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