**Online Analysis Research Seminar**

**Online Analysis Research Seminar**

We aim for talks on topics of current interest in harmonic analysis and adjacent areas that are accessible to a general mathematical audience with basic knowledge in analysis.

Time: Mondays, **12pm** Eastern Time, **biweekly**

Location: Zoom

*Please **subscribe** to receive Zoom link for talks (usually circulated the day before a talk).*

## Schedule Spring 2022

Jan 31: Jingrui Cheng (Stony Brooks)

Feb 14: Bodan Arsovski (Sheffield)

Feb 28: Sean Prendiville (Lancaster)

Mar 14: tbd.

Mar 28: tbd.

Apr 11: Diogo Oliveira e Silva (Instituto Superior Técnico & Birmingham)

## Schedule Fall 2021

Sep 20: Max Jahnke (Federal U. São Carlos) - Top-Degree Global Solvability in CR and Locally Integrable Hypocomplex Structures

Abstract: In this talk, I will discuss my recent work on top-degree global solvability for $\partial_b$ operator defined on a generic sub-manifold of the complex space as well as for the differential complex associated with a locally integrable structure over a smooth manifold. The main assumptions are that the locally integrable structure is hypocomplex and that the differential complex is locally solvable in degree one. One of the main tools is an adaptation of a sheaf theoretical argument due to Ramis-Ruget-Verdier. This is a joint work with Prof. Paulo Domingos Cordaro.

In recent work we construct a measure that is p-adic and q-adic doubling for any coprime p and q, yet not doubling overall. The proof involves an intricate interplay of number theory, geometry and analysis, and here we give an overview of some of the key features.

Let F be a finite subset of an additive group G, and let E be a subset of G. A (translational) tiling of E by F is a partition of E into disjoint translates a+F, a∈ A of F. The periodic tiling conjecture asserts that if a periodic subset E of G can be tiled by F, then it can in fact be tiled periodically; among other things, this implies that the question of whether E is tileable by F at all is logically (or algorithmically) decidable. This conjecture was established in the two-dimensional case G = Z^2 by Bhattacharya by ergodic theory methods; we present a new and more quantitative proof of this fact, based on a new structural theorem for translational tilings. On the other hand, we show that for higher dimensional groups the periodic tiling conjecture can fail if one uses two tiles F_1,F_2 instead of one; indeed, the tiling problem can now become undecidable. This is established by developing a "tiling language" that can encode arbitrary Turing machines.

This is joint work with Rachel Greenfeld.

Oct 11: Irina Holmes (Texas A&M) - A new proof of a weighted John-Nirenberg Theorem, via sparse operators

In this talk we revisit a result of Muckenhoupt and Wheeden, which gives a weighted version of the classical John-Nirenberg Theorem (specifically for Ap weights). We will discuss a modern proof of this result, using the recent machinery of sparse operators.

We show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set. This is joint work with Marianna Csörnyei.

Given a fractal set E on the plane and a set F of directions, can we find one direction \theta\in F such that the orthogonal projection \Pi_{\theta} E is large?

We will survey some classical and modern projection theorems and discuss their applications.

A natural 3-dimensional analogue of Bourgain’s circular maximal function theorem in the plane is the study of the sharp L^p bounds in R^3 for the maximal function associated with averages over dilates of the helix (or, more generally, of any curve with non-vanishing curvature and torsion). In this talk, we present a sharp result, which establishes that L^p bounds hold if and only if p>3. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.

Nov 8: Robert Fraser (Wichita) - Explicit Salem sets in R^n: an application of algebraic number theory to Euclidean harmonic analysis

Nov 15: Yuqiu Fu (MIT) - Decoupling for short generalized Dirichlet sequences

We will discuss some geometric similarities between the sequence {\log n}_{n=N+1}^{N+N^{1/2}} (and sequences with similar convexity properties) and the parabola from a decoupling point of view. Based on those observations we present decoupling inequalities for those sequences. The sequence {\log n}_{n=N+1}^{2N} is closely connected to a conjecture of Montgomery on Dirichlet polynomials but we see some difficulties in studying the sequence {\log n\}_{n=N+1}^{N+N^{\alpha}} for \alpha > 1/2. This is joint work with Larry Guth and Dominique Maldague.

Nov 22: *No Talk (Week of Thanksgiving)*

TBA

Nov 29: Terence Harris (Cornell) - The behaviour of Hausdorff dimension under curved 1-dimensional families of projections

Given a curve C with nonvanishing geodesic curvature in the unit sphere of R^3, it is an open problem whether the Hausdorff dimension of an arbitrary set A is almost surely preserved under projection onto the orthogonal complements of vectors in C. In this talk I will outline some recent progress on this problem, which makes use of some Fourier restriction tools such as decoupling and wave packet decompositions. Toward the end of the talk I will mention a couple of open problems suggested by the approach.

In this survey talk we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic

theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point

of view of oscillation estimates in both the single and several parameter settings. We establish a number of new oscillation inequalities

and give new proofs for known results with elementary arguments.