Grant: Discovery Grants Program - Individual

Scientists use mathematics to characterize all kinds of phenomena. Weather forecasting, political polling, the evolution of financial and consumer markets, traffic flow, are all modeled by specific mathematical equations. Solving the equations is often tantamount to seeing the future and gives insight into how one could influence it. Of course, predicting and manipulating the future is not quite so simple. For one thing, we can only build approximate models of reality, unable to capture all aspects. Generally, more complex models describe nature better, but the resulting equations become (hopelessly) hard to solve. One is then led to come up with effective equations, which focus on a small part of a system only, without having to describe every little detail of the complex surroundings with which it is in contact. My research focuses on systems obeying the Schrdinger equation of quantum theory , such as elementary particles interacting with each other (electrons, nuclei, atoms, molecules) and matter interacting with radiation (electromagnetic field, light). Quantum theory has long played an important role in mathematics, physics, chemistry and more recently also in biology. To reduce complexity, one often describes part of a system only, say a specific atom in a molecule or a few qubits pinned on a substrate (basic building blocks of a quantum computer). That part (atom, qubits) is called an open system , as it interacts with a complex `outside' system, called an environment (molecule, substrate). My continuing goal is to (1) derive effective equations for open quantum systems and (2) analyze the solutions of those equations . In collaboration with other scientists, I have been developing the quantum dynamical resonance theory to reach these goals over the past few years. Other than being motivated by progressing fundamental research, I hope to be able to find new techniques useful in applications. As an example, we recently derived and solved an effective quantum equation describing chemical processes happening in photosynthesis in algae. We found how exactly the process speed depends on certain parameters (temperature, light intensity...). So now one might try to optimize the speed to accelerate the reaction, which is relevant in biofuel production experiments. While I collaborate with theoretical physicists, chemists and biologists, my approach is based on mathematically rigorous methods. It involves tools from mathematical analysis, like the spectral theory of (infinite-dimensional, non-normal) operators, C*- and von Neumann algebra theory, quantum field theory and probability theory. Over the next five years, I will push the dynamical resonance theory further. The direction I will take is motivated by mathematical innovation, creating new techniques, and by physical relevance, addressing crucial issues identified in alliance with scientists of related disciplines.