Following Emily’s advice, I recently signed up to be mentor in the AWM Mentor Network. It’s been pretty good thus far (I recommend it to any of you who would like to do some menting), but I got a request from my mentee that I thought some of our audience might have better ideas about than me.

What math books would you suggest for relatively casual summer reading for an undergraduate math major finishing their third year? This is not the sort of thing I think about a lot, but I know a reasonable number of readers have a lot more experience with young mathematicians than I do.

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Maybe Hardy and Wright’s “An introduction to the theory of numbers”? For someone interested in number theory, it’s still very nice and because it has many independent chapters, it is quite suitable for casual reading. Some chapters are still, to my mind, the best introduction to their topic (for instance the one about the order of magnitude of arithmetic functions).

It is not very abstract (which may or may not be considered an advantage), and many important results are proved in three or four different ways, so one learns many nice tricks and techniques. And although I don’t really have much experience suggesting books for undergraduates, I can say that this was the first “real” math book I read, at a fairly similar age.

In general, for getting a broad and pleasant overview of “classical” mathematics, I like English textbooks of that period (e.g., Whittaker and Watson’s “A course on modern analysis” might also be interesting).

rudin’s real and complex analysis

(joking)

Journey Through Genius (William Dunham) is my favorite casual math book, but might be too easy. The Knot Book is a fun read. A little more difficult, I really enjoyed Silverman+Tate’s Rational Points on Elliptic Curves and Stewart+Tall’s Algebraic Number Theory.

One book I really liked as an undergrad is Computing the Continuous Discretely by Beck and Robins. It’s even freely available at http://math.sfsu.edu/beck/ccd.html

It doesn’t require too much mathematical knowledge. The topics are roughly: integer point enumeration related theorems for polytopes, variations of such things, discrete Fourier analysis, and Dedekind sums.

It might be a bit dry at first, but Korner’s “Fourier Analysis” branches off into some nicely chosen tangents and applications. In view of comment #1, perhaps worth noting that one of the toy applications given early on is the Weyl equidistribution theorem, and that the last theorem of the book is Dirichlet’s on primes in arithmetic progressions.

Am very fond of the book, whose structure is such that even if the reader is put off by the (relatively low-level but grindsome) hacking with continuity and epsilons, flicking ahead will bring one to little historical discursions – Bachelier, Gibbs, Watt, Kelvin, Pearson and Haldane all get mentions off the top of my head. And surely gratuitous use of Keats has to count for something?

Also, always good to remind people of Sir Cyril Burt and his uncannily precise correlation coefficients…

I remember greatly enjoying Manfred Schroeder’s “Number theory in science and communication”, but it may be too easy, and I may have the author wrong.

“Concrete Mathematics” by Graham, Knuth, and Patashnik is a great read.

I’d go with “Algebraic Curves and Riemann Surfaces” by Rick Miranda as a good start. Not going all the way through it, but the first three chapters are very example driven, and can be taught to undergrads over the summer (that’s how I learned them).

It really depends on what “casual reading” means and how much background the student has. A lot of the suggestions here are fantastic books (Concrete Mathematics is one of my very favorites, and I also loved Korner’s Fourier Analysis book), and they’d be perfect for an ambitious student who really wants to learn something serious. However, even very good undergraduates at most universities may find them pretty time-consuming (with the exception of Journey Though Genius). Here are some more accessible suggestions, which could definitely be considered casual reading:

Anything by William Dunham, Underwood Dudley, Constance Reid, Ian Stewart, or Martin Gardner.

The Book of Numbers, by Conway and Guy

The Pleasures of Counting, by Korner

The Man Who Knew Infinity, by Kanigel

Goedel, Escher, Bach, by Hofstadter

Flatland, by Abbott

Uncle Petros and Goldbach’s Conjecture, by Doxiadis

How to Solve It, by Polya

Hermann Weyl – On the Idea of Riemann Surfaces

Edward Nelson – Radically Elementary Probability Theory

Milnor,

Topology from the Differentiable Viewpoint. Short, clear, informative, and inspiring.Andreas Gathmann’s lecture notes on Algebraic Geometry. (Downloadable for free–check them out using Google.)

They require only a course in undergrad abstract algebra, and are surprisingly self-contained and carefully worked out. A real pleasure to dig through! And they bring together so many areas of mathematics, from geometry to topology to algebra to calculus.

I loved Ireland and Rosen’s A Classical Introduction to Modern Number Theory. It gives tastes of modern topics in a surprisingly low-tech way, while not forcing the reader to wade through a mess of computations.

Saunders Mac Lane and Garret Birkhoff’s Algebra makes for great light reading when algebra is still new – and then it makes for some rigorous re-reading when algebra is no longer new.

And anything

read on linear algebra will pay off handsomely in the last year of the standard undergraduate curriculum.Reuben Hersh’s “What is Mathematics Really?” or “A Mathematician’s Apology”. By the time you enter your senior year, I think it is good to be exposed to some non-standard philosophies of math. Both are fantastic and easy reads, and it might give the person a fighting chance at thinking about math in a non-Platonist way (personally, I feel this is a great hindrance to the progress of math).

Conceptual Mathematics: A first introduction to categories by Lawvere and Schanuel is definitely light and even conversational for most of the book. I won’t try to sell it anymore except to say that it was very useful and enjoyable for me a few years back.

I can’t say enough good things about Concrete Mathematics. It’s written in a very fun way, and once you have read it, you’ll never be intimidated by a binomial coefficient again. Almost a thousand problems, most of them fun and original.

Have to run, more ideas later.

I have a list of books which would be fine for undergraduates in Europe, but maybe not for americans (might be too difficult).

For algebraic geometry there’s a very VERY VERY beautiful book by Karen Smith and three more (sorry folks, I know Karen personally and the other names are mostly ten letters long and finnish).

If she likes exercises and hands-on computations (I don’t) there’s also Miles Reid’s undergraduate algebraic geometry – the last, historical chapter is worth reading anyway.

Cours d’arithmetique of Serre. I think there should be an english edition available.

I second Milnor. It was the textbook of my second year course and I really loved it.

Also Micheal Artin’s algebra book, which wasn’t there when I was a student.

I also loved Topology by Ja”nich (that I’m sure exists in english).

And I’m not sure what’s wrong with Real and Complex Analysis. It’s a lovely book, and I read it as a third year student.

There’s a somewhat dated, but wellwritten and elementary introduction to “all of mathematics” by Dieudonné, I think in english.

@Hoffnung (what a beautiful surname!) Thanks for the suggestion, I never know what to give to start undergraduates in category theory in a very soft way.

I would recommend

1. Princeton companion to mathematics. Ed. Tim Gowers.

2. Proofs from the Book. By Aigner and Ziegler.

However, if the mentee already has a preference, the advise would depend on it.

Two books I found quite inspiring as an undergraduate were “Elliptic Functions and Elliptic Integrals” by Prasolov and Solovyev and “Knots, links, braids, and 3-manifolds” by Prasolov and Sossinsky. The first is probably a bit more accessible, but both are very well-written and have minimal prerequisites.

A lot of the recommendations that people have made aim pretty high. They might work better at the graduate level, or for the very top undergraduates.

Here are some recommendations from me. They aren’t exactly easy either, but they may be a bit better, and in any case they are fun.

1) Winning ways for your mathematical plays, by Berlekamp, Conway, and Guy

2) Enumerative combinatorics, by Richard Stanley

3) Quantum computation and quantum information, by Nielsen and Chuang

4) Tilings and patterns, by Grunbaum and Shepard (this one could be the most accessible of the lot)

5) Knot theory, by Livingston (I haven’t read this one, but it looks good)

I second Alex Hoffnung (15.) re Conceptual Mathematics by Lawvere and Schanuel. It is a great introduction to category theory, full of examples and helpful exercises, and strong on motivation. I am not a mathematician, and I am making good progress, so maybe it would be considered too elementary. However, summer is here, the beach beckons, Wikipedia may not always be at hand, … so it might be just the thing.

For an advanced undergraduate or beginning graduate I would recommend some of the expository articles in the Bulletin, as a way of gaining a broader sense of various areas before diving too deeply into a particular one. My favorite in this category is George Mackey’s Harmonic Analysis as the Exploitation of Symmetry, a great historical introduction to representation theory.

The Smith-and-the-finns book on algebraic geometry is a very pleasant read, and the Knot book is supremely suitable even for more junior interested students than undergraduates.

My own personal favourite for algebraic geometry, though, was Ideals, Varities and Algorithms – which I read for a summer vacation read during my own undergraduacy; and for topology, I prefer Hatcher to Jänick.

Another very nice book for undergraduates would be Hurewitz’s book on ODE’s. It’s short and very easy to read, but it contains lots of nice things that don’t appear in the usual undergraduate courses. It’s similar in that way to Arnold’s book, but Hurewitz’s books is quite a bit more accessible.

When I was an undergraduate, I also did a reading course on Hurewitz and Wallman’s book on dimension theory. It’s probably too much for the casual student, but I found it to be wonderfully inspiring.

How about John Stillwell’s Naive Lie Theory?

Ahem… I will like to add :

Fulton : Introduction to algebraic curves

Serre: Course in Arithmetic

Arnold: Course in classical mechanics(if the student is sufficiently interested in geometry)

Ordinary differential equations

Guillemin Pollack: Introduction to Differential topology

Halmos : Finite dimensional vector spaces

Davenport : The higher arithmetic

Joe Harris : Algebraic Geometry

I must confess that may be except for Davenport, for anyone with a modest (undergrad) background, all other books aren’t really ‘casual’ reading but then they start from the basics and push you to the limits and that is definitely fun :)

Lie groups and physics by Sternberg

It’s a great intro to representation theory and also has lots of physics applications.

I’ll second Ireland and Rosen, but only if you already know the student has considerable interest in number theory. The Princeton companion makes fantastic light reading but is a little expensive if your library doesn’t have it. I also recommend Proofs that Really Count by Benjamin and Quinn; it’s a nice introduction to bijective combinatorics, especially as it spends a lot of time on identities which are not usually proven combinatorially.

Gabor Toth’s Glimpses of Algebra and Geometry is a good book on very (and many) interesting topics that never see the daylight in most undergrad curricula. It’s on an assortment of topics, for example: it talks about elliptic functions, introduces Fuchsian groups, Classification of finite Mobius groups, Klien’s theory of the icosahedron applied to get an analytic solution to the quintic. Tons of interesting problems (hand-held euler’s theorem on convex polyhedra). Little prereqs.

Or, how about J.W. Anderson’s Hyperbolic Geometry. It’s great for self-study. The book is short and can be leisurely completed in less than two months.

Or, Hartshorne’s Geometry: Euclid and Beyond.

spivak’s calculus on manifolds

ahlfors’ complex analysis

Visual Complex Analysis (Needham)

Fourier Analysis (Körner)

Winning Ways (mentioned earlier)

Surreal Numbers

or possibly something in game theory — the compleat strategist is a little dated but it’s an ok start, and maybe pitched at the right level (since the first three above may be a bit more technical than light summer reading).

If your student is at all interested in physics, Ted Frankel’s “The Geometry of Physics: An Introduction” is beautifully written and has a fun selection of topics. Baez’s “Gauge Fields, Knots, and Gravity” is also nice summer reading (imslw), although the last problem is rather difficult.

Also, one of my undergrad physics profs once described Misner, Thorne, and Wheeler’s general relativity textbook as “superb beach reading”.

Hi Ben,

This network sounds interesting, but do you know if there are any similar “mentor” groups that allow for male mentees as well?

I am not recommending any book unless it is a pleasure to read. It is the summer, after all!

I second “The Knot Book”. It’s a little easy, but it’s very fun and it’s not as if there are a shortage of difficult knot theory books after that one.

I think someone above already mentioned “Winning Ways for your Mathematical Plays”? If not, I will. Tons of pictures, a whimsical sense of humor, and applications of real math to somewhat realistic games.

I’ve got to say, I disliked “Calculus on Manifolds” intensely; I felt that it had no geometric motivation or intuition. But I like Spivak’s textbook “Calculus” very much; I think it is the best Calculus book for someone who plans to study math. Spivak starts with the axioms of complete ordered fields and develops all the results of a good single variable calc course. If your mentee hasn’t already had an intro to rigorous real analysis, she’ll get a great deal out of it.

People have recommended several different number theory books above. I don’t know what level your mentee is at. At different points in my undergraduate life, I got a great deal out of Hardy and Wright’s “Theory of Numbers” and Janusz’s “Algebraic Number Fields”. Both of these are very challenging. The former has no formal prerequisites; the latter assumes Galois theory and, in my opinion, should only be read after one has read an easier book on algebraic number theory. Unfortunately, I don’t have one to recommend.

Around that time in my life, I remember having my mind blown by Tate’s thesis. I learned it from the early chapters Weil’s “Basic Number Theory”. The consistent use of topological and analytic techniques in number theory was astonishing, although the later chapters were too much for me. (And the title is the biggest lie I’ve ever seen!) I believe that “Fourier Analysis on Number Fields”, by Ramakrishnan and Valenza, is considered a gentler introduction to the same material, but I didn’t read it until I was much more sophisticated.

In a totally different direction, I really liked “Numerical Methods that Work”, by Acton. This is a book from the 1970’s about how to do numerical computations of functions, integrals and so forth. Over and over again, he focuses on gaining intuitive understandings of which parts of an expression dominate, and how to control errors. Acton has a strong personality, and interrupts his solutions to rant about other people’s misguided approaches. I didn’t go into that field of research, and the book is quite old, so it probably isn’t a good guide to current research. But if she would like to see someone actually try to tackle the computations that get waved off as “routine”, she might enjoy this a great deal.

Since you’ve specified ‘relatively casual summer reading’, I can recommend Yandell’s “The Honors Class Hilbert’s Problems and Their Solvers”. It doesn’t shy away from discussing the actual mathematics involved, but you don’t need more than undergraduate mathematics to understand it. It also provides a lot of insight about the process of “doing mathematics”.

A lot of the books listed above are rather serious texts, and I have trouble seeing how they can be considered “relatively casual summer reading”. In particular, I agree with David Speyer about Calculus on Manifolds. It was quite dense, it failed to provide any big picture view, and as my first encounter with tensors, it made me horribly confused for a rather long time. Despite the intimidating size, I found Gravitation by Misner, Thorne, and Wheeler a much better introduction to a similar collection of topics, although I’d still hesitate to call it casual reading.

There seem to be a number of decent expository books on math history and neat theories or theorems. Marcus du Sautoy’s book on moonshine has a casual introduction to groups, along with interesting sketches of the personalities behind the classification of finite simple groups. I’ve heard that Constance Reid writes good history, although I haven’t read any of it myself. If you like romantic historical fiction with a big heap of sexism, you can try E.T. Bell’s Men of Mathematics – I’ve heard that a lot of mathematicians found it inspirational.

Despite the intimidating size, I found Gravitation by Misner, Thorne, and Wheeler a much better introduction to a similar collection of topics, although I’d still hesitate to call it casual reading.Yeah, reading MTW really is a pleasant way to learn the basic ideas of differential geometry and gravity. How many books begin by explaining geodesics with a parable about ants and apples?

And, of course, these kids today don’t have to suffer like we did, lugging around a library-bound 1200 page book, what with their internet dejavu and whatnot. Hrmmph.

It’s actually not so much a book about gravitation as it’s a case study in such.

Scott is certainly right that almost everyone’s recommendations have tended toward the hardcore. Here are some lighter ideas:

“Mathematical People”, by Albers. This is a series of interviews with mathematicians, mostly from the mid 20-th century. Less sexist than Bell, and also very inspiring.

“Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture,” by Bressoud. The history and personalities surrounding the most difficult enumerative result of the last century. This book is frustrating if you really want to get into the weeds of proofs, because it doesn’t quite give you enough information to follow the argument, but its a lot of fun as an overview.

“General Relativity from A to B” by Geroch. It’s been a while since I read it, but I recall it being an easier version of the other GR books being recommended above. I also got a lot out of “Discovering Relativity for Yourself” by Lilley, but that might be frustrating to a math major; Lilley goes to extraordinary lengths to avoid using calculus, even when it would make things shorter.

Nagel’s ‘Godel’s Proof’

How about GEB?

Klaus Janich’s “Topology” is really a pleasure to read. The entertaining style and the choice of proofs (i.e. the use of a bit of handwaving) keeps the reader happy and free from being bogged down in details. It is an undergraduate text, so it is not overly advanced, but it does include introductory remarks on algebraic topology. It is also small and lightweight so you can carry it wherever you go. Once you get home then I suppose you read PCM, the less mobile of the two.

There is a delightful book on inequalities by Michael Steele titled:

The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities (Maa Problem Books Series.) (Paperback)

It almost reads like a novel, with a wealth of information about how to think about inequalities. I heartily recommend this to every one.

A nice book for an easy introduction to geometry/topology is Differential Topology by Guillemin-Pollack. It is similar in topic to Milnor’s classic, but I think a bit easier to read.

Re: being blown away by Tate’s thesis.

I had the same experience at about the same time in my undergrad career. I’d recommend reading the original paper, from Cassells-Frohlich, as it is very well-written and entertaining. It’s also an inspiring source of side-projects at the undergraduate level which are suitably relaxed for summertime and can be accomplished with many books or internet sources. For example, learning about the analytic continuation and functional equation for the Riemann and Dedekind zeta functions, learning about basic analysis in the p-adic world, playing with the Gamma function, … the list goes on. None of it requires much in the way of prerequisites.

I’m at a similar stage having just finished my second year here in the U.K. I think “Algebraic Curves and Riemann Surfaces” is excellent, and can actually be read quite casually despite touching on some deeper subjects in latter chapters. I don’t really know how the uni system works in the U.S, but would someone who’s done three years of maths at uni really get anything from something like “The Man Who Knew Infinity” (not to mention it’s totally boring)? Milnor’s differential topology book is very readable as is Pugh’s analysis book (although this may be too basic for someone with 3 years undergraduate experience).

For a short fun read I would recommend Donald Knuth’s 1974 book “Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness.”

“Frobenius algebras and 2D topological quantum field theories”

by Joachim Kock is very readable!

algebraic topology by hatcher

and

k-theory by atiyah

these are cool books… i like them…

Alexander Soifer’s new Mathematical Colouring Book is an interesting book and very accessible.

I currently work in finance but devote most of my free time to mathematics. As I no longer attend lectures or get feedback on problems sets, it’s really important for me to find books that

1) Explain why and how the concept came about

2) Exercises are not all too challenging and solutions/hints are provided

3) Help relate the material to other areas of mathematics

4) Written by a master (this is the hardest one to find at introductory level but my faves are Atiyah/Milnor/Weyl/ Iwaniec/Serre)/Tao/Gowers)

I think it’s especially important for the undergrad to see how different areas of mathematics interrelate with each other, because the first couple years just seem like one is memorizing a dictionary. For example, I bet few undergrads can really appreciate the Lebesgue measure or the concept compactness. So here are my recommendations:

Princeton Companion to Mathematics, ed. Gowers

Moonshine Beyond the Monster – Gannon

Primes of the form x^2+ny^2 – Cox

Visual Complex Analysis – Needham

Undergraduate Commutative Algebra – Reid

Undergraduate Algebraic Geometry – Reid

Geometry and Topology – Reid

Radical Approach to Lebesgue’s Theory of Integration – Brossoud

A Classical Introduction to Modern Number Theory – Ireland and Rosen

Topology – Ja”nich (also get Munkres to do problems)

Galois Theory – by Cox

Idea of the Riemann Surface – Weyl

Rational Points on Elliptic Curves – Silverman/Tates

Hope this helps!

Redacted

I definitely enjoyed Ireland and Rosen’s _A Classical Introduction to Modern Number Theory;_ it does everything in a fairly algebraic context and in slightly more generality than most introductory number-theory books, which made the material much clearer to me.

I also enjoyed Guillemin and Pollack’s _Differential Topology,_ which has a clear expository style and plenty of good exercises. My only complaint would be the lack of generality; they demand that all manifolds live inside some euclidean space (and also don’t mention the possibility of using Banach spaces). Other than that, however, the geometric intuition the authors convey is really well done.

“An Introduction to Modern Number Theory” by Miller and Takloo-Bighash

It’s been 8 years since I got my math doctorate, and I left the field, so I appreciated hearing about some of these books, thanks! I have a couple of suggestions of my own:

The Road to Reality (Roger Penrose), is a crazy-ambitious book that somewhat naively strives to bring students from high school math through twistor theory and beyond, but it is written with a nice pedantic approach that is long on discussion, and nicely complements standard fare.

Representation Theory: A First Course (Fulton & Harris) is another somewhat conversational text, and this is a field that often turns up in many other fields, so the odds of it becoming useful in the future are quite good for any student. I’ve always meant to explore more of Springer’s “Readings in Mathematics” series to see if the quality is always as high.

I would add “A Course in Universal Algebra” by Burris & Sankappanavar. The first two chapters are especially recommended: a clear presentation of the basics; suitable for someone who has had an introduction to groups and rings, and would like to see how many concepts naturally generalize. Also, the book is freely available via

http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html

Algebra: Chapter 0

The usual undergraduate/graduate algebra shod in the unifying language of category theory.