• Speaker: Alex Waldron, Michigan State University
• Title: $G_2$-instantons on the 7-sphere
• Date: 09/05/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

I'll discuss a forthcoming paper studying families of $G_2$-instantons on $S^7$, focusing on those which are obtained by pulling back asd instantons on $S^4 $ via the quaternionic Hopf fibration. In the charge-1 case this yields a smooth and complete 15-dimensional family. The situation for higher charge is more complicated, but we are able to compute all the infinitesimal deformations.

• Speaker: David Boozer, UCLA
• Title: Holonomy perturbations of the Chern-Simons functional for lens spaces
• Date: 09/26/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

We describe a scheme for constructing generating sets for Kronheimer and Mrowka's singular instanton knot homology for the case of knots in lens spaces. The scheme involves Heegaard-splitting a lens space containing a knot into two solid tori. One solid torus contains a portion of the knot consisting of an unknotted arc, as well as holonomy perturbations of the Chern-Simons functional used to define the homology theory. The other solid torus contains the remainder of the knot. The Heegaard splitting yields a pair of Lagrangians in the traceless $SU(2)$-character variety of the twice-punctured torus, and the intersection points of these Lagrangians comprise the generating set that we seek. We illustrate the scheme by constructing generating sets for several example knots. Our scheme is a direct generalization of a scheme introduced by Hedden, Herald, and Kirk for describing generating sets for knots in $S^3$ in terms of Lagrangian intersections in the traceless $SU(2)$-character variety for the 2-sphere with four punctures.

• Speaker: Jesse Madnick , McMaster University
• Title: Bubble Tree Convergence of Parametrized Associative Submanifolds
• Date: 10/17/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

In symplectic geometry, part of Gromov's Compactness Theorem asserts that sequences of holomorphic curves with bounded energy have subsequences that converge to bubble trees, and that both energy and homotopy are preserved in this "bubble tree limit." In $G_2$ geometry, the analogues of holomorphic maps are the "associative Smith maps." In this talk, we'll see that familiar analytic features of holomorphic maps also hold for associative Smith maps. In particular, we'll describe how sequences of associative Smith maps give rise to bubble trees, and how energy and homotopy are again preserved in the limit. This is joint work with Da Rong Cheng and Spiro Karigiannis.

• Speaker: Boyu Zhang, Princeton University
• Title: Classification of links with Khovanov homology of minimal rank
• Date: 10/31/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

In this talk, I will present a classification of links whose Khovanov homology has minimal rank, which answers a question asked by Batson and Seed. The proof is based on an excision formula for singular instanton Floer homology that allows the excision surface to intersect the singularity. We will use the excision theorem to define an instanton Floer homology for tangles on sutured manifolds, and show that its gradings detect the generalized Thurston norm for punctured surfaces. This is joint work with Yi Xie.

• Speaker: Shelly Harvey, Rice
• Title: Pure braids and link concordance
• Date: 12/05/2019
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall

If one considers the set of m-component based links in R^3 with a 4-dimensional equivalence relationship on it, called concordance, one can form a group called the link concordance group, C^m. Questions in concordance are important in for classification questions in topological and smooth 4-manifolds It is well known that the link concordance group contains the isotopy class of pure braid with m strands, P_m. That is, two braids are concordant if and only if they are isotopic! In the late 90's Tim Cochran, Kent Orr, and Peter Teichner defined a filtration of the knot/link concordance group called the n-solvable filtration. This filtration gives a way to approximate whether a link is trivial in the group. We discuss the relationship between pure braids and the n-solvable filtration as well as various other more geometrically defined filtrations coming from gropes and Whitney towers. This is joint work with Aru Ray and Jung Hwan Park.