| 22-feb-2012 (wed) |
Richard Thomas
|
Counting curves on surfaces
|
14:00-15:00 |
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. |
| Speakers: |
Prof. Dr. Richard Thomas
(Imperial College, London, UK)
|
|
| 2-mar-2012 (fri) |
Aaron Pixton
|
Divisors on the moduli space of stable n-pointed curves of genus 0
|
16:00-17:00 |
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. |
| Speakers: |
|
|
| 9-mar-2012 (fri) |
Felix Janda
|
The housing theorem
|
16:00-17:00 |
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. |
| Speakers: |
|
|
| 16-mar-2012 (fri) |
Alex Massarenti
|
Automorphisms group of the moduli space of stable pointed curves
|
16:00-17:00 |
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). |
| Speakers: |
Alex Massarenti
(SISSA-ISAS, Trieste)
|
|
| 23-mar-2012 (fri) |
Jacopo Stoppa
|
A degeneration formula for quiver moduli and its Gromov-Witten equivalent
|
16:00-17:00 |
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. |
| Speakers: |
Jacopo Stoppa
(University of Cambridge, UK)
|
|
| 30-mar-2012 (fri) |
Johan Martens
|
Compactifications of reductive groups as moduli stacks of bundles
|
16:00-17:00 |
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). |
| Speakers: |
Johan Martens
(Aarhus University, Dänemark)
|
|
| 20-apr-2012 (fri) |
Andras Szenes
|
Cohomology of Higgs moduli
|
16:00-17:00 |
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. |
| Speakers: |
Prof. Dr. Andras Szenes
(Université de Genève)
|
|
| 18-may-2012 (fri) |
Sam Payne
|
Operational K-theory and localization for toric varieties
|
16:00-17:00 |
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. |
| Speakers: |
Prof. Dr. Sam Payne
(Yale / MPI Bonn)
|
|
| 8-jun-2012 (fri) |
Ben Young
|
|
16:00-17:00 |
HG G 43 |
| Speakers: |
Ben Young
(KTH Stockholm)
|
|
