My first paper was on the Cahn-Hilliard equation, which describes phase separation in metallic alloys when cooled to a certain temperature.

During my research, I discovered that the Cahn-Hilliard equation was used for image restoration, which involves replacing damaged parts of a digital image with suggested content from surrounding areas.

The Hamiltonian is a central concept in both classical and quantum mechanics, illustrating how perspectives in physics can evolve over time.

Mathematicians care about whether something holds true 100% of the time, unlike most fields where 99.99% suffices. This is crucial in fluid dynamics, where we need to understand if there are special initial states that could lead to blow-up scenarios.

The phenomenon of supercriticality versus criticality and subcriticality is a key qualitative feature that distinguishes equations that are predictable from those that are not, such as in fluid dynamics.

There's a fundamental connection between symmetry and conservation laws in physics, which holds true in both classical and quantum mechanics.

The project involves analyzing 22 million pairs of algebraic laws to determine if one law implies another, providing proofs or counterexamples as necessary. While many problems are solvable by undergraduate students, a few are quite challenging.

The splitting of positive and negative frequency components in quantum field theory is not conformally invariant, which presents a conceptual impasse.

The Navier-Stokes equations are sophisticated nonlinear equations that are challenging to solve, particularly in three dimensions where singularities can form.