Seminars (MOS)

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Event When Speaker Title Presentation Material
MOSW01 5th January 2011
10:00 to 11:00
Augmented bundles - I
The notion of augmented bundle comprises various extra structures on vector bundles. The lectures deal with examples like quasiparabolic bundles, coherent systems, framed bundles, Higgs bundles, and holomorphic triples, all on a compact Riemann surface. I'll explain their coarse moduli spaces and discuss the role of the stability parameters.
MOSW01 5th January 2011
11:30 to 12:30
T Gómez Principal bundles - I
This is an introduction to principal bundles in algebraic geometry. We will start with the basic definitions (associated fibrations, reduction and extension of structure group, stability, etc...) and then give a sketch of the construction of the moduli

Notes: The following are the notes of some lectures I gave in Warsaw. In that school I gave 5 lectures, so this time I won't be able to cover all this material, but this could serve as a reference:

T. Gomez, Lectures on principal bundles over projective varieties, Pragacz, P. (Ed.) Algebraic cycles, sheaves, shtukas, and moduli. Impanga lecture notes. Trends in Mathematics. Birkh\"auser Verlag, Basel, 2008. viii+236 pp., 45--68. Lectures on Principal Bundles over proyective varieties. arXiv: 1011.1801v1
MOSW01 5th January 2011
14:00 to 15:00
Geometric Invariant Theory - I
MOSW01 5th January 2011
15:30 to 16:30
N Nitsure Deformation Theory : an introduction to the algebraic approach - I
This series of three lectures aims to give an introduction to the algebraic approach to deformation theory, as developed by Grothendieck, Schlessinger, Illusie and Artin, illustrated with a few key examples.

The topics will include (i) infinitesimal theory over Artin local rings including Schlessinger's theorem, tangent-obstruction theories and use of cotangent complex, (ii) algebraization via the Grothendieck existence theorem for formal schemes and the Artin approximation theorem, and (iii) construction of global moduli as quotient algebraic spaces.
MOSW01 6th January 2011
10:00 to 11:00
Augmented bundles - II
The notion of augmented bundle comprises various extra structures on vector bundles. The lectures deal with examples like quasiparabolic bundles, coherent systems, framed bundles, Higgs bundles, and holomorphic triples, all on a compact Riemann surface. I'll explain their coarse moduli spaces and discuss the role of the stability parameters.
MOSW01 6th January 2011
11:30 to 12:30
T Gómez Principal bundles - II
This is an introduction to principal bundles in algebraic geometry. We will start with the basic definitions (associated fibrations, reduction and extension of structure group, stability, etc...) and then give a sketch of the construction of the moduli

Notes: The following are the notes of some lectures I gave in Warsaw. In that school I gave 5 lectures, so this time I won't be able to cover all this material, but this could serve as a reference:

T. Gomez, Lectures on principal bundles over projective varieties, Pragacz, P. (Ed.) Algebraic cycles, sheaves, shtukas, and moduli. Impanga lecture notes. Trends in Mathematics. Birkh\"auser Verlag, Basel, 2008. viii+236 pp., 45--68. Lectures on Principal Bundles over proyective varieties. arXiv: 1011.1801v1
MOSW01 6th January 2011
14:00 to 15:00
Geometric Invariant Theory - II
MOSW01 6th January 2011
15:30 to 16:30
N Nitsure Deformation Theory : an introduction to the algebraic approach - II
This series of three lectures aims to give an introduction to the algebraic approach to deformation theory, as developed by Grothendieck, Schlessinger, Illusie and Artin, illustrated with a few key examples.

The topics will include (i) infinitesimal theory over Artin local rings including Schlessinger's theorem, tangent-obstruction theories and use of cotangent complex, (ii) algebraization via the Grothendieck existence theorem for formal schemes and the Artin approximation theorem, and (iii) construction of global moduli as quotient algebraic spaces.
MOSW01 7th January 2011
10:00 to 11:00
P Gothen Representations of surface groups and Higgs bundles - I
Classical Hodge theory uses harmonic forms as preferred representatives of cohomology classes. A representation of the fundamental group of a Riemann surface gives rise to a corresponding flat bundle. A Theorem of Donaldson and Corlette shows how to find a harmonic metric in this bundle. A flat bundle corresponds to class in first non-abelian cohomology and the Theorem can be viewed as an analogue of the classical representation of de Rham cohomology classes by harmonic forms.
MOSW01 7th January 2011
11:30 to 12:30
Moduli spaces and physics
MOSW01 7th January 2011
14:00 to 15:00
Geometric Invariant Theory - III
MOSW01 7th January 2011
15:30 to 16:30
N Nitsure Deformation Theory : an introduction to the algebraic approach - III
This series of three lectures aims to give an introduction to the algebraic approach to deformation theory, as developed by Grothendieck, Schlessinger, Illusie and Artin, illustrated with a few key examples.

The topics will include (i) infinitesimal theory over Artin local rings including Schlessinger's theorem, tangent-obstruction theories and use of cotangent complex, (ii) algebraization via the Grothendieck existence theorem for formal schemes and the Artin approximation theorem, and (iii) construction of global moduli as quotient algebraic spaces.
MOSW01 10th January 2011
10:00 to 11:00
Geometric Invariant Theory - IV
MOSW01 10th January 2011
11:30 to 12:30
P Gothen Representations of surface groups and Higgs bundles - II
A Higgs bundle on a Riemann surface is a pair consisting of a holomorphic bundle and a holomorphic one-form, the Higgs field, with values in a certain associated vector bundle. A theorem of Hitchin and Simpson says that a stable Higgs bundle admits a metric satisfying Hitchin's equations. Together with the Theorem of Corlette and Donaldson, the Hitchin-Kobayashi correspondence generalizes the classical Hodge decomposition of the first cohomology of the Riemann surface, providing a correspondence between isomorphism classes of Higgs bundles and representations of the fundamental group of the surface.
MOSW01 10th January 2011
14:00 to 15:00
Introduction to derived categories and stability conditions - I
The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of t-structures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and Kontsevich-Soibelman) and discuss examples. The relevant literature is accessible on the arxiv.
MOSW01 10th January 2011
15:30 to 16:30
Introduction to moduli of varieties - I: Generalities on moduli spaces of varieties
The yoga of moduli spaces. Main methods of construction. Compactifications and Minimal Model Program. Stable pairs (slc & ample K+B). Honest work vs "accidental" examples.
MOSW01 11th January 2011
10:00 to 11:00
Introduction to stacks - I
Introduction to algebraic stacks as they are relevant to moduli theory.
MOSW01 11th January 2011
11:30 to 12:30
P Gothen Representations of surface groups and Higgs bundles - III
In the final lecture, we give some applications of Higgs bundle theory to the study of the geometry and topology of character varieties for surface groups, via the identification with moduli of Higgs bundles.
MOSW01 11th January 2011
14:00 to 15:00
Introduction to derived categories and stability conditions - II
The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of t-structures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and Kontsevich-Soibelman) and discuss examples. The relevant literature is accessible on the arxiv.
MOSW01 11th January 2011
15:30 to 16:30
Introduction to moduli of varieties - II: Stable toric and semiabelic varieties
Moduli of stable toric varieties vs toric Hilbert scheme. Convex tilings and secondary polytopes. Degenerations of abelian varieties. Semiabelian and semiabelic varieties. Compactifications of moduli spaces of abelian varieties.

Refs: http://arxiv.org/abs/math/9905103, http://arxiv.org/abs/math/0207272, http://arxiv.org/abs/math/0207274.
MOSW01 12th January 2011
10:00 to 11:00
Introduction to stacks - II
Introduction to algebraic stacks as they are relevant to moduli theory.
MOSW01 12th January 2011
11:30 to 12:30
Ringel-Hall algebras and applications to moduli - I
Moduli spaces of representations of quivers, parametrizing configurations of vector spaces and linear maps up to base change, provide a prototype for many moduli spaces of algebraic geometry.

The Hall algebra of a quiver, a convolution algebra of functions on stacks of its representations, can be used to obtain quantitative information on the moduli spaces: algebraic identities in the Hall algebra, proved by representation-theoretic techniques, yield identities for e.g. Betti numbers or numbers of points over finite fields.

We will develop several such identities and discuss more recent applications to wall-crossing formulae.

Notes: Several of the Hall algebra techniques which I would like to discuss are reviewed in the survey "Moduli of representations of quivers", arXiv:0802.2147. Although this paper was written for an audience of representation theorists, it might as well be helpful for the participants of the School on Moduli Spaces. The more recent applications to wall-crossing formulae are developed in "Poisson automorphisms and quiver moduli", arXiv:0802.2147.
MOSW01 12th January 2011
14:00 to 15:00
Introduction to derived categories and stability conditions - III
The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of t-structures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and Kontsevich-Soibelman) and discuss examples. The relevant literature is accessible on the arxiv.
MOSW01 12th January 2011
15:30 to 16:30
Introduction to moduli of varieties - III: Weighted hyperplane arrangements
$\bar M_{0,n}$. Hassett's weighted stable curves. Compact moduli of weighted hyperplane arrangements.

Refs: http://arxiv.org/abs/math/0205009, http://arxiv.org/abs/math/0501227, http://arxiv.org/abs/0806.0881
MOSW01 13th January 2011
10:00 to 11:00
Introduction to stacks - III
Introduction to algebraic stacks as they are relevant to moduli theory.
MOSW01 13th January 2011
11:30 to 12:30
Ringel-Hall algebras and applications to moduli - II
Moduli spaces of representations of quivers, parametrizing configurations of vector spaces and linear maps up to base change, provide a prototype for many moduli spaces of algebraic geometry.

The Hall algebra of a quiver, a convolution algebra of functions on stacks of its representations, can be used to obtain quantitative information on the moduli spaces: algebraic identities in the Hall algebra, proved by representation-theoretic techniques, yield identities for e.g. Betti numbers or numbers of points over finite fields.

We will develop several such identities and discuss more recent applications to wall-crossing formulae.

Notes: Several of the Hall algebra techniques which I would like to discuss are reviewed in the survey "Moduli of representations of quivers", arXiv:0802.2147. Although this paper was written for an audience of representation theorists, it might as well be helpful for the participants of the School on Moduli Spaces. The more recent applications to wall-crossing formulae are developed in "Poisson automorphisms and quiver moduli", arXiv:0802.2147.
MOSW01 13th January 2011
14:00 to 15:00
Introduction to derived categories and stability conditions - IV
The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of t-structures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and Kontsevich-Soibelman) and discuss examples. The relevant literature is accessible on the arxiv.
MOSW01 13th January 2011
15:30 to 16:30
Introduction to moduli of varieties - IV: Surfaces of general type related to abelian varieties and hyperplane arrangements
Compact moduli of surfaces of general type derived from (1) abelian varieties, (2) line arrangements. Campedelli, Burniat, Kulikov surfaces.

Refs: http://arxiv.org/abs/math/9905103, http://arxiv.org/abs/0901.4431
MOSW01 14th January 2011
10:00 to 11:00
Introduction to stacks - IV
Introduction to algebraic stacks as they are relevant to moduli theory.
MOSW01 14th January 2011
11:30 to 12:30
Ringel-Hall algebras and applications to moduli - III
Moduli spaces of representations of quivers, parametrizing configurations of vector spaces and linear maps up to base change, provide a prototype for many moduli spaces of algebraic geometry.

The Hall algebra of a quiver, a convolution algebra of functions on stacks of its representations, can be used to obtain quantitative information on the moduli spaces: algebraic identities in the Hall algebra, proved by representation-theoretic techniques, yield identities for e.g. Betti numbers or numbers of points over finite fields.

We will develop several such identities and discuss more recent applications to wall-crossing formulae.

Notes: Several of the Hall algebra techniques which I would like to discuss are reviewed in the survey "Moduli of representations of quivers", arXiv:0802.2147. Although this paper was written for an audience of representation theorists, it might as well be helpful for the participants of the School on Moduli Spaces. The more recent applications to wall-crossing formulae are developed in "Poisson automorphisms and quiver moduli", arXiv:0802.2147.
MOSW01 14th January 2011
14:00 to 15:00
Introduction to derived categories and stability conditions - V
The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of t-structures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and Kontsevich-Soibelman) and discuss examples. The relevant literature is accessible on the arxiv.
MOSW01 14th January 2011
15:30 to 16:30
Introduction to moduli of varieties - V: Surfaces of general type: the general case
Several versions of the moduli functor. Problems, and some solutions. Steps in the construction of the moduli space.

Ref: many older papers, http://arxiv.org/abs/0805.0576, http://arxiv.org/abs/1008.0621 (by Kollár)
MOS 18th January 2011
11:30 to 12:30
T Logvinenko Derived McKay correspondence in dimensions 4 and above
Given a finite subgroup G of SL_n(C) the McKay correspondence studies the relation between G-equivalent geometry of C^n and the geometry of a resolution of Y of C^n/G. In their groundbreaking work, Bridgeland, Kind, and Reid have established that for n = 2,3 the scheme Y = G-Hilb(C^n) is a crepant resolution of C^n/G and that the derived category D(Y) is equivalent to the G-equivalent derived category D^G(C^n). It follows that we also have D(Y) = D^G(C^3) for any other crepant resolution Y of C^3/G. In this talk, I discuss possible ways of generalizing this to dimension 4 and above.
MOS 20th January 2011
14:00 to 15:00
Stable spectral bundles on K3 fibered 3-folds
In this talk I will explain how to construct stable sheaves over K3 fibered threefolds using a relative Fourier-Mukai transform, which describes these sheaves in terms of spectral data. This procedure is similar to the spectral cover construction for elliptic fibrations, which I will also review.

On K3 fibered Calabi-Yau threefolds, the Fourier-Mukai transform induces an embedding of the relative Jacobian of line bundles on spectral covers into the moduli space of sheaves of given invariants. This makes the moduli space of spectral sheaves to a generic torus fibration over the moduli space of curves of given arithmetic genus on the Calabi-Yau threefold.

MOS 20th January 2011
15:30 to 16:30
Relative Fourier-Mukai transforms for Weierstrass fibrations, abelian schemes and Fano fibrations
Since its introduction by Mukai, the theory of integral functors and Fourier-Mukai transforms have been important tools in the study of the geometry of varieties and moduli spaces.

Working with a fibered scheme over a base $T$ it is quite natural to look at the group of $T$-linear autoequivalences. The description of this group seems a hard problem. We will restrict ourselves to the subgroup given by relative Fourier-Mukai transforms. In this talk, I will explain how for a projective fibration the knowledge of the structure of the group of autoequivalences of its fibres and the properties of relative integral functors provide a machinery to study that subgroup. I will work out the case of a Weierstrass fibrations and report about the results for abelian schemes and Fano or anti-Fano fibrations.

MOS 25th January 2011
10:00 to 11:00
Moduli Spaces of Gorenstein Quasi-Homogenous Surface Singularities
Gorenstein quasi-homogeneous surface singularities, studied by Dolgachev, Neumann and others, correspond to lifts of Fuchsian groups into the universal covering of PSL(2,R). I will show that the space of Gorenstein quasi-homogeneous surface singularities corresponding to a certain Fuchsian group is a finite affine space of Z/mZ-valued functions on the Fuchsian group, called m-Arf functions. Using m-Arf functions, I will count connected components of the space of Gorenstein quasi-homogeneous surface singularities and prove that any connected component is homeomorphic to a quotient of R^d by a discrete group. This work is connected to the earlier results of Atiyah and Mumford on spin structures on compact Riemann surfaces and of Jarvis, Kimura and Vaintrob on moduli spaces of higher spin curves. This is joint work with Sergey Natanzon.
MOS 25th January 2011
11:30 to 12:30
Kirwan blow-ups and moduli spaces
Kirwan's partial desingularization is a sequence of blow-ups which systematically resolves the strictly semistable points in geometric invariant theory. In this talk, I will discuss various examples of moduli spaces constructed by Kirwan blow-ups such as the moduli space of stable maps of degree 2, the moduli space of Hecke curves and the moduli space of weighted pointed rational curves. If time permits, I will also talk about a joint project with Jun Li in which we study wall crossing and (generalized) Donaldson-Thomas type invariants by using Kirwan blow-ups.
MOS 27th January 2011
14:00 to 15:00
Coupled Equations, cscK metrics and geodesic stability
We introduce a system of partial differential equations coupling a Kähler metric on a compact complex manifold X and a connection on a principal bundle over X. These equations generalize the constant scalar curvature equation for a Kähler metric on a complex manifold and the Hermitian-Yang-Mills equations for a connection on a bundle. They have an interpretation in terms of a moment map, where the group of symmetries is an extension of the gauge group of the bundle that moves the base X. We develop a natural "formal picture" for the problem which leads to analytic obstructions for the existence of solutions. When the structure group of the bundle is trivial we recover known obstructions for the theory of cscK metrics, in particular the Futaki invariant and the notion of geodesic stability. This is joint work with Luis Alvarez Consul and Oscar Garcia Prada.
MOS 27th January 2011
15:30 to 16:30
On the Calculus underlying Donaldson-Thomas theory
On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exteriour differentiation. This package is called a "calculus". The moduli space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.
MOS 1st February 2011
10:00 to 11:00
Poincare families and line bundles on moduli stacks of G-bundles
The talk deals with the following question: Which moduli spaces of principal G-bundles over a smooth projective curve carry Poincare families (or universal families)? The obstruction is the Brauer class of the moduli stack as a gerbe over the coarse moduli scheme. This obstruction is described in terms of the root system of G. The proof uses the Picard group of the moduli stack. This is joint work with I. Biswas.
MOS 1st February 2011
11:00 to 12:30
V Baranovsky A compactification for the moduli stack of bundles on a surface
We suggest a new moduli functor of quasibundles on a smooth projective surface, which is a stack theoretic counterpart of the Uhlenbeck compactification, and briefly outline its relation with other compactifications constructed by other authors (Balaji, Timofeeva, Schmitt and a few others).
MOS 3rd February 2011
14:00 to 15:00
Cell decompositions of moduli spaces and generalisations
MOS 3rd February 2011
15:30 to 16:30
Hodge polynomials of the moduli space of SL (2,C) - character varieties
Let X be a compact Riemann surface of genus g. SL(2,C)-character varieties of X are rich objects which lie in the intersection of algebraic geometry, complex geometry, and differential geometry. While they are diffeomorphic to moduli spaces of Higgs bundles, as algebraic varieties they are very different. Character varieties are affine, while Higgs moduli spaces are foliated by the fibers of the Hitchin map which are compact algebraic subvarieties. There has been much work investigating the mixed Hodge structures on the cohomology groups of these character varieties, and their Hodge polynomials have been computed using number theoretical techniques. Our goal is to compute the Hodge polynomial of SL(2,C)-character varieties, by stratifying these spaces in such a way that Hodge structure theory gives simpler formulas for the strata, so we are allowed to compute the whole polynomials in terms of the Hodge polynomials of the strata. This is work in progress with V. Muñoz and P. Newstead.
MOS 8th February 2011
11:30 to 12:30
E Izadi Counter-examples of high Clifford index to Prym-Torelli
For an \'etale double cover of smooth curves, the Prym variety is essentially the difference'' between the jacobians of the two curves. The Torelli problem for the Prym map asks when two double covers have the same Prym variety. It is known that the Prym map from the moduli space of double covers of curves of genus g at least 7 to principally polarized abelian varieties of dimension g-1 is generically injective. Counter-examples to the injectivity of the Prym map were, up to now, given by Donagi's tetragonal construction and by Verra's construction for plane sextics. It was conjectured that all counter-examples are obtained from double covers of curves of Clifford index at most 3. I will discuss counter-examples to this conjecture constructed by myself and Herbert Lange.
MOS 10th February 2011
14:00 to 15:00
The locus of intermediate Jabobians of cubic threefolds
We study the (closure of the) locus of intermediate Jacobians of cubic threefolds in the perfect cone compactification of the moduli space of principally polarized abelian fivefolds for which we obtain an expression in the tautological Chow ring. We also discuss possible generalizations of this locus in higher dimensions. This is joint work with S. Grushevsky.
MOS 10th February 2011
15:30 to 16:30
(k,l)-stability
Narasimhan and Ramanan introduced the concept of (k,l)-stability for vector bundles over projective curves. They used the (0,1) and (1,1)-stability to compute the deformations of the moduli space M(n,d) of stable vector bundles and to define the Hecke cycles. In this lecture I will present some properties of (k,l)-stability for any k and l and describe the set A(k,l) of (k,l)-stable in terms of the Segre invariants. For certain values of k and l I will give the relation between A(k,l) and the Hilbert scheme of M(n,d).
MOS 15th February 2011
10:00 to 11:00
Parahoric bundles and parabolic bundles
Let $X$ be a compact Riemann surface of genus $g \geq 2$ and let $G$ be a semisimple simply connected algebraic group. We introduce the notion of a {\em parahoric} $G$--bundle or equivalently a torsor under a suitable Bruhat-Tits group scheme. We also construct the moduli space of semistable parahoric $G$--bundles and identify the underlying topological space of this moduli space with certain spaces of homomorphisms of Fuchsian groups into maximal compact subgroup of $G$. These results generalize the earlier results of Mehta and Seshadri on parabolic vector bundles. (joint work with C.S Seshadri).
MOS 15th February 2011
11:30 to 12:30
O Serman On the singularities of moduli spaces of principal bundles
We show how local factoriality of GIT quotients can be used to give a description of the singular locus of moduli spaces of principal bundles on curves. We then compute the fundamental group of the smooth locus of these moduli spaces.
MOS 17th February 2011
14:00 to 15:00
T Tao The polynomial method in combinatorial incidence geometry
Combinatorial incidence geometry is concerned with controlling the number of possible incidences between a finite number of geometric objects such as points, lines, and circles, in various domains. Recently, a number of breakthroughs in the subject (such as Dvir's solution of the finite field Kakeya conjecture, or Guth and Katz's near-complete solution of the Erdos distance problem) have been obtained by applying the _polynomial method_, in which one uses linear algebra (or algebraic topology) to efficiently captures many of these objects inside an algebraic variety of controlled degree, and then uses tools from algebraic geometry to understand how this variety interacts with the other objects being studied. In this talk we give an introduction to these methods.
MOS 17th February 2011
15:30 to 16:30
N Nitsure Schematic Harder Narasimhan stratification
The Harder Narasimhan type (in the sense of Gieseker semistability) of a pure-dimensional coherent sheaf on a projective scheme is known to vary semi-continuously in a flat family, which gives the well-known Harder Narasimhan stratification of the parameter scheme of the family, by locally closed subsets. We show that each stratum can be endowed with a natural structure of a locally closed subscheme of the parameter scheme, which enjoys an appropriate universal property. As an application, we deduce that pure-dimensional coherent sheaves of any given Harder Narasimhan type form an Artin algebraic stack. As another application - jointly with L. Brambila-Paz and O. Mata - we describe moduli schemes for certain rank 2 unstable vector bundles on a smooth projective curve, fixing some numerical data.
MOS 22nd February 2011
10:00 to 11:00
Moduli of Symplectic Maximal Representations
Maximal representations of surface groups into symplectic real groups have been extensively studied in the last years. Beautiful results have been obtained using either the algebraic approach offered by the theory of Higgs bundles or a geometric approach based on a formula coming from bounded cohomology. After having recalled those results, we will construct, for a maximal representation $\rho: \pi_1(\Sigma_g) \to \mathrm{Sp}(2n, \mathbf{R})$, an open subset $\Omega \subset \mathbf{R} \mathbb{P}^{2n-1}$ where $\pi_1( \Sigma_g)$ acts properly with compact quotient. The topology of the quotient will then be determined. Finally we shall consider the problem of giving an interpretation of the moduli of maximal symplectic representations as a moduli space of $\mathbf{R} \mathbb{P}^{2n-1}$-structures and what are the questions that remain to give a complete answer to that problem.
MOS 22nd February 2011
11:30 to 12:30
Stable bundles and holonomy groups of smooth projective varieties.
MOS 24th February 2011
14:00 to 15:00
On the monodromy of the Hitchin connection.
In this talk I will show that the monodromy representation of the projective Hitchin connection on the sheaf of generalized theta functions on the moduli space of vector bundles over a curve has an element of infinite order in its image. I will explain the link with conformal blocks.
MOS 24th February 2011
15:30 to 16:30
M Speight The $L^2$ geometry of vortex moduli spaces
Let L be a hermitian line bundle over a Riemann surface X. A vortex is a pair consisting of a section of and a connexion on L satisfying a certain pair of coupled differential equations similar to the Hitchin equations. The moduli space of vortices is topologically rather simple. The interesting point is that it has a canonical kaehler structure, geodesics of which are conjectured to approximate the low energy dynamics of vortices. In this talk I will review what is known about this kaehler geometry, focussing mainly on the cases where X is the plane, sphere or hyperbolic plane.
MOS 1st March 2011
10:00 to 11:00
A Szenes Calculating Thom polynomials
There is a polynomial invariant, introduced by Thom in the 1950s, which links the enumerative characteristics of manifolds with the type of singularities which cannot be avoided in maps between them. In this talk, I will report on recent progress in calculating these polynomials.
MOS 1st March 2011
11:30 to 12:30
Higgs bundles for the non-compact dual of the unitary group
We show that the moduli space of U*(2n)-Higgs bundles over a compact Riemann surface is connected. We adopt the Morse-theoretic approach pioneered by Hitchin, and which has already been applied for several other groups, to reduce our problem to the study of connectedness of certain subvarieties of the moduli space. Joint work with O. García-Prada.
MOS 3rd March 2011
14:00 to 15:00
U Goertz Moduli spaces of abelian varieties with Iwahori level structure
We discuss moduli spaces of abelian varieties in positive characteristic $p$, with Iwahori level structure at $p$. In contrast to the moduli space of principally polarized abelian varieties these spaces are singular and are considerably more complicated. Their local structure can be described quite explicitly in terms of matrix equations. As to the global structure, the most interesting part is the supersingular locus (in other words the unique closed "Newton stratum"), whose structure is closely related to Deligne-Lusztig varieties.
MOS 3rd March 2011
15:30 to 16:30
Moduli of local systems on Deligne-Mumford stacks
Most aspects of the theory of moduli of local systems on smooth projective varieties extend to smooth (or even just normal) proper DM-stacks, via a little covering lemma. For other singularities or simplicial varieties we meet a phenomenon of weight filtration. The DM-stack case also allows us to approach the question of open varieties while avoiding the more difficult technical aspects there, and it provides a convenient formalism for finite group actions. In the example of a root stack over the projective line, the moduli space can have components containing no representations into a compact group.
MOS 8th March 2011
10:00 to 11:00
Morse theory and the moduli space of flat connections over a nonorientable surface
We studied the moduli space of flat connections over a nonorientable surface via a Morse theory approach adapted from Atiyah and Bott's work. We defined a Yang-Mills functional on the space of all connections over a nonorientable surface and obtained a Morse stratification of the space. We defined "super central extension" of the fundamental group of the surface. The representation varieties defined using super central extension correspond to gauge equivariant Yang-Mills connections and enable us to obtain reduction formulas for equivariant strata. To describe the phenomenon of the stratification here, we defined "anti-perfect Morse stratification", which is the case when the discrepancy (i.e. the difference between Morse series and Poincare series) reaches its maximal possible value while the perfect Morse stratification is when the discrepancy reaches its minimal possible value zero. This is a joint work with Chiu-Chu Melissa Liu.
MOS 8th March 2011
11:30 to 12:30
Moduli of real and quaternionic bundles over a curve
We examine a moduli problem for real and quaternionic vector bundles over a smooth complex projective curve, and we give a gauge-theoretic construction of moduli spaces for such bundles. These spaces are irreducible subsets of real points inside a complex projective variety. We relate our point of view to previous work by Biswas, Huisman and Hurtubise, and we use this to study the Gal(C/R)-action $[\mathcal{E}] \mapsto [\overline{\sigma^*\mathcal{E}}]$ on moduli varieties of semistable holomorphic bundles over a complex curve with given real structure $\sigma$. We show in particular a Harnack-type theorem, bounding the number of connected components of the fixed-point set of this action by $2^g +1$, where $g$ is the genus of the curve. In fact, taking into account all the topological invariants of a real algebraic curve, we give an exact count of the number of connected components, thus generalizing to rank $r \geq 2$ the results of Gross and Harris on the Picard scheme of a real algebraic curve.
MOS 10th March 2011
14:00 to 15:00
Introduction to a motivic point of view on the cohomology of moduli spaces of bundles on curves
For moduli spaces of vector bundles on curves and some moduli spaces of Higgs bundles, it is possible to compute their cohomology groups in a geometric way, i.e., one can describe the space by a cut-and-paste procedure in terms of cells and symmetric products of the base curve. This gives a rather explicit description of the "motive" of the space. For moduli space of vector bundles this is due to Behrend and Dhillon, relying on an argument of Bifet, Ghione, and Letizia. We will try to give an introduction to this point of view on cohomology calcuations for moduli spaces.
MOS 10th March 2011
15:30 to 16:30
A set up for the cathegorical Langlands Correspondence
MOSW05 11th March 2011
14:00 to 15:00
N Hitchin Poisson modules
Within the context of holomorphic Poisson geometry, there is a natural notion of a Poisson module, which is a holomorphic vector bundle with additional structure. For a symplectic structure, this is just a flat connection, but for a general Poisson structure there are a number of constructions and examples which we shall describe. Even the case of the zero Poisson structure is non-trivial as it leads to the notion of co-Higgs bundles.
MOSW05 11th March 2011
15:15 to 16:15
Nodal curves old and new
I will describe a classical problem going back to 1848 (Steiner, Cayley, Salmon,...) and a solution using simple techniques, but techniques that one would never really have thought of without ideas coming from string theory (Gromov-Witten invariants, BPS states) and modern geometry (the Maulik-Nekrasov-Okounkov-Pandharipande conjecture). In generic families of curves C on a complex surface S, nodal curves -- those with the simplest possible singularities -- appear in codimension 1. More generally those with d nodes occur in codimension d. In particular a d-dimensional linear family of curves should contain a finite number of such d-nodal curves. The classical problem -- at least in the case of S being the projective plane -- is to determine this number. The Göttsche conjecture states that the answer should be topological, given by a universal degree d polynomial in the four numbers C.C, c_1(S).C, c_1(S)^2 and c_2(S). There are now proofs in various settings; a completely algebraic proof was found recently by Tzeng. I will explain a simpler approach which is joint work with Martijn Kool and Vivek Shende.
MOSW02 14th March 2011
10:00 to 11:00
Higgs bundles and surface group representations in non-compact real groups
We will describe how Higgs bundles, inherently holomorphic objects, are related to surface group representations in non-compact real Lie groups and how this relationship can be used to answer questions about the corresponding representation varieties. The real group Sp(4,R) will be given special attention.
MOSW02 14th March 2011
11:30 to 12:30
Generalizations of parabolic bundles related to Higgs bundles
The first part of the talk will be devoted to work in progress together with O. Biquard and Ó. García-Prada on a version of parabolic Higgs bundles which correspond to (semistable) G-local systems on a punctured Riemann surface, where G is a real semisimple Lie group. In the second part we will talk about ongoing work with M. Logares on a generalization of parabolic bundles which correspond (via a construction which relates them to conic bundles) to Sp(6,R)-local systems on compact Riemann surfaces. Emphasis will be on the stability conditions; a unifying theme will be the interpretation of the local terms in the formula for the parabolic degree in terms of GIT.
MOSW02 14th March 2011
15:00 to 16:00
An Algebra of Observables for Cross Ratios
We define a Poisson Algebra called the swapping algebra using the intersection of curves in the disk. We interpret a subalgebra of the fraction swapping algebra -- called the algebra of multifractions -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of SL(n;R)-opers with trivial holonomy. We finally relate our Poisson structure to the Drinfel'd-Sokolov structure and to the Atiyah-Bott-Goldman symplectic structure for classical Teichmüller spaces and Hitchin components.
MOSW02 14th March 2011
16:30 to 17:30
Topology and singularities of free group character varieties
We will discuss some generalities of the geometry, topology and singularities of the the G-character variety of F, that is, the moduli space Hom(F,G)/G of representations of a finitely presented group F into a Lie group G.

Then, we concentrate on the case when G is a complex affine reductive Lie group with maximal compact subgroup K, and F is a free group of rank r. In this situation, it can be proved that Hom(F,K)/K is a strong deformation retract of Hom(F,G)/G; in particular, both spaces have the same homotopy type. In the case G=SL(n,C), one can explicitly describe the singular locus of these character varieties, showing that they have the homotopy type of a manifold only when F or G are abelian, or r+n
MOSW02 15th March 2011
10:00 to 11:00
BPS states, Donaldson-Thomas invariants, and Hitchin pairs
This is a report on recent work in collaboration with Wu-yen Chuang and Guang Pan relating Donaldson-Thomas theory and BPS to the cohomology of the Hitchin system. A string theoretic construction will be presented establishing a connection between curve counting invariants and the work of Hausel and Rodriguez-Villegas. A generalization to parabolic Hitchin pairs will be also briefly discussed.
MOSW02 15th March 2011
11:30 to 12:30
Self-duality of of reducible GL(n) Hitchin fibers
This is a report on a joint work in progress with D.Arinkin. We investigate the self-duality question for the compactified stacky Jacobian of a reducible curve with planar singularities. This is needed for extending the classical limit Langlands duality for GL(n) to the whole moduli of Higgs bundles. I will explain how the self-duality statement extends to a single reduced but possibly reducible curve and will discuss the technical issues that one needs to overcome to prove the statement in families. I will also discuss the way self-duality interacts with stability.
MOSW02 15th March 2011
15:00 to 16:00
On the motives of moduli spaces of Higgs bundles
We will explain an approach to the computation of the cohomology of moduli spaces of Higgs bundles on surfaces, that is closely related to the argument of Harder-Narasimhan for moduli spaces of vector bundles and give an application to the middle cohomology of the moduli space of SL_n Higgs bundles. This is joint work with O. Garcia-Prada and A. Schmitt.
MOSW02 15th March 2011
16:30 to 17:30
R Wentworth Topology of some representation varieties of surface groups
I will discuss recent generalizations of the techniques of Atiyah and Bott on equivariant Morse theory. These extend results on stable bundles to Higgs bundles and stable pairs. As a consequence, information is obtained on the topology of representation varieties into noncompact Lie groups.
MOSW02 16th March 2011
10:00 to 11:00
Domains of Discontinuity for Anosov Representations and Generalized Teichmüller Spaces
Many representations of surface groups (in particular those belonging to "generalized" Teichmüller spaces) are known to satisfy a strong dynamical property: they are Anosov representations. We shall first explain more fully this notion due to F. Labourie. Secondly we will explain how an Anosov representation $\Gamma \to G$ (for any group $\Gamma$) can be interpreted as the holonomy representation of a geometric structure by constructing a domain of discontinuity with compact quotient for $\Gamma$ into a homogenous $G$-space. At last we shall see to what extent this construction can be used in interpreting the generalized Teichmüller spaces as moduli of geometric structures.

This is a joint work with Anna Wienhard.
MOSW02 16th March 2011
11:30 to 12:30
D Toledo Convexity Properties of Energy on Teichmüller Space
Let M be a closed surface of genus at least two, N a manifold of non-positive Hermitian curvature (the Siu-Sampson condition) and fix a homotopy class of maps from M to N (or a representation of the fundamental group of M in the group of isometries of N). For each complex structure J on M there is a harmonic map f:M->N (or an equivariant harmonic map of the universal covers). In situations where this map is unique it depends smoothly on J and its energy E defines a smooth function on the Teichmüller space of M. We prove that this function is plurisubharmonic, and study conditions when it is strictly plurisubharmonic.

This result was suggested by Gromov as an alternative way of developing and strengthening the Siu-Sampson rigidity theory. Indications of these applications will be given as time permits.
MOSW02 16th March 2011
15:00 to 16:00
Local rigidity for complex hyperbolic lattices
I will explain how Hodge theory can be used to prove local rigidity results for complex hyperbolic lattices.
MOSW02 16th March 2011
16:30 to 17:30
Linear coverings of complex projective manifolds
This talk will survey the methods and applications of our joint work with Katzarkov Pantev and Ramachandran arxiv/0409.0693.
MOSW02 17th March 2011
10:00 to 11:00
A Iozzi Surfaces and bounded cohomology
We introduce the notion of causal representation of a surface group and relate it to that of maximal representation and of tight homomorphism. When the target is SL(2,R) we show that these are hyperbolizations. In the process we define and study the bounded fundamental class of a compact surface (with or without boundary) and establish a result characterizing it among all bounded classes. We relate this to the winding number of Chillingsworth and to work of Calegari on stable commutator length.
MOSW02 17th March 2011
11:30 to 12:30
M Burger Causal representations of surface groups
In this talk we will present a structure theorem concerning causal representations; in particular we will discuss the rationality of the Toledo invariant in the non compact case and explain its relation to the characterisation of non tube type domains in terms of the hermitian triple product.
MOSW02 17th March 2011
15:00 to 16:00
Asymptotics in TQFT
We will via the geometric quantization of moduli spaces of flat connections discuss various asymptotic properties of the associated representations of the mapping class groups.
MOSW02 17th March 2011
16:30 to 17:30
S Choi & K Choi Deforming convex real projective 3-orbifolds
A convex real projective 3-orbifold is the quotient orbifold of a convex domain in $RP^3$ by a discrete group of projective automorphisms in $PGL(4, R)$. Hyperbolic 3-orbifolds form a subclass. The convex real projective 3-manifolds were begun to be studied by Cooper, Long, and Thistlethwaite. We will summarize some of the recent results on deforming convex real projective structures on 3-dimensional orbifolds, including those of Benoist, myself, Marquis, Lee, Hodgson, Cooper, Tillman, and so on. In particular, a numerical study of real projective structures on Coxeter orbifolds is included. Finally, we discuss open problems in this area. Our topic is related to understanding the deformations of $SL(4,R)$-representations of discrete groups.
MOSW02 18th March 2011
10:00 to 11:00
P Boalch Irregular connections, Dynkin diagrams and fission
I'll survey some results (both old and new) related to the geometry of moduli spaces of irregular connections on curves. If time permits this will include: 1) new nonlinear geometric braid group actions, 2) new complete hyperkahler manifolds (including some gravitational instantons) [in work with O. Biquard], and 3) new ways to glue Riemann surfaces together to obtain (symplectic) generalisations of the complex character varieties of surface groups.
MOSW02 18th March 2011
11:30 to 12:30
Fibrations on the moduli of parabolic connections on P^1 minus 4 points
This reports on joint work with Frank Loray and Masa-Hiko Saito. Given a connection with parabolic structure, one can look at the limit as $t\rightarrow 0$ in Hitchin's twistor space. The limit is a $C^*$-fixed Higgs bundle. Breaking up the moduli space according to the isomorphism class of the limit leads to a decomposition in locally closed subvarieties. In the case of rank $2$ connections on $P^1-\{ t_1,t_2,t_3,t_4\}$ we are able to show that the subvarieties are closed. They are the fibers of fibrations, depending on the parabolic weights, which are already known: appearing for example in work of Arinkin and Lysenko, and of Iwasaki, Inaba, Saito. Katz's middle convolution is one of Okamoto's symmetries exchanging the different types of fibrations.
MOS 24th March 2011
14:00 to 15:00
Prym varieties of triple coverings
Classical Prym varieties are principally polarised abelian varieties associated to etale double coverings between curves. We study a special class of Prym-Tjurin varieties of exponent 3, coming from non-cyclicetale triple coverings of curves of genus 2. We show that the moduli space of such coverings is a rational threefold, mapping 10:1 via the Prym map to the moduli space of principal polarised abelian surfaces. The crucial ingredient used to obtain such an explicit description of the moduli space, is that any genus 4 curve which admits a non-cyclic triple cover over a genus 2 curve, is actually hyperelliptic. We also describe the extended Prym map from the moduli space of *admissible* S_3-covers onto A_2 . This is a joint work with Herbert Lange
MOS 24th March 2011
15:30 to 16:30
Koszul cohomology and higher rank vector bundles on curves
Some years ago V. Mercat proposed an interesting conjecture relating the Clifford index of a curve C (which measures the complexity of C in its moduli space) to stable vector bundles of higher rank on C. Even though some counterexamples have been found, Mercat's Conjecture is still expected to hold for a general curve, and the failure locus of the conjecture gives rise to new extremal divisors in the moduli space of curves. I will explain the general problem and discuss a Koszul-theoretic approach to Mercat's prediction.
MOS 4th April 2011
14:00 to 15:00
Bridgeland stability conditions on threefolds and birational geometry
I will explain a conjectural construction of Bridgeland stability conditions on smooth projective threefolds. It is based on a construction of new t-structures. They produce a stability condition if we assume a conjectural Bogomolov-Gieseker type inequality for the Chern character of certain stable complexes. In this talk, I will present evidence for our conjecture, as well as implications of the conjecture to the birational geometry of threefolds. In particular, it implies a weaker version of Fujita's conjecture. This is based on joint work with Aaron Bertram, Emanuele Macrì and Yukinobu Toda.
MOS 5th April 2011
11:30 to 12:30
Quadratic differentials as stability conditions
I will explain how certain moduli spaces of meromorphic quadratic differentials arising in Teichmuller theory are related to spaces of stability conditions on the Fukaya categories of some particular quasi-projective Calabi-Yau 3-folds. These Fukaya categories can be described via Ginzburg algebras associated to quivers defined by triangulations of a Riemann surface; suitable triangulations are obtained from the foliations defined by generic quadratic differentials. This is joint work with Tom Bridgeland.
MOS 5th April 2011
16:30 to 17:30
The stable pairs theory of local curves with stationary descendents
The counting function associated to the moduli space of stable pairs on a 3-fold is conjectured to give the Laurent expansion of a rational function. I will discuss a simple proof of this conjecture in the local curve case with only stationary descendent insertions. The proof uses a combination of reduction rules for descendents and some exact calculations involving the stable pairs vertex. This talk presents joint work with Rahul Pandharipande.
MOS 6th April 2011
11:30 to 12:30
E Segal The Pfaffian-Grassmannian correspondence via Landau-Ginzburg models
The Pfaffian-Grassmannian correspondence is an example of a derived equivalence between two Calabi-Yau threefolds that are not birational. Hori and Tong have given a physical explanation of the correspondence, I'll describe some work-in-progress that is trying to interpret their work mathematically, using categories of B-branes in Landau-Ginzburg models.
MOS 6th April 2011
16:30 to 17:30
The motivic Donaldson-Thomas invariants of fat points
Recent work of Behrend, Bryan and Szendroi gives a nice handle on the degree zero motivic Donaldson-Thomas invariants of threefolds, in which the calculation of the invariants of a single point in smooth three dimensional space plays the key role. I will explain how fat points fit naturally into the theory of 3-Calabi-Yau categories, and also what kind of contributions they make. In contrast with the smooth case, the story here throws up complicated motives and monodromy actions, and it is possible to see directly here the role played by the concept of orientation data.
MOS 7th April 2011
11:30 to 12:30
Postnikov stability for complexes
There are several approaches for extending (semi)stability from sheaves to complexes, or rather triangulated categories. In work with Georg Hein (arXiv 0704.2512, 0901.1554), we have introduced a notion of "Postnikov stability" for general triangulated categories, the motivation being Falting's observation that semistability of vector bundles on smooth, projective curves is characterised by the existence of orthogonal sheaves. Our main result is that for projective varieties, classical stability of sheaves can always be captured by an appropriate Postnikov stability. Some applications of this theory: compactifications of classical moduli spaces using genuine complexes; a more conceptual look on questions around "preservation of stability"; answer to a conjecture of Friedman on stable sheaves on elliptic surfaces (the latter is Bernadara/Hein, 1002.4986).
MOS 7th April 2011
16:30 to 17:30
T Logvinenko Relative spherical objects and spherical fibrations
Seidel and Thomas introduced some years ago a notion of a spherical object in the derived category D(X) of a smooth projective variety X. We introduce a relative analogue of this notion by defining what does it mean for an object E of the derived category D(Z x X) of a fiber product of two schemes Z and X to be spherical over Z. For objects of D(Z x X) which are orthogonal over Z (these are categorical equivalents of a subscheme of X fibered over Z) we show an object to be spherical over Z if and only if it possesses certain cohomological properties similar to those in the original definition by Seidel and Thomas. We then interpret this geometrically for the special case where our objects are actual flat subschemes of X flatly fibered over Y. This is a joint work with Rina Anno of UChicago."
MOS 8th April 2011
11:30 to 12:30
L Katzarkov Spectra and Gaps as cohomological theory
In this lecture we will introduce the theory of spectra and gaps of a category. We will make a connection with theory of algebraic cycles and other classical questions in algebraic geometry.
MOS 8th April 2011
14:00 to 15:00
L Katzarkov Homological Mirror Symmetry, Gaps and Spectra
This lecture will include a description of Homological Mirror Symmetry on examples of manifolds of general type and Fanos. The theory of gaps and spectra will be demonstrated in these examples.
MOSW03 11th April 2011
11:30 to 12:30
Categorified Heisenberg actions on Hilbert schemes
I will describe an action of a quantized Heisenberg algebra on the (derived) categories of coherent sheaves on Hilbert schemes of ALE spaces (crepant resolutions of C^2/G). This action essentially lifts the actions of Nakajima and Grojnowski on the cohomology of these spaces. (joint with Tony Licata)
MOSW03 11th April 2011
15:00 to 16:00
Quiver varieties and tensor products
I will explain how tensor products of representations of Yangian are realized in terms of geometry of quiver varieties.
MOSW03 11th April 2011
16:30 to 17:30
Trivertices and a corresponding class of hyperKahler spaces
Given a graph with lines and 3-valent vertices, one can construct, using a simple dictionary, a Lagrangian that has N=2 supersymmetry in 3+1 dimensions. The vacuum moduli space of such a theory is well known to give moment map equations for a HyperKahler manifold.

We will discuss the class of hyperkahler manifolds which arise due to such a construction and present their special properties. The Hilbert Series of these spaces can be computed and turns out to be a function of the number of external legs and loops in the graph but not on its detailed structure. The corresponding SCFT consequence of this property indicates a crucial universality of many Lagrangians, all of which have the same dynamics.

The talk is based on http://arXiv.org/pdf/1012.2119
MOSW03 12th April 2011
10:00 to 11:00
Stable pairs on local K3 surfaces.
I give a formula which relates Euler characteristic of moduli spaces of stable pairs on local K3 surfaces to counting invariants of semistable sheaves on them. The formula generalizes Kawai-Yoshioka's formula for stable pairs with irreducible curve classes to arbitrary curve classes. I also propose a conjectual multi-covering formula of sheaf counting invariants which, combined with the main result, leads to an Euler characteristic version of Katz-Klemm-Vafa conjecture for stable pairs.
MOSW03 12th April 2011
11:30 to 12:30
Why descendents?
I will discuss several motivations and results concerning descendents in 3-fold sheaf theories.
MOSW03 12th April 2011
15:00 to 16:00
K3 surfaces of genus 17
The moduli space M=M(2, h, 8) of semi-rigid vector bundles on a (polarized) K3 surface (S, h) of genus 17 is a K3 surface of genus 5. Moreover, the universal family gives an equivalence between the derived category of S and a twisted derived category of M. This equivalence induces us a rational map from S to the non-abelian Brill-Noether locus SU(2, K; 5F) of type II (see alg-geom/9704015) in the moduli space of 2-bundles on a curve of genus 5. We show that this map is an isomorphism when the modulus of (S, h) is general, using Thaddeus' formula. As a corollary the moduli space F17 of (S, h)’s is unirational.
MOSW03 12th April 2011
16:30 to 17:30
A Bondal Minuscule varieties and their degenerations
We will give a description of toric degenerations of minuscule varieties in terms of moduli spaces of representations of quivers. We apply this to description of the structure of derived categories of coherent sheaves on minuscule varieties and to constructing Landau-Ginzburg models for them.
MOSW03 13th April 2011
10:00 to 11:00
P Horja Categorical Matrix Factorizations
I will present a notion of matrix factorizations associated to an additive category endowed with a weak action of the integers. The role of the potential is played by a certain natural transformation compatible with the categorical weak action. Various applications of this framework to the understanding of Landau-Ginzburg models will be explored.
MOSW03 13th April 2011
11:30 to 12:30
Landau-Ginzburg/Calabi-Yau correspondence, analytic continuation and global mirror symmetry
Under Landau-Ginzburg/Calabi-Yau correspondence, the quantum cohomology of a weighted projective hypersurface is analytically continued to the FJRW quantum ring of the associated Landau-Ginzburg model. Via the Gamma integral structure, we will see that the analytic continuation is induced from Orlov's equivalence between the derived category of coherent sheaves and the category of matrix factorizations. We also mention to a relation to mirror symmetry. This is based on joint work with Alessandro Chiodo and Yongbin Ruan.
MOSW03 13th April 2011
15:00 to 16:00
A Kuznetsov Exceptional collections on Grassmannians of classical groups
I will describe a new approach to construction of exceptional collections on homogeneous varieties of semisimple algebraic groups. Using this approach I will construct exceptional collections of the expected length on all Grassmannians of classical groups. This is a joint work with Sasha Polishchuk.
MOSW03 13th April 2011
16:30 to 17:30
Ribbon Graphs and Mirror Symmetry
Beginning with a ribbon graph with some extra structure, I will define a dg category, the "constructible plumbing model," which serves as a stand-in for the Fukaya category of the Riemann surface associated to the ribbon graph. When the graph has a combinatorial version of a torus fibration with section, I will define a one-dimensional algebraic curve, and prove that the dg category of vector bundles on the curve is equivalent to the constructible plumbing model, a version of homological mirror symmetry in one-dimension. I will also discuss the higher-dimensional case.

This talk is based on joint work with Nicolo' Sibilla and David Treumann.
MOSW03 14th April 2011
10:00 to 11:00
S-duality and mirror symmetry in Chern-Simons theory
I will present joint work with S. Gukov on S-duality in three-dimensional Chern-Simons theories. Upon analytic continuation, the Chern-Simons "level" k can be taken to be an arbitrary complex number. Then, both Chern-Simons wavefunctions and certain operators that annihilate them acquire a certain symmetry under the inversion of k (S-duality). In terms of the operators, this can be understood in terms of mirror symmetry acting on branes in a hyperkahler space (a Hitchin moduli space).
MOSW03 14th April 2011
11:30 to 12:30
DE Diaconescu BPS states, Donaldson-Thomas invariants and Hitchin pairs
A string theoretic construction will be presented relating BPS states, Donaldson-Thomas invariants and the cohomology of the Hitchin system. In particular, the work of Hausel and Rodrgiguez-Villegas will be given a natural physical explanation based on geometric engineering.

This is work in collaboration with Wu-yen Chuang and Guang Pan.

MOSW03 14th April 2011
15:00 to 16:00
Hilbert schemes of singular plane curves and HOMFLY homology of their links
Intersecting a plane curve with the boundary of a small ball around one of its singularities yields a link in the 3-sphere. To any link may be attached a triply graded vector space, the HOMFLY homology. Taking its Euler characteristic with respect to a certain grading gives the HOMFLY polynomial, which in turn specializes variously to the Alexander polynomial, the Jones polynomial, and the other SU(n) knot polynomials.

We will present a conjecture recovering this invariant from moduli spaces attached to the singular curve. Specifically, we form the Hilbert schemes of points of the curve, and certain incidence varieties inside products of Hilbert schemes. Up to certain shifts of grading, we conjecture that the HOMFLY homology of the link of the singularity is the direct sum of the homologies of these spaces.

This talk presents joint work with J. Rasmussen and A. Oblomkov.
MOSW03 14th April 2011
16:30 to 17:30
D-manifolds, a new theory of derived differential geometry
I describe a new class of geometric objects I call "d-manifolds". D-manifolds are a kind of "derived" smooth manifold, where "derived" is in the sense of the derived algebraic geometry of Jacob Lurie, Bertrand Toen, etc. The definition draws on ideas of Jacob Lurie, David Spivak. The original aim of the project, which I believe I have achieved, is to find the "right" definition of the Kuranishi spaces of Fukaya, Oh, Ohta and Ono, which is the geometric structure on moduli spaces of J-holomorphic curves in a symplectic manifold.

The definition of d-manifolds involves doing algebraic geometry over smooth functions (C-infinity rings); roughly speaking, a d-manifold is a differential-geometric analogue of a scheme with a perfect obstruction theory. D-manifolds form a strict 2-category dMan. It is a 2-subcategory of the larger 2-category of "d-spaces" dSpa. The definition does not involve localization of categories, so we have very good control of what 1-morphisms and 2-morphisms are.

The 2-categories dMan and dSpa have some very nice properties. All fibre products exist in dSpa, and a fibre product of d-manifolds is a d-manifold under weak transversality condition. For example, any fibre product of two d-manifolds over a manifold is a d-manifold. You can glue d-manifolds by equivalences of open d-submanifolds (a kind of pushout in dMan) provided the glued topological space is Hausdorff. There is a notion of "virtual cotangent bundle" of a d-manifold, which lives in a 2-category of virtual vector bundles, and a 1-morphism of d-manifolds is etale (a local equivalence) iff it induces an equivalence of virtual cotangent bundles. And so on.

There are also good notions of d-manifolds with boundary and d-manifolds with corners, and orbifold versions of all this, d-orbifolds.

D-manifolds and d-orbifolds have applications to moduli spaces and enumerative invariants in both differential and algebraic geometry. Almost any moduli space which is used to define some kind of counting invariant should have a d-manifold or d-orbifold structure. Any moduli space of solutions of a smooth nonlinear elliptic p.d.e. on a compact manifold has a d-manifold structure. Any C-scheme with a perfect obstruction theory has a d-manifold structure. In symplectic geometry, Kuranishi spaces and polyfold structures on moduli spaces of J-holomorphic curves induce d-orbifold structures. So much of Gromov-Witten theory, Donaldson-Thomas theory, Lagrangian Floer cohomology, Symplectic Field Theory,... can be rewritten in this language.
MOSW03 15th April 2011
09:30 to 10:30
I Soibelman Integrable systems, Mirror Symmetry and Donaldson-Thomas invariants
This talk is about applications of some ideas from Mirror Symmetry to complex integrable systems. It is a joint work with Maxim Kontsevich, and it is related to our theory of Donaldson-Thomas invariants for 3d Calabi-Yau categories. In particular, wall-crossing formulas lead to a construction of exponential Hodge structure of infinite type".
MOSW03 15th April 2011
11:00 to 12:00
On the motivic class of the commuting variety and related problems
In 1960, Feit and Fine found a beautiful formula for the number of commuting n by n matrices over the finite field F_q. Their result can be reinterpreted as a formula for the motivic class of the commuting variety in the Grothendieck group. We will describe a simple new proof of their formula which allows us to generalize the result to several other settings with applications to motivic Donaldson-Thomas theory.
MOSW03 15th April 2011
13:00 to 14:00
Wall-crossing of the motivic Donaldson-Thomas invariants
We study the motivic Donaldson-Thomas invariants introduced by Kontsevich-Soibelman and Behrend-Bryan-Szendroi. A wall-crossing formula is proved for a certain class of mutations of quivers with potentials.
MOSW03 15th April 2011
14:30 to 15:30
Motivic Donaldson-Thomas theory of the conifold
The talk will explain the computation of the motivic refinement of Donaldson-Thomas theory and related enumerative theories on the resolved conifold geometry, in all chambers of the space of quiver stability conditions. The results are in full agreement with the expected answer from the refined topological vertex of Iqbal-Kozcaz-Vafa, and the wall-crossing computations of Jafferis-Chuang. This is joint work with Andrew Morrison and Kentaro Nagao.
MOS 21st April 2011
14:00 to 15:00
Generalisation of the tetragonal construction
The Donagi conjecture states that the Prym map is injective at a double cover of a curve if the curve does not admit a morphism of degree less or equal to 4 onto the projective line. The talk focusses on 2 subjects, I will explain why the existing proofs of the tetragonal construction do not generalize and then outline the proof of a generalization which give counterexamples to the conjecture. This is joint work with Elham Izadi.
MOS 21st April 2011
15:30 to 16:30
B Osserman Special determinants in higher-rank Brill-Noether theory
We give a brief survey of the role of special determinants in higher-rank Brill-Noether theory, including the original work of Bertram, Feinberg, and Mukai on smooth curves in the case of canonical determinant, generalizations to other determinants, and work (joint with Montserrat Teixidor i Bigas) on degeneration techniques in this context.
MOS 28th April 2011
14:00 to 15:00
C Voisin Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal
Given a smooth projective $3$-fold $Y$, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing $1$-cycles in $Y$ to the intermediate Jacobian $J(Y)$. We consider in this talk the existence of families of $1$-cycles in $Y$ for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When $Y$ itself is uniruled, we relate this property to the existence of an integral homological decomposition of the diagonal of $Y$.
MOS 28th April 2011
15:30 to 16:30
On symplectic hypersurfaces
The Grothendieck-Brieskorn-Slodowy theorem explains a relation between ADE-surface singularities $X$ and simply laced simple Lie algebras $g$ of the same Dynkin type: Let $S$ be a slice in $g$ to the subregular orbit in the nilpotent cone $N$. Then $X$ is isomorphic to $S\cap N$. Moreover, the restriction of the characteristic map $\chi:g\to g//G$ to $S$ is the semiuniversal deformation of $X$. We (j.w. Namikawa and Sorger) show that the theorem remains true for all non-regular nilpotent orbits if one considers Poisson deformations only. The situation is more complicated for non-simply laced Lie algebras. It is expected that holomorphic symplectic hypersurface singularities are rare. Besides the ubiquitous ADE-singularities we describe a four-dimensional series of examples and one six-dimensional example. They arise from slices to nilpotent orbits in Liealgebras of type $C_n$ and $G_2$.
MOSW06 5th May 2011
11:00 to 12:00
Moduli in derived categories
Classical moduli theory was born with a focus on objects we can easily see: varieties, vector bundles, morphisms, etc. In the last half-century, we have come to perceive a slew of subtler invariants, such as the derived category of coherent sheaves on a variety, that are decidedly murkier. Within the last decade, moduli spaces of objects in the derived category began to appear, drawing inspiration from birational geometry and mathematical physics. It turns out that a systematic approach to constructing these moduli spaces bears fruit in such disparate areas as Gromov-Witten theory, arithmetic geometry, and non-commutative algebra. I will describe some aspects of these moduli problems and a few of their principal applications.
MOSW06 5th May 2011
13:45 to 14:45
Moduli of sheaves and strange duality
Strange duality is a symmetry that spaces of sections of determinant line bundles over moduli spaces of sheaves are conjectured to obey. I will discuss the conjecture for moduli spaces of sheaves on K3 surfaces. Questions about the geometry of the moduli space of polarized K3s also arise.
MOSW06 5th May 2011
15:15 to 16:15
Y Toda Curve counting invariants on Calabi-Yau 3-folds
I give an introduction of Donaldson-Thomas type curve counting invariants on Calabi-Yau 3-folds, and explain the recent developments.
MOSW06 5th May 2011
16:30 to 17:30
Moduli space of bundles and Kloosterman sums
The relation between analytic properties of modular forms and arithmetic results has led to many famous results and conjectures. In the geometric analogue of this conjectural relation - called geometric Langlands correspondence quotients of the upper half plane are replaced by moduli spaces of bundles on the curve. We will try to motivate this analogy. Since the geometry of these spaces is complicated in general, very few explicit examples of such modular forms are known. In joint work with B.C. Ngô and Z. Yun - which was motivated by work of Gross and Frenkel - we found an explicit series of such forms which turn out to be closely related to classical Kloosterman sums. This gives an example of the (wild) geometric Langlands correspondence.
MOS 12th May 2011
10:30 to 11:30
Partially positive line bundles
Define a line bundle L on a projective variety to be q-ample, for a natural number q, if tensoring with high powers of L kills coherent sheaf cohomology above dimension q. Thus 0-ampleness is the usual notion of ampleness. Intuitively, a line bundle is q-ample if it is positive "in all but at most q directions". We prove some of the basic properties of q-ample line bundles. Related ideas have been used by Ottem to define what an "ample subvariety" of any codimension should mean.
MOS 12th May 2011
15:30 to 16:30
Metric properties of spaces of stability conditions
The space of Bridgeland stability conditions Stab(X) on the derived category D(X) of a variety X has a natural metric with respect to which the actions of the complex numbers C and of the automorphisms Aut(D(X)) of the category are isometries. Under mild assumptions this metric is complete. For example, when X is a complex projective curve with genus >0 one can compute directly that the quotient Stab(X)/C is isometric to the hyperbolic plane. I will discuss these results and some elementary consequences.
MOS 24th May 2011
10:00 to 11:00
A new approach for the Toledo invariant
I shall explain a new approach for the Toledo invariant from the Higgs bundle point of view, coming from group theoretic properties of Hermitian symmetric spaces. This approach covers all cases uniformly, including the exceptional ones. I shall give some applications. Joint work with O. García-Prada and R. Rubio.
MOS 24th May 2011
11:30 to 12:30
Vector bundles on the algebraic 5-sphere and punctured affine 3-space
The 5-dimensional complex sphere X is isomorphic to SL_3/S_2, which admits a fibration p: X-->Y to the 3-dimensional punctured affine space Y=C^{3}\{0} with C^{2} fibres. It was shown by Fabien Morel that vector bundles on a smooth affine variety are determined by A^{1}-homotopy classes of maps to BGL. The fibration p is an A^{1}-homotopy weak equivalence but the Y above is not affine, so it is natural to look for non-isomorphic vector bundles on Y with isomorphic pull-backs to X. We give interesting examples of such bundles of any rank bigger than 1. The examples are produced from vector bundles on the projective space P=P^2.
MOS 24th May 2011
15:30 to 16:30
I Cheltsov On simple finite subgroups in the Cremona group of rank 3
The Cremona group of rank N is the group of birational selfmaps of the projective space of dimension N. Recently Yura Prokhorov (Moscow) classified all finite simple subgroups in the Cremona group of rank 3 (this answers a question of Serre). I will show how to apply Nadel-Shokurov vanishing and Kawamata subadjunction to study conjugacy classes of the subgroups classified by Prokhorov. In particular, I give a partial answer to another question of Serre on normalizers of finite simple subgroups in the Cremona of rank 3. This is a joint work with Costya Shramov (Moscow).
MOSW07 26th May 2011
11:00 to 12:00
Algebraic approach to tensor product theorems
The aim of the talk is to give the general geometric invariant theoretic approach to proving tensor product theorems of semi-stable objects as highlighted in the work of Bogomolov and Ramanan & Ramanathan. This approach will be used to give algebraic proofs of the tensor product theorem for semi-stable Hitchin pairs over arbitrary ground fields. Towards this, one needs to develop a purely algebraic notion of a Hitchin scheme, an object dual in a certain sense to a Hitchin pair.
MOSW07 26th May 2011
13:45 to 14:45
Representations of the fundamental group and geometric loci of bundles
Starting with Weil's seminal work on vector bundles, the relation between representations of the fundamental group and vector bundles have been studied from various points of view. The work of Narasimhan and Seshadri made clear the relation of unitary representations to polystable bundles. One of Nigel Hitchin's results pertains to representations into the split real forms of semi-simple groups. I will be giving an overview of these ending up with my effort, in collaboration with Oscar Garcia-Prada, to understand these as relating to fixed point varieties under some natural involutions on the moduli of Higgs pairs.
MOSW07 26th May 2011
15:15 to 16:15
On the modular interpretation of the Nagaraj-Seshadri locus
We will survey constructions of moduli spaces for principal bundles on nodal curves over the complex numbers. This includes a moduli space for torsion free sheaves A of rank r and degree zero on an irreducible nodal curve X which are endowed with a homomorphism d: ∧^r A → O_X which is an isomorphism away from the node. It is a degeneration of the moduli space of SL_r(C)-bundles on a smooth curve. In many cases, this moduli space puts a scheme structure on the "Nagaraj-Seshadri locus" inside the moduli space of semistable torsion free sheaves of rank r and degree zero.
MOSW07 26th May 2011
16:30 to 17:30
New results in higher rank Brill-Noether theory
In the last 12 months, many new examples of rank 2 bundles (and some of rank 3 bundles) have been discovered. I will describe briefly the links with Koszul cohomology and the moduli spaces of curves, but most of the talk will be devoted to the construction of the bundles and their relevance to higher rank Brill-Noether theory.
MOS 31st May 2011
10:00 to 11:00
Orthogonal and symplectic parabolic bundles
In this work with S. Majumder and M. L. Wong, we investigate orthogonal and symplectic bundles, with parabolic structure, over a curve.
MOS 31st May 2011
11:30 to 12:30
A Bertram Birational models of the Hilbert scheme of points on $P^2$ are moduli of Bridgeland-stable complexes
The minimal model program applied to the Hilbert scheme of points on $P^2$ yields a series of birational models, followed by a Fano fibration. These birational models are themselves moduli spaces, but not (generally) of sheaves. Rather, they are moduli spaces of Bridgeland-stable objects in the derived category. Moreover, each of them may be identified with moduli of quiver representations of the quiver associated to $P^2$ and each wall-crossing is a GIT wall-crossing for a particular representation. This is joint work with Izzet Coskun and Daniele Arcara.
MOS 2nd June 2011
14:00 to 15:00
U Bruzzo Uhlenbeck-Donaldson compactification for framed sheaves
We study moduli spaces of framed sheaves on projective surfaces and introduce a "partial compactification" a la Uhlenbeck-Donaldson for them.
MOS 2nd June 2011
15:30 to 16:30
Stable Schottky relations
The Schottky problem is to find ways of distinguishing Jacobians from arbitrary principally polarized abelian varieties. From a classical viewpoint (that of this talk) the aim is to find Siegel modular forms that vanish along the Jacobian locus. In this talk we discuss what happens in the stable situation, that is, when the genus increases arbitrarily.
MOS 6th June 2011
15:30 to 16:30
S Keel Mirror symmetry for affine Calabi-Yaus
I will explain my recent conjecture joint with Hacking and Gross, and theorem in dimension two, which gives the mirror to an affine CY manifold of any dimensions as the Spec of an explicit algebra: The algebra comes with a basis parameterized by a Generalisation of Thurston's boundary sphere to Teichmuller space, and a multiplication rule in terms of counts of rational curves.
MOS 7th June 2011
10:00 to 11:00
S Keel Towards a geometric compactification of moduli of polarized K3 surfaces
I'll discuss my recent proof, joint with Hacking and Gross, of Tyurin's conjecture on canonical theta functions for polarized K3 surface, and our expectation that the construction determines a canonical toroidal compactification of moduli of polarized K3 surfaces, such that the universal family extends to a family of SLC Gorenstein K-trivial surfaces.
MOS 7th June 2011
11:30 to 12:30
Localization in quiver moduli spaces and the Refined GW/Kronecker correspondence
Gromov-Witten invariants of weighted projective lines and Euler characteristics of moduli spaces of representations of bipartite quivers are related via the tropical vertex. Using localization techniques it is possible to give explicit formulas for the Euler characteristics in several cases which allows us to give explicit formulas of Gromov-Witten invariants.
MOS 9th June 2011
14:00 to 15:00
Vector bundles and coherent systems on nodal curves
We study moduli spaces of coherent systems on an irreducible rational curve with one node. We determine the conditions for emptiness and nonemptiness of these moduli spaces. We study the properties like irreducibility, smoothness, seminormality and rationality of the moduli spaces.
MOS 9th June 2011
15:30 to 16:30
Linear stability and stability of dual span bundles
Dual span bundles have been constructed and used on many different purposes in algebraic geometry. The stability of these bundles has been proven in many cases and with different applications, from normal generation to the studying of the Picard sheaf, from the investigation on generalized theta-divisors, to that of coherent systems and Brill-Noether loci. We discuss about a work in progress with Lidia Stoppino, relating the stability of these bundles to the linear stability of linear systems on curves as defined by Mumford, providing some questions, some examples, few answers, and illustrating some connection with Butler's conjecture on stability of dual span bundles.
MOS 13th June 2011
17:00 to 18:00
Stability and its ramifications
The lecture will be concerned with the interaction between algebraic geometry and other major fields of mathematics and physics.The main theme will be the notion of "stability" which arose in moduli problems in algebraic geometry and its relationship with topics in partial differential equations,number theory and physics.
MOS 14th June 2011
10:00 to 11:00
L Alvarez-Consul Coupled equations for Kähler metrics and Yang-Mills connections
We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kähler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kähler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of the equations and study obstructions for the existence of solutions, generalizing the Futaki invariant, the Mabuchi K-energy and geodesic stability. We finish by giving some examples of solutions. This is joint work with Mario Garcia-Fernandez and Oscar Garcia-Prada (arXiv:1102.0991 [math.DG]).
MOS 14th June 2011
11:30 to 12:30
Bundles of rank 2 with 4 sections on curves
We study stable bundles with 4 sections on a smooth algebraic curve. We give geometric constructions for such bundles of lowest possible degree on curves of genus 10 and interpret the results in terms of generalized Clifford indices. We report on related examples and the current situations for rank 2 Brill-Noether theory.
MOS 16th June 2011
14:00 to 15:00
Virtual classes, Quot schemes and stable quotients
The theme of the talk is the construction of virtual fundamental classes for a Quot schemes of curves. First, I will explain the construction for smooth projective curves. Then, letting the curve vary in moduli, we obtain the space of stable quotients. Over nodal curves, a modification of the Quot scheme is necessary: a relative construction is made to keep the torsion of the quotient away from singularities. The resulting space can be used to study the Gromov-Witten theory of Grassmannians.
MOS 16th June 2011
15:30 to 16:30
A Ott Bounded cohomology and Higgs bundles
Over the past several years, representations of fundamental groups of compact complex hyperbolic manifolds into Hermitian Lie groups have been studied using bounded group cohomology on the one hand and Higgs bundles on the other hand. In this talk, we will discuss some aspects of the interplay between these two approaches. This is joint work with Tobias Hartnick.
MOS 17th June 2011
14:00 to 15:00
Logarithmic Gromov-Witten invariants
Relative Gromov-Witten invariants, which count curves with specified orders of tangency with a smooth divisor, have proven to be an extremely useful tool in Gromov-Witten theory: the gluing formula allows one to compute GW invariants of varieties by degenerating them to normal crossing unions of two varieties and then computing relative GW invariants on each of these varieties. Siebert and I propose a generalization of relative GW invariants which will allow tangency conditions with much more general divisors, and allow much more complicated degenerations.
MOS 21st June 2011
11:30 to 12:30
M Thaddeus Extensions of Grothendieck's theorem on principal bundles over the projective line
Let G be a split reductive group over a field. Grothendieck and Harder proved that any principal G-bundle over the projective line reduces (essentially uniquely) to a maximal torus. In joint work with Johan Martens, we show that this remains true when the base is a chain of lines, a football, a chain of footballs, a finite abelian gerbe over any of these, or the stack-theoretic quotient of any of these by a torus action.
MOS 23rd June 2011
14:00 to 15:00
Further Examples of Bridgeland Stable Moduli Spaces
Following recent work of Arcara and Bertram who give explicit constructions of moduli spaces of certain zero rank Bridgeland stable objects in the derived category of a surface with zero canonical class, we give a complete description of the Bridgeland stable objects with rank 1 and first Chern class equal to twice the polarization on an irreducible principally polarized abelian surface.
MOS 23rd June 2011
15:30 to 16:30
b-stability and blow-ups
In this lecture we will discuss some technical algebro-geometric questions which arise in connection with the existence problem for Kahler-Einstein metrics. We will begin by recalling the notion of "b-stability", which involves birational transformations of a degeneration. We will then explain how to adapt an argument of Stoppa to make progress towards proving a Kahler-Einstein manifold is b-stable.
MOSW04 27th June 2011
10:00 to 11:00
T Hausel Arithmetic and physics of Higgs moduli spaces
In this talk we discuss the connection between conjectures with Villegas on mixed Hodge polynomials of character varieties of Riemann surfaces achieved by arithmetic means and conjectures on the cohomology of Higgs moduli spaces derived by physicists Chuang-Diaconescu-Pan.
MOSW04 27th June 2011
11:30 to 12:30
Hyperkahler implosion
Symplectic implosion is a construction in symplectic geometry due to Guillemin, Jeffrey and Sjamaar, which is related to geometric invariant theory for non-reductive group actions in algebraic geometry. This talk (based on joint work in progress with Andrew Dancer and Andrew Swann) is concerned with an analogous construction in hyperkahler geometry.
MOSW04 27th June 2011
14:00 to 14:30
Hyperpolygons and moduli spaces of parabolic Higgs bundles
Given an $n$-tuple of positive real numbers $\alpha$ we consider the hyperpolygon space $X(\alpha)$, the hyperkähler quotient analogue to the Kähler moduli space of polygons in $\mathbb{R}^3$. There exists an isomorphism between hyperpolygon spaces and moduli spaces of stable, rank-$2$, holomorphically trivial parabolic Higgs bundles over $\mathbb{C} \mathbb{P}^1$ with fixed determinant and trace-free Higgs field. This allows us to prove that hyperpolygon spaces $X(\alpha)$ undergo an elementary transformation in the sense of Mukai as $\alpha$ crosses a wall in the space of its admissible values. We describe the resulting changes in the core of $X(\alpha)$ as well as the changes in the nilpotent cone of the corresponding moduli spaces of PHBs. If time permits, we will explain how to obtain explicit formulas for the computation of the intersection numbers of the core components of $X(\alpha)$ and of the nilpotent cone components of the corresponding moduli spaces of PHBs. As a final application, we describe the cohomology ring structure of these moduli spaces of PHBs and of the components of their nilpotent cone. This is joint work with Leonor Godinho.
MOSW04 27th June 2011
14:40 to 15:10
R Rubio Higgs bundles and Hermitian symmetric spaces
We study the moduli space of polystable G-Higgs bundles for noncompact real Lie groups G of Hermitian type. First, we define the Toledo character and use it to define the Toledo invariant, for which a Milnor-Wood type inequality is proved. Then, for the maximal value of the Toledo invariant, we state a Cayley correspondence for groups of so-called tube type and point out a rigidity theorem for groups of so-called non-tube type. The proofs of these results are based on the Jordan algebra structure related to the tangent space of the Hermitian symmetric space given by G and are independent of the classification theorem of Lie groups. (Joint work with O. Biquard and Ó. García-Prada.)
MOSW04 27th June 2011
15:20 to 15:50
J Martens Compactifications of reductive groups as moduli stacks of bundles
Given a reductive group G, we introduce a class of moduli problems of framed principal G-bundles on chains of projective lines. Their moduli stacks provide equivariant toroidal compactifications of G. All toric orbifolds are examples of this construction, as are the wonderful compactifications of adjoint groups of De Concini-Procesi. As an additional benefit, we show that every semi-simple group has a canonical orbifold compactification. We further indicate the connection with non-abelian symplectic cutting and the Losev-Manin spaces. This is joint work with Michael Thaddeus (Columbia U).
MOSW04 27th June 2011
16:30 to 17:30
Refined curve counting on algebraic surfaces
Let $L$ be ample line bundle on an an algebraic surface $X$. If $L$ is sufficiently ample wrt $d$, the number $t_d(L)$ of $d$-nodal curves in a general $d$-dimensional sub linear system of |L| will be finite. Kool-Shende-Thomas use the generating function of the Euler numbers of the relative Hilbert schemes of points of the universal curve over $|L|$ to define the numbers $t_d(L)$ as BPS invariants and prove a conjecture of mine about their generating function (proved by Tzeng using different methods).

We use the generating function of the $\chi_y$-genera of these relative Hilbert schemes to define and study refined curve counting invariants, which instead of numbers are now polynomials in $y$, specializing to the numbers of curves for $y=1$. If $X$ is a K3 surface we relate these invariants to the Donaldson-Thomas invariants considered by Maulik-Pandharipande-Thomas.

In the case of toric surfaces we find that the refined invariants interpolate between the Gromow-Witten invariants (at $y=1$) and the Welschinger invariants at $y=-1$. We also find that refined invariants of toric surfaces can be defined and computed by a Caporaso-Harris type recursion, which specializes (at $y=1,-1$) to the corresponding recursion for complex curves and the Welschinger invariants.

This is in part joint work with Vivek Shende.
MOSW04 28th June 2011
10:00 to 11:00
P Belkale Geometric unitarity of the KZ/Hitchin connection on conformal blocks in genus 0
We prove that the vector bundles of conformal blocks, on moduli spaces of genus zero curves with marked points, for arbitrary simple Lie algebras and arbitrary integral levels, carry geometrically defined unitary metrics (as conjectured by K. Gawedzki) which are preserved by the Knizhnik-Zamolodchikov/Hitchin connection. Our proof builds upon the work of T. R. Ramadas who proved this unitarity statement in the case of the Lie algebra sl(2) (and genus zero) and arbitrary integral level.
MOSW04 28th June 2011
11:30 to 12:30
M Teixidor i Bigas Brill-Noether theory for vector bundles with fixed determinant
Consider the set of vector bundles or rank r and fixed determinant L with at least k sections. The case of rank two and L canonical has been studied for a few years and some conjectures of Bertram Feinberg and Mukai are by now mostly proved. We will discuss some partial results for other determinants, mostly for rank two.
MOSW04 28th June 2011
14:00 to 14:30
Moduli of plane sheaves supported on curves of low multiplicity
We will classify the Gieseker semi-stable sheaves on the complex projective plane with support of dimension one and multiplicity four, five and, in some cases, multiplicity six. We will give natural stratifications for their moduli spaces. The strata are defined by means of cohomological conditions and have concrete geometric descriptions.
MOSW04 28th June 2011
14:40 to 15:10
FMR Schaffhauser Topology of moduli spaces of vector bundles on a real algebraic curve
Moduli spaces of real and quaternionic vector bundles on a curve can be expressed as Lagrangian quotients and embedded into the symplectic quotient corresponding to the moduli variety of holomorphic vector bundles of fixed rank and degree on a smooth complex projective curve. From the algebraic point of view, these Lagrangian quotients are irreducible sets of real points inside a complex moduli variety endowed with an anti-holomorphic involution. This presentation as a quotient enables us to generalise the equivariant methods of Atiyah and Bott to a setting with involutions, and compute the mod 2 Poincaré series of these real algebraic varieties. This is joint work with Chiu-Chu Melissa Liu (Columbia).
MOSW04 28th June 2011
15:20 to 15:50
Universal plane curve and moduli spaces of 1-dimensional coherent sheaves
We show that the universal plane curve M of degree d may be seen as a space of isomorphism classes of certain 1-dimensional coherent sheaves on the projective plane. The universal singular locus M' of M coincides with the subvariety of M consisting of sheaves that are not locally free on their support. It turns out that the blow up of M along M' may be naturally seen as a compactification of M_B=M\M' by vector bundles (on support).
MOSW04 28th June 2011
16:30 to 17:30
Rank two Brill-Noether theory and the birational geometry of the moduli space of curves
I shall discuss applications of Koszul cohomology and rank two Brill-Noether theory to the intersection theory of the moduli space of curves. For instance, one can construct extremal divisors in M_g whose points are characterized in terms of existence of certain rank two vector bundles. I shall then explain how these subvarieties of M_g can be thought of as failure loci of an interesting prediction of Mercat in higher rank Brill-Noether theory.
MOSW04 29th June 2011
10:00 to 11:00
N Manton Vortices on Riemann Surfaces
We will discuss the geometry and physics of U(1) vortex solutions on compact Riemann surfaces. The moduli space of N-vortex solutions has a natural Riemannian metric, for which there is a localised expression (Samols-Strachan) although this is not known explicitly. The volume of the moduli space is known, leading to an equation of state for a vortex gas. An asymptotic expression for the moduli space metric for one vortex on a large surface has been obtained, which could be developed further (Dunajski & Manton). The metric is also understood in the limit of a small surface, where the vortex dissolves (Manton & Romao).
MOSW04 29th June 2011
11:30 to 12:30
Monopoles on the product of a surface and the circle
One of the important ingredients of the Witten-Kapustin approach to the geometric Langlands program is the study of singular monopoles on the product of a Riemann surface and an interval; these mediate Hecke transforms. One special case of this, the self-transforms, corresponds to monopoles on the product of a Riemann surface and a circle. We study the moduli of these, and prove a Hitchin-Kobayashi correspondence. When the surface is a torus, there is in addition an interesting Nahm transform to instantons on the product of a three-torus and the line. (with Benoit Charbonneau).
MOSW04 30th June 2011
10:00 to 11:00
Kahler-Einstein metrics and Geometric Invariant Theory
I will discuss an approach to a version of Yau's conjecture, relating Kahler-Einstein metrics to notions of stability. The core of this approach involves estimates for the Chow invariant, obtained from asymptotic analysis. We will also describe progress on a variant of the set-up involving an anticanonical divisor, somewhat analogous to the theory of parabolic bundles. Another theme in the talk will be the importance of making progress on testing stability in explicit cases.
MOSW04 30th June 2011
11:30 to 12:30
Stability conditions for the local projective plane
Describing the space of Bridgeland stability conditions for the local projective plane turns out to be intimately related to classical results by Drezet and Le Potier on inequalities for Chern classes of slope-stable vector bundles on P2. I will describe how this allows one to relate the geometry of this space, and the group of autoequivalences, to the congruence subgroup Gamma1(3). I will also explain a mirror symmetry statement involving the moduli space of elliptic curves with Gamma1(3)-level structure. Time permitting, I will also discuss observations on the same problem for local del Pezzo surfaces. This is based on joint work with Emanuele Macrì.
MOSW04 30th June 2011
14:00 to 14:30
Derived categories and rationality of conic bundles
In this talk I present a joint work with Marcello Bernardara where we show that a standard conic bundle on a rational minimal surface is rational if and only if its derived category admits a semiothogonal decomposition via derived categories of smooth projective curves and exceptional objects. In particular, even if the surface is not minimal, such a decomposition allows to reconstruct the intermediate Jacobian as the direct sum of the Jacobian of those curves.
MOSW04 30th June 2011
14:40 to 15:10
M Khalid Derived equivalences of Azumaya algebras on K3 surfaces
We consider moduli spaces of Azumaya algebras on K3 surfaces. These correspond to twisted sheaves. We prove that when _(A) is zero and c2(A) is within 2r of its minimal bound, where r2 is the rank of A, then the moduli space if non empty is a smooth projective surface. We construct a moduli space of Azumaya algebras on the double cover of the projective plane. In some other special cases we prove a derived equivalence between K3 surfaces and moduli spaces of Azumaya algebras.
MOSW04 30th June 2011
15:20 to 15:50
Deligne-Hodge polynomials for SL(2,C)-character varieties of genus 1 and 2.
We give a method to compute Deligne-Hodge polinomials for various SL(2,C)-character varieties, with fixed conjugacy classes equal to Id, -Id, diagonal and Jordan type matrices. We will split them into suitable stratifications and analyze the behaviour of the polynomial for them. This is joint work with V. Mu~noz and P. Newstead.
MOSW04 30th June 2011
16:30 to 17:30
K Yoshioka Bridgeland stability conditions and Fourier-Mukai transforms
Bridgeland stability condition is preserved under the Fourier-Mukai transform by its definition. I will explain the relation with Gieseker stability. In particular, I will explain kown results on the birational maps of moduli spaces by using Bridgeland stability condition.
MOSW04 1st July 2011
10:00 to 11:00
Moduli spaces of locally homogeneous geometric structures
An Ehresmann structure on a manifold is a geometric structure defined by an atlas of local coordinate charts into a fixed homogeneous space. These structures form deformation spaces which themselves are modeled on the space of representations of the fundamental group. These deformation spaces admit actions of the mapping class group, whose dynamics can be highly nontrivial. In many cases the deformation space embeds inside the space of representations of the fundamental group, and geometric structures provide a powerful tool to study representation spaces of surface groups. This talk will survey several examples of these structures and relate them to other classification problems.
MOSW04 1st July 2011
11:30 to 12:30
Higgs bundles and quaternionic geometry
The circle action on the moduli space of Higgs bundles provides a link between hyperkahler geometry and quaternionic Kahler geometry. The lecture will discuss various aspects of this.