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.
The work is devoted to the investigation of the solvability of an integro-differential equation in the case of the double scale anomalous diffusion with a sum of two negative Laplacians in different fractional powers in $R^{3}$. The proof of the existence of solutions relies on a fixed point technique. Solvability conditions for the elliptic operators without the Fredholm property in unbounded domains are used.
The Heil-Ramanathan-Topiwala (HRT) conjecture is an open problem in time-frequency analysis. It asserts that any finite combination of time-frequency shifts of a non-zero function in $L^2(\mathbb{R})$ is linearly independent. Despite its simplicity, the conjecture remains unproven in full generality, with only specific cases resolved.
In this talk, I will discuss the HRT conjecture for a specific symmetric configuration of five points in the time-frequency plane, known as the $(3,2)$ configuration. Building upon restriction principles, we prove that for this specific setting, the Gabor system is linearly independent whenever the parameters satisfy certain rationality conditions (specifically, when one parameter is irrational and the other is rational). This result partially resolves the remaining open cases for such configurations. I will outline the proof methods, which involve an interplay of harmonic analysis and ergodic theory. This is joint work with Kasso A. Okoudjou.
Consider the following extremal problem: maximize the amplitude |X_T|, at time T, of a linear recurrent sequence X_1, X_2,... of order N < T, under natural constraints: (I) the initials are uniformly bounded; (II) the characteristic polynomial is R-stable, i.e., its roots are in the origin-centered disc of radius R. While the maximum at time T = N essentially follows from the classical Gautschi bound (1960), the general case T > N turns out to be way more challenging to handle. We find that for any triple (N,R,T), the amplitude is maximized when the roots coincide and have modulus R, and the initials are chosen to align the phases of fundamental solutions. This result is striking for two reasons. First, the same configuration of roots and initials is uniformly optimal for all T, i.e. the whole envelope is maximized at once. Second, we are not aware of any purely analytical proof: ours uses tools from algebraic combinatorics, namely Schur polynomials indexed by hook partitions.
In the talk, I will sketch the proof of this result, making it as self-sufficient as possible under the circumstances. If time permits, we will discuss a related conjecture on the optimal error bounds in complex Lagrange interpolation.
The talk is based on the work https://arxiv.org/abs/2508.13554.
The Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem extends the classical restriction theorem for measures on smooth manifolds to fractal measures. We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions. The proof uses number fields to construct fractal measures in R^d. This work is joint with Robert Fraser and Kyle Hambrook.
The Heisenberg projection problem asks whether there is an analogue of the Marstrand projection theorem in the first Heisenberg group, namely whether Hausdorff dimension of sets generically decreases under projection, for a natural family of projections arising from the group structure. This problem is still open, but I will discuss a recent improvement to the known bound obtained through a variable coefficient local smoothing inequality.
Rather than going through the proof in detail, I will spend most of the talk introducing the problem and explaining the connection to averaging operators over curves, and explaining why these operators are Fourier integral operators satisfying Sogge's cinematic curvature condition. This condition was originally introduced by Sogge to generalise Bourgain's circular maximal theorem, but it turns out to have useful applications to projection theory.
