Abstract Maths 101: From Zero to Hero
Finish the year with a mathematical transformation. A combined, bi‑weekly live programme on Topology, Linear Algebra and C*-Algebras designed for ambitious researchers and industry professionals.
$$\array{ && im (\phi_n) &&\to&& ker (\phi_{n-1}) \\ & \nearrow && \searrow && \swarrow \\ \Omega_{n+1} &&\stackrel{\phi_n}{\to}&& \Omega_n &&\stackrel{\phi_{n-1}}{\to}&& \Omega_{n-1} \\ & && \swarrow && \searrow && \nearrow \\ && coker (\phi_n) &&\stackrel{}{\to}&& im (\phi_{n-1}) }$$Master the Fundamentals
Go beyond recipes and build a rigorous, proof-based understanding of topology and linear algebra from the ground up.
Become AI & Quantum Ready
Grasp the mathematical language behind ML generalisation, quantum computing, and modern algorithms.
Develop Problem-Solving Skills
With bi-weekly homework and fully worked solutions, you'll learn to think like a mathematician and solve complex problems like a mathematician.
Go from Foundation to Application
Equivalent to Two Full Semesters
This intensive programme is designed to provide a comprehensive foundation equivalent to two full semesters of university-level abstract mathematics. Upon completion, you will feel very comfortable with the topics covered and be able to confidently apply these advanced concepts to your own research and industry projects.
Syllabus at a Glance
Track A — Topology 101 (6 lectures)
L1: Topological spaces, bases & subbases
L2: Continuity, homeomorphisms, product/subspace
L3: Separation axioms; Hausdorff & uniqueness of limits
L4: Connectedness & path‑connectedness
L5: Compactness (Heine–Borel, metric spaces)
L6: Quotients & glimpses of algebraic topology
Track B — Linear Algebra (6 lectures)
L1: Vector spaces, subspaces, span, basis, dimension
L2: Linear maps, matrices, rank–nullity
L3: Inner products, norms, orthogonality, projections
L4: Eigenvalues/eigenvectors, diagonalisation, SVD
L5: Normal, self‑adjoint, unitary/orthogonal; spectral theorem
L6: Groups of matrices; applications to ML & quantum
Track C — Introduction to C*-Algebras (6 lectures)
L1: Banach algebras, involutions, and the C*- Algebras axioms
L2: Spectrum, self-adjoint/normal/unitary elements
L3: Positive elements, order structure, and approximate units
L4: Continuous Functional Calculus for normal elements
L5: States, positive linear functionals, and the GNS construction
L6: Irreducible representations, pure states, & Gelfand-Neumark Theorem
Learn With a Global Community
Tap into Our 8,000+ Member Network
Your subscription doesn't just unlock this course. You'll join our vibrant Substack community of over 8,000 researchers, engineers, and enthusiasts. Ask questions, share insights, and connect with peers who are as passionate about advanced mathematics as you are.
Frequently Asked Questions
What time are the live lectures?
To accommodate learners from around the world, live lectures are now held three times. You can join the session that works best for you: 1 PM GMT (for those based in Europe & Asia), or 8 PM GM (for those based in Europe & North America ).
What are the prerequisites for this programme?
A good grasp of basic set theory and a level of mathematical maturity is expected. Nice to be comfortable with basic proof techniques such as induction and contradiction. No prior exposure to topology or linear algebra is required.
What if I miss a live lecture?
No problem. All live lectures are recorded and will be available to paid Substack subscribers within 24 hours, along with the corresponding homework assignment.
Is this course theoretical or applied?
Both. The programme is rigorously proof-based to build a solid theoretical foundation. However, the content is specifically chosen for its relevance and application to machine learning and quantum computing.
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