My mathematical interests lie in the analysis of nonlinear partial differential equations and the calculus of variations with a keen eye towards applications. I find the subtle interplay of analysis, geometry and physics fascinating: energy minimization

My mathematical interests lie in the analysis of nonlinear partial differential equations and the calculus of variations with a keen eye towards applications. I find the subtle interplay of analysis, geometry and physics fascinating: energy minimization often interacts with geometry to yield intricate structures underlying physical phenomena. Specifically, my research program centers around rigorously deriving effective energetic models as asymptotic limits of phase field models, and studying related equilibrium and dynamics questions. Two canonical examples illustrate this philosophy. First, the Modica-Mortola sequence of functionals modeling phase transitions asymptotically yield the surface area functional of the phase boundaries, [29, 33]. Second, the Ginzburg-Landau sequence of functionals modeling superconductors asymptotically yield a finite dimensional renormalized energy functional determining point vortex locations [2]. Effective variational models are useful because they often reduce the analysis of infinite-dimensional phase field models to the study of simpler low dimensional structures such as point vortices or curves to leading order. Such singular structures abound in applications. The projects I have been involved in during graduate school all display these features: (1) Time periodic vortex solutions to the Gross-Pitaevskii equations [35]: The Gross-Pitaevskii equations (GP) arise in modeling quantized vortices in superfluid Helium, and are approximated in a sense to be explained, by a finite dimensional Hamiltonian flow known as the Point Vortex flow (PVF). I desingularize certain periodic orbits to (PVF) called “vortex crystals” to construct periodic orbits for (GP) posed in the unit disc; see Section 1 for details. (2) A Ginzburg Landau type energy for highly anisotropic nematic liquid crystals [17]: Motivated by questions in modeling liquid crystal defects in the nematic-isotropic phase boundaries [24], we study an anisotropic variation of the GinzburgLandau energy that penalizes divergences much more than other combinations of derivatives. We prove a sort of Gamma convergence result to identify a limiting energy. The analysis of minimizers of the limiting energy in a number of examples reveals the emergence of both point defects, as well as line defects; see Section 2. (3) Micromagnetics of Galfenol [9]: We study the peculiar zig-zag Landau state observed in recently reported single crystal Galfenol samples [6]. We employ the micromagnetics functional for cubic ferromagnets appended with the magnetostriction term, that accounts for large magnetoelastic coefficients of this material; see Section 3. (4) Stability of Turing instabilities for the Brusselator system [34]: Using the Lyapunov-Schmidt reduction approach, we show existence and stability of small-amplitude periodic solutions, i.e. Turing patterns bifurcating off the uniform state, for the Brusselator system arising in modeling autocatalytic chemical reactions. Our analysis confirms that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg-Landau equations, rigorously validating the so-called Eckhaus criterion. Being at an early stage of my career, I am very much open to learning and exploring new directions in the mathematical analysis of Partial Differential Equations and its applications.