Research Interests
My research interests fall into the areas of applied mathematics, dynamical systems, mathematical & computational modelling and probability theory & stochastic processes. Currently, I am involved in the Dynamical Systems Working Seminar . In May 2013, I attended Houston Summer School on Dynamical Systems and in July 2014, I attended a summer school at MSRI on Stochastic Partial Differential Equations.
In Spring 2018, I received the Nancy L. Mergler Dissertation Completion Fellowship for the academic year 2018-19. The purpose of this award is to recognize especially meritorious projects and to support their completion. It is granted for each academic year on a competitive basis to seven most outstanding proposals from Ph.D. candidates in Norman-Campus programs who are in the final phases of dissertation writing.
In Summer 2017, I completed the National Science Foundation's (NSF) Mathematical Sciences Graduate Internship NSF-MSGI . The aim of this project is to develop a high-resolution, high-speed 3D microscope that enables the dynamic studies of molecular and cellular processes in biological systems using multifocal plane microscopy. My profile and highlights about the project are featured on SIAM's (Society for Industrial and Applied Mathematics) website and published in their online SIAM NEWS BLOG .
Publications
In: Pinto A., Zilberman D. (eds), Springer Proceedings in Mathematics & Statistics: Modeling, Dynamics, Optimization and Bioeconomics III, 224: 273 - 284, 2018.
Previous Work
Since epidemiology modeling can suggest crucial data that should be collected, identify trends, make general forecasts, and estimate the uncertainty in forecasts, my Master's thesis was motivated by studying an age-structured SIR epidemic model of the McKendrick-Von Foerster type. In this work, I derived a weak formulation of an age-structured population model and for numerical validation, we introduced the Galerkin approximation using Laguerre spectral functions to compute the density of the population and derive convergence results.
Tick-borne diseases (theleriosis, re-lapsing fever, TBE (tick-borne encephalitis)) are serious health problems affecting humans as well as domestic animals in many parts of the world. These infections are generally transmitted through a bite of an infected tick, and it appears that most of these infections are widely present in some wildlife species; hence, an understanding of tick population dynamics and its interaction with hosts is essential to understand and control such diseases. Therefore, my second research project involved studying a simulation model for the tick life cycle model that accounts for different classifications within the population and various parameters that are weather and climate dependent. For this model, we derived a Galerkin approximation using finite element semi-discretizations, tested it using data available on tick populations and analyzed the numerical results to investigate and understand the dynamics of the population. >
Current Research
Using statistical mechanics to study pattern formation in hydrodynamic systems has increasingly gained interest over the years. Some common, everyday examples are the fascinating growing patterns found in snowflakes or bacteria colonies, ripples in sandy deserts and the complex turbulent flow patterns found in the atmosphere. As much as the shape of the patterns formed might be exotic, it is in fact more intriguing to study the governing conditions that cause the onset of these instabilities in such simple systems. A huge proportion of the early work on pattern formation was motivated by the study of convection , which is the overturning of a fluid that is heated from below.
My current research lies in the intersection of Dynamical Systems, Partial Differential Equations and Fluid Mechanics. In particular, I am applying tools from spectral theory, dynamical systems and statistical mechanics to study instabilities in a hydrodynamic system modelled using Navier-Stokes equations for compressible fluid flow. The project I am currently working on stems from modelling the turbulent flow found in the atmosphere and studying the onset of complex convective patterns caused mainly by the interactions of solar radiation with the Earth's surface.