I am interested in uncovering the underlying mathematical principles in applied fields. I therefore started from an interdisciplinary undergraduate programme, analysed systems of PDEs which model phenomena in atmospheric and ocean dynamics, and am now working with experimental biologists and meteorologists. I aspire for my research to always have three components: experimental results, simulations that involve all the model complexity, and a simple toy model that captures and reveals the underlying mathematical structure responsible for the system dynamics.
I am interested in how cells move in tissue, since it is a very basic effect present in wound healing and embryogenesis. However, to understand it one needs a detailed understanding of the subcellular processes. Cells move due to changes in adhesion and tension between the adjacent cells. Cell-cell adhesion is achieved via targeted delivery of adhesion molecules (E-cadherins) onto the cell membrane. This is done by molecular motors walking along microtubules, which are dynamic polar filaments (see the microtubules on the right). Both processes are highly stochastic. In addition, motors drive flow inside cells, which was recently shown to affect the microtubule dynamics. Membrane tension is achieved via the acto-myosin complex (which is like a rubber band inside the cell at its boundary), however, microtubule plus-ends also affect the complexes tension. Hence to understand cell motion, one needs to develop a multi-scale multi-physics model hierarchy, with coupled stochastic processes modelling transport inside cells, couple it with continuous fluid mechanics inside cells, and include the microtubule effects on a model of the elastic cell boundary.
I started at the smallest scales and developed a highly interdisciplinary collaboration with the lab of N. Bulgakova (U. Sheffield). In our work experiments suggest modelling approaches, and modelling results give new experimental objectives. We discovered that microtubule self-organisation is driven by cell geometry, that it is robust on the tissue scale, and does not depend on the details of parameters of the microtubule dynamics (w/Aleks Plochocka and Sandy Davie). We further showed how it is affected by cellular crowding (w/Aleks Plochocka). Finally, in a mathematical model we discovered a rule of thumb - that in bi-polar microtubule networks, the two motor types have distinct functions: kinesin mixes cellular components, while dynein reliably delivers them to the cell boundary (w/Gleb Zhelezov and V. Alfred). Mathematically, these discoveries rely on the scale-separation in cell dynamics, namely, that the rates of microtubule growth and shrinking are much larger than the rates of switching between these two states. The next step is to include the fluid dynamics of the cytoplasm, which is a highly non-trivial modelling challenge, and to aid the data-analysis with machine learning.
In the field of atmospheric sciences, I am interested in improving our understanding and modelling capabilities of the equatorial dynamics. Inaccurate representations of moisture and its effect on equatorial dynamics is one of the largest sources of uncertainty in numerical weather and climate prediction.
Before working with data, I have mostly worked on simple models. First, I worked on the onset of mixing in stratified flows, namely reintroducing the concept of shear instability as a point where the equations change type from hyperbolic to elliptic and become ill-posed. This extended the classical Richardson number 1/4 criterion to unsteady non-planar flows. Then I worked on the damping in atmospheric models due to wave radiation into the atmosphere. Mathematically, it is a problem of finding a functional basis in problems with the radiation boundary condition. Practically, it involved working with complex exponential waves, which required extension of the classic concept of group velocity to their case.
I am now starting to work with data. Recent machine learning techniques allow one to approach a multitude of classic problems in weather and climate modelling in a novel way. Many turbulent processes are underresolved and require more accurate data-driven subgrid-scale parameterisations, while for others, such as the precipitation-humidity relation, we still need to learn the physics from data. I am interested in addressing these problems with machine learning, with a goal of developing data-driven parameterisations and learning large-scale equations for the dynamics from data. I hope that these experiments will help to uncover mathematical structures, e.g. separation of scales, that will help improve our basic understanding of the processes in the tropical dynamics.