Volkan Demir

Undergraduate Roadmaps

An opinionated guide for students navigating an undergraduate degree in mathematics or mathematical physics. Boxes are courses or topics, arrows are suggested prerequisites, and colors mark areas. Start from the left of each strand and follow the arrows — underlined courses are clickable and have book recommendations below the map.

These maps are working drafts — I am completing them, and they will keep growing. Suggestions are welcome by e-mail.

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Reading guide — Pure Mathematics

Click an underlined course on the map to jump to its books and notes here.

Metrics & Topology

  1. Metric Spaces — Satish Shirali & Harkishan Vasudeva
  2. Topology — James Munkres

Manifolds

  1. An Introduction to Manifolds — Loring W. Tu

Category Theory

  1. Very Introductory Notes on Category Theory — Matteo Paganin
  2. An Introduction to Category Theory — Harold Simmons
  3. Categories for the Working Mathematician — Saunders Mac Lane

(Example note — replace with your own.) Start with Paganin’s notes, continue with Simmons; Mac Lane is the reference to grow into.

Linear Algebra I

  1. Linear Algebra Done Right — Sheldon Axler
  2. Linear Algebra — Kenneth Hoffman & Ray Kunze

Real Analysis I

  1. Understanding Analysis — Stephen Abbott

Reading guide — Mathematical Physics

Click an underlined course on the map to jump to its books and notes here.

Metrics & Topology

  1. Metric Spaces — Satish Shirali & Harkishan Vasudeva
  2. Topology — James Munkres

Manifolds

  1. An Introduction to Manifolds — Loring W. Tu

Category Theory

  1. Very Introductory Notes on Category Theory — Matteo Paganin
  2. An Introduction to Category Theory — Harold Simmons
  3. Categories for the Working Mathematician — Saunders Mac Lane

(Example note — replace with your own.) Start with Paganin’s notes, continue with Simmons; Mac Lane is the reference to grow into.

Linear Algebra I

  1. Linear Algebra Done Right — Sheldon Axler
  2. Linear Algebra — Kenneth Hoffman & Ray Kunze

Real Analysis I

  1. Understanding Analysis — Stephen Abbott

Quantum Physics I

  1. Principles of Quantum Mechanics — R. Shankar
  2. Quantum Mechanics: Concepts and Applications — Nouredine Zettili
  3. Quantum Physics — Stephen Gasiorowicz