Arithmetic Combinatorics

Fall 2016

(642:588)

 

Course Info

Instructor: Swastik Kopparty (swastik.kopparty@rutgers.edu)

Class Time and Place: Tuesdays and Thursdays, 3:20pm – 4:40pm, in Hill 425

Office Hours: Thursday 11 am – 12 noon (Hill 432)

Prerequisites: combinatorics, probability, algebra, mathematical maturity.

References: Tao & Vu (Additive Combinatorics).

 

 

Syllabus

 

Arithmetic Combinatorics is the study of combinatorial questions involving arithmetic operations.

This course will cover some classical and modern aspects of the subject.

Possible topics include:

·       sumsets

·       the sum-product phenomenon

·       Ramsey questions

·       Szemeredi's theorem

·       probabilistic methods

·       geometric methods

·       graph theoretic methods

·       algebraic methods

·       Fourier and group-representation methods

·       analytic methods

 

There will be 2-3 problem sets.

 

Lecture Schedule

·       September 6: the Cauchy-Davenport theorem (notes)

·       September 8: NO CLASS (makeup class to be scheduled)

·       September 13: the Erdos Heilbronn conjecture, Schnirerlmann density (notes)

·       September 15: Mann’s theorem

·       September 20: sumset inequalities

·       September 22: additive energy, the Balog-Szemeredi Gowers theorem

·       September 27: introduction to Fourier analysis (notes from finite fields course)

·       September 29: Gauss sums, BLR linearity test (same notes as above)

·       October 4: the sum-product theorem (notes)

·       October 6: NO CLASS (Avi Wigderson’s birthday conference)

·       October 11: NO CLASS (makeup class to be scheduled)

·       October 13: NO CLASS (makeup class to be scheduled)

·       October 18: sum-product over the reals, Szemeredi-Trotter, crossing numbers

·       October 20: sum-product over reals again, Sidon sets

·       October 25: perfect difference sets, sum-free sets, Schur’s theorem

·       October 27: van der Waerden’s theorem, Roth’s theorem part 1

·       November 1: Roth’s theorem part 2

·       November 3: capsets and the Croot-Lev-Pach-Ellenberg-Gijswijit theorem

·       November 8: Bohr sets, APs and subspaces in sumsets part 1

·       November 10: APs and subspaces in sumsets part 2

·       November 15: the Croot-Sisask sampling method

·       November 17: Freiman homomorphisms, Chang’s lemma

·       November 22: additive structure in Fourier coefficients

·       November 24: NO CLASS (Thanksgiving)

·       November 29: Gauss sums for small multiplicative subgroups part 1

·       December 1: Gauss sums for small multiplicative subgroups part 2

·       December 6: Waring’s problem in integers part 1

·       December 8: Waring’s problem in integers part 2

·       December 13: Hindman’s theorem

·       December 16: (SPECIAL MAKEUP CLASS) representation theory, Gowers’ theorem on product-free sets (notes)