Everything, in the order it should be learned

How to use VibeFormer, how the three learning tracks differ, and the notation conventions used throughout.

• How to Use This Site
• Choosing Your Track
• The Prerequisite Map
• Mathematical Notation Guide

Vector spaces through SVD. The language every model in this curriculum is written in, built from first principles with worked numeric examples.

• Vectors and Vector Spaces
• Subspaces, Span and Basis
• Linear Independence, Rank and Nullity
• Matrices and Their Operations

Single- and multi-variable calculus, convexity, Lagrange multipliers and KKT — the machinery behind every training loop.

• Functions, Limits and Continuity
• Differentiability and Rules of Differentiation
• Rolle's and the Mean Value Theorems
• Taylor and Maclaurin Series

Counting through Markov chains: the complete probability syllabus, with every distribution derived and applied to worked numeric problems.

• Counting, Permutations and Combinations
• Inclusion–Exclusion and Pigeonhole
• Sample Spaces and the Axioms of Probability
• Events, Independence and Mutual Exclusivity

From descriptive summaries to hypothesis tests, estimation theory, experiment design and causal reasoning.

• Descriptive Statistics
• Skewness and Kurtosis
• Populations, Samples and Sampling Methods
• Sampling Distributions and Standard Error

Linear and integer programming, duality, first-order and interior-point methods, metaheuristics and Bayesian optimisation.

• The Optimisation Landscape
• Formulating an Optimisation Problem
• Linear Programming
• The Simplex Method

Strategic interaction from Nash equilibrium to mechanism design, Shapley values and the games hidden inside GANs and multi-agent systems.

• What Game Theory Studies
• Normal-Form Games
• Dominance and Best Response
• Nash Equilibrium

Python for data work, then the full DSA syllabus: complexity, linear structures, trees, hashing, sorting, searching and graph algorithms.

• Python for Data Science
• NumPy Essentials
• Pandas Essentials
• Complexity Analysis

Relational modelling, algebra, SQL, normalisation and indexing, then OLAP, dimensional schemas and the preprocessing pipeline.

• The Entity–Relationship Model
• The Relational Model and Keys
• Relational Algebra
• Tuple Relational Calculus

A principled approach to charts and exploratory analysis: what to plot, why it works perceptually, and how charts mislead.

• The Grammar of Graphics
• Choosing the Right Chart
• Visualising Distributions
• Visualising Relationships

Propositional through higher-order logic, proof systems, decidability, and the modal, temporal, description and non-classical families.

• What Logic Is
• Propositional Logic
• Logical Equivalence and Laws
• Normal Forms: NNF, CNF and DNF

Search, adversarial games, planning, knowledge representation, and exact and approximate inference in Bayesian networks.

• Agents and Environments
• Problem Formulation and State Spaces
• Uninformed Search
• Informed Search and A*

The concepts every algorithm shares: risk minimisation, generalisation, the bias–variance trade-off, validation and metrics.

• What Machine Learning Actually Is
• Formulating a Learning Problem
• Empirical Risk Minimisation
• Generalisation, Overfitting and Underfitting

Every supervised algorithm on the syllabus, each derived from its objective, traced on small numeric data, then coded from scratch.

• Simple Linear Regression
• Multiple Linear Regression
• Regression Assumptions and Diagnostics
• Polynomial and Basis Expansion Regression

Clustering, dimensionality reduction, association rules and anomaly detection, each traced numerically and derived where it matters.

• The Unsupervised Landscape
• Distance and Similarity Measures
• k-Means Clustering
• Initialisation and Choosing k

Bandits and MDPs through to PPO: the full progression from tabular dynamic programming to deep policy-gradient methods.

• The Reinforcement Learning Problem
• Multi-Armed Bandits
• Exploration Strategies
• Markov Decision Processes

Backpropagation derived and computed by hand, then optimisers, CNNs, RNNs, autoencoders, VAEs, GANs and diffusion models.

• From Perceptron to Deep Networks
• The Universal Approximation Theorem
• Forward Propagation
• Backpropagation

Spectral graph theory, PageRank and node embeddings through to GCN, GraphSAGE, GAT and graph transformers.

• Graph Theory Refresher
• Representing Graphs for Learning
• The Graph Laplacian
• Spectral Graph Theory

Tokenisation and n-grams through embeddings, attention and the complete transformer, with shapes traced end to end.

• The NLP Pipeline
• Text Normalisation
• Regular Expressions for Text
• Stemming and Lemmatisation

How modern LLMs are built, aligned, decoded, evaluated, served and turned into agents — with the mechanics, not the hype.

• What an LLM Actually Is
• Decoder-Only Architecture in Detail
• Pretraining Objectives
• Pretraining Data Pipelines

Full fine-tuning, LoRA and QLoRA, quantisation, distributed training and data curation — with the maths behind each.

• Prompt, RAG or Fine-Tune?
• Full Fine-Tuning
• Catastrophic Forgetting
• Instruction Tuning

Chunking, embeddings, vector indexes, hybrid retrieval, reranking and evaluation, through to agentic and graph RAG.

• Why RAG Exists
• RAG Architecture
• Document Ingestion and Parsing
• Chunking Strategies

Stationarity, ACF/PACF, the ARIMA family, exponential smoothing, state-space models and deep forecasting.

• Components of a Time Series
• Stationarity
• Unit Root Tests
• ACF and PACF

Getting models into production and keeping them honest: versioning, monitoring, drift, fairness, privacy and governance.

• Framing an ML Problem
• Data and Model Versioning
• Experiment Tracking
• Model Registry and Packaging

Formula sheets, banks of fully solved problems, GATE-DA style question sets and interview preparation.

• Formula Sheets
• Solved Problems: Probability
• Solved Problems: Linear Algebra
• Solved Problems: Statistics