Mathematical Notes

On this page I have archived some notes from lecture courses that I (or occasionally others) have taken and then typed up, as well as some notes of my own on various topics.

• MA482 Stochastic Analysis
• MA3F4 Linear Analysis
• MA475 Riemann Surfaces
• MA377 Rings and Modules
• MA359 Measure Theory
• MA453 Lie Algebras
• MA3B8 Complex Analysis

MA482 Stochastic Analysis

Notes from the University of Warwick's third-year undergraduate course on stochastic analysis, as lectured by David Elworthy in the 2005–06 academic year. The main points are:

1. Re-Cap of Measure Theory
2. Fourier Transforms of Measures
3. Gaussian Measures on Finite-Dimensional Spaces
4. Gaussian Measures on Banach Spaces
5. Cylinder Set Measures
6. The Paley–Wiener Map and the Structure of Gaussian Measures
7. The Cameron–Martin Formula: Quasi-Invariance of Gaussian Measures
8. Stochastic Processes and Brownian Motion in ℝⁿ
9. Itō Integrals as Divergences

 Download the notes in PDF format (47 pp., 436 kB).

MA3F4 Linear Analysis

Notes from the University of Warwick's third-year undergraduate course on linear functional analysis, as lectured by Stefan Teufel in the 2003–04 academic year; the original notes were taken by James Beardwood. These are condensed notes: definitions and theorems only, with no proofs. The main points are:

1. Normed Spaces
2. Functional Operators
3. The Hahn–Banach Theorem
4. Baire's Theorem and its Consequences
5. Fréchet Spaces

 Download the notes in PDF format (16 pp., 152 kB).

MA475 Riemann Surfaces

Notes from the University of Warwick's fourth-year undergraduate course on Riemann surfaces, as lectured by Mark Gross in the 2003–04 academic year. These are condensed notes: definitions and theorems only, with no proofs. The main points are:

1. Preliminary Definitions and Results
2. Maps of Riemann Surfaces
3. Covering Maps and Spaces
4. Analytic Continuation
5. Integration of Differential One-Forms

 Download the notes in PDF format (12 pp., 162 kB).

MA377 Rings and Modules

Notes from the University of Warwick's third-year undergraduate course on rings and modules, as lectured by Charudatta Hajarnavis in the 2003–04 academic year. The main points are:

1. Rings
2. Modules
3. Zorn's Lemma
4. Completely Reducible Modules
5. Chain Conditions
6. Semi-Simple Artinian Rings
7. Wedderburn's Theorem on Finite Division Rings
8. Some Elementary Homological Algebra

 Download the notes in PDF format (61 pp., 332 kB).

MA359 Measure Theory

Notes from the University of Warwick's third-year undergraduate course on measure theory, as lectured by Omri Sarig in the 2002–03 academic year. The main points are:

1. Elementary definitions and properties of general measures.
2. The construction and properties of Lebesgue measure on the real line ℝ.
3. The construction and properties of the Lebesgue integral.
4. Product measure spaces and Fubini's theorem.
5. An introduction to L^p spaces.

As a side note, the horrific experience of typing up this course and MA453 Lie Algebras in Microsoft Word was the straw that broke the camel's back: I ditched Word, converted to LaTeX, and have never looked back.

 Download the notes in PDF format (78 pp., 573 kB).

MA453 Lie Algebras

Notes from the University of Warwick's fourth-year undergraduate course on Lie algebras, as lectured by Roger Carter in the 2002–03 academic year. The main points are:

1. Definitions and Basic Principles
2. Representations and Modules of Lie Algebras
3. Abelian, Nilpotent and Soluble Lie Algebras
4. Representations of Nilpotent Lie Algebras
5. Cartan Subalgebras
6. The Killing Form
7. The Lie Algebra 𝖘𝖑_n(ℂ)
8. The Cartan Decomposition
9. The Root System and the Weyl Group
10. The Dynkin Diagram
11. The Indecomposable Root Systems
12. The Semisimple Lie Algebras over ℂ

As a side note, the horrific experience of typing up this course and MA359 Measure Theory in Microsoft Word was the straw that broke the camel's back: I ditched Word, converted to LaTeX, and have never looked back.

 Download part 1 of the notes in PDF format (49 pp., 377 kB).

 Download part 2 of the notes in PDF format (43 pp., 380 kB).

MA3B8 Complex Analysis

Notes from the University of Warwick's third-year undergraduate course on measure theory, as lectured by Young-Eun Choi in the 2002–03 academic year. The main points are:

1. Differentiability and the Cauchy–Riemann Equations
2. Complex Contour Integration
3. Poles, Residues and Integrals
4. Conformal Maps
5. Harmonic Maps

 Download the notes in PDF format (56 pp., 530 kB).