Georgia Institute of Technology College of Sciences

We study the sum of the negative eigenvalues of the Dirichlet Laplace operatoron a bounded domain in the semiclassical limit. We give a new proof thatyields not only the Weyl term but also the second asymptotic term involvingthe surface area of the boundary of the domain.The proof is valid under weak smoothness assumptions on the boundary and theresult can be extended to non-local, non-smooth operators like fractionalpowers of the Dirichlet Laplacian.(This is joint work with Rupert L. Frank.)

By means of examples, I will illustrate the connection between orthogonal hypergeometric polynomials which satisfy interesting spectral and self-dual properties and representations of Lie algebras.

I will talk about some progress in proving the Degree Conjecture for torus knots. The conjecture states that the degree of a colored Jones polynomial colored by an irreducible representation of a simple Lie algebra g is locally a quadratic quasi-polynomial. This is joint work with Stavros Garoufalidis.

A set~$A$ of integers is a \textit{Sidon set} if all thesums~$a_1+a_2$, with~$a_1\leq a_2$ and~$a_1$,~$a_2\in A$, aredistinct. In the 1940s, Chowla, Erd\H{o}s and Tur\'an determinedasymptotically the maximum possible size of a Sidon set contained in$[n]=\{0,1,\dots,n-1\}$. We study Sidon sets contained in sparserandom sets of integers, replacing the `dense environment'~$[n]$ by asparse, random subset~$R$ of~$[n]$.Let~$R=[n]_m$ be a uniformly chosen, random $m$-element subsetof~$[n]$. Let~$F([n]_m)=\max\{|S|\colon S\subset[n]_m\hbox{ Sidon}\}$. An abridged version of our results states as follows.Fix a constant~$0\leq a\leq1$ and suppose~$m=m(n)=(1+o(1))n^a$. Thenthere is a constant $b=b(a)$ for which~$F([n]_m)=n^{b+o(1)}$ almostsurely. The function~$b=b(a)$ is a continuous, piecewise linearfunction of~$a$, not differentiable at two points:~$a=1/3$and~$a=2/3$; between those two points, the function~$b=b(a)$ isconstant.