The seminar meets Fridays 3:30-4:30 pm in Science Center 507. The organizers are Peter Kronheimer, Fan Ye, and Gage Martin. If you'd like to get future announcements, you could send email to gagemartin@math.harvard.edu.
For those without Harvard IDs, the 5th floor of the Science Center can be accessed by elevator during 3:00-6:00 pm.
Title: Twisted Knots and the Perturbed Alexander Invariant
Abstract: The perturbed Alexander invariant, defined by Bar-Natan and van der Veen, is an infinite family of polynomial invariants of knots in the three-sphere. The first polynomial, rho_1, is quick to calculate and may be better at distinguishing knots than practically any other computable invariant; it also has deep connections to both classical and quantum topology. We will discuss the perturbed Alexander invariant and rho_1 in particular, and give results on the behavior of rho_1 and the classical Alexander polynomial under the operation of applying full twists to a knot. Our arguments use a model of random walks on a knot diagram.
Title: The Seiberg-Witten Equations and Einstein Metrics on Finite Volume 4-Manifolds
Abstract: Irreducible solutions to the Seiberg-Witten equations give a priori estimates for the total scalar curvature of the underlying 4-manifold. This was used by LeBrun in the late 90s to construct the first examples of closed 4-manifolds that satisfy the strict Hitchin-Thorpe inequality yet do not admit any Einstein metrics. In this talk, I will describe an extension of this story to the finite volume setting, where we consider complete metrics with asymptotically hyperbolic cusps. As an application we will construct an infinite family of finite volume 4-manifolds with $T^3$ ends that do not admit any asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei.
Title: Spectral flow and reducible solutions to the massive Vafa-Witten equations
Abstract: The Vafa-Witten equations (with or without a mass term) constitute a non-linear, first order system of differential equations on a given oriented, compact, Riemannian 4-manifold. Because these are the variational equations of a functional, the linearized equations at any given solution can be used to define an elliptic, first order, self-adjoint differential operator. This talk will describe bounds (upper and lower) for the spectral flow between respective versions of this operator that are defined by the elements in diverging sequences of reducible solutions. (The spectral flow is formally the difference between the respective Morse indices of the solutions when they are viewed as critical points of the functional.) In some cases, the absolute value of the spectral flow is bounded along the sequence, whereas in others it diverges. This is a curious state of affairs.