I enjoy problems with a good mix of abstraction and applicability. I work on a variety of problems coming from
Dispersive equations model the propagation of waves in various media. Nonlinearities make the study of dispersive equations more realistic, as well as mathematically interesting. For example, when the effect of nonlinearity opposes the effect of dispersion (focusing case), these equations exhibit peculiar solutions called solitons that retain their spacial localization for a long time. John Scott Russell observed such solitary-waves in a shallow river in 1834. Here is an experiment.
My work on nonlinear dispersive equations has been about the low-regularity theory, especially in the critical regime, where we can observe a dichotomy between global existence and blow-up of the solution in finite time, depending on the profile of the initial data. With my advisor Svetlana Roudenko and collaborator Luiz Gustovo Farah, I have studied the Zakharov-Kuznetsov equation
and its modifications, which model propagation of ion-accoustic waves in magnetized plasma. It is also one of the higher dimensional generalizations of the celebrated Korteweg-de Vries equation that models shallow water waves.
The main tool used in my work on asymptotics of low-regularity data is known as the I-method, which was featured in 2006 Fields Medal citations for Terrence Tao. Using harmonic analytic tools like Fourier analysis and Littlewood-Payley analysis, we carefully investigate of the interaction of the constituent frequencies of the solution to establish an almost conservation law for the energy functional of a mollified solution. Using this law, we iterate the local existence theory finitely many times to establish the global well-posedness for rough data. On the other hand, we prove that low-regularity solutions which blow up in finite time have the property that the mass of the solutions concentrates inside a ball of shrinking radius. The main tools here are the I-method and an almost-orthogonal profile decomposition.
Even though I was first exposed to Harmonic analysis as a tool to study dispersive equations, I became interested in its application in other areas of science, in particular signal processing, when I took a class on frames, wavelets, time series analysis and compressed sensing with Prof. John Benedetto.
During Summer 2018, I had the opportunity to work with the ANITA collaboration, a research group conducting balloon experiments in the antarctic region with a goal to detect ultra-high energy neutrinos. Filtering out electro-magnetic noise picked up by the highly sensitive antennas of ANITA payload and de-blurring the signals is known as the deconvolution problem in signal processing. I worked with the astrophysicists at the University of Hawai’i at Monoa under the supervision of Peter Gorham on the deconvolution problem using Fourier and Wavelet analysis and implemented a C++ library called libWTools. I am currently working on a multi-antenna generalization of this algorithm.
A blurred, noisy signal (left) is deconvolved (right) using the ForWaRD algorithm [code]
Understanding how solid materials deform and fail under external loading conditions has been a long-standing area of research for scientists and engineers for centuries. The classical approach is to treat the solids as a continuum and model the displacements of material points as a solution to a differential equation, known as the Cauchy Momentum equation. However, due to the differential formulation, the classical theory fails to describe material behavior when the deformation field is non-differentiable at certain material points, for example, when a fracture is formed.
Introduced by Stewart Silling in 2000, peridynamics has become useful to address this limitations. Peridynamics assumes that every material point interacts with its neighbors via a bond force and reformulates material deformation using an integral equation, thus accommodating discontinuous deformations, such as fractures. Peridynamics has been used to model crack formation and crack branching, among many other fracture problems.
Simulation of crack propagation and branching in soda-lime glass with a pre-notch under external force in the outward vertical direction [code]
Working with Pablo Seleson and Jeremy Trageser at the Oak Ridge National Laboratory, I considered three-dimensional axisymmetric problems, where the geometry of the material is symmetric about an axis of symmetry and the external loading conditions are such that the deformation fields are symmetric about the same axis of symmetry. This is an important class of problem in solid mechanics where the symmetry can be exploited to reduce dimension of the problem. We derived the two-dimensional model that exactly represent the full 3D axisymmetric linear peridynamic model by incorporating out-of-plane bond forces into the representative 2D plane, thus reducing the computational cost significantly.
During Summer 2018, I was learning to use popular machine lerning techniques using scikit-learn as yet another problem-solving tool. While documenting my understanding, I became more interested in the underlying mathematics of it. With Radu Balan and Naveed Haghani, I am working on a permutation-invariant encoding of finite-dimensional data points.