Scientific Interests

I originally started out as a physicist, so a lot of my research is guided by me wanting to understand particular physical phenomena. That typically involves tools from very different mathematical disciplines — what matters is that you solve a problem and not that you stay within the confines of one particular mathematical field.

Hence, the collection of mathematical techniques I have employed is quite eclectic and ranges from (functional) analysis, pseudodifferential theory, operator algebras and K-theory.

Derivation of Effective Dynamics with Adapted Pseudodifferential Calculi

One of my main interests is understanding how to derive simpler, approximate models from more fundamental equations. The main technical tool are very often pseudodifferential calculi adapted to the problem at hand. For example, I have derived non- and semirelativistic quantum dynamics for a spin-1/2 particle from the Dirac equation with the exact same construction, the only difference being the choice of momentum scale. That changes the way we split the Dirac operator and how we reorder the terms in the asymptotic expansions.

Understanding mathematical and physical mechanisms of why certain approximations work and what assumptions are necessary improves the understanding of the mechanisms behind physical phenomena. Very often one assumes a special choice or class of state(s) or that the system possesses two scales.

Perturbed periodic systems, e. g. crystalline solids subjected to an external electromagnetic field or slowly modulated photonic crystals, possess two scales. In a crystalline solid, the microscopic scale is given by the crystal lattice, whereas the macroscopic scale is defined by the externally applied electromagnetic field which drives the current.

For those I have derived effective tight-binding models, assuming the system is at zero temperature — a choice of state. These tight-binding operators are much simpler to analyze as they discretize space and allow the particle to hop from one unit cell to another.

Alternatively, one can derive effective semiclassical equations of motion, where properties of the full operator enter into the hamiltonian equations of motion. In photonic crystals, this leads to ray optics equations with the help of semiclassical techniques. Mathematically, this is done with a (magnetic) pseudodifferential calculus for equivariant (“periodic up to a phase”) operator-valued functions.

Semiclassical Dynamics for Dissipative and Open Systems: A Pseudodifferential Calculus for Super Operators

Recently, my interest has turned to deriving semiclassical dynamics for dissipative and open systems. While the form of the quantum evolution equations has been well-understood for 50 years, there is no uniformly accepted mathematical framework for dissipative classical dynamics (akin to hamiltonian mechanics, which builds on symplectic geometry).

As a starting point Gihyun Lee and I have introduced the notion of magnetic pseudodifferential super operator, operators that act on other operators, and we plan on using the semiclassical limit as a way to decide which is the right mathematical structure for the classical equations.

Quantum-Wave Analogies: Mathematics Enforces Conceptual Clarity

Viewing different physical theories through a mathematical lens makes it easier to spot commonalities and adapt techniques from one field to another. What differentiates mathematical physics from applied mathematics, though, is the conceptual rigor that is necessary to correctly interpret the same mathematical results in the context of different physical theories. One specific instance are similarities between quantum mechanics and classical wave equations for e. g. electromagnetic or acoustic waves. I am interested in making such quantum-wave analogies conceptually and mathematically precise.

Mathematically speaking, (perturbed) periodic Schrödinger and (perturbed) periodic Maxwell-type operators can be treated with the same mathematical techniques, which allows us to apply techniques initially developed for quantum mechanics to classical electromagnetism in media. A proper understanding of the physics is necessary to formalize the analogy between quantum mechanics and classical waves. To name two differences, electromagnetic fields are directly accessible to measurement whereas the quantum wave function — as a matter of principle — is not. A second is that classical fields are real-valued, which leads to an unbreakable constraint in the complexified equations. This amounts to removing superfluous, unphysical degrees of freedom.

Making these analogies mathematically precise opens the door to realizing many phenomena first proposed for quantum systems with electromagnetic and other classical waves. Ray optics equations emerge from Maxwell's equations via the same mathematical mechanism as classical mechanics can be derived from quantum mechanics. From the point of view of physics, this opens the door to realizing proof-of-principles experiments with classical waves, which might be difficult or impossible to realize with quantum systems. One of the reasons is that periodic media for classical waves can be engineered to have certain properties and fabricated with modern methods.

At the Intersection of Analysis and Algebra: Topological Phenomena

One such quantum-wave analogy I have work on since 2012 are topological phenomena in classical waves. This has first been proposed by Raghu and Haldane in 2005 (published in 2008), then experimentally verified by Joannopoulos and Soljacic in 2009. The idea behind topological phenomena is that discrete symmetries enforce certain phase relationships, which need to be met. These tend to be very robust under perturbations, a feature that makes them very useful in applications.

In the long term, I aim to prove the bulk-boundary correspondence for Maxwell's equations and other classical wave equations. Even though these have simply been accepted by the physics community, mathematically, there has been no proof due to the behavior of very long-wavelength waves. Partial results include a bulk classification and an understanding of the effective dynamics. These ideas have been extended to dynamically stable BdG systems.

A Modern Take on Linear Response Theory

Also the analytic-algebraic framework for linear response theory developed with Giuseppe De Nittis lives in the intersection of analysis and algebra. It can be used to obtain macroscopic conductivity properties for many microscopic quantum mechanical models. Concretely, we apply an electric field E to drive a current through a sample and add control parameters such as temperature and an externally applied magnetic field B. This linear response formalism allows us to justify this “Taylor expansion” in |E|.

Thanks to its grounding in algebra, our formalism is very flexible, it can be applied to discrete and continuous models and incorporate effects of disorder. On the other, we obtain very precise hypotheses posed in the language of analysis that one needs to check.