Analytic Geometry, Brief Course.

Differential Geometry Course Notes

1.1. Review of topology. Definition 1.1. A topological space is a pair (X, T) consisting of a set X and a collection T = {U α } of subsets of X, satisfying the following: (1) ∅, X ∈ T , (2) if U α , U β ∈ T , then U α ∩ U β ∈ T , (3) if U α ∈ T for all α ∈ I, then ∪ α∈I U α ∈ T. (Here I is an indexing set, and is not necessarily finite.) T is called a topology for X and U α ∈ T is called an ope...

Overview of Rigid Analytic Geometry

The idea is simple: we want to develop a theory of analytic manifolds and spaces over fields equipped with an arbitrary complete valuation. Of course, it is a standard fact that such a field must be either R, C, or a field with a nonarchimedean valuation, so what we really mean is that we want to develop a theory of nonarchimedean analytic spaces. Doing this näıvely (i.e., defining manifolds in...

Geometry and Proof: Course Summary

This course is designed to explicate the following motto attributed to Hilbert. ”It must always be possible to substitute ’table’, chair’ and ’beer mug’ for ’point’, line’ and ’plane’ in a system of geometrical axioms.” We have considered this theme where ‘geometrical axioms’ are statements in Tarski’s world, various systems of geometry, and axioms for algebraic systems like the real numbers an...

A course in Algebraic Geometry

A Course in Riemannian Geometry