## Information

This Mini-Symposium is a one day event in various areas of Harmonic Analysis and PDEs and will take place at the International Centre for Mathematical Sciences
ICMS on the 5th floor of the Bayes Centre (located at 47 Potterrow) Edinburgh. The event is sponsored by the LMS Harmonic Analysis and PDEs research network and the European Research Council (grant no. 637995 “ProbDynDispEq”).

There will be a wine reception after the workshop.

There is no registration fee for this workshop. However if you are planning to attend, please register by clicking the "Registration" link above.

## Speakers

Cristina Benea (Université de Nantes)

Tim Candy (Bielefeld University)

Silouanos Brazitikos (University of Edinburgh)

Robert Fraser (University of Edinburgh)

## Programme

 10.30 - 11.30 Robert Fraser Large Sets Avoiding Algebraic Polynomial Configurations Abstract: A 2017 result of András Máthé states that, given any degree $d$ polynomial $p : \mathbb{R}^{nv} \to \mathbb{R}$ with rational coefficients, there is a subset $E \subset \mathbb{R}^n$ of Hausdorff dimension $\frac{n}{d}$ that does not contain any $v$ distinct points $x_1, \ldots, x_v$ such that $p(x_1, \ldots, x_v) = 0$. We discuss a version of this result that applies when the coefficients of $p$ are assumed only to be algebraic over the rational numbers. 11.30 - 12.00 Coffee and Tea 12.00 - 13.00 Tim Candy Bilinear restriction estimates for general phases Abstract: Bilinear restriction (or extension) estimates for free waves give an efficient way to exploit both transversality and curvature, and essentially optimal estimates for the wave and Schrodinger equations are known. However, for general phases at very different scales, the essentially sharp bounds have only recently been obtained. We give an overview of these estimates, as well as some extensions of the bilinear theory from free waves to the adapted function spaces U^p. These extensions give a way to connect the bilinear restriction theory with the global well-posedness problem for dispersive PDE. In particular, they can be used to give another solution to the division problem for the wave maps equation. 13.00 - 15.00 Lunch break 15.00-16.00 Cristina Benea On Rubio de Francia's square function and the variational Carleson Abstract: The variational Carleson operator measures the r-variation of a function's Fourier series. Very much like Carleson's operator, any approach requires a good understanding of the interaction between spatial and frequential data, i.e. a time-frequency analysis. Rubio de Francia's operator is a square function associated to Fourier projections onto arbitrary disjoint intervals, and it can be represented as a vector-valued Calderon-Zygmund operator. We will present a time-frequency approach to Rubio de Francia's square function, which eventually lead to a significant simplification (and better understanding) in the variational Carleson proof. 16.00 - 16.30 Coffee and Tea 16.30 - 17.30 Silouanos Brazitikos Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies Abstract: The classical Loomis-Whitney inequality and the uniform cover inequality of Bollob´as and Thomason provide upper bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer’s dual Loomis-Whitney inequality. 17.30 - Wine Reception