All of my work can be described as an attempt to answer the following meta question: 

in a system with a very large number of components, how do small scale changes relate to large scale changes?

In finance or economics, small scale can mean an individual, while large scale can correspond to a market. There is a lot of variance in individual choices, but some regularity in their aggregated implications.  In fluid mechanics, small scale means individual molecules while large scale describes the fluid as a whole. Small scale behaviour is driven by erratic collisions between molecules, while large scale behaviour is driven by pressure differences. Understanding the relation between the two scales is key to solving the millennium Navier-Stokes problem. In its most extreme form, the meta question asks for a theory of everything in physics: small scale corresponds to particles, described by quantum mechanics, while large scale corresponds to all of space-time, described by general relativity.

My work aims to gain insight on these grand problems through the resolution of more modest mathematical problems. 

These problems can be organised in three categories: diffusion (where erratic miscroscopic behaviour aggregates to a regular macroscopic behaviour), wave oscillation (where groups of particles positively or negatively interfere with each other), and uncertainty principles (describing which groupings of particles are meaningful).

My work contributes to the understanding of three types of phenomena that appear, in different forms, across all the sciences: diffusion (D), wave interactions (W), and uncertainty principles (U). These phenomena (described below) are quite different, but the mathematical techniques required to investigate them have much in common. The goal of my work is to free these mathematical techniques from some of their current limitations, so that they can shine light on phenomena (D), (W), and (U) in much more realistic scenarios than the highly idealised models they are currently restricted to.

Topics

(D) Diffusion refers to the change over time of the distribution of a quantity, as it tends to move from areas of high density to areas of lower density, while being subjected to some external driving forces. Examples include heat diffusion (warming of cold regions and cooling of hot regions, with heat being added by heat sources) or contagious disease diffusion (contamination of areas with low disease incidence, immunity developing in areas of high incidence). Less obvious examples include AI learning mechanisms or modelling of stock market prices.

(W) Wave interactions refers to the fact that waves (of any kind: ocean waves, sound, light, economic fluctuations) can cancel or reinforce each other depending on their speed and direction.

(U) Uncertainty principles refer to quantum mechanical principles of the form ”the more precisely one knows the momentum of a particle, the less precisely one can know its position (and vice versa)”. These principles are fundamentally microscopic (at the quantum level), but do have macroscopic consequences. For instance, a wave that is localised in a small region of space has to have components that oscillate arbitrarily rapidly (albeit with a very small amplitude when oscillation is very rapid).

Questions and Applications

Specifically, my work aims to answer the following questions and has the following potential applications.

(D) How does the distribution of the quantity diffusing evolve over time, and how does this evolution depend on the environment?

Answers to this question help design new materials (the “environment”) better capable of trapping heat, canceling sound, or maximising photovoltaic output (by concentrating light energy in specific regions).

(W) How intense and concentrated can waves be, depending on the environment they propagate into?

Answers to this question help improve the accuracy of medical imaging (e.g. x-ray, MRI) and other forms of non-invasive detection (e.g. radar, sonar). This is because we can use our understanding of how the waves are affected by the medium they traverse to deduce what the medium is from the behaviour of waves sent through it.

(U) How do fundamental observables (such as position in space, momentum, time, and energy) relate to each other in complex quantum systems?

Answers to this question help design quantum computers. For instance, an understanding of the nature of time in quantum systems is critical to the running of fully quantum algorithms and to the recording of quantum information.

My work contributes to the understanding of three types of phenomena that appear, in different forms, across all the sciences: diffusion (D), wave interactions (W), and uncertainty principles (U). The common technical theme in my study of these three phenomena is the extension of a range of analysis techniques to rougher settings, i.e. to settings where some smoothness assumptions do not hold, or where certain invariances are not present.

For instance, Fourier series are used to study (D) at the macroscopic level. Indeed, the heat equation was Fourier's original motivation. These series are expansions in a basis of eigenfunctions for the usual partial derivative operators. This corresponds to studying (D) in an empty euclidean space. Working in more realistic environments requires replacing partial derivatives with more complicated differential operators. Harmonic analysis is a set of techniques to do so, exploiting spectral and algebraic properties of these operators. Currently though, it is still too closely connected to the usual partial derivative operators. I am thus working on tailoring its techniques to different operators.

At the microscopic level, the movement of particles (or agents) in a diffusion problem is best described using stochastic differential equations. This involves a generalised form of calculus that takes randomness into account. Such a calculus has features that can look surprising at first. For instance, the fundamental theorem of calculus takes a form that involves not one, but two derivatives! The term involving the second derivative is multiplied by the variance of the underlying random variable. That's why it disappears in the deterministic case. Consequently, some key results available in deterministic calculus are still missing in stochastic calculus (for instance, the existence theory for equations with particularly discontinuous coefficients). One of my aims is to develop such results to obtain a microscopic understanding of diffusion equations in irregular media.

For (W), microlocal analysis describes waves at the microscopic level using a system of differential equations for the position and momentum of the relevant particles. It also describes waves at the macroscopic level using partial differential equations, and relates the two levels through a form of classical-quantum correspondence. Microlocal analysis can be thought of as a form of harmonic analysis on phase space that exploits geometric and dynamical information. It is more constrained by regularity assumptions than pure harmonic analysis. For instance, coefficients in heat equations don't need to be continuous, while wave equation theory requires, at least, Lipschitz continuous coefficients. I work to import the ideas of rough harmonic analysis into microlocal analysis.

Last but not least, functional analysis provides the framework for everything I do. In particular, it allows us to formalise (U) in mathematical terms, and to quantify diffusion and interference in (D) and (W). Functional analysis can be thought of as calculus and linear algebra in infinite dimension (for instances in the spaces of functions or random variables used to model (D), (W) and (U)). I often need to refine or generalise functional analysis frameworks to suit my needs (for instance, by adapting function spaces to match differences in the harmonic or the stochastic analysis).

I am an analyst. I work on diverse problems, but my research is, at the technical level, fundamentally about one question:

how should analysis be adapted when the standard Laplace operator is replaced by a differential operator with varying coefficients? 

I aim to approach this question in all relevant forms of analysis: harmonic, stochastic, functional, and microlocal. The key technical phenomenon that needs to be understood is roughness. When the coefficients (or the underlying space, or the functions that the operator acts upon) are not smooth enough, classical methods become inefficient (because the problem cannot be effectively treated as a perturbation of the smooth case, and because algebraic properties become more elusive). On the other hand, roughness can also bring advantages, such as regularisation by noise. I look for the effects of roughness, and for answers to the overarching question of my research, by working on three types of problems: diffusion (D), wave interactions (W), and uncertainty principles (U). 

For each of these types of problems, I provide below an example of what I have done recently, a dream objective, and an indication of the additional theories that I want to add to my perspective (I seek collaborations and supervision opportunities for this purpose). What I am currently working on is somewhere between what I have already done and what the dream objective is. 

(D) Well posedness for linear parabolic PDE and SPDE with rough coefficients, drivers, and initial data (see, in particular, [Auscher, P. 23]).

Dream objective: to prove an optimal short-time well posedness result for general quasilinear parabolic SPDEs, not relying on ad-hoc regularity or structural assumptions, and allowing for general singular drivers. The corresponding deterministic result is a pillar of parabolic PDE theory. The lack of a complete stochastic analogue is a key limitation to the development of SPDE theory in the direction of geometric flows.

Additional theories: Paracontrolled distributions and/or regularity structures to capture the idea of rough scales of regularity, and to handle singular terms. Probabilistic scaling to determine and use critical function spaces.

(W) Fixed time Lp estimates for wave equations with rough coefficients and initial data (see, in particular, [Frey, P. 21] and [Hassell, Rozendaal, P., Yung 24]).

Dream objective: prove the local smoothing conjecture (optimal Lp mapping properties of the solution map for the wave equation) for general (rough) coefficients. Even for constant coefficients, the local smoothing conjecture implies many well-known conjectures in harmonic analysis (e.g. Kakeya and Bochner-Riesz conjectures), see [Terry Tao's description].

Additional theories: More decoupling theory, as in [Guth, Wang, Zhang 20], and more microlocal analysis (including the analysis of rough Hamiltonian systems).

(U) Spectral multiplier estimates for various analogues of the quantum harmonic oscillator (e.g. the Ornstein-Uhlenbeck operator and the harmonic oscillator on quantum euclidean spaces), and construction of positive operator valued measures associated with the time/energy uncertainty (see [Arhancet, Hagedorn, Kriegler, P. 24] and [van Neerven, P. 23]).

Dream objective: design a unified theory of quantum time, allowing observation of time from any quantum system. 

Additional theories: Some operator algebra (around Tomita-Takesaki theory), more Lie group theory, and some non-commutative functional/harmonic analysis (related to [Junge, Mei, Parcet, Xia 21]).