Mathematics

Monitoring marked individuals is a common strategy in studies of wild animals (referred to as mark-recapture or capture-recapture experiments) and hard to track human populations (referred to as multi-list methods or multiple-systems estimation). A standard assumption of these techniques is that individuals can be identified uniquely and without error, but this can be violated in many ways. In some cases, it may not be possible to identify individuals uniquely because of the study design or the choice of marks. Other times, errors may occur so that individuals are incorrectly identified. I will discuss work with my collaborators over the past 10 years developing methods to account for problems that arise when are only individuals are only partially identified. I will present theoretical aspects of this research, including an introduction to the latent multinomial model and algebraic statistics, and also describe applications to studies of species ranging from the golden mantella (an endangered frog endemic to Madagascar measuring only 20 mm) to the whale shark (the largest know species of fish, measuring up to 19m).

With the advancement in the high performance computing (HPC), it has become feasible to simulate various physical processes and phenomena. Such processes have applications ranging from energy & transportation sector to biological research. The process of liquid fuel injection and atomization forming fuel drops in aircraft engines is central to the formation of pollutants, therefore, it is crucial to study and control this process. The atomization is a physical process in which bulk liquid breaks up into small drops, further breaking up into even smaller drops finally leading to their evaporation. Quite often these drops are studied in an Eulerian fashion. Another approach to investigate the drops or deformable capsules is in a Lagrangian fashion. In this approach, each drop/capsule is tracked separately and is assumed to be either a rigid sphere or a deformable thin membrane. The latter has the direct application to the investigations of red blood cells (RBC) in biological systems. In fact, a RBC has a visco-hyperelastic thin membrane rendering it to be transported through capillary blood vessels of two times smaller its own size. By studying the dynamics of deformation of this membrane, it is possible to extract vital mechanical properties and develop a generalizable numerical model. This model has the potential to be employed to predict blocks in blood vessel the knowledge of which is helpful in improving the measurement of blood pressure. In this talk, I will be presenting two accurate, efficient, and robust numerical methods for simulating liquid fuel atomization process along with showcasing their engineering applications for subsonic & supersonic aircrafts. Furthermore, I will be giving a brief introduction to my current research work on the development of a numerical membrane model (NMM) for studying RBC deformation dynamics.

Convective storms are highly intermittent and intense, making their occurrence and strength difficult to predict. This is especially true for climate models, which have grid resolutions much coarser (e.g., 100 km) than the scale of a storm’s microphysical and dynamical processes (< 1 km). Physically-based parameterizations struggle to account for this scale mismatch, causing large model errors in rain and lightning. This talk will explore some avenues of using statistical techniques (such as generalized linear and log-Gaussian Cox process models) and machine learning methods (such as random forests and neural networks) that are trained by satellite observations of thunderstorms to see how well they can improve upon existing physical parameterizations in producing accurate rain and lightning characteristics given a set of large-scale environmental conditions.

Exposure of bacteria to cidal stresses typically select for the emergence of stress-tolerant cells refractory to killing. Stress tolerance has historically been attributed to the regulation of discrete molecular mechanisms, including though not limited to regulating pro-drug activation or pumps abrogating antibiotic accumulation. However, fractions of mycobacterial mutants lacking these molecular mechanisms still maintain the capacity to broadly tolerate stresses. We have sought to understand the nature of stress tolerance through a largely overlooked axis of mycobacterial-environmental interactions, namely microbial biomechanics. We developed Long-Term Time-Lapse Atomic Force Microscopy (LTTL-AFM) to dynamically characterize nanoscale surface mechanical properties that are otherwise unobservable using other established advanced imaging modalities. LTTL-AFM has allowed us to revisit and redefine fundamental biophysical principles underlying critical bacterial cell processes targeted by a variety of cidal stresses and for which no molecular mechanisms have previously been described. I aim to highlighting the disruptive power of LTTL-AFM to revisit dogmas of fundamental cell processes like cell growth, division, and death. Our studies aim to uncover new molecular paradigms for how mycobacteria physically adapt to stress and provide expanded avenues for the development of novel treatments of microbial infections.

An important problem in machine learning and computational statistics is to sample from an intractable target distribution, e.g. to sample or compute functionals (expectations, normalizing constants) of the target distribution. This sampling problem can be cast as the optimization of a dissimilarity functional, seen as a loss, over the space of probability measures. In particular, one can leverage the geometry of Optimal transport and consider Wasserstein gradient flows for the loss functional, that find continuous path of probability distributions decreasing this loss. Different algorithms to approximate the target distribution result from the choice of the loss, a time and space discretization; and results in practice to the simulation of interacting particle systems. Motivated in particular by two machine learning applications, namely bayesian inference and optimization of shallow neural networks, we will present recent convergence results obtained for algorithms derived from Wasserstein gradient flows.