We describe the solution to two problems concerning indefinite integral ternary quadratic forms. The first about anisotropic forms was popularized by Margulis following his solution of the Oppenheim Conjecture. The second about the density of isotropic forms was raised by Serre. Joint work with A. Gamburd, A. Ghosh and J. Whang.

This talk by Peter Huston gives an overview of the motivation for and classification of fusion 1-categories and 2-categories. In particular, we will review how fusion 1-categories naturally arise in operator algebras from the subfactor classification programme, which furnishes exotic examples of fusion category, such as the Haagerup subfactor, which are inaccessible by other approaches. Fusion 2-categories are a categorification of fusion 1-category, arising naturally from the study of TQFT in 4D, or as quantum symmetries of fusion 1-categories. We will outline the classification of fusion 2-categories. In particular, we will see that, while fusion 1-categories are wild in the sense that they cannot be constructed from lower dimensional data like finite groups, fusion 2-categories are comparatively tame, expressible in terms of braided fusion 1-categories and extension theory.

In the context of Fourier analysis on the real line, a \textit{one-sided problem} involves deducing properties of a function $f$ from some information about the restriction of its Fourier transform $\widehat{f}$ to a half-line, for instance to $\mathbb{R}_- := (-\infty, 0)$. A prototypical result, which is foundational to the theory of Hardy spaces on $\mathbb{R}$, asserts that if $f \in L^2(\mathbb{R})$ is non-zero and $\widehat{f}$ vanishes on a half-line, then $f$ satisfies the \textit{Szeg\H{o} condition} $\int_{-\infty}^\infty \frac{\log |f(x)|}{1+x^2} \, dx > -\infty$. 

Various problems in operator theory involve the study of functions $f$ satisfying a weaker condition of decay of $\widehat{f}$ on a half-line. In this setting, simple examples show that the Szeg\H{o} condition need not be satisfied. However, the following local Szeg\H{o}-type conditions hold: if the decay of $\widehat{f}$ is strong enough on a half-line, then the mass of the function $f \in L^2(\mathbb{R})$ must concentrate enough for the integral $\int_E \log |f(x)| dx$ to converge on a "massive" set $E$. 

In his talk, Bartosz Malman will describe this mass condensation phenomenon and its applications to operator-theoretic problems.