Georgia Institute of Technology College of Sciences

Every knot in S^3 bounds a PL (piecewise-linear) disk in the four ball. But this is no longer true for knots in other three manifolds, as demonstrated first by Akbulut, who constructed a knot which does not bound any PL disk in a specific contractible four manifold. Then Levine showed that there exist knots that do not bound a PL disk in any homology four ball. What happens if we relax the condition of bounding PL disk to bounding a PL surface with some given genus? I will discuss the joint work with Hom and Stoffregen, where we proved that for each n, there exists a knot K_n in an integer homology sphere that does not bound a PL surface of genus n in any homology four ball. This talk is meant to be accessible to a broad audience.  

A line arrangement is a collection of lines in the projective plane.  The intersection lattice of the line arrangement is the set of all lines and their intersections, ordered with respect to reverse inclusion.  A line arrangement is called free if the Jacobian ideal of the line arrangement is saturated.  The underlying motivation for this talk is a conjecture of Terao which says that whether a line arrangement is free can be detected from its intersection lattice.  This raises a question - in what ways does the saturation of the Jacobian ideal depend on the geometry of the lines and not just the intersection lattice?  A main objective of the talk is to introduce planar rigidity theory and show that 'infinitesimal rigidity' is a property of line arrangements which is not detected by the intersection lattice, but contributes in a very precise way to the saturation of the Jacobian ideal.  This connection builds a theory around a well-known example of Ziegler.  This is joint work with Jessica Sidman (Mt. Holyoke College) and Will Traves (Naval Academy).

We will continue our discussion of the key ingredients of a multi-scale analysis, namely resolvent decay and the Wegner type estimate. After briefly discussing how the Wegner estimate is obtained in the regime of regular noise, we will discuss the strategy used in Bourgain-Kenig (2005) and Ding-Smart (2018) to obtain analogues thereof using some form of unique continuation principle.

From here, we'll examine the quantitative unique continuation principle used by Bourgain-Kenig, and the lack of any even qualitative analogue on the two-dimensional lattice. From here, we'll discuss the quantitative probabilistic unique continuation result used in Ding-Smart.

 I will discuss the soliton resolution and asymptotic stability problems for the sine-Gordon equation. It is known that the obstruction to the asymptotic stability for the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds. This is joint work with Jiaqi Liu and Bingying Lu.