• Topology of Metric Spaces: Continuity definitions, open and closed sets, compactness (Heine-Borel, sequential compactness), connectedness, completeness, Cauchy sequences, and Baire Category Theorem.
• Real Analysis: Uniform vs. pointwise convergence, Arzelà-Ascoli theorem, Stone-Weierstrass theorem, Riemann-Stieltjes integration, power series, and radius of convergence.
• Complex Analysis: Cauchy-Riemann equations, Cauchy's Integral Theorem and Formula, Laurent series, Residue Theorem, conformal mappings, and analytic continuation.
• Linear Operators & Canonical Forms: Spectral Theorem for normal operators, Jordan Normal Form, Schur Decomposition, Singular Value Decomposition (SVD), and low-rank approximations.
• Matrix Analysis: Matrix norms (operator, Schatten, Lp,q), Courant-Fischer Min-Max Theorem, Gershgorin Circle Theorem, Perron-Frobenius Theorem, condition numbers, and matrix stability.
• Differential Forms & Vector Calculus: Exterior algebra, wedge product, exterior derivative, differential k-forms, pullbacks, and Generalized Stokes' Theorem on Manifolds.

📚 Primary Reading: Walter Rudin — Principles of Mathematical Analysis | Lars Ahlfors — Complex Analysis | Sheldon Axler — Linear Algebra Done Right | Gene H. Golub, Charles F. Van Loan — Matrix Computations.

• Group Theory: Normal subgroups, quotient groups, First/Second/Third Isomorphism Theorems, group actions, Sylow Theorems, solvable groups, and symmetric or alternating groups.
• Ring & Ideal Theory: Commutative rings, prime and maximal ideals, Principal Ideal Domains (PID), Unique Factorization Domains (UFD), Euclidean domains, and quotient rings.
• Field Extensions & Galois Theory: Algebraic vs. transcendental extensions, splitting fields, finite fields (Galois Fields), Fundamental Theorem of Galois Theory, and cyclotomic polynomials.
• Computational Number Theory: Extended Euclidean Algorithm, Chinese Remainder Theorem, Euler's Totient function, Discrete Logarithm Problem, Quadratic Reciprocity, Miller-Rabin primality testing, and Pollard's rho algorithm.

📚 Primary Reading: David S. Dummit, Richard M. Foote — Abstract Algebra | Victor Shoup — A Computational Introduction to Number Theory and Algebra.

• Measure Theory: Sigma-algebras, Borel sets, outer measures, Carathéodory Extension Theorem, Lebesgue measure construction, and Vitali non-measurable sets.
• Lebesgue Integration: Measurable and simple functions, Monotone Convergence Theorem, Fatou's Lemma, Dominated Convergence Theorem, Lp spaces, and completeness of Lp (Riesz-Fischer Theorem).
• Product Measures & Absolute Continuity: Product sigma-algebras, Fubini-Tonelli Theorems, Radon-Nikodym Theorem and derivatives, and Lebesgue Decomposition Theorem.
• Probability Foundations: Probability spaces, random variables as measurable mappings, independence, Borel-Cantelli Lemmas, and Kolmogorov's 0-1 Law.
• Martingale Theory: Filtrations, conditional expectation as L2 projection, super/sub-martingales, stopping times, Doob's Optional Stopping Theorem, Doob's Convergence Theorem, and Azuma-Hoeffding inequality.
• Stochastic Calculus: Brownian motion (Wiener process), Itô integral, Itô's Lemma, Stochastic Differential Equations (SDEs), Fokker-Planck equation, and Ornstein-Uhlenbeck processes.

📚 Primary Reading: Patrick Billingsley — Probability and Measure | R. M. Dudley — Real Analysis and Probability | Bernt Øksendal — Stochastic Differential Equations.

• Convex Analysis: Convex sets, Supporting Hyperplane Theorem, proper/closed/lower-semicontinuous functions, Fenchel Conjugate, subgradients, and subdifferentials.
• Optimality & Duality: Karush-Kuhn-Tucker (KKT) conditions, Slater's constraint qualification, Lagrangian duality, strong duality, dual ascent, and primal-dual algorithms.
• First-Order Optimization: Nesterov Accelerated Gradient Descent, Proximal Gradient Descent (ISTA/FISTA), ADMM, Mirror Descent, and Frank-Wolfe algorithms.
• Stochastic & Non-Convex Optimization: SGD convergence under smoothness, Polyak-Łojasiewicz inequality, variance reduction (SVRG, SAGA), escaping saddle points via Hessian-free or perturbed SGD, and Adam/RMSProp stability limits.
• Manifold Optimization: Riemannian Gradient Descent, retractions, vector transport, and optimization on Grassmannian and Stiefel manifolds.

📚 Primary Reading: Stephen Boyd, Lieven Vandenberghe — Convex Optimization | Yurii Nesterov — Lectures on Convex Optimization | Nicolas Boumal — An Introduction to Optimization on Smooth Manifolds.

• Smooth Manifolds & Tensors: Smooth maps, immersions, submersions, tangent and cotangent spaces, tangent bundles, vector fields, Lie brackets, and exterior algebra.
• Riemannian Geometry: Riemannian metrics, Levi-Civita connection, parallel transport, geodesics, exponential maps, Riemann curvature tensor, Ricci curvature, and scalar curvature.
• Lie Groups & Lie Algebras: SO(3), SE(3), SU(N) groups, exponential mapping from Lie algebras to groups, adjoint representations, and Lie algebra actions on manifolds.

📚 Primary Reading: John M. Lee — Introduction to Smooth Manifolds | Manfredo P. do Carmo — Riemannian Geometry.

• Structural Complexity: Turing machines (Deterministic, Non-deterministic, Oracle), classes P, NP, coNP, Polynomial Hierarchy, PSPACE, Ladner's Theorem, and Baker-Gill-Solovay Relativization barrier.
• Space & Circuit Complexity: Savitch's Theorem, Immerman-Szelepcsényi Theorem (NL = coNL), NC and AC circuit classes, and Razborov-Smirnov Natural Proofs barrier.
• Interactive Proofs & Zero-Knowledge: IP = PSPACE Theorem (Shamir), Interactive Proof Systems, Arthur-Merlin games (AM, MA), zero-knowledge definitions (Computational, Statistical, Perfect), PCP Theorem, and Hardness of Approximation.
• Quantum Complexity: Postulates of Quantum Mechanics, quantum circuits, class BQP, Shor's factoring algorithm, Grover's search, and Quantum Supremacy frontiers.
• Information Theory: Shannon entropy, joint and conditional entropy, mutual information, KL divergence, Rényi entropy, channel capacity, Fano's Inequality, and Data Processing Inequality.

📚 Primary Reading: Sanjeev Arora, Boaz Barak — Computational Complexity: A Modern Approach | Thomas M. Cover, Joy A. Thomas — Elements of Information Theory | Michael A. Nielsen, Isaac L. Chuang — Quantum Computation and Quantum Information.

• Lattice Geometry: Geometry of numbers, full-rank lattices, determinant, dual lattice, fundamental parallelotope, successive minima, and Minkowski's Theorems.
• Lattice Reduction Algorithms: Gram-Schmidt Orthogonalization, LLL reduction algorithm, Hermite factor, BKZ reduction, and Babai's Closest Vector algorithms (Nearest Plane, Rounding).
• Lattice Hard Problems: Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Bounded Distance Decoding (BDD), Shortest Independent Vectors Problem (SIVP), and worst-case to average-case reductions.
• Learning With Errors (LWE) Mechanics: LWE formulation, Search-LWE to Decision-LWE reduction, Ring-LWE, and Module-LWE.
• PQC Standards & Implementations: NIST PQC Standards (ML-KEM/Kyber, ML-DSA/Dilithium), Number Theoretic Transform (NTT) for fast polynomial multiplication, constant-time side-channel mitigations, and power analysis (SPA/DPA) defenses.
• Zero-Knowledge Proof Construction: Rank-1 Constraint Systems (R1CS), Algebraic Intermediate Representation (AIR), Polynomial Commitment Schemes (KZG, IPA, FRI), SNARKs (Groth16, PLONK), and STARKs.

📚 Primary Reading: Daniele Micciancio, Oded Regev — Lattice-Based Cryptography | Oded Goldreich — Foundations of Cryptography (Volumes 1 & 2) | Steven Galbraith — Mathematics of Public Key Cryptography.

• Lambda Calculus: Untyped lambda calculus (beta/eta reduction, Church-Rosser theorem, Y-combinator), Simply Typed lambda calculus (Strong Normalization), System F, and System F-omega.
• Dependent Type Theory: Calculus of Constructions (CoC), Pure Type Systems, dependent types, Martin-Löf Type Theory (MLTT), and Homotopy Type Theory (HoTT) basics including the Univalence Axiom.
• Curry-Howard Isomorphism: Propositions-as-Types, Proofs-as-Programs, classical vs. intuitionistic logic, and constructive mathematics.
• Formal Program Semantics: Small-step and Big-step operational semantics, denotational semantics (domain theory, complete partial orders), axiomatic semantics (Hoare logic, invariants, weakest preconditions), and Separation Logic.
• Verification Toolchains & Solvers: SMT solving algorithms (DPLL(T), Congruence Closure, Simplex for LIA/LRA, Bit-vector theories), Abstract Interpretation (Galois Connections, Widening/Narrowing), and Theorem Provers (Coq / Lean 4 proof automation).

📚 Primary Reading: Benjamin C. Pierce — Types and Programming Languages (TAPL) | Benjamin C. Pierce et al. — Software Foundations (Coq Series) | Bradley & Manna — The Calculus of Computation: Decision Procedures with Applications to Verification.

• PAC Learning Framework: Probably Approximately Correct (PAC) learning, Agnostic PAC learning, Sample Complexity bounds, and Empirical Risk Minimization (ERM).
• Combinatorial & Function Class Complexity: Growth function, Sauer-Shelah Lemma, VC-Dimension, Vapnik-Chervonenkis bounds, Rademacher and Gaussian complexities, and fat-shattering dimension.
• Concentration of Measure: Sub-Gaussian and Sub-Exponential random variables, Hoeffding's Lemma/Inequality, Chernoff bounds, McDiarmid's Inequality, Bernstein's Inequality, and Hansen-Wright Inequality.
• Overparameterization Mechanics: Double Descent curve (interpolation threshold), benign overfitting in high dimensions, Neural Tangent Kernel (NTK) regime (infinite-width limit, ODE dynamics of gradient flow), and lazy training vs. feature learning regimes (muP parametrization).

📚 Primary Reading: Shai Shalev-Shwartz, Shai Ben-David — Understanding Machine Learning: From Theory to Algorithms | Roman Vershynin — High-Dimensional Probability | Martin J. Wainwright — High-Dimensional Statistics: A Non-Asymptotic Viewpoint.

• Symmetry and Equivariance: Invariance and equivariance principles, Group Equivariant CNNs (G-CNNs), Spherical CNNs, and Gauge Equivariant Mesh CNNs.
• Graph & Mesh Theory: Spectral Graph Theory (Graph Laplacian, Normalized Laplacian, Graph Fourier Transform), Message Passing Neural Networks (MPNNs), oversmoothing, and oversquashing phenomena.

📚 Primary Reading: Michael M. Bronstein et al. — Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges.

• Adversarial Threat Formalization: L-infinity, L2, and L1 threat models, first-order attacks (FGSM, PGD, C&W, AutoAttack), black-box attacks (query-based, transferability, substitute models), and backdoor/data poisoning attacks.
• Certified Defense Mechanics: Randomized smoothing (L2 certified radius via Neyman-Pearson Lemma), Interval Bound Propagation (IBP), bound propagation via abstract interpretation (alpha/beta-CROWN, LiRPA), and exact verification via SMT/MILP solvers (Reluplex, Marabou).
• Distributional Robustness & Domain Generalization: Wasserstein Distributionally Robust Optimization (WDRO), Optimal Transport distance (W1), Invariant Risk Minimization (IRM), Domain Adversarial Neural Networks (DANN), and Out-of-Distribution (OOD) detection.

📚 Primary Reading: Pin-Yu Chen, Cho-Jui Hsieh — Adversarial Robustness for Machine Learning | Cédric Villani — Optimal Transport: Old and New.

• Preference Learning: Bradley-Terry and Plackett-Luce preference models, RLHF objective formulation (PPO with KL-penalty), Direct Preference Optimization (DPO), Kahneman-Tversky Optimization (KTO), and Identity-PO.
• Control-Theoretic Safety: Control Barrier Functions (CBF), Control Lyapunov Functions (CLF), Safe Reinforcement Learning, and Constrained Markov Decision Processes (CMDPs) with Lagrangian multipliers.
• AI Agent Security & Red Teaming: Direct/Indirect prompt injection, Representation Engineering, jailbreak mechanics (GCG algorithm), guardrail system design, automated multi-agent red-teaming, and formal specification drift.

📚 Primary Reading: Dimitri P. Bertsekas — Reinforcement Learning and Optimal Control | NeurIPS / ICML / ICLR / IEEE S&P / USENIX Security — Current Safety & Alignment Proceedings.

• Semiconductor Physics & MOSFETs: P-N junctions, bandgap mechanics, NMOS/PMOS operation, CMOS logic gates, parasitic capacitance, leakage current, and dynamic power consumption.
• Digital Logic & Sequential Circuitry: Boolean algebra minimization (Karnaugh maps, Quine-McCluskey), setup/hold times, clock jitter/skew, metastability, synchronizers, and Finite State Machines (Mealy & Moore).
• Computer Organization & RTL: Register-Transfer Level (RTL) design, ALU architectures, carry-lookahead adders, Wallace tree multipliers, pipelining hazards, forwarding, stall units, and branch predictors (Gshare, TAGE).
• Memory Subsystem Microarchitecture: SRAM topology (6T) vs DRAM cell (1T1C), cache line architecture, direct-mapped vs set-associative caches, MESI/MOESI cache coherence, write-buffers, and store-to-load forwarding.

📚 Primary Reading: Neil Weste, David Harris — CMOS VLSI Design | David Harris, Sarah Harris — Digital Design and Computer Architecture.

• Advanced CPU Microarchitecture: Out-of-Order execution (OoO), Reorder Buffer (ROB), Reservation Stations, Register Renaming (RAT), Branch Target Buffers (BTB), SIMD vectorization (AVX-512, ARM SVE), and cache bottlenecks (false sharing, NUMA effects).
• GPU Architecture & Compute Pipelines: SIMT execution model, Streaming Multiprocessors (SMs), Warps (32 threads), warp divergence, register pressure, shared memory bank conflicts, Tensor Cores (MMA execution paths), and HBM3 vs SRAM bandwidth constraints.
• Roofline Performance Modeling: Arithmetic intensity (FLOP/Byte), operational boundaries, memory-bound vs compute-bound classification, and latency hiding via concurrency.
• Custom Kernel Engineering (CUDA & Triton): Shared memory tiling, coalesced global memory access, warp shuffle primitives, double buffering, asynchronous data transfers, Triton compiler IR (Triton-GPU MLIR dialect), and PTX/SASS assembly inspection.

📚 Primary Reading: John L. Hennessy, David A. Patterson — Computer Architecture: A Quantitative Approach | David B. Kirk, Wen-mei W. Hwu — Programming Massively Parallel Processors: A Hands-on Approach.

• Compiler Frontend & Middle-End: Abstract Syntax Trees (AST), Control Flow Graphs (CFG), Static Single Assignment (SSA) form construction, dominance frontiers, and dominator trees.
• Optimization Passes: Loop Invariant Code Motion (LICM), Dead Code Elimination (DCE), Common Subexpression Elimination (CSE), alias analysis, inline expansion, loop unrolling, and vectorization.
• Polyhedral Compilation Model: Polyhedral representation of nested loops, affine transformations, loop tiling, skewing, interchange, fusion, and Pluto Algorithm.
• MLIR Infrastructure: Dialect Ecosystem (Linalg, Vector, GPU, Arith, LLVM), Operation Definition Specification (ODS), Declarative Rewrite Rules (DRR), passes, lowering pipeline design, and JIT compilation via LLVM ORC.
• Advanced Runtimes & Memory Management: Garbage collection algorithms (tri-color marking, generational collectors), lock-free memory reclamation (Epoch-Based Reclamation, Hazard Pointers, Read-Copy-Update).

📚 Primary Reading: Alfred V. Aho et al. — Compilers: Principles, Techniques, and Tools | Chris Lattner et al. — MLIR: Scaling Compiler Infrastructure for Domain Specific Computation.

• Virtual Memory Subsystem: Multi-level page tables (4-level / 5-level), page walks, Translation Lookaside Buffer (TLB), TLB shootdowns, HugePages, Memory-Mapped I/O (MMIO), page fault handling, and Kernel Page Table Isolation (KPTI).
• Kernel Memory Allocation: Buddy Allocator, Slab/Slub/Slob allocators, memory overcommit mechanics, and Out-Of-Memory (OOM) killer internals.
• Async I/O Subsystems: epoll architecture, io_uring ring-buffer design (Submission Queue / Completion Queue), zero-copy networking, and direct I/O (O_DIRECT).
• eBPF (Extended Berkeley Packet Filter): Bytecode verification, JIT compilation to native machine code, kprobes, uprobes, tracepoints, and XDP packet processing at the NIC driver level.
• Microarchitectural Vulnerabilities & Hardware Security: Transient execution attacks (Spectre, Meltdown, Foreshadow), Rowhammer DRAM bit-flips, fault injection (voltage/clock glitching), and hardware enclaves (Intel SGX, AMD SEV, ARM Realm/CCA).

📚 Primary Reading: Daniel P. Bovet, Marco Cesati — Understanding the Linux Kernel | Brendan Gregg — Systems Performance: Enterprise and the Cloud.

• Fundamental Theorems: FLP Impossibility Theorem, CAP Theorem, PACELC Theorem, and synchronous vs. asynchronous network bounds.
• Logical Time & Ordering: Lamport Timestamps, Vector Clocks, Matrix Clocks, causal consistency, total order broadcast, and Chandy-Lamport distributed snapshot algorithm.
• Formal Specification & Verification: TLA+ (Temporal Logic of Actions), PlusCal, model checking via TLC, verifying safety invariants and liveness properties, and refinement mapping.

📚 Primary Reading: Leslie Lamport — Specifying Systems: The TLA+ Language and Tools | Nancy A. Lynch — Distributed Algorithms.

• Crash Fault Tolerant (CFT) Consensus: Paxos (Single-decree, Multi-Paxos), Raft, Viewstamped Replication (VR), and State Machine Replication (SMR).
• Byzantine Fault Tolerant (BFT) Consensus: PBFT, HotStuff, Narwhal & Tusk (mempool and consensus separation), Async BFT (HoneyBadgerBFT, Verifiable Secret Sharing), and DAG-based consensus mechanics.
• Distributed Data Structures & Transactions: Consistent hashing (Rendezvous, Chord), Distributed Hash Tables (Kademlia), Two-Phase Commit (2PC), Three-Phase Commit (3PC), SAGA Pattern, and Spanner architecture (TrueTime API, external consistency, MVCC).

📚 Primary Reading: Martin Kleppmann — Designing Data-Intensive Applications | Christian Cachin, Rachid Guerraoui, Luís Rodrigues — Introduction to Reliable and Secure Distributed Programming.