Schedule for: 24w5264 - On the Interface of Geometric Measure Theory and Harmonic Analysis

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Abstract for talk: Let $x$ be a point in the plane. The radial projection $P_x$ maps any point $y \neq x$ in the plane to the unit vector pointing in the direction of $y-x$. We show that for any set $E$ not contained in any line, there exists a point $x \in E$, such that $dim_H P_x(E)= \min\{ 1, \dim_HE\}$. This is an analogue of Marstrand's projection theorem. Joint work with Tuomas Orponen and Pablo Shmerkin.

The Fourier spectrum is a continuously parametrised family of dimensions living between the Fourier and Hausdorff dimensions. I will briefly motivate this concept and discuss applications to distance sets and projections.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Find abstract and notes here: https://drive.google.com/file/d/1s8qX0ZBPBQjTgO0D8iYeHg2USqWR1Y8a/view. The notes can also be accessed from the page https://sites.google.com/view/tuomaths/teaching.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

More than 30 years ago, Jean Bourgain proved a ground-breaking result that continues to exert considerable influence in harmonic analysis and beyond. He proved that if $ {\{\phi_i\}}_{i=1}^n$ is a sequence of orthogonal functions in a locally compact abelian group G, with $ {||\phi_i||}_{\infty} \leq 1$, then for a generic set $ S \subset \{1,2 \dots, n\}$ of size $\sim n^{\frac{2}{p}},$ there exists a constant C(p) depending only on p such that $$ {\left|\left| \sum_{i=1}^n a_i\phi_i \right|\right|}_{L^p(G)} \leq C(p) {\left( \sum_{i=1}^n {|a_i|}^2 \right)}^{\frac{1}{2}}.$$ We will describe some connections between this result and signal recovery. We shall also discuss some continuous manifestations of Bourgain's theorem and connections with the classical restriction phenomenon.

Research groups will be formed during the problem session on Monday and rooms will be assigned at that time.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Find abstract and notes here: https://drive.google.com/file/d/1s8qX0ZBPBQjTgO0D8iYeHg2USqWR1Y8a/view. The notes can also be accessed from the page https://sites.google.com/view/tuomaths/teaching.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Find abstract and notes here: https://drive.google.com/file/d/1s8qX0ZBPBQjTgO0D8iYeHg2USqWR1Y8a/view. The notes can also be accessed from the page https://sites.google.com/view/tuomaths/teaching.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We are given a self-similar Iterated Function System (IFS) on the real line $$\mathcal{S}:=\{S_1,\dots, S_m\}.$$ We fix a sufficiently large interval $\widehat{I}$ which is sent into itself by all mappings of $\mathcal{S}$. For an arbitrary $n \geq 1$, and $\mathbf{i}=(i_1,\dots,i_n)\in\{1,\dots, m\}^n$ the corresponding level $n$ cylinder interval is \begin{equation} \label{2} I_{ i_1,\dots,i_n }: =S_{i_1}\circ\dots\circ S_{i_n}(\widehat{I}). \end{equation} The collection of level $n$ cylinder intervals is $$ \mathcal{I}_n := \left\{ I_{ i_1,\dots,i_n }: (i_1,\dots,i_n)\in\{1,\dots, m\}^n \right\}. $$ The attractor is \begin{equation} \label{1} \Lambda :=\bigcap _{n=1}^{\infty }\bigcup_{I\in\mathcal{I}_n}I. \end{equation} We say that $\Lambda$ is self-similar set. \bigskip {\bf{Open Problem}} Is there a self-similar set of positive Lebesgue measure and empty interior on the line? \bigskip We consider this problem for randomly perturbed self-similar sets, which are obtained in the following way: In the randomly perturbed case, the $n$-cylinder interval $\widetilde{I}_{ i_1,\dots,i_n }$ corresponding to the indices $\mathbf{i}=(i_1,\dots,i_n)\in\{1,\dots, m\}^n$ is obtained by replacing $S_{i_k}$ in formula \eqref{2} by a random and independent of everything translation $\widetilde{S}_{i_k}$ of $S_{i_k}$ for all $k=1,\dots, n$. Then we build the randomly perturbed attractor in an analogous way to formula \eqref{1} from the randomly perturbed cylinder intervals $\widetilde{I}_{i_1\dots i_n}$. That is $$\widetilde{\mathcal{I}}_n := \left\{ \widetilde{I}_{ i_1,\dots,i_n }: (i_1,\dots,i_n)\in\{1,\dots, m\}^n \right\},$$ and the randomly perturbed self-similar set is $$\widetilde{\Lambda }:= \bigcap _{n=1}^{\infty }\bigcup_{\widetilde{I}\in\widetilde{\mathcal{I}}_n}\widetilde{I}. $$ First, I review results related to the Lebesgue measure and Hausdorff dimension of these randomly perturbed self-similar sets. Then, I turn to our new result (joint with M. Dekking, B. Szekely, and N. Szekeres) about the existence of interior points in these randomly perturbed self-similar sets.

We derive $L^{3/2}$ estimates for restricted families of projections onto planes and lines in $\mathbb{R}^3$. Such estimates have application in understanding generic intersections of sets in $\mathbb{R}^3$ with light rays and light planes, and imply a sharp $L^{3/2}$ Kakeya maximal inequality for $SL_2$ lines. This is joint work with John Green and Terry Harris.

Research groups will be formed during the problem session on Monday and rooms will be assigned at that time.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

In the last decade it was realized that many classical problems in geometric measure theory are more tractable for Ahlfors regular sets (perhaps in an approximate or finitary sense) than for general sets. I will survey some recent progress in this direction, obtained in (separate) joint work with H. Wang and with T. Orponen. The main goal of the talk will be to indicate how the Ahlfors regularity assumption comes into play.