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📐 Foundational Mathematics Dojo

Fundamentals of Mathematics & Physics for Materials Informatics

📚 18 Series | 📖 90 Chapters | 🎯 Cross-Domain

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🎓 About Foundational Mathematics Dojo

The Foundational Mathematics Dojo is a cross-cutting series that provides mathematical foundations across all domains (PI/MS/MI/ML). Students learn mathematical tools essential to materials science, process engineering, and machine learning—including mathematical physics, statistical mechanics, probability theory, and numerical computing—pairing theory with implementation (Python code).

Key Features: Each series is structured in cycles of "theory → examples → implementation → exercises," deepening understanding not just through equations but through hands-on Python coding. Provided in Jupyter Notebook format, learners can study at their own pace interactively. Progressive instruction spans from foundational basics for beginners to advanced applications useful in research.

📐 Mathematical Physics Fundamentals Series (4 Series)

📘

Introduction to Calculus and Vector Analysis

Basics of differentiation and integration, multivariable calculus, vector fields, gradient/divergence/curl, numerical calculus implementation

Beginner 90-110 min 5 Chapters, 35 Examples

📘

Linear Algebra and Tensor Analysis

Matrices and determinants, eigenvalues and eigenvectors, tensor fundamentals, NumPy/SymPy implementation, ML/MS applications

Beginner 100-120 min 5 Chapters, 40 Examples

📘

Complex Analysis and Special Functions

Complex numbers and complex plane, holomorphic functions and residue theorem, Fourier transform and Laplace transform, Bessel functions, signal processing applications

Intermediate 90-110 min 5 Chapters, 35 Examples

📘

Partial Differential Equations and Boundary Value Problems

Wave equation and heat equation, Laplace equation, variational methods and functionals, finite element method, process simulation applications

Intermediate 100-120 min 5 Chapters, 40 Examples

⚛️ Quantum Mechanics Series (6 Series)

📗

Introduction to Quantum Mechanics

Wave function and Schrödinger equation, quantum harmonic oscillator, angular momentum and hydrogen atom, perturbation theory, solid-state quantum theory and materials science applications

Intermediate 90-110 min 5 Chapters, 35 Examples

📗

Introduction to Quantum Field Theory

Field quantization, free and interacting fields, Feynman diagrams, and renormalization — foundations of particle physics and many-body theory

Advanced 200-250 min 5 Chapters, 40 Examples

⚛️

Introduction to Quantum Computing

Quantum computing for materials and chemistry research: qubits and gates, the variational quantum eigensolver applied to electronic structure, and a realistic assessment of NISQ-era limits — with every simulator built from scratch in NumPy, no quantum SDK required

Advanced 185-210 min 5 Chapters, 36 Examples

🔧

Introduction to Quantum Hardware

Quantum hardware as a materials problem: superconducting qubits, trapped ions, neutral atoms, photons, semiconductor spins and topological modes, each traced from its Hamiltonian to the material that limits it — two-level-system defects in surface oxide, the dielectric participation ratio, electrode noise, isotopic purity — with every mechanism verified numerically in NumPy

Advanced 215-240 min 5 Chapters, 33 Examples

🧮

Intermediate Quantum Algorithms

Where provable speedups live and what each one assumes: Grover and amplitude amplification, the QFT and phase estimation, Shor's algorithm factoring 15 and 21 end to end, block encoding and qubitization, and QAOA compared against classical heuristics at equal budget — every algorithm implemented on the same NumPy state-vector simulator, with each speedup stated together with the assumptions it needs

Intermediate to Advanced 225-250 min 5 Chapters, 36 Examples

🧱

Introduction to the Quantum Software Stack

Not an SDK tutorial but a course on what SDKs do: a miniature quantum software stack built from nothing in NumPy — a circuit IR with a unitary-equivalence checker, a peephole optimizer with Euler and KAK gate synthesis, layout and SWAP routing onto lattice and heavy-hex connectivity, a three-level pulse simulator with Rabi, Ramsey and DRAG calibration loops, and readout correction, zero-noise extrapolation, probabilistic error cancellation and resource estimation — with every rewrite verified to preserve the circuit's meaning

Advanced 230-255 min 5 Chapters, 38 Examples

🔥 Statistical Mechanics and Thermodynamics Series (4 Series)

📕

Introduction to Classical Statistical Mechanics

Fundamentals of statistical ensembles (micro, canonical, grand canonical), partition function and free energy, thermodynamic quantities calculation, introduction to Monte Carlo method, materials properties applications

Intermediate 100-120 min 5 Chapters, 35 Examples

📕

Equilibrium Thermodynamics and Phase Transitions

Thermodynamic potentials, Maxwell relations, phase equilibrium and phase diagrams, critical phenomena and scaling, materials science applications

Intermediate 90-110 min 5 Chapters, 35 Examples

📕

Non-Equilibrium Statistical Mechanics

Boltzmann equation and H-theorem, stochastic processes and master equation, Brownian motion and Langevin equation, linear response theory and fluctuation-dissipation theorem, applications to chemical reactions and diffusion processes

Intermediate 90-110 min 5 Chapters, 35 Examples

📕

Computational Statistical Mechanics

Monte Carlo methods (Metropolis, Wang-Landau), molecular dynamics, replica exchange method, free energy calculations, materials properties prediction

Intermediate 100-120 min 5 Chapters, 35 Examples

📊 Probability Theory and Statistics Series (2 Series)

📗

Probability Theory and Stochastic Processes

Random variables and probability distributions, law of large numbers and central limit theorem, Markov processes and Poisson processes, stochastic differential equations, process control applications

Intermediate 100-120 min 5 Chapters, 35 Examples

📗

Inferential Statistics and Bayesian Statistics

Estimation theory (maximum likelihood, interval estimation), hypothesis testing and statistical power, Bayesian inference fundamentals, hierarchical Bayesian models, quality control and ML applications

Intermediate 100-120 min 5 Chapters, 35 Examples

🖥️ Numerical Computation Series (2 Series)

📙

Fundamentals of Numerical Analysis

Numerical differentiation and integration, solving systems of linear equations, ordinary differential equations (Runge-Kutta, Adams), nonlinear equations (Newton's method), SciPy implementation

Beginner 100-120 min 5 Chapters, 35 Examples

📙

Numerical Methods for Partial Differential Equations

Finite difference method (FTCS, BTCS, Crank-Nicolson), finite element method fundamentals, spectral methods, Monte Carlo method, practical process simulation

Intermediate 100-120 min 5 Chapters, 35 Examples

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