Georgia Institute of Technology College of Sciences

Satellite operations are one of the most basic operations in knot theory. Many researchers have studied the behavior of knot Floer homology under satellite operations. Most of these results use Lipshitz, Ozsvath and Thurston's bordered Heegaard Floer theory. In this talk, we discuss a new technique for studying these operators, and we apply this technique to a family of operators called L-space operators. Using this theory, we are able in many cases to give a simple formula for the behavior of the concordance invariant tau under such operators. This formula generalizes a large number of existing formulas for the behavior of tau under satellite operations (such as cabling, 1-bridge braids and generalized Mazur patterns), and also has a number of topological applications. This is joint work with Daren Chen and Hugo Zhou.

Suppose $f$ is a univariate polynomial with integer coefficients of absolute value at most $H$, exactly $t$ monomial terms, and degree $d$. Suppose also that $p,q$ are integers of absolute value at most $H$. Then one can determine the sign of $f(p/q)$ in time $(d^3 \log H)^{2+\epsilon}$, by combining work of Liouville from about 170 years ago, work of Mahler from about 60 years ago, work of Neff, Reif, and Pan from about 30 years ago, and more recent refinements.

However, when $t$ is small, one can do much better: When $t=2$, one can determine the sign of $f(p/q)$ in time $\log^5(dH)$, via work of Baker from about 55 years ago. Koiran asked, around 2016, about the $t=3$ case, after proving new bounds on the minimal separation of complex roots of univariate trinomials.

We make progress on Koiran's question by giving a new, dramatically faster algorithm for solving univariate trinomials over the real numbers. The key innovation is a new family of non-hypergeometric series for the roots of $f$ when $f$ is close to having a degenerate root. This is joint work with Emma Boniface and Weixun Deng.

We characterize global minimizers below the so-called first critical field of the inhomogeneous version of the Ginzburg-Landau energy functional in a three-dimensional setting. Minimizers of this functional describe the behavior of type-II superconductors exposed to an external magnetic field, which is characterized by the presence of codimension 2 singularities called vortices where superconductivity is locally suppressed. We will talk about how to adapt the results from the standard Ginzburg-Landau theory into an inhomogeneous framework and present results from a recent work in collaboration with Carlos Roman (Pontificia Universidad Catolica de Chile).

Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t) + \gamma(x) w_t(x,t) + (-\Delta + 1)^{s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier Bessel transform. In particular, if $E \subset \mathbb{R}^+$ is $\mu_\alpha$-relatively dense (where $d\mu_\alpha(x) \approx x^{2\alpha+1}\, dx$) for $\alpha > -1/2$, and $\operatorname{supp} \mathcal{F}_\alpha(f) \subset [R,R+1]$, then we show $$\|f\|_{L^2_\alpha(\mathbb{R}^+)} \lesssim \|f\|_{L^2_\alpha(E)},$$ for all $f\in L^2_\alpha(\mathbb{R}^+)$, where the constants in $\lesssim$ do not depend on $R > 0$. Previous results on PLS theorems for the Fourier-Bessel transform by Ghobber and Jaming (2012) provide bounds that depend on $R$. In contrast, our techniques yield bounds that are independent of $R$, offering a new perspective on such results. This result is applied to derive decay rates of radial solutions of the damped wave equation. This is joint work with Ben Jaye.