507- Plünnecke inequalities and sumset estimates

February 10, 2011

George Petridis, a student of Gowers, has found very nice new arguments for the Plünnecke-Ruzsa sumset inequality (If A is a finite subset of an Abelian group G, and {}|A+A|\le C|A|, then {}|kA-lA|\le C^{k+l}|A| for any k,l\in{\mathbb N}) and for the Plünnecke graph inequalities.

In lecture we went through the nice standard argument. But the new proofs are significantly simpler. For example, the graph inequalities are no longer needed for the sumset ones, and Menger’s theorem is no longer needed fro the graph inequalities. Gowers has posted the nice proof in his blog, with links to Petridis’s papers on the ArXiv.

507- Problem list (IV)

January 21, 2011

For Part III, see here.

(Many thanks to Robert Balmer, Nick Davidson, and Amy Griffin for help with this list. If you have corrections/updates, please email me. Sorry for the delay with posting this.)

507- Problem list (III)

January 20, 2011

For Part II, see here.

(Many thanks to Robert Balmer, Nick Davidson, and Amy Griffin for help with this list.)

Part IV.

507 – Homework 3

October 24, 2010

This set is due Monday, November 8.

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507 – Problem list (II)

October 16, 2010

For the beginning of the list, see here.

Part III.

507 – Homework 2

September 29, 2010

This set is due Monday, October 18.

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507 – Problem list (I)

September 21, 2010

This is the list of “problems of the day” mentioned through the course.

(Thanks Nick Davidson and Summer Hansen.)

Part II.

507 – Homework 1

September 2, 2010

This set is due Monday, September 13.

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507 – Advanced Number Theory (Syllabus)

May 14, 2010

Mathematics 507: Advanced number theory.

Section 1.
Andrés E. Caicedo.
Contact Information: See here.
Time: MWF 12:40-1:30 pm.
Place: M/G 124.
Office Hours: MF 11:40 am-12:30 pm 10:40-11:30 am (or by appointment).

The (admittedly impossible) goal is to discuss the following topics:

  1. Divisibility
  2. Congruences
  3. Primitive roots
  4. Quadratic reciprocity
  5. Prime numbers
  6. Waring’s problem
  7. Geometry of numbers
  8. Transcendental number theory
  9. Additive combinatorics
  10. Integer partitions

Recommended background: Mathematics 306: Number theory, or equivalent. It is highly desirable that you have taken prior courses in analysis and abstract algebra.

Textbook: Elementary methods in number theory. By Melvin Nathanson. Springer-Verlag (2000), ISBN: 0-387-98911-9.

At the end, this was a matter of personal taste, as my list of topics leans more towards the “analytic” than the “algebraic” side of things. This list is somewhat nonstandard. Here are some additional suggested references, both for the course and for number theory in general. I will add suggestions through the course, depending on the topic being covered:

  1. A classical introduction to number theory. By Kenneth Ireland and Michael Rosen. Springer-Verlag (1990), ISBN: 0-387-97329-X. Highly recommended, this could have been our textbook.
  2. Algebraic number theory and Fermat’s last theorem. By Ian Stewart and David Tall. A K Peters (2002), ISBN: 978-1568811192. This would be a nice textbook for a first course in algebraic number theory, but it requires background in Galois theory.
  3. Multiplicative number theory. By Harold Davenport, revised by Hugh Montgomery. Springer-Verlag (2000), ISBN: 0-387-95097-4. This is a very nice book, but it definitely requires background in complex analysis.
  4. Making transcendence transparent: an intuitive approach to classical transcendental number theory. By Edward Burger and Robert Tubbs. Springer-Verlag (2004), ISBN: 978-0387214443.
  5. Additive number theory: The classical bases. By Melvyn Nathanson. Springer-Verlag (1996), ISBN: 978-0387946566.
  6. Additive number theory: Inverse problems and the geometry of sumsets. By Melvyn Nathanson. Springer-Verlag (1996), ISBN: 0-387-94655-1.
  7. Integer partitions. By George Andrews and Kimmo Eriksson. Cambridge University Press (2004), ISBN: 0-521-60090-1. Andrews also authored a more advanced textbook on partition theory, that requires complex analysis.

Grading: Based on homework.

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