09:00 to 10:00 On the geometry of some Hyperkaehler manifolds I will discuss the geometry of some hyperkaehler manifolds : the QALE geometry of the Hilbert scheme of n-points in the complex plane or the QAC geometry of the cotangent bundle of Grassmannian. INI 1 10:00 to 11:00 X Zhu (Massachusetts Institute of Technology)Nodal degeneration of hyperbolic metrics and application to Weil-Petersson metric on the moduli space This is joint work with Richard Melrose. We analyze the behavior of the Laplacian on the fibres of a Lefschetz fibration and use it to describe the behavior of the constant curvature metric on a Riemann surface of genus $>1$ undergoing nodal degeneration. We apply this to deduce the asymptotics of the Weil-Petersson metric on the moduli space $\mathcal{M}_g$. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Asymptotics of hyperboilic, Weil-Peterssen and Takhtajan-Zograf metrics This will be a continuation of the talk by Xuwen Zhu on our joint work concerning the regularity of the fibre hyperbolic metrics up to the singular fibres for Lefschetz fibrations. In particular this applies to the universal curve over moduli space. I will discuss the marked case with the moduli space $\mathcal{M}_{g,n}$ of surfaces of genus $g$ with $n$ ordered distinct points in the stable range, $2g+n\ge3.$ As in the unmarked case the description of the regularity of the fibre hyperbolic metrics, up to the divisors forming the `boundary' of the Knudsen-Deligne-Mumford compactification, implies boundary regularity for the Weil-Peterssen metric. In this case it also leads to an asymptotic description of the Takhtajan-Zograf metric which contributes to the Chern form of the determinant bundle for $\bar\partial$ on the fibres of the universal curve. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 Discussion INI 1 14:30 to 15:30 Renormalized volume on the Teichmuller space of punctured Riemann surfaces We define and study the renormalized volume for geometrically finite hyperbolic 3-manifolds that may have rank-1 cusps. We prove a variation formula, and show that for certain families of convex co-compact hyperbolic metrics degenerating to a geometrically finite hyperbolic metric with rank-1 cusps, the renormalized volume converges to the renormalized volume of the limiting metric. INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 17:00 LD Saper (Duke University)Perverse sheaves on compactifications of locally symmetric spaces Perverse sheave have important applications to representation theory, number theory, and algebraic geometry. I will discuss work in progress to understand the category of perverse sheaves on the Baily-Borel compactification of a Hermitian locally symmetric space; the method is to first work on the less singular reductive Borel-Serre compactification and then push down. Along the way I will introduce these various compactifications and give examples. INI 1
 09:00 to 10:00 R Bielawski (Leibniz Universität Hannover)Asymptotics and compactification of monopole moduli space I shall give several (equivalent) descriptions of asymptotic metrics on various regions of SU(2)-monopole moduli spaces, as well as of their gluing which provides an asymptotic description of SU(2)-monopole metric. I shall also describe the common compactification of the monopole moduli space and of the above asymptotic approximation. INI 1 10:00 to 11:00 Coulomb branches of 3-dimensional $\mathcal N=4$ gauge theories Let $M$ be a quaternionic representation of a compact Lie group $G$. Physicists study the Coulomb branch of the 3-dimensional supersymmetric gauge theory associated with $(G,M)$, which is a hyper-Kaehler manifold, but have no rigorous mathematical definition. When $M$ is of a form $N\oplus N^*$, we introduce a variant of the affine Grassmannian Steinberg variety, define convolution product on its equivariant Borel-Moore homology group, and show that it is commutative. We propose that it gives a mathematical definition of the coordinate ring of the Coulomb branch. If time permits, we will discuss examples arising from quiver gauge theories. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 ALG and the SU($\infty$) Toda equation The purpose of the talk is to pose a question about 4-dimensional moduli spaces of Higgs bundles: what are the minimal assumptions one needs to prove that the moduli space is ALG? INI 1 12:30 to 13:30 Lunch at Wolfson Court 16:00 to 17:00 'Breakout' Session Format: 2 x (10 min introduction + 5-10min discussion) M. Singer - Spectral Curves: What are they good for? B. Schroers - Monopole Clouds: What are they? Problems: For those who have a problem that they can clearly state in less than 3mins and explain why it is interesting in less than (a further) 2mins INI 1 19:30 to 22:00 Conference Dinner at Emmanuel College