# Summer Term 2014

David Witt-Nystrom (Cambridge). Homogeneous Monge-Ampere equations and canonical tubular neighbourhoods in Kahler geometry. Friday May 2nd, Huxley 139, 1.30-2.30pm.

Abstract: In this talk I will describe some joint work with Julius Ross. By solving the Homogeneous Monge-Ampere equation on the deformation to the normal cone of a complex submanifold of a Kahler manifold, we get a canonical tubular neighbourhood adapted to both the holomorphic and the symplectic structure. If time permits I will describe an application, namely an optimal regularity result for certain naturally defined plurisubharmonic envelopes.

Friday May 9th. No seminar due to King’s College Geometry day.

Friday May 16th. No seminar.

Liam Watson (Glasgow). Khovanov homology and the symmetry group of a knot. Friday May 23rd, Huxley 130, 1.30-2.30pm.

This talk will introduce a new invariant of tangles derived from Khovanov homology. As application, we construct an invariant of strong inversions of knots in the three-sphere and, in turn, produce an object that is quite sensitive to non-amphicheirality. Surprisingly, this new invariant picks up information that is not detected by the Jones polynomial or, more generally, the Khovanov homology of the original knot.

András Stipsicz (Budapest). Knot Floer homologies.  Friday May 30th, Huxley 139, 1.30-2.30pm.

Knot Floer homology (introduced by Ozsvath-Szabo and independently by Rasmussen) is a powerful tool for studying knots and links in the 3-sphere. In particular, it gives rise to a numerical invariant, which provides a nontrivial lower bound on the 4-dimensional genus of the knot. By deforming the definition of knot Floer homology by a real number t from [0,2], we define a family of homologies, and derive a family of numerical invariants with similar properties.  The resulting invariants provide a family of homomorphisms on the concordance group. One of these homomorphisms can be used to estimate the unoriented 4-dimensional genus of the knot.  We will review the basic constructions for knot Floer homology and the deformed theories and discuss some of the applications. This is joint work with P. Ozsvath and Z. Szabo.

Thomas Walpuski (Imperial). The Seiberg–Witten equation with n spinors in dimension three and gauge theory on G_2–manifolds. Friday June 6th, Huxley 340, 1.30-2.30pm.

In 1998 Donaldson––Thomas proposed to construct enumerative invariants for G_2––manifolds by counting associative submanifolds and/or G_2––instantons. Over the last years it has become clear that such an invariant (if it exists at all) has to be a weighted count of a range of objects between associative submanifolds and G_2––manifolds. I will describe an approach to constructing a coherent system of weights using a version of the Seiberg––Witten equation in dimension three, and discuss some work in progress with Andriy Haydys on the analysis of this equation.

Paul Feehan (Rutgers). Global existence and convergence of smooth solutions to Yang-Mills gradient flow over compact four-manifolds. Friday June 13th, Huxley 139,  1.30-2.30pm.

Given a compact Lie group and a principal bundle over a closed Riemannian manifold, the quotient space of connections, modulo the action of the group of gauge transformations, has fundamental significance for algebraic geometry, low-dimensional topology, the classification of smooth four-dimensional manifolds, and high-energy physics.

The quotient space of connections is equipped with the Yang-Mills energy functional and Atiyah and Bott (1983) had proposed that its gradient flow with respect to the natural Riemannian metric on the quotient space should prove to be an important tool for understanding the topology of the quotient space via an infinite-dimensional Morse theory. The critical points of the energy functional are gauge-equivalence classes of Yang-Mills connections. However, thus far, smooth solutions to the Yang-Mills gradient flow have only been known to exist for all time and converge to critical points, as time tends to infinity, in relatively few cases, including (1) when the base manifold has dimension two or three (Rade, 1991 and 1992, in dimension two and three; G. Daskalopoulos, 1989 and 1992, in dimension two), (2) when the base manifold is a complex algebraic surface (Donaldson, 1985), and (3) in the presence of rotational symmetry in dimension four (Schlatter, Struwe, and Tahvildar-Zadeh, 1998). Global existence of solutions with up to finitely many point singularities (caused by the “bubbling” phenomenon) at up to finitely many times was proved independently by Struwe (1994) and Kozono, Maeda, and Naito (1995). However, the question of global existence and convergence of smooth solutions over general compact, Riemannian, four-dimensional base manifolds has thus far remained unresolved.

In this talk we shall describe our proof of the following result: Given a compact Lie group and a smooth initial connection on a principal bundle over a compact, Riemannian, four-dimensional manifold, there is a unique, smooth solution to the Yang-Mills gradient flow which exists for all time and converges to a smooth Yang-Mills connection on the given principal bundle as time tends to infinity.

Konni Rietsch (King’s College London). On Mirror Symmetry for Grassmannians.  Friday June 20th, Huxley 139, 1.30-2.30pm.

The small quantum cohomology ring of a Grassmannian X encodes enumerative information about holomorphic maps from CP^1 to X, and has a well known presentation and associated combinatorics involving Young diagrams (quantum Schubert calculus). Considering the quantum cup product with the hyperplane class gives rise to a D-module via the small Dubrovin connection. The goal of this talk is to write down a regular function defined on a ‘mirror dual’ Grassmannian with an anti-canonical divisor removed, which we can show encodes the same D-module in the form of a Gauss-Manin system. This regular function also restricts, in suitable coordinates, to the Laurent polynomial superpotential first proposed by Eguchi Hori and Xiong, and our mirror theorem implies a formula for the constant term of the J-function conjectured by Batyrev, Ciocan-Fontanine, Kim and van Straten in 1998. A central role in our proofs (joint work with Robert Marsh) is played by the cluster algebra structure of the homogeneous coordinate ring of the mirror Grassmannian and works of Postnikov, Scott and Marsh-Scott

Lorenzo Foscolo (Stony Brook). MODULI SPACES OF MONOPOLES AND GRAVITATIONAL INSTANTONS. Friday June 27th, Huxley 139, 1.30-2.30pm.

It is well-known that moduli spaces of anti-self-dual (ASD) connections on hyperkähler 4–manifolds are themselves hyperkähler. Using argument from physics, Cherkis and Kapustin suggested that “moduli spaces of solutions to dimensional reductions of the ASD equations are a natural place to look for gravitational instantons”, i.e. complete hyperkähler 4-manifolds with decaying curvature at infinity. The talk will focus on moduli spaces of monopoles with singularities on $\mathbb{R}^3$ and $\mathbb{R}^2 \times S^1$. Thanks to work of Cherkis–Kapustin and others, when 4–dimensional these (are expected to) yield examples of gravitational instantons of type ALF and ALG, respectively. I will discuss a gluing construction in these two settings and show how it can be exploited to understand the asymptotic geometry of the moduli spaces.