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November 2017
by Professor Adam Jacob

Adam Jacob joined our Department in 2015, and was awarded a Hellman Fellow in 2017. He is a pure mathematician who works in the area of differential geometry. In general, he studies nonlinear partial differential equations that arise from the geometry of manifolds, with a special emphasis on manifolds which admit a complex structure. This current project focuses on the Yang-Mills equations, which were originally formulated in context of particle physics, yet can be studied from a purely geometric viewpoint.

Yang-Mills theory was developed in the 1950’s and 60’s and plays a key role in our understanding of particle physics. The equations of motion given by this theory are known as the Yang-Mills equations, which can be formulated purely from a geometric viewpoint. Remarkably, their study has yielded profound insight into the mathematical structure of the geometric spaces where solutions exist. Most prominently, Simon Donaldson analyzed the space of solutions of the Yang-Mills equations to compute new invariants of 4-manifolds, work for which he was awarded the Fields Medal in 1986. In higher dimensions, study of the Yang-Mills equations becomes much more difficult. However, working on a complex manifold, if the connection is compatible with the complex structure, the equation can be simplified in many important ways, which can help overcome some of these difficulties.

I plan on using my Hellman grant to study two distinct problems related to the Yang-Mills equation and complex geometry. The first problem involves understanding the analytic deformation theory of a Yang-Mills connection with isolated singularities. A singularity of the Yang-Mills equations is the analogue of a black hole in the theory of general relativity, in other words it is a point where magnitude of the curvature approaches infinity. Despite the relative prominence the deformation space of smooth Yang-Mills connections plays in in many important problems, the singular theory is not well understood. I intend to focus on the case of complex dimension three (real dimension six), which has applications to the construction of singular solutions to the Yang-Mills equations on G2 manifolds, which are seven dimensional manifolds of exceptional holonomy. It would also be interesting if this analytic deformation theory relates to the algebraic deformation theory of reflexive sheaves. The second problem I intend to study involves investigating how Yang-Mills connections behave under a certain degeneration. Specifically, I will focus on elliptically fibered K3 surfaces, which can be described as a family of holomorphically varying elliptic curves fibered over the Riemann sphere, with 24 singular fibers. If X denotes such a surface, consider a sequence of Riemannian metrics on X collapsing the elliptic fibers. On a vector bundle over X, suppose there exists a sequence of Yang-Mills connections corresponding to this degeneration. Once can then ask: What is the limiting behavior of this sequence? Analytically, this is a very difficult question, since ellipticity of the equation completely degenerates as the fibers shrink. However, there is hope that a robust understanding of the geometry can help overcome these difficulties. In order to identify the limit, one can look to a conjecture of Fukaya, who postulates that after a coordinate dilation the connections will converge to a limiting connection, which restricts to a flat connection on each fiber. From the point of view of Strominger-Yau-Zaslow mirror symmetry, this limiting property is important, as it defines a dual Lagrangian fibration on the mirror K3 surface. Although Fukaya’s conjecture is stated in a much more general setting, I hope to understand this picture better in the surface case.