### Excellent students

If you are:
* An undergraduate student learning mathematics,
* One of the best students in your class,
* You enjoy mathematics, and
* You (secretly) consider an option of an academic career,

First of all, you must recognize that:
* What you study in class is not enough.
* Not even close.

What should you be doing?
* Join the "Supervised reading" courses, 88-198 and 88-199.
* Join the specialized courses: "Problems workshop" 88-300 or "Undergraduate research workshop" 88-522.
* Find an advisor (who will send you to read even more).
* Establish a learning group or a students seminar.
* Read mathematical blogs, such as this one or this one. * Read Wikipedia articles on mathematics. (Then write them).
* If you are seeking challenging and beautiful problems, you should visit the world of Mathematical Olympiads. An excellent Israeli site is that of Taharut Ha'arim.
* Keep an eye open for the yearly competition for students held in Bar-Ilan.

When looking for textbooks to support their courses, students do not always look for the most challenging (and most informative) books. Check with your lecturer what textbooks are recommended if you are willing to spend some more effort. This list was prepared with the help of Boaz Tsaban and Michael Schein.
Here are a few suggestions:
* Infinitisimal calculus: Rudin, "Principles of Mathematical Analysis"; Saks, "Theory of Integrals".
* Linear algebra: Halmos; Jacobson, "Basic Algebra I".
* Group theory: Rotman, "An Introduction to the Theory of groups".
* Ring theory: Herstein, "Topics in Algebra"; Jacobson, "Basic Algebra II".
* Galois theory: Edwards.
* Set theory: Kunen, "Set Theory An Introduction To Independence Proofs", or Jech, "Set Theory",
* Topology: Munkers, "Topology, a first course".
* Advanced analysis: Rudin, "Real and Complex Analysis".
* Complex functions: Ahlfors, "Complex analysis".
* Differential geometry: de Carmo.
* Commutative algebra: Atiyah and Macdonald; Zariski and Samuel.
* Number Theory: Hardy and Wright.
* ...

The library is rich and full of options. One thing is crucial: accept that you will be reading in English; get used to it. Here are just a few examples.
* "I Want to Be a Mathematician", Paul Halmos;
* "On Numbers and Games", John Conway;
* "An Introduction to the Theory of Numbers", Hardy and Wright;
* "Proofs from the Book";
* "The mathematics of Fermat-Wiles";
* "How to solve it", George Polya;
* (to be continued)

"Behind any successful research mathematician will be thousands of hours spent pondering mathematics, only very few of which will have directly led to breakthroughts. It is strange, in a way, that anybody is prepared to put out those hours. ... Principle 8.3: If you are truly interested in mathematics, then hard mathematical work does not feel like a chore: it is what you want to do." (W. Timothy Gowers, How do IMO Problems Comapre with Research Problems?, in D. Schleicher and M. Lackmann, eds., An Invitation to Mathematics, From Competitions to Research (Springer, 2011)

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Last updated: 26 June 2009