Georgia Institute of Technology College of Sciences

I will present on recent work  - joint with John Green, Terence Harris, Kevin Ren, and Yumeng Ou -  towards proving lower bounds for the dimensions of Furstenberg sets of circles and sine curves in the plane.  A circular $(u,v)$-Furstenberg set is a set that contains a $u$-dimensional subset of each circle from a $v$-dimensional family of circles.  One can approach the circular Furstenberg problem by proving estimates for the number of incidences between families of $\delta$-disks and $\delta$-annuli that satisfy certain dimension conditions.  For different values of $u$ and $v$, we prove incidence estimates using local smoothing and using trilinear restriction estimates for the cone in $\mathbb{R}^3$.  As time permits, I will discuss work relevant to proving dimension estimates for Furstenberg sets of sine curves (which satisfy all of the bounds we prove for circular Furstenberg sets) and/or work for Furstenberg sets of curves that satisfy a more general cinematic curvature condition.

This talk provides an elementary introduction to skein algebras of surfaces, which serve as quantizations of $SL_n$-character varieties. For surfaces with boundary, we extend this framework to stated skein algebras, demonstrating how they provide simple and transparent geometric interpretations of various quantum group structures.

Specifically, we present a geometric realization of the dual canonical basis of $\mathscr{O}_q(\mathfrak{sl}_n)$ using skeins for $n=2$ and $n=3$. If time permits, we will also show how the skein algebra framework can be used to recover the Shapiro–Schrader embedding of the quantized enveloping algebra into a quantum torus algebra.

In work in progress with Ryan Stees, we show that every closed, oriented 3-manifold can be obtained by 0-surgery on a link. Since the 0-surgery of a link can capture the data of many of the typical isotopy and concordance invariants of a link, particularly in the pairwise linking number 0 case, this result gives us a nice lens through which to study both 3-manifolds and links. However, 0-surgery on a link is certainly not a complete link invariant, and we also give multiple constructions for non-isotopic (and even non-concordant) links with homeomorphic 0-surgeries.

The capillary energy functional is used to model the equilibrium shape of a liquid drop meeting a substrate at a prescribed interior contact angle. We will discuss a rigidity theorem for volume-preserving critical points of the capillary energy in the half-space: among all sets of finite perimeter, every such critical configuration corresponding to a prescribed contact angle between 90 degrees and 120 degrees must be a finite union of spheres and spherical caps with the correct contact angle. Assuming that the tangential part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity extends to the full hydrophobic range of contact angles between 90 degrees and 180 degrees. We will also present an anisotropic counterpart, establishing rigidity under suitable lower density assumptions. This talk is based on joint work with A. De Rosa and R. Resende.