A hyperbolic monopole is a certain configuration of $SU(2)$ fields in hyperbolic $3$-space, and has two basic numbers associated to it: a positive integer $k$, called the magnetic charge, and a positive real $m$ called the monopole mass. To a hyperbolic monopole, one can associate a spectral curve, which is (generically) a compact Riemann surface of genus $(k-1)^2$ encoding all the information about the fields. In this talk, I will describe joint work with Paul Norbury clarifying how one can compute the mass $m$ from a given spectral curve. I shall be discussing two classes of examples for which this calculation can be made quite explicit: (i) monopoles of charge $k=2$; (ii) monopoles with the symmetry of a platonic solid.

The problem of Riemann Ellipsoids has an long and important history, going back to Newton, MacLaurin, Dirichlet, Riemann and Poincaré. It has its origins in the attempt to provide an explanation to the rotating figure of the Earth. In this talk, we will review this problem from the point of view of Differential Geometry and discuss how the geometric perspective can give some insight into the nonlinear stability of some of its classical solutions.

I will discuss pure spinors and Dirac structures on Lie groups, and explain how they can be used in the study of group-valued moment maps, a theory introduced by Alekseev-Malkin-Meinrenken which is parallel to the usual theory of hamiltonian actions and offers a finite-dimensional approach to the study of certain moduli spaces. This is joint work with A. Alekseev and E. Meinrenken.

For a connected reductive group $G$, and a finite dimensional $G$-module $V$, Alexeev and M. Brion have built the invariant Hilbert scheme: it parametrizes $G$-stable closed subschemes of $V$ affording a fixed, multiplicity-finite representation of $G$ in their coordinate ring. We shall describe this scheme in the simplest case, where it parametrizes invariant deformations of the cone of primitive vectors of a simple $G$-module. The classification we get is related to those of simple Jordan algebras and of wonderful varieties of rank one whose open orbit is affine.

We give a notion of stability on the infinite Grassmannian $Gr(k((z))^{\oplus r})$ for the action of the group $Sl(r,k[[z]])$ coherent with the classic stability notion for vector bundles over a curve. We give a numerical criterium and prove the existence of some geometric quotients.

Using the presentation of orbifolds by proper etale groupoids we first explain how a deformation quantization of an orbifold induces a formal deformation of the convolution algebra. We then construct a twisted trace density, which gives rise to a kind of universal trace on the deformed convolution algebra. The algebraic index formula for orbifolds expresses this trace in terms of characteristic classes of the orbifold. In the case of a reduced orbifold, our index formula proves a conjecture by Fedosov, Schulze and Tarkhanov. Moreover, it also entails the index theorem by Kawasaki for elliptic operators on orbifolds. (Joint work with N. Neumaier, H. Posthuma and X. Tang).

We determine all the Fourier-Mukai transforms for coherent systems consisting of a vector bundle over an elliptic curve and a subspace of its global sections, showing that these transforms are indexed by the positive integers. We prove that the natural stability condition for coherent systems, which depends on a parameter, is preserved by these transforms for small and large values of the parameter. By means of the Fourier-Mukai transforms we prove that certain moduli spaces of coherent systems corresponding to small and large values of the parameter are isomorphic. Finally we draw some conclusions about the possible birational type of the moduli spaces.

In this talk I will attempt to familiarize the audience with the notion of an asymptotically hyperbolic manifold and give a rough outline of the most current joint work of myself and Jie Qing. Briefly, we take an approach similar to that developed by Miao in the asymptotically flat setting to establish a positive mass theorem for asymptotically hyperbolic spin manifolds admitting corners along a smooth hypersurface. Our main analytic achievement uses an integral representation of a solution to a perturbed eigenfunction equation to obtain an asymptotic expansion for a conformal factor prescribing scalar curvature with appropriate scalar curvature lower bound. This allows us to understand the change of the so-called “mass aspect” function under a conformal change of asymptotically hyperbolic metrics and ultimately the conclusion of our result.

Legendrian knots and the Whitney-Graustein Theorem

The first lecture will be a colloquium-style talk. I shall give a very gentle introduction to some basic concepts of contact geometry, notably concerning knots in contact 3-manifolds. This will be used to give a contact geometric proof of the Whitney-Graustein theorem in planar geometry: immersions of the circle in the 2-plane are classified, up to regular homotopy, by their rotation number.

An orbifold is a paracompact Hausdorff space which can locally be described as the quotient of an open subset of $R^n$ by the action of a finite group [4]. For example, the orbit space of the action of a compact Lie group on a manifold, which acts with finite isotropy groups, is an orbifold. Orbifolds which can be obtained in this way are called representable.

In order to study orbifold homotopy theory, we will use a representation by Lie groupoids. Such representations are unique up to essential equivalence. An orbifold is representable precisely when it has a representation by a translation groupoid. We will describe a notion of generalized morphism between Lie groupoids, which takes this notion of essential equivalence into account. These morphisms correspond precisely to what Adem, Chen, Ruan, and others have called good maps of orbifolds. They have been used to study orbifold $K$-theory and various kinds of orbifold cohomology with applications to mathematical physics.

For representable orbifolds, one may also want to use the techniques of equivariant homotopy theory to obtain invariants, since such an orbifold is a $G$-space. However, this representation is not unique, one may be able to describe the same orbifold as both $M/G$ and as $N/H$. In this case the two translation groupoids would be essentially equivalent. So our goal will be to describe an orbifold Bredon cohomology which is invariant under essential equivalence.

In this talk I will give a description of generalized maps between translation groupoids in terms of equivariant morphisms, and give a precise characterization of the essential equivalences between such groupoids. As an application I will show that equivariant Bredon cohomology can be viewed as an orbifold invariant. In the special case when one takes Bredon cohomology with coefficients in the representation rings of the isotropy subgroups, one obtains orbifold $K$-theory. This was considered by Adem and Ruan [1] with coefficients tensored over the rational numbers, and by Honkasalo [3] over a general ground ring. However, our proof is more widely applicable, and much simpler than the proof by Adem.

1. A. Adem, Y. Ruan, Twisted orbifold $K$-theory, Comm. Math. Phys., 237 (2003), pp. 533-556.
2. W. Chen, Y. Ruan, A new cohomology theory of orbifold, Comm. Math. Phys., 248 (2004), pp. 1-31.
3. H. Honkasalo, Equivariant Alexander-Spanier cohomology for actions of compact Lie groups, Math. Scand., 67 (1990), pp. 23-34.
4. I. Satake, On a generalization of the notion of manifold, Proc. of the Nat. Acad. of Sc. U.S.A., 42 (1956), pp. 359-363.

Tight contact structures and Cerf's Theorem

Contact Dehn surgery

Nekrasov has shown how the low-energy behaviour of certain super-symmetric quantum field theories can be derived from calculating equivariant volumes of moduli-spaces of instantons. From a mathematical point of view this work generates a number of remarkable conjectures in algebraic geometry. We review this and related recent work, and then explain an alternative method in symplectic geometry for calculating the equivariant volumes, using varying compactifications of the ADHM spaces and the Jeffrey-Kirwan-Witten method of non-abelian localization. This cohomological method parallels earlier work in K-theory of Nekrasov-Shadchin, in each case expressing the volumes as the iterated residue of a single rational function. This talk is based upon the preprint math.SG/0609841.

Symplectic fillings and property \(\mathcal{P}\) for knots

Constructions of higher-dimensional contact manifolds

A singular foliation $F$ on a complete Riemannian manifold $M$ is said to be Riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets . The singular foliation is said to admit sections if each regular point is contained in a totally geodesic complete immersed submanifold that meets every leaf orthogonally and whose dimension is the codimension of the regular leaves. A typical example of a singular Riemannian foliation with section (s.r.f.s for short) is the partition by orbits of a polar action, i.e. a proper isometric action which admit sections. The most well known example is the partition by orbits of the adjoint action of a compact Lie group on itself. Other examples of s.r.f.s are the partition formed by parallel submanifolds of an isoparametric submanifold and examples constructed by suspension of homomorphism, suitable changes of metric and surgery. In this talk we review some author's results about singular holonomy of s.r.f.s and also some results of a joint work with Toeben and a joint work with Gorodski.

After recalling some classical results on the rigidity of compact group actions on differentiable manifolds, we study the actions preserving a Poisson structure on the manifold. In particular, we state the rigidity of Hamiltonian actions of compact semisimple type.

We construct the Nahm transform for Higgs bundles over a Riemann surface of genus at least 2 as hyperholomorphic connections on the total space of the tangent bundle of its dual Jacobian. Roughly speaking, the Nahm transform is a nonlinear analogue of the Fourier transform, transforming anti-self-dual connections on the Euclidean $\mathbb{R}^{4}$ which are invariant under a lattice $\Lambda\subset\mathbb{R}^{4}$ into anti-self-dual connections on the dual space $(\mathbb{R}^{4})^{\vee}$ which are invariant under the dual lattice $\Lambda^{\vee}$. Although the construction is in principle well understood for any subgroup of translations $\Lambda$, analytical details vary for each $\Lambda$. We generalize the Nahm transform of solutions of Hitchin's equations on a 2-dimensional torus $T^{2}$ to solutions of Hitchin's equations on a Riemann surface $\Sigma$ of genus $g\ge2$. More precisely, denoting by $J^{\vee}$ the dual to the Jacobian of $\Sigma$ we prove that the Nahm transform of an irreducible solution of rank at least 2 of Hitchin's equations over a Riemann surface $\Sigma$ of genus $g\ge2$ is a Hermitian vector bundle $\widehat{\mathcal{E}}\to J^{\vee}\times H^{0}(K_{\Sigma})$ equipped with a hyperholomorphic unitary connection $\widehat\nabla$ without flat factors. Moreover, the holomorphic structure induced by $\widehat\nabla$ on $\widehat{\mathcal{E}}$ with respect to a product complex structure on $J^{\vee}\times H^{0}(K_{\Sigma})$ extends to a holomorphic bundle over $J^{\vee}\times\mathbb{P}(H^{0}(K_{\Sigma})\oplus\mathbb{C})$. Closely related to this result is Bonsdorff's Fourier-Mukai transform for stable Higgs bundles on $\Sigma$; in fact our construction is the differential geometric analogue of theis algebraic geometric construction, with the advantage of providing one additional piece of information: the existence of the hyperholomorphic connection without flat factors on $J^{\vee}\times H^{0}(K_{\Sigma})$. This connection and its curvature are explicitly computed and analysed. This is joint work with M. Jardim.

After some background information on coherent systems on algebraic curves, I will show how the classical theorem of Clifford and some of its refinements can be extended to semistable coherent systems. This is joint work with Herbert Lange.

For any vector $r=(r_1,\ldots,r_n)$, let $M_r$ denote the moduli space (under rigid motions) of polygons in $\mathbb{R}^3$ with $n$ sides whose lengths are $r_1,\ldots,r_n$. We give an explicit characterization of the oriented $S^1$-cobordism class of $M_r$ which depends uniquely on the length vector $r$. The main ingredients to prove our result are the bending action (Kapovich-Millson) and results presented by Ginzburg, Guillemin and Karshon that, under suitable hypothesis, link the $S^1$-cobordism class of an even-dimensional manifold $M$ to data associated to the connected components of the fixed point set $(M)^{S^1}$. Also we prove a formula expressing the volume of $M_r$ as a piecewise polynomial function in the $r_i$ (in accordance with the Duitermaat-Heckman Theorem). The main tool to prove this result are localization theorems in equivariant cohomology (Martin, Jeffrey-Kirwan, Guillemin-Kalkman) together with an equivariant integration formula for symplectic quotients by non-abelian groups (Martin).

The signature of a closed oriented 4-manifold is an important topological invariant. In the case of complex algebraic surfaces holomorphically fibred over a compact Riemann surface, the signature localizes around several “degenerate” fibres under an extra condition. I talk about how an algebraic method explains such a topological phenomena, when the fibre genus is small.

For any vector $r = (r_1,...,r_n)$, let $M_r$ denote the moduli space (under rigid motions) of polygons in $\mathbb{R}^3$ with n-sides with lengths $r_1,..., r_n$. $M_r$ can also be described as the symplectic reduction for the natural action of the torus $U^n_1$ of diagonal matrices in the unitary group $U_n$, on the complex Grassmannian of $2$-planes $Gr_2(\mathbb{C}^n)$ (Hausmann-Knutson). An interesting application of the volume formula for $M_r$ (Vu The Khoi; ---) is the calculation of the cohomology ring $H^*(M_r)$. In fact we show that the coefficients of the Duistermaat-Heckman polynomial encode all the necessary information on the generators and relators of $H^*(M_r)$. The ring $H^*(M_r)$ has already been determined by Hausmann and Knutson, but the technique that we present involves a thorough analysis of how the diffeotype of $M_r$ changes as $r$ crosses a wall in $\mu_{U^n_1}(Gr_{2,n})$ (which we believe has an independent interest) and perhaps gives a geometrically more direct comprehension of $H^*(M_r)$.

The moduli of representations from a surface group into a reductive algebraic group $G$ is a poisson variety. However if two surfaces with boundary have the same Euler characteristic then their moduli are algebraically identical. They are distinguished by their symplectic leaves. When $G=SL(3,\mathbb{C})$, we show that the symplectic leaves for the one-holed torus are generically transverse to the symplectic leaves of the three-holed sphere. The projection mapping the three-holed sphere to the one-holed torus induces a mapping of the coordinate rings of the moduli which kills the poisson bivector of the one-holed torus. Consequently, since the tangent space of a representation breaks up into a direct sum of two symplectic vector spaces, the induced bivector projects to the bivector of the three-holed sphere under projection. This talk reflects work in progress.

In 1987 Miyaoka gave a criterion for the semistability of a vector bundle on a curve in terms of the numerical properties of a suitable divisor. In the last few years several generalizations of this criterion have appeared, dealing with bundles on higher dimensional projective varieties, Kaehler manifolds, principal bundles and Higgs bundles. In this talk I wish to present a version of Miyaoka’s criterion for principal Higgs bundles on complex projective manifolds, relating it to the semistability of the relevant bundles restricted to arbitrary curves and to their numerical properties.