Differential Geometry With Finite Fields - Differential Geometry

Explore differential geometry in finite fields through this 50-minute lecture. Learn the basics of finite prime fields, working mod p for a fixed prime p, and discover how the study of tangent conics to cubic polynomials extends to finite fields. Delve into arithmetic over F_p, primitive roots, Fermat's theorem, and polynomial arithmetic. Examine tangent conics to a cubic over F_11, illustrating Ghys' observation on their disjointness. Understand the differences between sub-derivatives and derivatives in finite fields, and explore Taylor expansions using sub-derivatives. Gain insights into the distinction between polynomials and polynomial functions in finite fields. Discover how calculus extends beyond traditional settings, offering rich combinatorial and algebraic patterns in finite calculus.

Introduction
Finite fields
exponentiation
primitive roots
polynomial arithmetic
calculus over finite fields
power function example
Graphing polynomials
Graphing quadratic equations
Natural questions
Generalizing