Here you will find my reading list with annotations on the content and a suggested order in which to read. The first section pertains to my research interests. The second section is a mathematics and physics undergraduate bibliography: an annotated reading list for the ambitious undergraduate wishing to choose books for self-study.

Here is a selection of some of the papers and books that I've read with the goal in mind of getting someone up to speed in the topics that I take an interest in. It is biased by my opinions on what is good writing and which points of view are easier to understand. I have also included some "aspirational reading".

Work in progress; last updated 2 Jan 2015.

**Characteristic Classes** Milnor and Stasheff

This is the book which all of my PhD advisor's students are made to read and do exercises from. It takes a topological point of view.

**Differential Forms in Algebraic Topology** Bott and Tu

Bott and Tu's amazing book on de Rham cohomology continues where differential geometry classes left off. I never understood cohomology until Bott and Tu's book. They also teach you Cech cohomology. The last chapter deals with characteristic classes. If you had to read just one book on cohomology and characteristic classes, this would be it!

**The moment map and equivariant cohomology** Atiyah, Bott

This paper is a tour de force in the technique of localisation in equivariant cohomology. One of the applications is the Duistermaatâ€“Heckman formula which plays an important part in physics.

**Classical solutions in quantum field theory** E. Weinberg

This is an introductory book on kinks, vortices, monopoles and instantons. It's written for a physicist who knows good maths or for mathematicians who want physical motivation. It's a very good introduction to the field of solitonic particles, the book I wish I had read first.

**Geometry of Yang-Mills fields** Atiyah

Colloquially referred to by my thesis advisor as "the Italian Lectures", this is a complete introduction to the theory of instantons. The first few chapters are fairly simple and very geometrical, almost written for a physics audience but the later chapters employ a lot more algebraic geometry. Like a lot of Atiyah, I find that it goes from opaque to blindingly clear when you have stared at it for long enough. I wish I had read this first but I'm not sure that it isn't because I have spent enough time with it.

**Instantons and Geometric Invariant Theory** Donaldson

This is a short and "unashamedly computational" paper expressing the ADHM construction of instantons very simply. With GIT, Donaldson shows that the moduli of holomorphic bundle on ℂℙ^{2} trivial on a line are equivalent to the moduli of instanton bundles on ℂℙ^{3}.

**The Geometry of Four Manifolds** Donaldson and Kronheimer

I prefer Donaldson to Atiyah in the matter of clarity and this is true here. This book is an amalgamation of the different viewpoints in the study of instantons. In particular, it talks about the generalised Fourier transform viewpoint that is missing in Atiyah and plays a role in the Geometric Langlands/S-duality story. Donaldson wrote it as a grounding for his later work on invariants of four manifolds.

**Classical solutions in quantum field theory** E. Weinberg

See the section on instantons.

**The Geometry and Dynamics of Magnetic Monopoles** Atiyah, Hitchin

**Magnetic Monopoles in Hyperbolic Space** Atiyah

This is a reading list/curriculum for ambitious undergraduates seeking to teach themselves mathematics and physics. It was inspired by the Chichago Undergraduate Mathematics Bibliography which is another good resource with a wider scope. The main differences are that my list includes physics reading and is more recent. You will find books here which were had not been written when the Chicago bibliography was written. It is always dicey when one develops an opinion. My list reflects my opinions as well as the fact that I am a geometer and a mathematical physicist. Reader beware.

Like the Chicago bibliography, my choices are somewhat "honours" stream. You may roughly assume that the physics topics are in ascending order of difficulty. Likewise, the maths topics go from early undergraduate to graduate. There are gradations within each topic and once again, roughly ascending order of difficulty/depth. I have marked "second course" books with an asterisk.

Feedback is welcome and I can be contacted by email. Work in progress; last updated 2 Jan 2015.

**Fundamentals of Physics** Halliday, Resnick, Walker

This is *the* first year physics textbook. It goes through derivations of formulae in reasonable detail. Doing the exercises will familiarise you with the "physical way of thinking". Some older editions of this book are used in Physics Olympiad training.

**Six Easy Pieces** Feynman and Sands

This isn't really an intro physics textbook but I felt that it deserved mentioning. It's six of the easier lectures from the Feynman Lectures. I was given this book by my father at age twelve and I wanted to be a physicist ever since, though I did not know it at the time.

**Feynman Lectures** Feynman and Sands

What is there to say? The Feynman lectures remain, decades after his death the clearest introductory text on physics. Feynman is an idol of mine and one of his greatest strengths is his ability to explain - to make simple yet inspire imagination. These lectures are Feynman's attempt at reinventing the way that physics is taught. While it has not quite become the standard textbook, his ability to explain shines from its pages. The cherry on top is that it is available online for free, thanks to the generosity of Caltech.

**Introduction to Electrodynamics** Griffiths

This was my first Griffiths book and it was love at first sight. Griffiths is very clear and the examples and exercises are well-paced.

***Modern Electrodynamics** Zangwill

On my to-read list. This book has recently come up on my radar as a contender modern alternative to Jackson.

***Classical Electrodynamics** Jackson

This is the graduate level electrodynamics book that everyone appears to have been scarred by. I've never properly read it. It's got some good bits and covers some advanced stuff that Griffiths does not such as Cerenkov radiation and the treatment of electromagnetism as a gauge field theory. A lot of the material is about how to do calculations in lots of different geometries. My feeling is that unless you are an electrical engineer, this will mainly serve as a reference.

**Introduction to quantum mechanics** Griffiths

Like the other Griffiths books, this is very clearly written. It covers quantum mechanics from a very elementary point of view and if you intend to do work in quantum mechanics/optics, this won't be enough. However, if you want to learn particle physics, go straight to Griffith's intro to elementary particles.

**Quantum Mechanics and Path Integrals** Feynman, Hibbs

Feynman discovered the path integral approach and Feynman explains it best.

**Principles of Quantum Mechanics** Dirac

I have not read this but I love Dirac and the reviews are generally very good. Aspirational reading.

I have neglected some classics in GR because I prefer books that take a more modern approach and are reasonable in length (goodbye MTW). Books which I haven't read for these reasons include Schutz, D'inverno, Misner-Thorne-Wheeler and Weinberg.

**Classical mechanics** Goldstein, Poole, Safko

Known as Goldstein. I picked this up in high school, struggled with it for a time and then put it down when my teacher told me to work on exams instead. Looking back at it now, it's not a bad book but the words "old fashioned" comes to mind.

**Lectures on Classical Dynamics** Tong

These course notes are conversational but don't waffle too much and explain and motivate well. Highly recommended and it's free.

**Spacetime and Geometry** Carroll

This is one of (if not straight out) the best modern introduction texts on GR. The book is "baby Wald" in that it covers a lot of the early chapters of Wald in a far more accessible way. However, it is not just "baby Wald": there are also chapters of gravitational waves, cosmology and quantum field theory in a curved spacetime (which Wald leaves for his second tome Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics). There is also a great short explanation of tensors, fields and classical mechanics.

**General Relativity** Wald

This is the modern definitive text on general relativity. It is sufficiently rigorously written to use as training in mathematical GR (there is always Hawking and Ellis if this does not suffice). Graduate level.

**Gravity: An Introduction to Einstein's General Relativity** Hartle

If Carrolls's dazzling introduction is too challenging, this is a step down in difficulty. Hartle's book is aimed at the undergraduates with little special relativity and no differential geometry under their belts. The book has an emphasis on the physics: there is good coverage of tests of GR, which the previous two books neglect.

**Problem book in relativity and gravitation** Lightman

**Cosmology** Weinberg

My choice of texts could be said to be "modern". Classics such as Mandl-Shaw, Sakurai and Coleman not mentioned in here probably still holds great value but as an novice eager for the frontiers of research, I've preferred texts which incorporate the latests viewpoints in the field.

Before learning QFT, you will need some prerequisites from other fields. A bare minimum non-canonical example of pre-QFT reading might be Feynman lectures, Griffiths' electrodynamics and quantum mechanics, Carroll's book/Tong's classical mechanics notes, calculus/analysis, Artin's Algebra and something on tensor calculus. A modern choice for a theoretician might be Griffiths -> Schwartz -> Elvang and Huang.

**Introduction to elementary particles** Griffiths

This is the great achievement of Griffiths' opus. He provides a very accessible introduction to quantum field theory and particle physics, teaching you special relativity and classical mechanics in the process. It's a zeroth course in QFT. Read this first!

**Quantum Field Theory and the Standard Model** Schwartz

The most modern first course on QFT. The subject is motivated very well and the context for the subject's idiosyncracies are given. The first section is quite standard material: a quantum mechanics review, an intro to Lagrangians and Hamiltonians, Greens functions and Feynman diagrams. A standout is part IV which covers modern methods like spin helicity formalism and BCFW recursion which are absent from older texts. This book is perfect for self-study.

**Quantum Field Theory** Srednicki

Another modern book and a faster/steeper alternative to Schwarzt, this book takes quantum field theory from a spin number by spin number approach. I really like that traditionally "later" topics like renormalisation are arrived at quite quickly. However, the price is that photon/QED calculations are left for much later. Notably, instantons and monopoles are covered towards the end of the book as well as Grand Unification.

**Lectures on Quantum Field Theory** Tong

These are lecture notes rather than a textbook so it can be a bit terse at times. He has taken pieces out of the traditional texts ("Peskin and Schroeder explained") and made a fairly coherent and clear exposition. Also, free.

**Quantum Field Theory in a Nutshell** Zee

I did not take a liking to this one. I felt that it lacked detail. However, it receives rave reviews so it might be worth looking at.

***Gauge Field Theories** Frampton

More of a second course in QFT. He doesn't spend much time on motivation but it's fast, concise and clear. And short. One of my favourites.

***Scattering Amplitudes** Elvang and Huang

A second course in QFT with a focus on computing massless amplitudes via the spinor helicity formalism (Srednicki and Schwartz do a bit of this too). It starts out very basic and you only need to know what scattering amplitudes, Feynman diagrams, propagators and Feynman rules are (so you could really read this after Griffiths). I think of this as a guide for calculating massless amplitudes. Their goal is to "provide a practical introduction to some on-shell methods, taking as a starting point what you know after a first introductory course on quantum field theory. Indeed, much of that material in Sections 2 and 3 could be part of any modern course on quantum field theory, but as it is generally not..." (Schwartz remedies this.)

***An Introduction to Quantum Field Theory** Peskin and Schroeder

A generation of quantum field theorists and collider physicists trained off this manual.

***Quantum Theory of Fields** Weinberg

This three volume behemoth is *the complete book*. It's reputation is the book where you find stuff you can't find elsewhere. The writing is somewhere in between what would please a mathematician and what would please a physicist. Not really an early text on QFT.

**Fundamentals of Calculus** Stewart

That expensive but standard textbook. It's good practice for basic skills.

**Calculus** Spivak

Spivak's book comes from a time when calculus was not taught in secondary/high school. It is aimed at ambitious or capable students and teaches entry level calculus from a rigorous point of view (essentially analysis). His writing is very easy to follow and the exercises are in the style of a calculus text, so not very intimidating and suited to self-study.

**Real Analysis** Tao

Australian Fields medalist and superstar Terry Tao wrote this two volume book as notes for his calculus class. The early part of the book attempts to train students in proof-writing with the exercises based around foundational material (set theory, ZFC and Peanos axioms, Cauchy construction of the real numbers). The second volume is a course in multivariable analysis and measure theory.

**Real Mathematical Analysis** Pugh

I think of this as a replacement to baby Rudin. It moves faster than Tao and Spivak. The writing is very clear and economical. The book also covers a second course in analysis on function spaces, multivariable calculus and measure theory. He has some interesting topics in there like space filling curves and Antoine's necklace. The exercises are a standout: they are graded by difficulty and include questions from UC Berkeley's exams for first year grad students.

***Real and complex analysis** Rudin

The first half is on measure theory and function spaces. Graduate level!

**Algebra** Artin

Artin's well-written albeit wordy book integrates linear algebra with abstract algebra. Two for the price of one. The symmetries motivation for the introduction of groups is particularly well done.

**Introduction to Linear Algebra** Lang

Not to be mistaken for Lang's Linear Algebra, this book assumes almost no prerequisites and is a good place to learn to write proofs. As with any other Lang book, none of the material is motivated, there isn't much in the way of practical applications and the writing is elegant and crystal clear.

***Introduction to Linear Algebra** Lang

A second course in linear algebra, done with much abstraction, eschewing much of the matrix viewpoint. A good follow up to Introduction to Linear Algebra. As with the other book, not much motivation nor application is given.

**Linear Algebra** Friedberg, Insel, Spence

I never got very far with this before I lost my copy. Strikes a good balance between concreteness and abstraction, but not really a first "proofs course".

**Algebra** Artin

Artin's well-written albeit wordy book integrates linear algebra with abstract algebra. Two for the price of one. The symmetries motivation for the introduction of groups is particularly well done.

**A first course in abstract algebra** Fraleigh

An alternative to Artin. Almost too elementary. Good place to go if you find that you're just not grasping abstract algebra.

***Abstract Algebra** Dummit and Foote

Supposed to be a good book for breaching the gap between Artin and Lang. Sadly, I read Lang before I read this so I can't tell you if what the difficulty level actually is.

***Algebra** Lang

Graduate level (it was used as the class text in the graduate algebra course I took at Berkeley) but don't let this put you off from reading it early, say after Artin. Puts the abstract in abstract algebra. The point of view is very modern with category theory introduced fifty pages in. I used to think that it was merely a reference but I have come around to thinking that it is a great book to learn from.

Point set topology is a generalisation of the material of real analysis and you need it for the rest of the topics past this point. These books also happen to contain some material on algebraic topology, especially on the first of the homotopy groups: the fundamental group.

**Topology** Munkres

This is the traditional text on point set topology and intro algebraic topology. It's very written but still challenging to the beginner. The first four chapters is adequate point set topology for the other fields of maths.

**Introduction to topological manifolds** Lee

This is a newer alternative to Munkres - and spelled out a bit more. Might be easier for self-study.

**Schaum's Tensor calculus** Kay

For practice with tensors and a very elementary explanations of forms at the end. I will probably be shunned by all the purists out there for recommending a Schaum's but it gets the job done.

**Introduction to smooth manifolds** Lee

**A comprehensive introduction to differential geometry vol 1.** Spivak

He "translates" Gauss and Riemann for you, which is a treat. Requires more mathematical maturity than the writing style alludes to.

**Riemannian geometry** Do Carmo

**Lectures on Differential geometry** Chern, Chen, Lam

Shiing Shen Chern was one of the masters of differential geometry and it shows in this book. Clear and brisk. Sadly, no exercises.

***Foundations of differential geometry** Kobayashi and Nomizu

I use this as a research reference.

You will need to know some point set topology before proceeding down this rabbit hole.

**Algebraic Topology** Hatcher

The elementary course in algebraic topology. And free!

**Differential Forms in Algebraic Topology** Bott and Tu

I never had a good understanding of cohomology until I read this. This book introduces the essential tools of cohomology via de Rham and Cech cohomology which are both in my opinion, the best for computing examples.

***Topology and Geometry** Bredon

Elegant, modern and readable. Second course in algebraic topology.

**Complex Variables and Applications** Churchill and Brown

**Complex Analysis** Ahlfors

**A course in complex analysis and riemann surfaces** Schlag

**Partial differential equations** Strauss

**Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic methods and perturbation theory** Bender and Orzag

**Schaum's Laplace Transforms** Spiegel

***Partial differential equations** Evans

**Commutative Algebra** Atiyah and MacDonald

Brisk, terse and elegant. Work through the exercises and secretly, you'll learn algebraic geometry in the process. Not much in terms of motivation.

***Local Algebra** Serre

An alternative to Atiyah-MacDonald. Assumes that you already know some basic commutative algebra (ideals and that sort of thing.) Even brisker, if that were possible.

**Commutative algebra with a view toward algebraic geometry** Eisenbud

It's supposed to be a textbook but I found it a bit too busy. Works as a reference though. The aim of the book is to cover all the commutative algebra outsourced by Hartshorne's Algebraic geometry.

You really need to know some commutative algebra before this point. Alternatively, read Lang's Algebra's section on rings.

**Algebraic geometry: A first course** Harris

**Computational algebraic geometry** Schenck

**The geometry of schemes** Eisenbud and Harris

**Algebraic geometry** Hartshorne

**Lie groups, Lie algebras and Representation Theory** Hall

Hall takes an elementary approach. All of the Lie groups he deals with are matrix Lie groups and this is very helpful if you have just finished a linear algebra course. The focus is on the representation theory of SU(2) and SU(3), the prototypical examples and the most important in particle physics. This makes it very concrete. The book for self-study.

**Representations of compact Lie groups** Brocker and tom Dieck

**An Introduction to Homological Algebra** Rotman