03 / math & research

Math & Research

Faculty-guided research and self-study — the theory underneath the projects above.

Symplectic Reconstruction Self-Taught Math & Physics

Research — Dr. Sandeep Singh, RPI

Symplectic Reconstruction Research

Fall 2024 – Spring 2025

Phase space vector field of a Hamiltonian system — Phase-space flow of the target Hamiltonian system

Developed a method for reconstructing an unknown Hamiltonian system's parameters from time-series data, by identifying a connection between the Lie group structure of transformation matrices and the time-evolution operators of a Hamiltonian ODE.

That connection produced universal constraints that sharpened a regression algorithm approximating the system's Jacobian matrix locally — turning a fairly abstract piece of group theory into a practical system-identification tool.

Symplectic matrix constraint equations — Symplectic constraint equations, sp(2n,ℝ)

Original vs reconstructed Rössler attractor trajectories — Original vs. reconstructed attractor trajectory

Independent Study

Self-Taught Math & Physics

With guidance from RPI professors, I've been working through Quantum Field Theory, General Relativity, and Statistical Mechanics on my own. The mathematics behind QFT changed how I look at engineering problems that seem unrelated on the surface: understanding Gibbs free energy gave me a much deeper handle on phase transitions when designing cryogenic tanks, and studying the Lie groups behind spinor transformations is what led directly to the symplectic reconstruction project above.

I've also hand-derived neutron diffusion theory and criticality conditions for a reactor core on a whiteboard — not aerospace in the traditional sense, but the same transport-equation and eigenvalue-problem machinery shows up in compressible flow stability and in the reactor designs behind nuclear thermal propulsion.

Whiteboard derivation of neutron transport and probability equations — Neutron transport / criticality derivation, whiteboard

Explaining gauge theory on a blackboard — Explaining gauge theory to a friend, blackboard