Home » Lectures » Joel Fine: Lectures

Joel Fine: Lectures

September 12, 2023
TITLE: Knot invariants from hyperbolic, SU(3) and G_2 geometries

ABSTRACT: I will describe a conjectural programme for defining invariants of knots or links in the 3-sphere. Whilst the approach is purely mathematical, there should be a connection with both holography and twistor string theory which I would be very grateful if someone could explain to me! Let L be an oriented link in the 3-sphere, which we think of as the ideal boundary of hyperbolic 4-space. The first conjecture is that the number of connected oriented minimal surfaces in H^4 with ideal boundary L is an isotopy invariant of L. The second conjecture is that these counts of minimal surfaces can be used to recover the HOMFLYPT polynomial of L. The minimal surfaces correspond to J-holomorphic curves in the twistor space Z of H^4 (with the NON-integrable Eells-Salamon almost complex structure). So the (conjectural) minimal surface counts are a type of Gromov-Witten invariant. Now, (Z,J) has a natural SU(3) structure, for which the both the symplectic form and the real part of the complex volume form are closed (an “almost Calabi-Yau” if you will). From here, we can consider the Floer-type picture due to Donaldson-Thomas: the product ZxR caries a closed G_2 structure; there is a functional F on the space of all surfaces in Z whose critical points are J-holomorphic curves; moreover solutions to the gradient flow of F are associative submanifolds in ZxR. This leads to a third (admittedly very speculative!) conjecture: the count of minimal fillings of a link L can be categorified by counting associatives in ZxR which interpolate between appropriate J-holomorphic curves. And, if minimal surfaces lead to the HOMFLYPT polynomial, perhaps counting associatives is related to the Khovanov-Rozansky homology, which categorifies the HOMFLYPT polynomial.

September 13, 2017
TITLE: Hypersymplectic 4-manifolds and the G2 Laplacian flow

ABSTRACT:The G 2 Laplacian flow (introduced independently by Bryant and Hitchin) is designed to deform a closed G2 structure to one with vanishing torsion. I will describe joint work with Chengjian Yao on this flow in the special case when the underlying 7-manifold is the product of a 4-manifold X and a 3-torus and the G2 structure is torus invariant. The G2 structure can be described in terms of a triple of symplectic forms on X, called a hypersymplectic structure, and this leads to a problem in 4-dimensional symplectic topology which is of interest in it’s own right. Our main result is that in this situation the G2 Laplacian flow can be extended for as long as the scalar curvature remains bounded. The proof involves ideas from Ricci flow, but it is interesting that the result is stronger than what is currently known for the 4-dimensional Ricci flow. The proof exploits the hypersymplectic structure to carry out a more refined blow-up analysis than is directly possible in the Ricci flow case.

January 9, 2017
TITLE: Adiabatic limits and constant scalar curvature Kähler metrics

ABSTRACT: I will describe how to use adiabatic limits to find constant scalar curvature Kähler metrics, emphasising aspects which carry over to adiabatic limits in general. I will focus on the cases when all fibres are non-singular, as these are completely understood (due to Hong, Brönnle and also my PhD thesis).