Research

On the Spectral Properties of Discrete Landscape Functions

With Dr. Sudhir Ranjan Jain. The third paper in the landscape-functions programme unifies the discrete interval picture from Paper I with the arithmetic dynamics of Paper II. We give an explicit closed form for the discrete landscape, compute its spectral coefficients exactly, and show that these coefficients lie in explicit abelian extensions of the rationals.

The central result is the Chandra-Jain Conjecture: for the first spectral coefficient, exact computation for every N from 3 to 30 attains the degree bound in every case, and we conjecture this for all N. The paper also connects parity, the Arnold cat map, and Lefschetz numbers to this arithmetic degree question.

Read the PDF on arXiv or the abstract.

A Mathematically Perfect Society

Arithmetic Landscape Functions for the Discrete Catmap

This paper studies how the geometry of a dynamical system is encoded in a spectral object called the diagonal landscape function, defined on a finite discrete torus equipped with the cat map. Also available on Zenodo.

The main result gives an exact closed formula for the landscape in terms of the orbit structure of the map. The landscape is maximized uniquely at the origin, the unique fixed point of the system, whose existence follows from a single arithmetic identity that holds for every modulus simultaneously. This produces sharp spatial localization driven purely by arithmetic, with no disorder and no broken symmetry, a mechanism absent from classical localization theory. Further results include a trace formula linking the landscape to the dynamical zeta function, a complete spectral description of the operator, and a perturbation theorem showing how the localization degrades under an added Laplacian. All results are verified computationally to machine precision.

This is the second paper in a three-part series. Paper I (doi:10.5281/zenodo.17866404) established the one-dimensional instance of the programme. Paper III (arXiv:2608.11372) proves spectral bounds and states the Chandra-Jain Conjecture.

Sigma Xi Student Research Showcase

I presented Can a Shape Hear Itself? — a year-long project on landscape functions under IIT Delhi mentorship — at the Sigma Xi Student Research Showcase, an international competition founded by Cornell University. The showcase drew 311 student participants across 13 disciplinary categories.

I received the People's Choice Award: the single project, across the High School, Undergraduate, and Graduate divisions, with the best abstract, the most technical showcase, and the highest engagement among fellow participants. My project received 2.6k likes — the highest engagement in Sigma Xi history — an overall grade of 47/50, and a $250 cash prize. I was featured in the official Sigma Xi press release.

The Spectral Content of a Sagging String

I wrote my paper with Dr. Sudhir Ranjan Jain.

This paper develops a precise correspondence between the static deformation of an interval and the spectral structure of the Dirichlet Laplacian. The torsion function generated by a uniform load decomposes into local quadratic profiles that coincide with the nodal intervals of each eigenfunction, while the reciprocal of each local profile identifies the unique extremum of the corresponding vibrational mode. These local descriptions are unified by a global spectral expansion that expresses the torsion function as a uniformly convergent superposition of all odd eigenfunctions. The results provide a complete one dimensional framework in which equilibrium geometry, nodal structure, and vibrational behaviour are fully aligned.

Research with Dr. Sudhir Ranjan Jain, IIT Delhi

I met Dr. Sudhir Ranjan Jain after he gave a talk in Gurgaon related to fluid dynamics. I had the opportunity to go to lunch with Dr. Jain, Dr. Nischal Dwivedi, and Ms. Niamat (Oxford University). Soon, I was exploring research ideas with Dr. Jain that developed into a year-long apprenticeship.

Euler Circle

Within eighteen months, between eighth and ninth grade, I completed my school's four-year high school mathematics curriculum under the guidance of Dr. Saugata Ghosh. Looking for opportunities to continue studying mathematics beyond the classroom, I applied to Euler Circle's Cryptography 2024 and IPRW 2024 programs while I was in the tenth grade.

My research journey began with a rigorous study of real analysis under Ari Krishna. Working through selected chapters of Princeton Lectures in Analysis, alongside An Infinitely Large Napkin, I developed the mathematical foundation necessary to explore modern analysis.

From there, I moved into measure theory, studying the construction of the Lebesgue measure, Borel σ-algebras, measurable functions, and introductory stochastic processes. This provided the theoretical framework for my first independent research project.

My paper explored methods of understanding fractal geometry through the Hausdorff dimension, introducing both the underlying measure-theoretic ideas and their geometric applications. You can read my paper here. Slides are available upon request.

Building on this work, I extended my research to Brownian motion, deriving its Hausdorff dimension using the energy method. I later presented this work to an audience of more than thirty Euler Circle students, teaching assistants, and faculty members. Since then, I have remained an active member of the Euler Circle community and continue to stay in touch with Professor Simon.