updates?

content/Math401/Extending_thesis/Math401_S5.md

@@ -1,4 +1,4 @@
-# Math 401, Fall 2025: Thesis notes, S4, Differential Forms
+# Math 401, Fall 2025: Thesis notes, S5, Differential Forms
 
 This note aim to investigate  What is homology and cohomology?
 
@@ -235,3 +235,7 @@ $$
 $$
 
 A tangent vector at $p\in M$ is the 
+
+[2025.12.03]
+
+Goal: Finish the remaining parts of this book

content/Math401/Extending_thesis/Math401_S6.md

@@ -0,0 +1,11 @@
+# Math 401, Fall 2025: Thesis notes, S6, Algebraic Geometry
+
+## Affine algebraic variety
+
+The affine algebraic variety is the common zero set of a collection $\{F_i\}_{i\in I}$ of complex polynomials on complex n-space $\mathbb{C}^n$. We write
+
+$$
+V=\mathbb{V}(\{F_i\}_{i\in I})=\{z\in \mathbb{C}^n\vert F_i(z)=0\text{ for all } i\in I\}
+$$
+
+Note that $I$ may not be countable or finite.