Learn ML / Going deeper
Going deeper
The mathematical thinking behind the algorithms and equations of computation and ML: for when you want to know why things work, or found you like maths more than you thought. Plus specialist topics.
Mathematical foundations
These build on each other, like the core of a maths degree.
Specialist and graduate topics
Stand-alone topics, each with what it builds on.
- Information theoryBuilds on Probability theory
- Learning theoryBuilds on Probability theory and Real analysis
- Diffusion and flow matchingBuilds on Stochastic processes, plus Deep learning (stage 4)
- Geometric deep learningBuilds on Algebra and group theory, plus Deep learning (stage 4)
- Game theoryBuilds on Foundations (stage 1)
- Discrete optimisationBuilds on Discrete mathematics
- Physics and engineeringBuilds on Differential equations (Computational modelling)
Resources per subject
proof-based you learn by proving results · practical you learn by computing and applying
Mathematical foundations
Proofs and mathematical thinking
- Elliot Nicholson, LogicClear, worked proof-writing exercises, from logic and quantifiers to proof techniques. Then practise on Real analysis, the first full course built on proofs.★ recommended · proof-basedvideos · proof-writing exercises
- Elliot Nicholson, Set Theoryproof-basedvideos · proofs
- Keith Devlin, Introduction to Mathematical Thinking, Stanford University (Coursera)proof-basedvideos · exercises · free to audit
4 more options
- Andrew Stacey, The Nature of Proof (Mathforge)proof-basednotes
- Art of Problem Solving, Proof writingproof-basednotes · exercises
- Burak Kaya, Set Theory, Middle East Technical Universityproof-basedset theory · videos
- Set Theory for GATE, PacketPrep (exam prep)practicalvideos
Discrete mathematics
- Tom Leighton, MIT 6.042J: Mathematics for Computer ScienceCounting, graphs, number theory and discrete probability. Also the standard place where computer scientists learn to write proofs: induction, invariants and counting arguments.★ recommended · proof-basedvideos · notes · exercises
Theory of computation
- Michael Sipser, MIT 18.404J: Theory of ComputationAutomata, computability and complexity: the most proof-heavy part of computer science, taught by the author of the standard textbook.★ recommended · proof-basedvideos · notes · exercises
Real analysis and measure theory
- Francis Su, Real Analysis, Harvey Mudd CollegeA much-loved lecture series that teaches you to read and write proofs in analysis.★ recommended · proof-basedvideos
- Elliot Nicholson, Real Analysisproof-basedvideos · worked proofs
- Francis Su, Real Analysis course websiteproof-basednotes
7 more options
- Paul Seidel, MIT 18.100C: Real Analysisproof-basednotes · exercises
- Katrin Wehrheim, MIT 18.100B: Analysis Iproof-basednotes · exercises
- Victor Guillemin, MIT 18.101: Analysis IIproof-basednotes · exercises
- Arthur Parzygnat, Analysis IIproof-basedvideos
- Jeff Viaclovsky, MIT 18.125: Measure and Integrationproof-basedmeasure theory · notes
- Joel Feinstein, Functional Analysis, University of Nottinghamproof-basedvideos
- Elliot Nicholson, Measure Theoryproof-basedmeasure theory · videos
Probability theory and mathematical statistics
- MIT 18.175: Theory of ProbabilityProbability built rigorously on measure theory: the natural step after real analysis.★ recommended · proof-basednotes · exercises
- Rick Durrett, Probability: Theory and Examplesproof-basedfree book
- MIT 18.655: Mathematical Statisticsproof-basednotes · exercises
Complex analysis
- Alar Toomre, MIT 18.04: Complex Variables with ApplicationsThe official MIT course.★ recommended · practicalvideos · notes · exercises
- Petra Bonfert-Taylor, Introduction to Complex Analysis, Wesleyan University (Coursera)practicalvideos · exercises · free to audit
- Fabio Vlacci, Complex Analysis (ICTP)proof-basedvideos
2 more options
- James Cook, Complex Analysis (supermath.info)videos · notes · exercises
- Rodolfo Rosales, MIT 18.04 (1999): problem setspracticalexercises
Algebra and group theory
- Lothar Göttsche, Abstract Algebra (ICTP)A full university course from the International Centre for Theoretical Physics. Group theory underpins symmetry in geometric deep learning.★ recommended · proof-basedvideos
- Sheldon Axler, Linear Algebra Done Rightproof-basedfree book · proof-based linear algebra
- Benedict Gross, Abstract Algebra, Harvard Extension School (lecture recordings, re-uploaded)proof-basedvideos
3 more options
- J.S. Milne, Group Theoryproof-basednotes · exercises
- Group Theory Methods in Physics, IIT Bombay (NPTEL)practicalvideos · notes · exercises
- Ladislau Fernandes, Group Theoryvideos
Topology and geometry
- James Munkres, MIT 18.901: Introduction to TopologyThe standard first course, from the author of the classic textbook.★ recommended · proof-basednotes · exercises
- J. Rasmussen, Metric and Topological Spaces, Cambridge (Part IB notes)proof-basednotes
- Daniel Chan, Point-set topology, UNSWproof-basedvideos
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- Elliot Nicholson, Metric Spaces and Topologyproof-basedvideos
- Bruno Zimmermann, Topology (ICTP)proof-basedvideos
- E-Academy, Topologyvideos
- Jonathan Evans, Topology and Groupsproof-basedvideos
- Tadashi Tokieda, Topology and Geometry (AIMS)videos
- Haynes Miller, MIT 18.905: Algebraic Topology Iproof-basedalgebraic topology · notes · exercises
- Pierre Albin, Algebraic Topology, University of Illinoisproof-basedalgebraic topology · videos
- Fernando Rodriguez Villegas, Algebraic Topology (ICTP)proof-basedalgebraic topology · videos
- Isabel Darcy, Algebraic Topology, University of Iowaalgebraic topology · videos · notes
- N.J. Wildberger, Algebraic Topology, UNSWalgebraic topology · videos
- Claudio Arezzo, Differential Geometry (ICTP)proof-baseddifferential geometry · videos
- N.J. Wildberger, Differential Geometry, UNSWdifferential geometry · videos
- N.J. Wildberger, Universal Hyperbolic Geometry, UNSWgeometry · videos
Stochastic processes
- Robert Gallager, MIT 6.262: Discrete Stochastic ProcessesMarkov chains, random walks and Poisson processes; the base for SDEs and diffusion models.★ recommended · proof-basedvideos · notes · exercises
- quantpie, Stochastic Calculus and SDEsproof-basedvideos · the maths behind diffusion models
- Jeff Gore, MIT, Stochastic modeling (Systems Biology lecture)practicalvideo
3 more options
- Steven Lalley, Brownian Motion, University of Chicagoproof-basednotes
- MathWorks, Stochastic differential equation modelspracticaldocs · MATLAB
- Chris Rackauckas, Solving stochastic PDEs on GPUs in Julia (Stochastic Lifestyle blog)practicalblog post · Julia
More pure mathematics
- Lothar Göttsche, Algebraic Geometry (ICTP)proof-basedalgebraic geometry · videos
- Kiran Kedlaya, MIT 18.726: Algebraic Geometryproof-basedalgebraic geometry · notes
- Ravi Vakil, Foundations of Algebraic Geometry, Stanfordproof-basedalgebraic geometry · notes
- Abhinav Kumar, MIT 18.781: Theory of Numbersproof-basednumber theory · notes · exercises
1 more option
- Rich Cochrane & Andrew McGettigan, Euclidean Geometry (mathcentre)proof-basedEuclidean geometry · notes
Specialist and graduate topics
Information theory
- Information Theory, Inference, and Learning Algorithms (MacKay, Cambridge)Information theory and Bayesian inference in one free classic.★ recommended · free book
- Seth Lloyd, Introduction to Information Theory (Santa Fe Institute)videos · exercises
Learning theory
- Understanding Machine Learning: From Theory to Algorithms (Shalev-Shwartz & Ben-David)Why and when learning works: PAC learning, VC dimension and generalisation.★ recommended · proof-basedfree book
Diffusion and flow matching
- Peter Holderrieth & Ezra Erives, MIT 6.S184: Introduction to Flow Matching and Diffusion ModelsThe maths of diffusion models from stochastic differential equations, with labs.★ recommended · videos · lecture notes · labs
Geometric deep learning
- Geometric Deep Learning (Bronstein, Bruna, Cohen & Veličković)Symmetry and invariance as the unifying idea behind CNNs, graph networks and transformers.★ recommended · free book · lectures
Game theory
- Ben Polak, Game Theory, Yale UniversityA complete, entertaining Yale course.★ recommended · practicalvideos · notes
- Jackson, Leyton-Brown & Shoham, Game Theory I, Stanford & UBC (Coursera)practicalvideos · exercises · free to audit
- Jackson, Leyton-Brown & Shoham, Game Theory II, Stanford & UBC (Coursera)practicalvideos · exercises · free to audit
2 more options
- Justin Grana, Game Theory I: Static Games (Santa Fe Institute)practicalvideos · exercises
- Justin Grana, Game Theory II: Dynamic Games (Santa Fe Institute)practicalvideos · exercises
Discrete optimisation
- Peter Stuckey, Basic Modeling for Discrete Optimization, University of Melbourne (Coursera)Where the series starts: modelling scheduling and constraint problems.★ recommended · practicalvideos · exercises · free to audit
- Jimmy Lee & Peter Stuckey, Advanced Modeling for Discrete Optimization (Coursera)practicalvideos · exercises · free to audit
- Jimmy Lee & Peter Stuckey, Solving Algorithms for Discrete Optimization (Coursera)practicalvideos · exercises · free to audit
Physics and engineering
- Dennis Freeman, MIT 6.003: Signals and SystemsThe engineering view of signals and systems, the base for control and signal processing.★ recommended · practicalvideos · notes · exercises
- Gilbert Strang, MIT 18.085: Computational Science and Engineering Ipracticalvideos · notes · exercises
- Emilio Frazzoli, MIT 6.241J: Dynamic Systems and Controlpracticalcontrol · notes · exercises
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- Vikram Gadre, Signals and Systems, IIT Bombay (edX)practicalvideos · exercises
- Nitin Upadhyaya, Perturbation theorypracticalperturbation theory · videos
- John Hunter, Asymptotic analysis and perturbation theory, UC Davispracticalperturbation theory · notes
- Erika May, Perturbation theory, Occidental Collegepracticalperturbation theory · notes
- Kripa Varanasi, MIT 2.06: Fluid Dynamicspracticalfluid dynamics · notes · exercises
- Allan Adams, MIT 8.04: Quantum Physics Ipracticalquantum mechanics · videos · notes