# Workshop Programme

## for period 27 June - 1 July 2011

### Closing conference

27 June - 1 July 2011

Timetable

 Monday 27 June 09:00-09:55 Registration and welcome 09:55-10:00 Opening remarks and welcome from Sir David Wallace (INI Director) 10:00-11:00 Hausel, T (Oxford) Arithmetic and physics of Higgs moduli spaces Sem 1 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. 11:00-11:30 Morning coffee 11:30-12:30 Kirwan, F (Oxford) Hyperkahler implosion Sem 1 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. 12:30-13:30 Lunch at Wolfson Court 14:00-14:30 Mandini, A (Técnica de Lisboa) Hyperpolygons and moduli spaces of parabolic Higgs bundles Sem 1 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. 14:40-15:10 Rubio, R (Oxford) Higgs bundles and Hermitian symmetric spaces Sem 1 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.) 15:20-15:50 Martens, J (Aarhus) Compactifications of reductive groups as moduli stacks of bundles Sem 1 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). 16:00-16:30 Afternoon Tea 16:30-17:30 Goettsche, L (ICTP) Refined curve counting on algebraic surfaces Sem 1 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. 17:30-18:30 Welcome wine reception 18:45-19:30 Dinner at Wolfson Court (residents only)
 Tuesday 28 June 10:00-11:00 Belkale, P (North Carolina) Geometric unitarity of the KZ/Hitchin connection on conformal blocks in genus 0 Sem 1 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. 11:00-11:30 Morning Coffee 11:30-12:30 Teixidor i Bigas, M (Tufts) Brill-Noether theory for vector bundles with fixed determinant Sem 1 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. 12:30-13:30 Lunch at Wolfson Court 14:00-14:30 Maican, M (IMRA) Moduli of plane sheaves supported on curves of low multiplicity Sem 1 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. 14:40-15:10 Schaffhauser, FMR (Los Andes) Topology of moduli spaces of vector bundles on a real algebraic curve Sem 1 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). 15:20-15:50 Iena, O (SISSA) Universal plane curve and moduli spaces of 1-dimensional coherent sheaves Sem 1 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). 16:00-16:30 Afternoon Tea 16:30-17:30 Farkas, G (Humboldt) Rank two Brill-Noether theory and the birational geometry of the moduli space of curves Sem 1 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. 18:45-19:30 Dinner at Wolfson Court (residents only)
 Wednesday 29 June 10:00-11:00 Manton, N (Cambridge) Vortices on Riemann Surfaces Sem 1 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). 11:00-11:30 Morning Coffee 11:30-12:30 Hurtubise, J (McGill) Monopoles on the product of a surface and the circle Sem 1 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). 12:30-13:30 Lunch at Wolfson Court 13:30-17:30 Free time 19:30-22:00 Conference Dinner at Christ's College
 Thursday 30 June 10:00-11:00 Donaldson, S (Imperial) Kahler-Einstein metrics and Geometric Invariant Theory Sem 1 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. 11:00-11:30 Morning Coffee 11:30-12:30 Bayer, A (Connecticut) Stability conditions for the local projective plane Sem 1 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. 12:30-13:30 Lunch at Wolfson Court 14:00-14:30 Bolognesi, M (Rennes 1) Derived categories and rationality of conic bundles Sem 1 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. 14:40-15:10 Khalid, M (St. Patrick's College Drumcondra) Derived equivalences of Azumaya algebras on K3 surfaces Sem 1 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. 15:20-15:50 Logares, M (ICMAT) Deligne-Hodge polynomials for SL(2,C)-character varieties of genus 1 and 2. Sem 1 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. 16:00-16:30 Afternoon Tea 16:30-17:30 Yoshioka, K (Kobe) Bridgeland stability conditions and Fourier-Mukai transforms Sem 1 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. 18:45-19:30 Dinner at Wolfson Court (residents only)
 Friday 1 July 10:00-11:00 Goldman, W (Maryland) Moduli spaces of locally homogeneous geometric structures Sem 1 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. 11:00-11:30 Morning Coffee 11:30-12:30 Hitchin, NJ (Oxford) Higgs bundles and quaternionic geometry Sem 1 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. 12:30-13:30 Lunch at Wolfson Court 13:30-17:00 Free time 18:45-19:30 Dinner at Wolfson Court (residents only)