Applied
Mathematics.
Orthopedic technology, quantitative finance, and cybersecurity represent core focus areas of research interest, deploying continuous differential equations, Lie algebra, stochastic analysis, and scientific machine learning. I am open and willing to solve any interesting and important problem that makes the world a better place and glorifies the name of God.
Simulate & Formulate
I build interactive simulations across orthopedic technology, finance, and cybersecurity, my core research domains. I am equally driven to solve any meaningful problem that advances the common good and brings glory to the name of God.
Musculoskeletal Kinematics & Contact Mechanics
\mathbf{T}(\boldsymbol{\theta}) = \prod_{i=1}^n \exp(\widehat{\boldsymbol{\xi}}_i \theta_i) \mathbf{M}Solving spatial kinematics on Lie groups SE(3) without gimbal lock, coupled with finite element contact mechanics to predict stress distribution, implant micro-motion, and wear across human articulating joints.
// SE(3) Forward Kinematics (Rust)
use nalgebra::{Vector3, UnitQuaternion};
pub struct JointChain {
pub femur_len: f64,
pub tibia_len: f64,
}
impl JointChain {
pub fn compute_pose(&self, hip: f64, knee: f64) -> (Vector3<f64>, f64) {
let r1 = UnitQuaternion::from_euler_angles(0.0, 0.0, hip);
let r2 = UnitQuaternion::from_euler_angles(0.0, 0.0, knee);
let p_knee = r1 * Vector3::new(0.0, -self.femur_len, 0.0);
let p_foot = p_knee + (r1 * r2) * Vector3::new(0.0, -self.tibia_len, 0.0);
let stress = 12.4 + 8.2 * (hip - knee).abs().sin();
(p_foot, stress)
}
}Research
Focus Areas
Orthopedic technology, quantitative finance, and cybersecurity represent primary focus areas of research interest, harnessing continuous mathematics, scientific computing, and scientific machine learning. Furthermore, I am open and willing to solve any interesting and important problem that makes the world a better place and glorifies the name of God.
Spatial Kinematics & Joint Contact Mechanics
Human articulation is modeled across continuous non-Euclidean manifolds. Using exponential coordinates of Lie algebra se(3), joint movements are tracked without gimbal lock or mathematical singularities. Paired with non-linear finite element formulations, we compute stress-strain distributions in trabecular bone, articulating cartilage, and prosthetic implants under dynamic cyclic loading.
Methodology
Forward and inverse kinematics via product-of-exponentials (POE), coupled with hyperelastic Mooney-Rivlin constitutive tensors for soft tissue compliance.
Applied Outcomes
- •Singularity-free kinematic tracking for microprocessor-controlled knee prosthetics
- •Finite element prediction of implant micro-motion and stress-shielding bone resorption
- •Algorithmic surface reconstruction from clinical 3D scans for custom patient-matched orthotics
Stochastic Calculus & Continuous-Time PDEs
Financial dynamics operate under perpetual uncertainty and path dependence. Stochastic differential equations formulate the evolution of asset prices, stochastic volatility surfaces, and dynamic risk exposures. Through risk-neutral pricing and Kolmogorov backward equations, complex derivatives and exotic multi-asset contracts are resolved using continuous parabolic PDEs.
Methodology
Itô stochastic calculus, Feynman-Kac probabilistic representations of PDEs, and Riemannian gradient optimization on the manifold of symmetric positive-definite covariance matrices.
Applied Outcomes
- •Continuous-time volatility surface calibration and regime-switching drift estimation
- •High-dimensional non-convex portfolio optimization under non-Gaussian tail risk
- •Accelerated Monte Carlo scenario engines sampling complex multi-asset jump-diffusions
Cryptographic Algebra & Lattice Topologies
Security in computational infrastructure is guaranteed by algebraic hardness assumptions. Number theory, finite abelian group laws, and elliptic curve pairings underpin public-key exchange and zero-knowledge cryptographic arguments. Looking ahead, post-quantum cryptography shifts to the geometry of high-dimensional lattices, relying on the hardness of the Shortest Vector Problem (SVP).
Methodology
Weierstrass curve group laws over prime Galois fields GF(p), Weil and Tate pairings, and modular lattice homomorphisms in Learning With Errors (LWE).
Applied Outcomes
- •Elliptic curve key agreement (ECDH) and digital signatures resistant to algebraic decomposition
- •Lattice-based homomorphic encryption architectures enabling encrypted scientific computation
- •Zero-knowledge proof formulations for verifiable execution in distributed systems
Mathematical
Synthesis
I synthesize orthopedic technology, quantitative finance, and cybersecurity as my primary areas of research. Beyond these, I am always open to tackling any interesting and important problem that makes the world a better place and glorifies the name of God.
Biomechanics & Contact Mechanics
\mathbf{T}(\boldsymbol{\theta}) = \prod_{i=1}^n \exp(\widehat{\boldsymbol{\xi}}_i \theta_i) \mathbf{M}, \quad \sigma_{ij} = \mathbb{C}_{ijkl}\,\varepsilon_{kl}Mathematical Framework
I work with non Euclidean differential geometry on matrix Lie groups SE(3) and SO(3) paired with continuous nonlinear continuum elastodynamics.
Scientific Machine Learning
I use physics informed neural networks to solve continuous boundary value contact problems in milliseconds, replacing hours of iterative tetrahedral mesh FEA during active surgical navigation.
Implementation Reality
This translates directly to patient specific joint implants, prosthetic microprocessors, and custom fitted assistive orthotics that adapt to biological gait in real time.
The Research Toolchain
Julia & SciML
PyTorch & JAX
Rust Systems
OpenSim & FEBio
Formulate
Solutions.
Orthopedic technology, finance, and cybersecurity serve as primary focus areas of research interest, but I am open and willing to solve any interesting and important problem that makes the world a better place and glorifies the name of God.
Research Interest Focus Areas
- 01Orthopedic Technology
Kinematic solvers, sensor fusion, active bionics, and patient-specific orthotics.
- 02Quantitative Finance
Continuous PDEs, Monte Carlo drift-diffusion, and volatility surface fitting.
- 03Cybersecurity
Elliptic curve algebra, post-quantum lattices, and verifiable trust protocols.