Mathematics Curriculum · tensor-to-tenant

Track 1 · Modules 1–10

Mathematical Foundations

Set theory, proofs, information and measure-theoretic foundations.

#	Module	Focus
1	Notations and structures	Sets, functions, sequences, big-O
2	Probability and combinatorics	Counting, pigeonhole, discrete probability
3	Proofs and structures	Induction, proof by contradiction, well-ordering
4	Information and probability I	Entropy, conditional entropy, mutual information
5	Information and probability II	KL divergence, MDL, Pinsker, data processing
6	Topology and metric spaces I	Open/closed sets, continuity, compactness
7	Topology and metric spaces II	Metrics, completeness, Banach space, contraction
8	Measure and integration I	Sigma-algebras, measures, measurable functions
9	Measure and integration II	Lebesgue integral, convergence theorems
10	Functional analysis	Normed spaces, bounded linear operators, spectra

Track 2 · Modules 11–20

Numerical Methods & Applied Analysis

Floating-point reality, conditioning, factorization and iterative methods.

#	Module	Focus
11	Floating point arithmetic	IEEE 754, rounding, ulp, cancellation
12	Numerical conditioning	Condition numbers, stability, error
13	Root finding and optimization	Bisection, Newton, quasi-Newton
14	Interpolation and approximation	Polynomials, splines, least squares
15	Numerical integration	Quadrature, adaptivity, high-dimensional
16	Eigenvalues and iterative methods	Power iteration, Lanczos, shift-invert
17	SVD and matrix factorizations	LU, QR, Cholesky, SVD, rank
18	Dense vs sparse methods	Storage, bandwidth, iterative solvers
19	Krylov methods	CG, GMRES, preconditioning
20	Randomized numerical linear algebra	Random projection, randomized SVD

Track 3 · Modules 21–30

Calculus, Probabilities & Statistics

Differentiation, optimization, probability and the classical inference toolkit.

#	Module	Focus
21	Automatic differentiation	Reverse-mode, Jacobian-vector products, vector-Jacobian products
22	Unconstrained optimization	Gradient descent, momentum, adaptive methods
23	Constrained optimization	Lagrange multipliers, KKT, convex duality
24	Convex analysis	Convex sets, functions, subgradients
25	Probability theory	Random variables, expectations, variance
26	Distributions and moments	Bernoulli, Gaussian, exponential, Poisson, moments
27	Multivariate probability	Covariance, correlation, joint/marginal/conditional
28	Limit theorems	LLN, CLT, delta method
29	Statistical inference	MLE, MAP, estimators, bias/variance
30	Hypothesis testing	p-values, Type I/II errors, power, MDE

Track 4 · Modules 31–40

Experimentation & Analytics

The quantitative machinery behind trustworthy product experiments.

#	Module	Focus
31	Experimental design	Randomization, control, blocking
32	Sampling and resampling	Bootstrap, permutation, jackknife
33	Confidence intervals	Construction, interpretation, coverage
34	Multiple testing	Bonferroni, FDR, q-values
35	Bayesian inference	Priors, posteriors, conjugate, credible intervals
36	Regression analysis	Linear/logistic, GLM, residuals
37	Causal inference	Potential outcomes, confounding, DAGs
38	Sequential analysis	Group sequential, alpha spending, peeking
39	A/B testing fundamentals	Absolute/relative lift, pooled variance
40	Modern ML metrics	Calibration, NDCG, AUC, fairness

Track 5 · Modules 41–50

ML Theory & Foundation Models

Learning theory and the mathematics of modern foundation models.

#	Module	Focus
41	Learning theory	PAC learning, bias-variance, VC dimension
42	Empirical risk minimization	Loss functions, regularization, generalization
43	Optimization in ML	SGD, mini-batch, convergence
44	Kernel methods	Kernels, SVM, RKHS
45	Probabilistic graphical models	Bayes nets, Markov, inference
46	Bayesian deep learning	Uncertainty, dropout, VI, ensembles
47	Representation learning	Autoencoders, contrastive learning
48	Transformer theory	Attention, positional encodings, scaling
49	Scaling laws	Compute, data, parameters
50	Emergent behavior	Zero-shot, few-shot, capabilities, limitations

Track 6 · Modules 51–60

Distributed Systems Mathematics

Hashing, consensus, consistency and causality for distributed systems.

#	Module	Focus
51	Complexity and cost models	Amortized analysis, communication cost
52	Hash functions	Consistent hashing, minhash, SimHash
53	Routing and load balancing	Weighted round robin, rendezvous hashing
54	Consensus and leader election	Paxos, Raft, leader lease
55	Vector clocks and causality	Happens-before, concurrent events
56	CRDTs and state merging	LWW, G-Counters, conflict resolution
57	Quorums and replication	Read/write quorums, quorum intersection
58	Distributed transactions	Two-phase commit, saga, idempotency
59	Gossip protocols	Failure detection, epidemic broadcast
60	Distributed consistency	Linearizability, sequential, eventual, CAP

Track 7 · Modules 61–70

Applied Probability for Inference

Queueing, processes, sketches and tail behavior behind serving systems.

#	Module	Focus
61	Queueing theory	M/M/1, M/M/c, Little’s law, tail behavior
62	Renewal theory	Renewal processes, residual life
63	Markov processes	Discrete/continuous time, stationarity
64	Poisson processes	Arrivals, thinning, superposition
65	Branching processes	Galton–Watson, extinction probability
66	Statistical process control	Control charts, CUSUM
67	Randomized algorithms	Sampling, hashing, load balancing
68	Tail bounds	Markov, Chebyshev, Chernoff, Hoeffding
69	Streaming and sketching	Count-Min, HyperLogLog, t-digest
70	Heavy hitters	Misra-Gries, Space-Saving

Track 8 · Modules 71–80

Optimization & Inference

Convex and discrete optimization, graph algorithms and approximation.

#	Module	Focus
71	Convex optimization	Duality, interior point, KKT
72	Non-convex optimization	Saddle points, landscape, heuristics
73	Linear and integer programming	Simplex, LP relaxation, branch and bound
74	Dynamic programming	Optimal substructure, state design
75	Network flows	Max flow, min cut, min cost flow
76	Greedy algorithms	Exchange arguments, matroids
77	Graph algorithms	Shortest paths, MST, Euler tour
78	Advanced graph structures	Euler Tour Trees, heavy-light decomposition
79	Spectral methods	Graph Laplacians, eigenvectors
80	Approximation algorithms	NP-hardness, approximation ratios

Track 9 · Modules 81–90

Information, Inference & AI Foundations

Information theory, decision theory, RL and generative modeling.

#	Module	Focus
81	Information theory	Entropy, mutual information, channel capacity
82	Statistical learning theory	Generalization, Rademacher complexity
83	Bayesian statistics	Priors, posteriors, computation, MCMC
84	Decision theory	Utility, expected utility, minimax
85	Game theory	Nash equilibria, zero-sum games
86	Reinforcement learning	MDPs, value iteration, Q-learning
87	Sequential decision making	Bandits, exploration/exploitation
88	Generative modeling	VAEs, GANs, diffusion
89	Causal inference II	Instrumental variables, do-calculus
90	Fairness and safety	Bias, fairness metrics, alignment, robustness

Track 10 · Modules 91–100

Production AI Systems & Advanced Inference

Scheduling, cost, reliability and the cutting edge of inference systems.

#	Module	Focus
91	Scheduling	Priority queues, fair scheduling, work stealing
92	Autoscaling and capacity	Utilization, SLO-based scaling, throttling
93	Cost modeling	Hardware cost, efficiency, pricing
94	Reliability engineering	SLOs, error budgets, SLA, incident response
95	Advanced inference	KV cache, paging, continuous batching
96	Advanced quantization	FP8, AWQ, GPTQ, KV-cache quantization
97	Speculative decoding	Draft models, verification, acceptance rates
98	Long context	RoPE, eviction, memory compression
99	Distributed inference	Prefill/decode disaggregation, expert parallelism
100	Advanced parallel training	Tensor, pipeline, expert parallelism

Category = Learning Module

Nearly every entry is a learning module to derive, prove, implement and explain - not a lecture to watch. The proof of mastery is the artifact you can hand over.