Tropical Geometry - Lecture 1 - Plane Curves | Bernd Sturmfels

Twelve lectures on Tropical Geometry by Bernd Sturmfels (Max Planck Institute for Mathematics in the Sciences | Leipzig, Germany)
We recommend supplementing these lectures by reading the book "Introduction to Tropical Geometry" (Maclagan, Sturmfels - 2015 - American Mathematical Society)

Lecture I - Plane Curves | August 19, 2020

Chapters:
00:00 Introduction
00:41 Tropical Polynomials
07:03 Proposition 1.3.1
14:48 Figure 1.3.4. | Biquadratic Curves
17:55 Bézout's Theorem
28:14 Theorem 1.3.3. | Stable Intersection
33:51 1.4 | Amoebas and their Tentacles
44:18 1.5 | Implicitization
47:28 Questions

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts.