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.