# Antha-PDE

## Warwick University, Coventry, 24-27 July 2023

*Antha-PDE is grateful to the Leverhulme Trust for their support.*

**Organisers: Jean-Claude Cuenin (Loughborough), Simon L. Rydin Myerson (Warwick), Vedran Sohinger (Warwick)**

**Keynote speakers: Hajer Bahouri (CNRS/Sorbonne), Jeffrey Galkowski (University College London), Jonathan Hickman (Edinburgh), Junxian Li (Bonn), Igor Wigman (King’s College London)**

A small, relaxed invitation-based workshop to get PhD students and postdocs interested in the overlap between harmonic analysis, PDE and analytic number theory. Registration is open and all applications are considered, although places are limited. Some travel and accommodation funds are available for invited speakers.

**Contents:**

- Schedule
- Talks or mini-courses by keynote speakers
- Talks by invited speakers
- List of attendees
- Acknowledgements

**Schedule:**

| Monday | Tuesday | Wednesday | Thursday |

9:15-9:45 | Registration | |||

10-11h | Li | Invited talks: | ||

11-12h |
Office hour | |||

12-13h | Lunch in Zeeman common room (all days) | |||

13-14h | Invited talks: | Invited talk: | ||

14-15h | Coffee in Zeeman common room (all days) | |||

15-16h | Wigman | Office hour | Conference ends | |

16-17h | Cuenin | |||

circa 18:30-20:30 | Reception with |

**Talks or mini-courses by keynote speakers:**

- Hajer Bahouri (CNRS/Sorbonne)
- 'Dispersion phenomena and nonlinear evolution PDEs'
- Dispersion phenomena, which express that waves with different frequencies move at different velocities, play a central tool in the study of semilinear and quasilinear equations. These lectures intend to quantify these phenomena for the linear Schrödinger and wave equations in several frameworks with applications to nonlinear PDEs arising in physics and fluid mechanics.

- Jeffrey Galkowski (University College London)
- 'An introduction to the Weyl law'

- Jonathan Hickman (Edinburgh; PDF lecture notes)
- 'Non-concentration phenomena for the Schrodinger equation: a taster of modern methods in harmonic analysis'
The linear Schrodinger equation is a prototypical example of a dispersive PDE: roughly speaking, solutions tend to spread out over time. This phenomenon means that it is hard for solutions to concentrate in small portions of space-time, but measuring this non-concentration precisely is a delicate task. In these lectures, I will discuss the theory of fractal Strichartz estimates, which measure how solutions can concentrate in porous, fractal-like sets. The development of this theory was central to breakthrough work of Du—Zhang (Ann. Math. 2019) which resolved a long-standing problem of Carleson concerning almost everywhere convergence of Schrodinger waves back to their initial conditions. The theory has a variety of additional applications, and notably gave new bounds on the Falconer distance problem from geometric measure theory.

The lectures will (at least partially) follow my exposition of the topic for the Séminaire Bourbaki. Rather than focusing on the most recent developments, I will build the theory from the ground up, describing basic underlying principles which apply to vast swathes of harmonic analysis and PDE and have found fundamental applications to areas beyond such as analytic number theory. In this sense, the minicourse should be interesting and accessible to a wide audience.

- Junxian Li (Bonn)
- 'Correlations of values of random diagonal forms'
- We consider the value distribution of diagonal forms in k variables and degree d with random real coefficients and positive integer variables. We show that the l-correlation of almost all such forms is Poissonian when k is large enough depending on l and d. The key is to estimate the number of certain matrices with small determinants.

- Igor Wigman (King’s College London)
- 'Arithmetic random waves'
- The "arithmetic random waves" are random exponential sums, whose frequencies are lattice points lying on a circle of a given radius. These could also be thought as random Laplace eigenfunctions on the standard torus, admitting spectral multiplicities. It was found that some important aspects of the geometry of the arithmetic random waves could be expressed in terms of some arithmetic properties of the lattice points lying on circles. A particularly important such property, namely, their correlations (that is, tuples of lattice points summing up to 0) will be put forward during this mini-series. The best known to date unconditional upper bound, due to Bombieri-Bourgain, for the number of length-6 correlations, valid for the full sequence of radii, will be discussed.

- Jean-Claude Cuenin (Loughborough)
- 'Open problems'
- This is an open problem session, which to which contributions are welcome, either by email in advance or spontaneously in the moment.

We propose that two ‘office hours' will be held where speakers who choose to participate, take possession of a study space in or near the common room specifically to get questions and discuss their course with students and postdocs.

**Talks by invited speakers:**

- Christian Bernert (Goettingen)
- 'Cubic forms in many variables: what have number fields ever done for us?'

- Elena Danesi (Padova/École Polytechnique)
- 'Generalized Strichartz estimates for the Dirac-Coulomb equations'

- Michele Ferrante (Birmingham)
- 'Mizohata-Takeuchi problems for general measures'
- We prove a simple variant of the Mizohata-Takeuchi conjecture, holding for a wide class of measures, inspired by the numerology of Sobolev embeddings. We consider the validity of the classical Mizohata-Takeuchi conjecture for general measures, giving some counterexamples and positive results.

- Anthony Gauvan (Orsay/ENS Paris)
- 'Sharp weak-type estimate for maximal operators and an arithmetic condition'
- I would like to discuss about a problem in Harmonic/geometric analysis concerning the behavior of maximal operators defined in the Euclidean space. It turns out that this problem might be related to some arithmetic or combinatorics issues... This problem is question is traditionally called Zygmund's problem.

- Anouk Greven (Goettingen)
- 'Asymptotics for integral points of bounded height on a log Fano variety'

- Tanja Kuefner (Goettingen)
- 'Beyond Waring: Freiman-Scourfield's Theorem and related results'

**List of attendees:**

*
Hajer Bahouri;
Christian Bernert;
Sebastián Carrillo Santana;
Jean-Claude Cuenin;
Elena Danesi;
Michele Ferrante;
Jeffrey Galkowski;
Anthony Gauvan;
Anouk Greven;
Jonathan Hickman;
Tanja Kuefner; Chung-Hang (Kevin) Kwan;
Jungwon Lee;
Junxian Li;
Matthew Northey;
Donnell Obovu;
Simon L Rydin Myerson;
Alisa Sedunova;
Vedran Sohinger;
Igor Wigman;
Hollis Williams.*

I must have your permission to publish your name. If your name is missing email me.

**Acknowledgements:**

The organisters thank Alison Humphries and the MRC team for their practical and considerate support. This workshop is made possible by the Leverhulme Trust through grant ECF-2020-200.

See also:

Mathematics Research Centre

Mathematical Interdisciplinary Research at Warwick (MIR@W)

Past Events

Past Symposia

**Internet Access at Warwick**:

Where possible, visitors should obtain an EDUROAM account from their own university to enable internet access whilst at Warwick.

**WiFi**whilst at Warwick, click here for instructions (upon arrival at Warwick)

**Registration**:

You can register for any of the symposia or workshops online. To see which registrations are currently open and to submit a registration, please click hereLink opens in a new window.

**Contact:**

Mathematics Research Centre

Zeeman Building

University of Warwick

Coventry CV4 7AL - UK

**E-mail:**

MRC@warwick.ac.uk