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.

Partial Diff. Eq.

  1. Partial Differential Equations — Christopher C. Tisdell

Low-Dimensional Geometry

  1. Low-Dimensional Geometry — Francis Bonahon

Elementary Diff. Geo

  1. Elementary Differential Geometry — Andrew Pressley

Metric Spaces

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

Topological Manifolds

  1. Introduction to Topological Manifolds — John M. Lee
  2. An Introduction to Manifolds — Loring W. Tu

Riemannian Manifolds

  1. Introduction to Riemannian Manifolds — John M. Lee

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.

Vector Calculus

  1. Vector Calculus — Jerrold Marsden & Anthony Tromba

Real Analysis

  1. Understanding Analysis — Stephen Abbott

Smooth Manifolds

  1. Introduction to Smooth Manifolds — John M. Lee
  2. An Introduction to Manifolds — Loring W. Tu

Complex Manifolds

  1. Introduction to Complex Manifolds — John M. Lee

Axiomatic Geometry

  1. Axiomatic Geometry — John M. Lee

Reading guide — Mathematical Physics

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

A Logical Introduction to Proof

  1. A Logical Introduction to Proof — Daniel W. Cunningham

Elements of Modern Algebra

  1. Elements of Modern Algebra — Jimmie Gilbert & Linda Gilbert

A First Course in Abstract Algebra

  1. A First Course in Abstract Algebra — John Fraleigh

Linear Algebra Done Right

  1. Linear Algebra Done Right — Sheldon Axler

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

Partial Diff. Eqs.

  1. An Introduction to Partial Differential Equations — Daniel Arrigo
  2. Partial Differential Equations — Christopher C. Tisdell

Vector Calculus

  1. Vector Calculus — Jerrold Marsden & Anthony Tromba

Elementary Differential Geometry

  1. Elementary Differential Geometry — Andrew Pressley

Low-Dimensional Geometry

  1. Low-Dimensional Geometry — Francis Bonahon

Introduction to Manifolds

  1. An Introduction to Manifolds — Loring W. Tu

Diff. Geometry

  1. Differential Geometry — Loring W. Tu

Si Li QM

  1. Quantum Mechanics and Geometry — Si Li

Si Li Classical Mechanics

  1. Classical Mechanics and Geometry — Si Li

Si Li Electromagnetism

  1. Electromagnetism and Geometry — Si Li

Si Li QFT

  1. Crash Course on QFT and Symmetry — Si Li

Visual Diff. Geometry

  1. Visual Differential Geometry and Forms — Tristan Needham

Complex Analysis

  1. Complex Variables and Applications — James W. Brown & Ruel V. Churchill

Visual Complex Analysis

  1. Visual Complex Analysis — Tristan Needham

Real Analysis

  1. Understanding Analysis — Stephen Abbott

Measure Theory

  1. Measure, Integration & Real Analysis — Sheldon Axler

Metric Spaces

  1. Metric Spaces — Satish Shirali & Harkishan Vasudeva

Topology

  1. Topology — James Munkres

Hilbert Spaces

  1. Introduction to Hilbert Spaces with Applications — Lokenath Debnath

QM for Math

  1. Mathematical Aspects of Quantum Mechanics — Roland Speicher

QFT for Math

  1. What is a Quantum Field Theory? — Michel Talagrand