Together with K. U. Kristiansen, a project partner based at the Technical University of Copenhagen, we began by focusing on a classical problem in combustion theory, namely, the propagation of planar laminar premixed flames. In this context, flame propagation is mathematically described as a traveling wave in a reaction-diffusion equation known as the Zeldovich-Kamenetskii equation. We provided a rigorous and geometrically informative characterization of formally correct but non-rigorous results which date back to early pioneering works in mathematical combustion theory in the late 1930's. In addition to providing a modern, geometric and mathematically rigorous lens on this historically significant problem, we contributed novel mathematical methods - based on the so-called geometric blow-up method - which are expected to be transferrable and applicable to a wide range of related problems in combustion theory and beyond. The findings were published in a leading dynamical systems journal.
In subsequent collaborations with the same project partner, we initiated a investigations into the possibility of using similar mathematical methods as a mathematically rigorous approach to model reduction of gene regulatory networks. Due to the early termination of the project after approximately 9 months (due to the fact that the Fellow accepted an offer for a permanent academic position), we were unable to complete our investigations within the reduced time-frame of the project. Nevertheless, we have continued this collaboration and expect to publish key results on the validity and invalidity of 'combined' approaches to model reduction that are commonly used in the modeling of gene regulatory dynamics. In particular, we expect to publish our findings for a simple but representative gene regulatory network by May 2025, and for the larger network classes more generally by early 2026.
In addition to the work and achievements outlined above, the Fellow published two additional articles in leading dynamical systems journals, which focused on (i) the geometric description characterization of an experimentally identified multi-scale oscillation in a model for intracellular calcium dynamics, and (ii) mathematical theory for the study of tipping phenomena. Both of these works leveraged mathematical techniques which closely related to the project aims (particularly via the use of geometric blow-up techniques), and demonstrated the broad applicability of these ideas in the mathematical sciences.
