# Analysis Seminar

The Analysis Seminar usually takes place on Tuesdays at 2pm during term time in LTB Watson.

## Spring 2020

### The Interplay of Two Euler-Lagrange Equations in the Context of Self Maps of the Annulus

• George Morrison, New York University Shangha
• Tuesday 10 March 2020, 14:00
• WATN-R17 18, Watson Building
• Tea and coffee will be provided after the talk at the common room

The objective of this talk is to classify all critical points of an energy functional associated to an incompressible elastic deformation of an annulus. We principally consider a sub-class of self maps of the domain in the form of whirls. After a study of the relationship between two Euler-Lagrange systems we resolve the main question in the context of a weighted Dirichlet energy. Of particular interest here is a striking discrepancy in the solution sets depending on whether the underlying spatial dimension is odd or even. This is joint work with Dr. Ali Taheri.

### Weak universalities for some singular stochastic PDEs

• Weijun Xu, Mathematical Institute University of Oxford
• Tuesday 03 March 2020, 15:00
• WATN-R17 18, Watson Building
• Tea and coffee will be provided after the talk at the common room

Abstract not available

### Ito type Chain rule for measure dependent random fields under full and conditional measure flows

• Gonçalo dos Reis (Edinburgh)
• Tuesday 03 March 2020, 14:00
• WATN-R17 18, Watson Building
• Tea and coffee will be provided after the talk at the common room

We present several Itô-Wentzell formulae on Wiener spaces for real-valued functionals random field of Itô type depending on measures. We distinguish the full- and marginal-measure flow cases. Derivatives with respect to the measure components are understood in the sense of Lions. This talk is based on joint work with V. Platonov (U. of Edinburgh), see https://arxiv.org/abs/1910.01892.

### Rough solutions of the $3$-D compressible Euler equations

• Qian Wang (Oxford)
• Tuesday 25 February 2020, 14:00
• WATN-R17 18, Watson Building
• Tea and coffee will be provided after the talk at the common room

I will talk about my work arxiv:1911.05038. We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \omega) \in Hs\times Hs\times Hs'$, $2< s' <s$. At the opposite extreme, in the incompressible case, i.e. with a constant density, the result is known to hold for$\omega\in Hs$, $s>3/2$ and fails for $s\le 3/2$, see the work of Bourgain-Li. It is thus natural to conjecture that the optimal result should be $(v,\varrho, \omega) \in Hs\times Hs\times H^{s’}$, $s>2, \, s’>\frac{3}{2}$. We view our work here as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustic spacetime.

### Distance sets and a nonlinear version of Bourgain's projection theorem

• Pablo Shmerkin (Universidad Torcuato Di Tella)
• Tuesday 18 February 2020, 14:00
• WATN-R17 18, Watson Building
• Tea and coffee will be provided after the talk at the common room

Bourgain’s projection theorem is an extension of his celebrated discretized sum-product estimate that has found striking applications in many areas. I will discuss a generalization of the projection theorem from the family of linear projections to parametrized families satisfying a technical but mild condition. I will present some applications, particularly to the Falconer distance set problem. The proofs are based on applying the original version of the theorem to measures in a suitable multiscale decomposition, that I will describe if time allows.

### Local maximizers for Fourier adjoint restriction estimates

• Giuseppe Negro (Birmingham)
• Tuesday 11 February 2020, 14:00
• WATN-R17 18, Watson Building
• Tea and coffee will be provided after the talk at the common room

Restricting the Fourier transform to curved hypersurfaces improves its continuity properties in Lebesgue spaces. We consider the estimates obtained by such restriction to the paraboloid, to the cone and to the sphere, which are due to Strichartz, Tomas and Stein. These involve a multiplicative constant, whose exact value has been computed only in a few cases, corresponding to low spatial dimension. There is a conjecture, due to Foschi, concerning what the constant and the functions that attain it should be, in arbitrary dimension. We prove that there is an open set of functions on which this conjecture does hold. This is joint work with Felipe Gonçalves (Univ. Bonn).

### Bilinear proof of the decoupling theorem for the moment curve

• Pavel Zorin-Kranich (Bonn)
• Tuesday 04 February 2020, 14:00
• WATN-R17 18, Watson Building
• Tea and coffee will be provided after the talk at the common room

The decoupling theorem for the moment curve, due to Bourgain, Demeter, and Guth, implies the Vinogradov mean value theorem with the sharp exponent.

I will present a bilinear proof of this result that avoids Brascamp-Liebinequalities.

Joint work with S. Guo, Z. Li, and P.L. Yung.

### Restricted Loomis-Whitney type inequalities

• Silouanos Brazitikos (Edinburgh)
• Tuesday 28 January 2020, 14:00
• WATN-R17 18, Watson Building
• Tea and coffee will be provided after the talk at the common room

We consider restricted versions of the Loomis Whitney inequality for geometric and functional setting. We discuss also the reverse ones.

### Explicit Salem Sets in R^n

• Rob Fraser (Edinburgh)
• Tuesday 21 January 2020, 14:00
• WATN-R17 18, Watson Building
• Tea and coffee will be provided after the talk at the common room

In 1981, R. Kaufman showed that the $\tau$-approximable numbers in $\mathbb{R}$ support a measure $\mu$ satisfying

$|\hat{\mu}|\leq C_\epsilon (1+|\xi|)-1/(1+\tau)+\epsilon$

for any $\epsilon>0$. The exponent $-1/(1+\tau)$ is optimal for a set with Hausdorff dimension $2/(1+\tau)$. Sets supporting a measure with nearly optimal pointwise Fourier decay for their Hausdorff dimension are called Salem sets. We show that a higher-dimensional analogue of the $\tau$-approximable numbers is a Salem set. This provides the first explicit example of a Salem set of dimension other than 0, n-1, or n in $\mathbb{R}n$. (Joint work with Kyle Hambrook)

### Large sets without Fourier restriction theorems

• Constantin Bilz (Birmingham)
• Tuesday 14 January 2020, 14:00
• WATN-R17 18, Watson Building
• Tea and coffee will be provided after the talk at the common room

It was recently established that Fourier restriction theorems have implications for the structure of Lebesgue sets of Fourier transforms. On the other hand, no methodical study of these sets is available. In this talk, we construct a function that lies in Lp(Rd)for every p>1 and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We combine this with recent results in restriction theory to prove a lack of valid relationsbetween the Hausdorff dimension of a set and the range of restriction exponents for measures supported in the set.