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)