Georgia Institute of Technology College of Sciences

The Ozsvath-Szabo tau-invariant is a concordance invariant coming from knot Floer homology. The tools of bordered Heegaard Floer homology provide a way to study the knot Floer homology of satellite knots, and for many patterns have given formulas for the behavior of tau under satelliting. We give formulas for tau and epsilon of satellite knots whose patterns are braided, meaning they wind around the solid torus without reversing, and we do this without the use of bordered Heegaard Floer homology. Our methods lead us to define the class of squeezed patterns, analogous to squeezed knots as defined by Feller-Lewark-Lobb. We show that all braided patterns are squeezed, and we give a tau formula for squeezed patterns as well. Also, towards a conjecture of Hedden, we show that no squeezed pattern, and thus no braided pattern, with winding number at least 2 induces a homomorphism on the concordance group.

We study the effect of time-periodic forcing on the edge state of the semi-infinite Su–Schrieffer–Heeger (SSH) model, a 1D tight-binding model. Numerical simulations and an asymptotic expansion demonstrate that if the frequency of forcing is in resonance with the continuous spectrum of the unforced Hamiltonian, then on a time scale proportional to the inverse square of the forcing amplitude, the edge state decays in amplitude due to the radiation of its energy into the bulk. A proof is work in progress, and makes use of a new dispersive decay estimate for the time-evolution induced by the Hamiltonian. 

Heegaard Floer homology is a modern algebraic package that has seen utility in many areas of low dimensional topology. In this introductory talk, we will give a birds eye view of its construction and properties, and present a highlight reel on the problems it helps solve. As many of our beloved faculty find use for Floer homology in their research, this talk intends to ease the newcomer into the vernacular they may hear in a Seminar talk, and at the same time, convince them that it is a beautiful and rich field of study. We do not assume knowledge in low dimensional topology. It is also rumoured that some Dunkin Donuts will be in attendance.

For $x$ real, let $ \{ x \}$ be the fractional part of $x$ (i.e. $\{x\} = x - \lfloor x \rfloor $).  The lonely runner conjecture can be stated as follows: for any $n$ positive integers $ v_1 < v_2 < \dots < v_n $ there exists a real number $t$ such that $ 1/(n+1) \le \{ v_i t\} \le n/(n+1) $ for $ i = 1, \dots, n$.  In this paper we prove that if $ \epsilon >0 $ and $n$ is sufficiently large (relative to $\epsilon$) then such a $t$ exists for any collection of positive integers $ v_1 < v_2 < \dots < v_n$ such that $ v_n < (2-\epsilon)n$.  This is an approximate version of a natural next step for the study of the lonely runner conjecture suggested by Tao.  
    
The key ingredient in our proof is a result on coprime mappings.  Let $A$ and $B$ be sets of integers.  A bijection $ f:A \to B$ is a coprime mapping if $ a $ and $f(a)$ are coprime for every $ a \in A$. We show that if $A,B \subset [n]$ are intervals of length $2m$ where $ m = e^{ \Omega({(\log\log n)}^2)}$ then there exists a coprime mapping from $A$ to $B$.