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.
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.
The boundedness of Fourier multipliers on non-commutative $L_p$-spaces ($1 < p < \infty$) is a fundamental problem in non-commutative analysis. Building on the non-commutative Cotlar identity introduced by Mei and Ricard (2017), which yields $L_p$-boundedness ($1 < p < \infty$) of Hilbert transforms on amalgamated free products of finite von Neumann algebras, their approach relies heavily on freeness in the underlying free product structure.
In this talk, Xiaoqi Lu introduces a new strategy that overcomes this limitation. Our approach combines a generalized Cotlar identity, which holds on suitable subspaces and captures non-freeness information, with an additional condition related to the property of Rapid Decay to control the remaining components. From this framework, we establish the $L_p$-boundedness ($1 < p < \infty$) of Rademacher-type Hilbert transforms on graph products of finite von Neumann algebras. This unified framework extends earlier results for free products of finite von Neumann algebras and for graph products of groups acting on right-angled buildings. This is a joint work with Runlian Xia.
It is known for a long time, due to a celebrated theorem of Ornstein and Weiss, that (classical/plain) orbit equivalence offers no information about ergodic probability measure preserving actions of amenable groups. On the other hand, conjugacy is too intractable, and effectively hopeless to study in full generality. Quantitative orbit equivalence aims to bridge this gap by adding intermediate layers of rigidity— a strategy that has borne fruit already in the late 1960s but was used as a general framework only semi-recently. In this talk, Spyridon Petrakos will introduce aspects of quantitative orbit equivalence and present a complete picture of it for integer odometers. This is joint work with Petr Naryshkin.
The notion of invariant random subgroups (IRS) is a fruitful, well-studied concept in dynamics on groups. In this talk, Hanna Oppelmayer will explain what it is and how to extend this notion to group von Neumann algebras LG, where G is a discrete countable group. We call it invariant random sub-von Neumann algebra (IRA). As an application, Hanna will provide a result concerning amenable IRAs, which generalises (in the discrete setup) a theorem of Bader-Duchesne-Lécureux about amenable IRSs. This is joint work with Tattwamasi Amrutam and Yair Hartman.