And, in addition, some notes of expository talks I gave when I was a graduate student.

## Kinetic theory

Introduction. Kinetic theory provides a mesoscopic description of gases, fluids, plasmas and other many-particle systems such as galaxies. This means that the unknown quantity is a function $$f=f(t,x,v)$$ measuring the density of particles that at a given time $$t$$ are located at the point $$x$$ and have momentum $$v$$. Assuming that the full time derivative of $$f$$ equals zero, we obtain the Vlasov equation $\tag{V} \frac{\partial f}{\partial t}+v\cdot\nabla_{x}f+F\cdot\nabla_{v}f=0,$ where we used that $$\dot{x}=v$$ and $$\dot{v}=a$$, and Newton's second law $$F=ma$$, taking the mass $$m=1$$. The zero on the right hand side expresses a no collision assumption: particle motion is only governed by the driving force $$F=F(t,x,v)$$ (which we discuss below) but not due to collisions or any other effect.

The driving force $$F$$ captures the physics of the problem. I am mainly interested in plasma models, where the force is electromagnetic and the particles are ions and electrons for which gravitational forces are negligible. Hence, the force is the  Lorentz force, given by $F=q \left(E+\frac{v}{c}\times B\right)$ where $$q=\pm 1$$ is the particle charge, $$E=E(t,x)$$ is the electric field, $$B=B(t,x)$$ is the magnetic field and $$c$$ is the speed of light (which we usually take to be $$1$$ for simplicity). We find the fields using Maxwell's equations \begin{align} &\nabla\cdot{E}=\rho\qquad\qquad\qquad \nabla\cdot{B}=0\\ &\nabla\times{E}=-\frac{1}{c}\frac{\partial{B}}{\partial t}\qquad \nabla\times{B}=\frac{1}{c}{j}+\frac{1}{c}\frac{\partial{E}}{\partial t}, \end{align} where $$\rho=\rho(t,x)$$ is the charge density and $${j}={j}(t,x)$$ is the current density, which are defined as $$\rho=4\pi q\int f\;\text dv$$ and $${j}=4\pi q\int v f\;\text dv.$$ We therefore obtain a system of coupled equations called the Vlasov-Maxwell system. [There are some modifications that need to be made to the velocity in order to preserve relativistic symmetries, but this is omitted here]. In the most general case there are few existence and uniqueness results. Perhaps the most important result is due to Glassey and Strauss (ARMA 1986). In the electrostatic case where magnetic effects are assumed to be negligble (then the system is called Vlasov-Poisson) there is a robust theory due to Pfaffelmoser (JDE 1992), Lions and Perthame (Invent. Math. 1991) and Schaeffer (CPDE 1991).

Stability analysis. I am mainly interested in plasma instabilities. That is, I am interested in the following question: given an equilibrium $$f^0=f^0(x,v)$$ of the Vlasov-Maxwell system, how does a small perturbation $$g(t,x,v)=f^0(x,v)+\varepsilon f(t,x,v)$$ behave? (of course, we must first verify that the necessary existence and uniqueness theory is established). The first step is to linearize the Vlasov equation and obtain $$\tag{LV} \frac{\partial f}{\partial t}+v\cdot\nabla_{x}f+F^0\cdot\nabla_{v}f=-F\cdot\nabla_{v}f^0,$$ where $$F^0$$ is the equilibrium Lorentz force and $$F$$ is its first-order perturbation. My approach to finding an instability, going back to Lin (MRL 2001), is to convert the problem into a spectral problem. First, we make a growing mode ansatz: we assume that $$f$$ and $$F$$ have time dependence proportional to $$e^{\gamma t}$$ where $$\gamma > 0$$. This allows us to replace the time derivative in $$\text{(LV)}$$ by multiplication by $$\gamma$$. Then, this equation is inverted, and one obtains an expression for $$f$$ in terms of some integral expression of $$F$$ and $$f^0$$, depending on $$\gamma$$ as a parameter. By plugging the expression for $$f$$ into Maxwell's equations, we obtain a selfadjoint problem which we need to solve. That is, very generally speaking, we obtain a selfadjoint operator depending upon the parameter $$\gamma$$ for which we need to find a nontrivial kernel. This is done by writing the problem for $$\gamma=0$$ and $$\gamma=+\infty$$ and tracking the spectrum of the operator as $$\gamma$$ varies between these two values.

Publications:

1. Concentrating solutions of the relativistic Vlasov-Maxwell system
 Commun. Math. Sci., 17, 377-392 (2019) | preprint | arXiv | journal | doi with S. Calogero and S. Pankavich

2. Arbitrarily large solutions of the Vlasov-Poisson system
 SIAM J. Math. Anal., 50, 4311-4326 (2018) | pre | post | arXiv | journal | doi with S. Calogero and S. Pankavich

3. Instabilities of the relativistic Vlasov-Maxwell system on unbounded domains
 SIAM J. Math. Anal., 49, 4024-4063 (2017) | preprint | postprint | arXiv | journal | doi with T. Holding

4. Instabilities in kinetic theory and their relationship to the ergodic theorem
Contemp. Math., 653, 25-40 (2015) | preprint | arXiv | journal | doi

5. Instability of nonsymmetric nonmonotone equilibria of the Vlasov-Maxwell system
J. Math. Phys., 52, 123703 (2011) |
6. Instability of nonmonotone magnetic equilibria of the relativistic Vlasov-Maxwell system
Nonlinearity, 24, 3353-3389 (2011) | preprint | arXiv | journal | doi

## Spectral properties of first order differential operators and applications to ergodic theory

In the preceding discussion, after applying the growing-mode ansatz and plugging into $$\text{(LV)}$$, the left hand side becomes $$(\gamma+D)f$$ where $$D=(v\cdot\nabla_x,F^0\cdot\nabla_v)$$ is a vector field in phase space. When inverting $$\text{(LV)}$$ one has to average the right hand side along the trajectories of $$D$$. This average is very similar to the one familiar from the ergodic theorem, and as $$\gamma\to0$$ it converges (in the strong operator topology) to a spatial average. This raises the following question: under what circumstances, if at all, is this convergence in fact uniform?

One of the very first proofs of the ergodic theorem, due to John von Neumann (the other is due to Birkhoff), relied on the spectral theorem and a detailed analysis of the spectral family (resolution of the identity) of the associated selfadjoint operator. In a nutshell, von Neumann's theorem and proof are as follows:

Theorem (von Neumann). Let $$\mathcal{H}$$ be a separable Hilbert space and let $$G_t:\mathcal{H}\to\mathcal{H}$$ be a continuous one-parameter group of unitary transformations. Let $$P$$ be the orthogonal projection onto $$\{v\in\mathcal{H}\ |\ \forall t,\ G_tv=v\}$$. Then for any $$f\in\mathcal{H}$$ $\lim_{T\to\infty}\frac 1{2T}\int_{-T}^T G_tf\ \text dt=Pf.$

Sketch of proof. The proof uses the spectral decomposition $$\{E(\lambda)\}_{\lambda\in{\mathbb R}}$$ of the generator of $$G_t$$ (Stone's theorem) to get $$\frac 1{2T}\int_{-T}^T G_t \text dt=\frac{1}{2T}\int_{-T}^T\int_{\mathbb R} e^{it\lambda}\text dE(\lambda)\text dt=\int_{\mathbb R}\frac{\sin{T\lambda}}{T\lambda}\text dE(\lambda)$$. Von Neumann then shows that when $$T\to\infty$$ only the contribution from $$\lambda=0$$ remains, which precisely corresponds to the projection $$P$$.
$$\Box$$

Now, consider the simple case of a one-dimensional flow governed by the operator $$H=-i\frac{\text d}{\text dx}$$. Then $$G_t=e^{itH}$$ and the spectral family can be explicitly identified via the Fourier transform, relating $$H$$ to a multiplication operator by $$\xi$$. This allows us to get the formula $\left(E(\lambda)f,g\right)_{L^2(\mathbb R)}=\int_{\xi\leq\lambda}\widehat f(\xi)\overline{\widehat{g}(\xi)}\;\text d\xi$ which can be differentiated with respect to $$\lambda$$ when restricted to the absolutely continuous subspace. In turn, this allows us to make the substitution $$\text dE(\lambda)=\frac{\text dE}{\text d\lambda}\text d\lambda$$ where $$\frac{\text dE}{\text d\lambda}$$ is called the density of states. Estimates on the density of states allow us to obtain uniform convergence in Von Neumann's argument on certain weighted-$$L^2$$ subspaces.

In higher dimensions we consider the operator $$H=-i(u\cdot\nabla)$$ where $$u:\mathbb R^d\to\mathbb R^d$$ is divergence-free. The main complication is understanding the form of the Fourier transform in this case. This requires finding a global rectification for $$u$$. Another complication stems from the benefit of working in weighted-$$L^2$$ spaces, where the Fourier transform is no longer so "clean'' and identifying the density of states is more difficult.

Publications:
1. Averaging along degenerate flows on the annulus
 Submitted, 15 pages (2019) | preprint | arXiv with B. Morisse and J. Zhang

2. Uniform convergence in von Neumann’s ergodic theorem in the absence of a spectral gap
 Ergod. Theor. Dyn. Syst., to appear (2020) | preprint | arXiv | journal | doi with B. Morisse

3. Weak Poincaré inequalities in the absence of spectral gaps
 Ann. Henri Poincaré, 21, 359-375 (2020) | preprint | arXiv | journal | doi with A. Einav

4. On the spectrum of shear flows and uniform ergodic theorems
J. Funct. Anal., 267, 299-322 (2014) | preprint | arXiv | journal | doi

### Infinite Dimensional Approximation Theory

A sensible approach to analysing the spectrum of an operator $$H$$ is to first study the truncated operator $$P_nHP_n$$ where $$P_n$$ is a projection onto an $$n$$-dimensional subspace, and then let $$n\to\infty$$. However, this is not always so straightforward, and it raises fundamental questions related to complexity and computability:
• Can the spectrum of an operator always be calculated from finite-dimensional approximations?
• If so, are some types of operators simpler to treat than others?
• Can one quantify how difficult such a calculation is?
In this work with my coauthors we address these questions, generalising the concept of a tower of algorithms, introduced by Doyle and McMullen (Acta Math. 1989) in their seminal paper showing that finding roots of quartic and quintic polynomials is more complex than finding roots of cubic polynomials, and that finding roots of sextic polynomials is infinitely more complex. The basic idea is that some calculations cannot be performed by an iterative algorithm that processes a finite amount of information at each iteration. A tower of algorithms is a sequence of algorithms, where each link in the sequence relies on the information provided by the preceding one. Hence, instead of a single iterative algorithm, several limits must be taken. In our work we treat several problems, but most importantly we discuss the computation of
• spectra of Schrödinger operators of the form $$H=-\Delta+V$$ acting in $$L^2(\mathbb R^d)$$,
• spectra of bounded infinite matrices of the form $$A=(a_{ij})_{i,j=1}^\infty$$ acting in $$\ell^2(\mathbb N)$$.

For these problems we show that a tower is required, and provide actual algorithms explicitly. We show that these algorithms are essential (in the sense that the tower cannot be shortened by one level) by providing suitable counterexamples.

Publications:
1. Computing Spectra - On the Solvability Complexity Index Hierarchy and Towers of Algorithms
 Submitted, 95 pages (2020) | preprint | arXiv with M. Colbrook, A. Hansen, O. Nevanlinna and M. Seidel

2. Approximations of strongly continuous families of unbounded operators
 Commun. Math. Phys., 345, 615-630 (2016) | preprint | arXiv | journal | doi | erratum with T. Holding

3. The Solvability Complexity Index - Computer Science and Logic Meet Scientific Computing
 Submitted, 15 pages (2015) | preprint with A. Hansen, O. Nevanlinna and M. Seidel

4. New barriers in complexity theory: On The Solvability Complexity Index and Towers of Algorithms
 C. R. Acad. Sci., 353, 931-936 (2015) | preprint | journal | doi with A. Hansen, O. Nevanlinna and M. Seidel

### Other

Publications:
1. Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations
 Commun. PDE, 42, 179-234 (2017) | preprint | arXiv | journal | doi with D. Marahrens and S. Neukamm