The **Virtual Maxwell Analysis Seminar** is a joint seminar series with between the analysis groups in the University of Edinburgh and Heriot-Watt University. It is coorganised by Heiko Gimperlein and Jonathan Hickman.

Due to the COVID-19 outbreak, we have moved all our seminars online. Information about how to join is emailed each week on the analysis mailing list (follow the mailing list link to join).

# Next talk

Friday, 15th October 2021

### Boundedness of spectral projectors on Riemannian manifolds

### Simon Myerson (University of Warwick)

**Abstract:**

Given the Laplace(-Beltrami) operator on a Riemannian manifold, consider the spectral projector on (generalized) eigenfunctions whose eigenvalue lies in – where and . We ask the question of optimal to bounds for this operator. I will present new results in this direction for the Euclidean torus (joint with Pierre Germain).

# Full schedule

## Autumn 2021

Friday, 1st October 2021

### Mathematical aspects of nanophotonics

### Matias Ruiz (University of Edinburgh)

**Abstract:**

In the last couple of decades, the field of nanophotonics, i.e. the study of light-matter interactions at the nanoscale, has gained enormous importance in different disciplines of science and engineering. A large portion of this field concerns the study of electromagnetic resonances in nanoparticles which are, in turn, the building blocks of many applications in energy, healthcare, material science, to name a few.

In this applied analysis talk I will introduce some of the mathematically interesting problems arising in nanophotonics and discuss recent analytical results in the analysis of resonances in metallic nanoparticles (also known as plasmonic resonances), which include the use of layer potential techniques and asymptotic analysis.

Friday, 8th October 2021

### Harmonic analysis tools in spectral theory

### Jean-Claude Cuenin (Loughborough University)

**Abstract:**

I will give a survey of how methods from harmonic analysis, particularly those related to Fourier restriction theory, can be used in spectral and scattering theory. Applications include eigenvalue estimates for Schrödinger operators with complex potentials and almost sure scattering for random lattice Schrödinger operators in 3d with slowly decaying potential.

Friday, 15th October 2021

### Boundedness of spectral projectors on Riemannian manifolds

### Simon Myerson (University of Warwick)

**Abstract:**

Given the Laplace(-Beltrami) operator on a Riemannian manifold, consider the spectral projector on (generalized) eigenfunctions whose eigenvalue lies in – where and . We ask the question of optimal to bounds for this operator. I will present new results in this direction for the Euclidean torus (joint with Pierre Germain).

Friday, 22nd October 2021

**TBA**

### Guido De Philippis (CIMS)

**Abstract:**

TBA

Friday, 29th October 2021

**TBA**

### Myrto Maria Galanopoulou (Heriot-Watt University)

**Abstract:**

TBA

Friday, 5th November 2021

**TBA**

### Gigliola Staffilani (MIT)

**Abstract:**

TBA

Friday, 19th November 2021

**TBA**

### Laure Saint-Raymond (IHES)

**Abstract:**

TBA

Friday, 26th November 2021

### Polynomial progressions in continuous fields

### Jim Wright (University of Edinburgh)

**Abstract:**

Friday, 3rd December 2021

**TBA**

### Jan Sbierski (University of Edinburgh)

**Abstract:**

TBA

# Previous semesters

## Spring 2021

Friday, 22nd January 2021

**Topological properties of minimizers in Landau-de Gennes theory of nematic liquid crystals**

### Federico Luigi Dipasquale (Università di Verona)

**Abstract:**

I will talk about recent results on the topology of minimizers in the Landaude Gennes (LdG) theory of nematic liquid crystals which shed light on the socalled “biaxial torus minimizers”. According to many numerical studies, these are the minimizers of the most studied LdG energy in very relevant physical conditions. Such minimizers are axially symmetric and characterized by remarkable topological properties (where the topology is to be sought in the level set of an appropriate indicator function, the signed biaxiality). However, they eluded precise mathematical description for years. In this talk, I will introduce some new key ideas which allow to study the topology of minimizers (but also of some classes of more general configurations). I will try to explain how to implement them and how this leads in a natural way to the definition of an appropriate asymptotic regime in which, in fact, topological structures are found in the biaxiality sets of minimizers. The topological results match qualitatively many features expected from the numerical simulations. We will also see that, restricting to axially symmetric configurations, we have still better accordance with simulations and much more can be said, although several interesting problems remain open. Joint work with V. Millot (Paris XIII) and A. Pisante (Sapienza).

Friday, 29th January 2021

**-quasiconvexity, function spaces and regularity**

### Franz Gmeineder (Universität Bonn)

**Abstract:**

By Morrey’s foundational work, quasiconvexity displays a key notion in the vectorial Calculus of Variations. A suitable generalisation that keeps track of more elaborate differential conditions is given by Fonseca & Müller’s -quasiconvexity. With the topic having faced numerous contributions as to lower semicontinuity, in this talk I give an overview of recent results for such problems with focus on the underlying function spaces and the (partial) regularity of minima. The talk is partially based on joint work with Sergio Conti (Bonn), Jan Kristensen (Oxford) and Lars Diening (Bielefeld).

Friday, 5th February 2021

**Landis’ conjecture on the decay of solutions to Schrödinger equations on the plane**

### Eugenia Malinnikova (Stanford)

**Abstract:**

We consider a real-valued function on the plane for which the absolute value of the Laplacian is bounded by the absolute value of the function at each point. In other words, we look at solutions of the stationary Schrödinger equation with a bounded potential. The question discussed in the talk is how fast such function may decay at infinity. We give the answer in dimension two, in higher dimensions the corresponding problem is open. The talk is based on the joint work with A. Logunov, N. Nadirashvili, and F. Nazarov.

Friday, 12th February 2021

**Global invertibility via invertibility on boundary and applications to Nonlinear Elasticity**

### Stefan Krömer (Czech Academy of Sciences)

**Abstract:**

Globally invertible Sobolev maps are of particular interest in models for the elastic deformation of solids, because invertibility corresponds to non-interpenetration of matter. By a result of John Ball (1981), a locally orientation preserving Sobolev map is almost everywhere globally invertible provided that its boundary values admit a homeomorphic extension. We will see that the conclusions of Ball’s theorem and related results can be reached while completely avoiding the (hard) problem of homeomorphic extension. The proof heavily relies on Brouwer’s topological degree. Among other things, the result can be applied to justify more efficient numerical approximation respecting global invertibility constraints.

Friday, 19th February 2021

**The nonlinear stability of the Schwarzschild black hole without symmetry**

### Mihalis Dafermos (Cambridge/Princeton)

Friday, 26th February 2021

**Hölder continuous Euler flows with local energy dissipation**

### Hyunju Kwon (IAS)

**Abstract:**

The Euler equations describe the behavior of ideal fluids. It is well-known that smooth (spatially periodic) Euler flows conserve total kinetic energy in time. In the theory of turbulence, on the other hand, a famous conjecture of Lars Onsager asserts that kinetic energy conservation may fail when an Euler flow belongs to the Hölder spaces with Hölder exponent less than 1/3, which was proved recently by Isett. In light of these developments, I’ll discuss a stronger version of the Onsager conjecture: the existence of Hölder continuous Euler flows which locally dissipate kinetic energy.

Friday, 5th March 2021

**Regularity of solutions of complex coefficient elliptic systems: the -ellipticity condition**

### Jill Pipher (Brown)

**Abstract:**

Formulating and solving boundary value problems for divergence form real elliptic equations has been an active and productive area of research ever since the foundational work of De Giorgi – Nash – Moser established H\”older continuity of solutions when the operator coefficients are merely bounded and measurable. The solutions to such real-valued equations share some important properties with harmonic functions: maximum principles, Harnack principles, and estimates up to the boundary that enable one to solve Dirichlet problems in the classical sense of nontangential convergence. Weak solutions of complex elliptic equations and elliptic systems do not necessarily share these good properties of continuity or maximum principles. In joint work with M. Dindos, we introduced in 2017 a structural condition (-ellipticity) on divergence form complex elliptic equations that was inspired by a condition related to contractivity due to Cialdea and Maz’ya. The -ellipticity condition was also simultaneously discovered by Carbonaro-Dragicevic, who used it to prove a bilinear embedding result. Subsequently, the condition has proven useful in the study of well-posedness of a degenerate elliptic operator associated with domains with lower-dimensional boundary. In this talk we discuss -ellipticity for complex divergence form equations, and then describe recent work, joint with J. Li and M. Dindos, extending this condition to elliptic systems. In particular, we give applications to solvability of the Dirichlet problem for the Lame systems.

Friday, 12th March 2021

**Rigidity for measurable sets**

### Ilaria Fragalé (Politecnico di Milano)

**Abstract:**

We discuss the rigidity of measurable subsets in the Euclidean space such that the Lebesgue measure of their intersection with a ball of radius , centred at any point in the essential boundary, is constant. Based on a joint work with Dorin Bucur.

Friday, 19th March 2021

### A -dimensional Analyst’s Travelling Salesman Theorem for general sets in Euclidean space

### Matthew Hyde (University of Edinburgh)

**Abstract:**

In the nineties, Peter Jones proved the Analyst’s Travelling Salesman Theorem (TST), which gives a necessary and sufficient condition for when a set in Euclidean space can be contained in a curve of finite length. This condition is stated in terms of beta-numbers, which give a measure of how flat a set is at each location and scale. Several higher dimensional analogues of the TST have been proven since. These, along with Jones’ original result, have found many applications in fields such as complex analysis, harmonic analysis, and harmonic measure. Unlike Jones’ TST, these higher dimensional variants require additional assumption on the set. In this talk we will discuss these results in more detail and introduce a higher dimensional TST that holds for every set in Euclidean space.

Friday, 26th March 2021

### Discrete analogues of maximally modulated singular integrals of Stein-Wainger type

### Joris Roos (University of Massachusetts Lowell & University of Edinburgh)

**Abstract:**

Stein and Wainger introduced an interesting class of maximal oscillatory integral operators related to Carleson’s theorem. The talk will be about joint work with Ben Krause on discrete analogues of some of these operators. These discrete analogues feature a number of substantial difficulties that are absent in the real-variable setting and involve themes from number theory and analysis.