Given a set in some manifold M, what is the probability that a quantum particle freely traveling in M spends a positive amount of time in the given set? Sets for which there is a positive (uniform) probability after some long time will be called observation sets. We will survey the well-studied case of compact manifolds and discuss our recent extension of these results to the non-compact setting of the whole Euclidean space. This is joint work with Perry Kleinhenz.
We completely characterize the range of $L^p$-boundedness of certain multilinear Radon-like transforms involving vertical projections in the Heisenberg group. This result is now available on arXiv:2603.17147.
I will present on recent work - joint with John Green, Terence Harris, Kevin Ren, and Yumeng Ou - towards proving lower bounds for the dimensions of Furstenberg sets of circles and sine curves in the plane. A circular $(u,v)$-Furstenberg set is a set that contains a $u$-dimensional subset of each circle from a $v$-dimensional family of circles. One can approach the circular Furstenberg problem by proving estimates for the number of incidences between families of $\delta$-disks and $\delta$-annuli that satisfy certain dimension conditions. For different values of $u$ and $v$, we prove incidence estimates using local smoothing and using trilinear restriction estimates for the cone in $\mathbb{R}^3$. As time permits, I will discuss work relevant to proving dimension estimates for Furstenberg sets of sine curves (which satisfy all of the bounds we prove for circular Furstenberg sets) and/or work for Furstenberg sets of curves that satisfy a more general cinematic curvature condition.
We discuss a general philosophy that loosely states that if the matrix representation of a linear operator is concentrated on its diagonal, then the operator’s compactness is characterized by the decay of its matrix representation along the diagonal. We formulate rigorous versions of this idea and apply them to study the compactness of (bi-parameter) Calderón-Zygmund operators, pseudodifferential operators, and Fourier integral operators. These applications recover and unify various earlier works and provide new results.
Please Note: The Hilbert transform maps L¹ functions into weak-L¹ ones. In fact, this estimate holds true for any operator T(m) defined by a bounded Fourier multiplier m with singularity only in the origin. Tao and Wright identified the space replacing L¹ in the endpoint estimate for T(m) when m has singularities in a lacunary set of frequencies, in the sense of the Hörmander-Mihlin condition. In this talk we will quantify how the endpoint estimate for T(m) for any arbitrary m is characterized by the lack of additivity of its set of singularities . This property of the set of singularities of m is expressed in terms of a Zygmund-type inequality. The main ingredient in the proof of the estimate is a multi-frequency projection lemma based on Gabor expansion playing the role of Calderón-Zygmund decomposition. The talk is based on joint work with Bakas, Ciccone, Di Plinio, Parissis, and Vitturi.
