Research Identity

My Research is a Cycle. Various Phases in the Cycle are listed below. These are guided by fundamental questions that have emerged repeatedly in my natural experience over the years. In usual Language, these are the Areas which I keep revisitng, these areas keep coming back to me. In the Language of Places, these are my homes. In the Language of Light, these are my reflections. In the Language of Arts, these are my Self Portraits. In the Language of Sound these are my Fundamental frequencies (I Resonate with these ideas). In the Language of Groups, these are my Identity. In the Language of Play, these are my Expressions. In the Language of Acting, these are my Characters. In the Language of Family, these are my Relatives. In the Language of Dance, these are my Complementary Pairs. In the Language of Writing, these are chapters of my Autobiography. In the Language of Photography, these are my pictures. In the Language of Languages, these are my Translations. In Natural Language, these are the elements of my Nature. In the Language of Devotion, I see how God is expressed as these forms. Every work shows how I make my own Universe. To be Interpreted as you wish.

Here we go:

Provide a unified formalism for transformation media theory based on differential forms. Based on the work of Greenleaf-Uhlmann, Leonhardt, H. Ammari on inverse problems and Milton’s Theory of Composites, using asymptotic expansions for Steklov eigenvalues and Gohberg-Sigal theory (Argument principle extended to infinite dimensional spaces via Fredholm operators), I with my advisor Prof. Harsha Hutridurga jointly with Prof. Habib Ammari) develop novel schemes of cloaking and analyze it with numerical simulations. This is done by the reformulation of cloaking in terms of non-unique solution to inverse problems such as Calderon’s problem or electrical impedance tomography. Experimental part of this is being done with my advisor Prof. Shobha Shukla’s NEMO research group. For design optimization, the study of Girouard-Nadirashvili-Polterovich’s work on Steklov spectral geometry led to study of Shape Analysis using Steklov operators.

Surveyed literature (16 page pdf summary) on spectral geometry of the Laplacian, including works of Lichneroqicz-Obata, Gromov, Li-Yau (8 page notes pdf), Hoffman-Ostenhoff-Nadirashvili. Particularly interested in the nodal geometry of Laplace eigenfunctions in the compact setting, including works of Mangoubi (inner radius estimates), Colding-Minicozzi, Georgiev-Mukherjee. Highly interested in local elliptic techniques, as exemplified in the recent work of Logunov-Malinnikova on the size of nodal sets. Interested in seeing how such frequency function based local elliptic techniques carry over to the Steklov setting as well, and read some works of Bellova-Lin, Zhu, Georgiev-Roy-Fortin, in that direction. Inspired from my thesis, seeing how frequency function based local elliptic techniques apply in Steklov setting and studying global methods of semiclassical analysis in eigenfunction mass concentration and quantum ergodicity. For instance, billiard flow problems as in Marzuola, Hassell-Hillairet-Marzuola, Cekic-Georgiev-Mukherjee.

Studied relationship of extremal Steklov eigenvalues with minimal surfaces, Hersch’s theorem and Harmonic maps (5 page pdf).

To be able to build materials for quantum computing, studied BCS theory and high temperature superconducting materials. Surveyed literature on solid state theory including BCS theory of superconductivity. Studying Josephson junctions and Google’s implementation of qubits for building Quantum computers. This builds on background in topological quantum computing for which course project was done on Kitaev Model and Majorana Fermions covering Jordan-Wigner Transforms, braiding and Non-Abelian Statistics. Studied Pontryagin’s maximum principle and Optimal Control on Lie groups, theory of Cartan, KAK decomposition, Birkhoff, Schur convexity for SU(N) which extends to Kostant convexity in the general case of compact semisimple Lie Algebras. Learning spectroscopy techniques including NMR in optimal quantum control and error correction for quantum systems.

Performed \(\bf{Density Functional Theory (DFT)}\) based IR and Raman spectrum calculations using Quantum Espresso. DFT calculations of thermoelectric properties of doped \(Mg_2Si\) using VASP, BoltzTrap.

Intend to combine theoretical techniques with computational methods to solve major problems. For instance, using second quantization, tight binding, Wannier excitons, optical Bloch equations, perturbation theory, Green’s functions, Bethe-Salpeter equation with DFT simulations for calculations of excitonic spectrum of ZnO nanorods led to my undergraduate research award. Aim is to work on the GW approximation and many body perturbation theory to get closer to the experimental readings.

Increasing use of data analysis, optimization and statistical methods involving deep neural networks has led to undertaking following courses(click to see content covered via my lecture notes): Optimization lecture notes 1 and 2, Computational Many Body Physics, Numerical Methods in PDEs, Modern Theory of PDEs, Machine Learning Notes, Wedderburn Theory Birkhoff Algebra in Hyperplane Arrangements, Advanced Image Processing (aka Inverse Problems in Imaging), Differential Geometric Methods in Control, Physics and Control, Solid State Systems and Control and Quantum Control.

Please click on the links for detailed description.

Plasmonics and Metamaterials

Steklov Eigenvalues

Density Functional Theory (DFT)

  • Density Functional Theory based IR and Raman spectrum calculations using Quantum Espresso.

  • DFT Calculations for Thermoelectric Properties of Doped Semiconductors using VASP, BoltzTrap.

Course Projects

Hypoellipticity of Elliptic operators

Modern Theory of PDEs (MA540), Prof. Mayukh Mukherjee, Dept. of Mathematics, IITB

  • Presented Proof, including the construction of Parametrix for constant coefficient elliptic operators. Prof. Melrose’s notes were my primary reference.

Kitaev Model & Majorana Femions

Computational Many Body Physics (PH513), Prof. Soumya Bera, Dept. of Physics, IITB

Mie Scattering

Electromagnetic Theory 1 (PH308), Prof. Archana Pai, Dept. of Physics, IITB

  • Presented Slides on Mie theory and scattering by a dielectric sphere, solution of scalar Helmholtz equation by wave vector formalism.

Rashba Spin Orbit Coupling in Graphene and Zigzag Nanoribbons

Introduction to Nanoscience and Nanotechnology (EP 425), Prof. Anshuman Kumar, Dept. of Physics

  • Reproduced plots from this paper using Matlab and presented (Slides) their analysis.

Poincare-Hopf Index Theory

Nonlinear Dynamics (PH542), Prof. Punit Parmananda, Dept. of Physics, IITB

Solitons and Solitary Waves

Nonlinear Dynamics (PH542), Prof. Punit Parmananda, Dept. of Physics, IITB

Nirvana and Anattavada (No-Self theory) in Buddhist Philosophy via Topology

Introduction to Philosophy (HS301), Professor Amrita Banerjee, Dept. of HSS, IITB

  • Term Paper relating Jordan Brouwer Separation Theorem to Anattavada in Buddhist philosophy, Buddha’s logic to Topos theory.

SPPs on thin films and Leakage Radiation Microscopy

Physics of Nanostructures and Nanodevices (EP 432), Professor Anshuman Kumar, Department of Physics, IITB

  • Presented (slides) analysis of SPPs in thin lossy films, application to Leakage Radiation Microscopy and reproduced dispersion plots from Drezet et.al. and Burke et.al..

Geometric Spectral Theory

Harmonic Analysis

Seeing the following sources: