynomial of a quiver moduli space. We discuss this formula, its proof, and its interaction with localization techniques. Finally we use the work of Gross, Pandharipande and Siebert on the tropical vertex to show how (for Euler characteristics, and a large class of quivers) the MPS formula is equivalent to a standard degeneration formula in Gromov-Witten theory. Joint work with M. Reineke and T. Weist.
30 March 2012
Johan Martens
Aarhus University, Dänemark
Compactifications of reductive groups as moduli stacks of bundles  HG G 43 
Abstract: We will introduce a class of moduli problems for arbitrary reductive groups G, whose moduli stacks provide us with (toroidal) equivariant compactifications of G. Morally speaking, the objects in the moduli problem could be thought of as stable maps of a twice-punctured sphere into the classifying stack BG. More precisely, they consist of G_m equivariant G-principal bundles on chains of projective lines, framed at the extremal poles. The choice of a fan determines a stability condition. All toric orbifolds are special cases of these, as are the "wonderful compactifications" of semi-simple groups of adjoint type constructed by De Concini - Procesi. Our construction further provides a canonical orbifold compactification for any semi-simple group. From a symplectic point of view, these orbifolds can be understood as non-abelian cuts of the cotangent bundle of a maximal compact subgroup of G. This is joint work with Michael Thaddeus (Columbia).
20 April 2012
Prof. Dr. Andras Szenes
Université de Genève
Cohomology of Higgs moduli  HG G 43 
Abstract: The moduli spaces of Higgs bundles on a Riemann surface are a remarkable complex manifolds, which have played a central role in several recent developments in Geometry. The cohomology ring of these moduli spaces have a rich structure, conjecturally linked to integrable systems, representation theory of Hecke algebras, combinatorics of orthogonal polynomials, etc. In this talk, I will describe some of these conjectures, and recent progress towards their proof.
18 May 2012
Prof. Dr. Sam Payne
Yale / MPI Bonn
Operational K-theory and localization for toric varieties  HG G 43 
Abstract: The Grothendieck rings of ordinary and equivariant vector bundles on a smooth complete toric variety are well-understood and can be described through localization in terms of ``piecewise Laurent polynomials"; this is the K-theory analogue of the standard description of the cohomology rings in terms of piecewise polynomials on fans. A satisfactory understanding of Grothendieck rings of vector bundles on singular toric varieties, however, remains out of reach. I will discuss joint work with Dave Anderson exploring an ``operational equivariant K-theory" that agrees with the Grothendieck ring of equivariant vector bundles on a smooth variety with torus action and can be described in terms of localization and piecewise polynomials on an arbitrary singular toric variety.
8 June 2012
Dr. Ben Young
KTH Stockholm
Combinatorics of DT and PT invariants  HG G 19.1 
Abstract: I will discuss a combinatorial problem which comes from algebraic geometry. The problem, in general, is to show that two theories for "counting" curves in a complex three dimensional space X (Pandharipande-Thomas theory and reduced Donaldson-Thomas theory) give the same answer. I will prove a combinatorial version of this correspondence in a special case (X is toric Calabi-Yau), where the difficult geometry reduces to a study of the "topological vertex'' (a certain generating function) in these two theories. The combinatorial objects in question are plane partitions, perfect matchings on the honeycomb lattice and related structures. There will be many pictures. This is a combinatorics talk, so no algebraic geometry will be used once I explain where the problem is coming from.
29 June 2012
Prof. Dr. Mark Gross
UC San Diego
Mirror Symmetry and Cluster Varieties  HG G 43 
Abstract: I will talk about the intersection of the theory of cluster varieties with recent work of myself with Paul Hacking and Sean Keel. The cluster X- and A-varieties as defined by Fock and Goncharov have natural interpretations in the world of mirror symmetry, and canonical bases of the associated cluster algebras can then be constructed via tropical methods.
* 9 July 2012
Prof. Dr. Michael Thaddeus
Columbia University, USA
Moduli spaces of Higgs bundles, their cohomology rings, and the duality of their torus fibrations  HG G 43 
Abstract: I will discuss joint work with T. Hausel that appeared around 2001, showing that that the Hitchin system for Higgs bundles with structure groups PGL(n) and SL(n) are mirror partners in the sense of SYZ, and computing the cohomology ring in the case of PGL(2).

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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