Instructor: Swastik
Kopparty (__swastik.kopparty@rutgers.edu__)

Class Time and Place: Mondays and
Wednesdays, 5:00pm – 6:30pm, in Hill 425

Office Hours: Wednesday 3:30-4:30
(Hill 432)

Prerequisites: undergraduate
level abstract algebra, mathematical maturity.

References: Lidl & Niederrieter (Finite Fields), Schmidt (Equations over
Finite Fields), Tao & Vu (Additive Combinatorics).

__Syllabus__

This course will
cover some important classical and modern themes in the study of finite fields.
These will include:

·
Solutions
of equations

·
Pseudorandomness

·
Exponential
sums and Fourier techniques

·
Algebraic
curves over finite fields, the Weil theorems

·
Additive
combinatorics and the sum-product phenomenon

·
Many
applications to combinatorics, theoretical computer science and number theory

There will be 2-3 problem sets.

__Homework__

·
Homework
1 (due September 25)

·
Homework
2 (due November 4)

__Lecture Schedule__

·
September 4:
finite field basics (notes)

·
September 9:
finite field basics, continued

·
September 11:
finite field basics, introduction to Fourier analysis on finite abelian groups

·
September 16:
more Fourier analysis, the Gauss sum (notes)

·
September 18:
character sums over algebraic sets

·
September 23:
the Waring problem

·
September 25:
character sums with polynomial arguments (notes)

·
September 30:
character sums with polynomial arguments, continued

·
October 2:
polynomials over finite fields: basic properties

·
October 7:
irreducible polynomials, zeta and L functions (notes)

·
October 9:
irreducible polynomials in arithmetic progressions

·
October 14: (3
hour class) the Weil bound (notes)

·
October 16: the
Weil bound, continued

·
October 21: (3
hour class) the Weil bound, continued

·
October 23: the
Weil bound, continued

·
October 28: no
class

·
October 30:
applications of the Weil bound

·
November 4: (3
hour class) some nuggets of additive combinatorics

·
November 6:
additive energy

·
November 11: sumsets

·
November 13: the
sum product theorem

·
November 18: (3
hour class) additive character sums over multiplicative subgroups

·
November 20:
counting integer solutions to polynomial equations, compactness

·
November 25: first-order
theory, Ax’s theorem, pseudo-finite fields, course wrap-up