MATH530: Differential Geometry - Comprehensive Solutions

These are visualizations of gauss maps for a cylinder and a plane. Do you want to know what they are and why they look like this?- Then read through this file!

Repository Overview

This repository is a comprehensive collection of my solutions for the MATH530: Differential Geometry course, a 500-level advanced mathematics course. The repository is intended to showcase my analytical and problem-solving skills in the field of differential geometry, particularly for graduate admissions committees and potential employers.

Course Content

The course covered a wide array of topics within differential geometry, including:

• Tensor Calculus: Deep dives into the fundamentals and applications of tensor calculus in differential geometry.
• Gauss Maps: Exploration and visualization techniques for Gauss maps, integral to understanding geometric surfaces.
• Weingarten Maps: Detailed analysis of Weingarten maps and their role in computing curvatures.
• Minimal Surfaces: Investigative approaches to finding and understanding minimal surfaces.
• Gauss's Theorem: Comprehensive studies on Gauss's Theorem and its implications in geometry.
• Spectral Theorem for Hermitian Operators: Examination of the spectral theorem in the context of Hermitian operators.
• Dual Spaces and Symmetric Polynomials: Insights into the structures and applications of dual spaces and symmetric polynomials.
• Differential Forms: In-depth discussions on the use and significance of differential forms in geometry.

Special Focus: General Relativity and Differential Geometry

A significant portion of the course was dedicated to applying concepts of differential geometry to general relativity. Key achievements in this area include:

• Solutions to Geodesic Equations: Detailed solutions and interpretations of geodesic equations in the context of general relativity.
• Schwarzschild Metric Tensors: Problems and solutions involving Schwarzschild metric tensors, contributing to a deeper understanding of spacetime geometry.

Repository Content

• PDF Solutions: The repository primarily contains the comprehensive pdf of my carefully thought-out solutions to the course problems. These documents reflect my methodical approach to complex mathematical problems in differential geometry

Purpose and Audience

This repository serves as a testament to my analytical abilities and my proficiency in advanced mathematical concepts, tailored for:

• Graduate Admissions Committees: Demonstrating my readiness and suitability for advanced studies in mathematics and related fields.
• Prospective Employers: Showcasing my problem-solving skills and my ability to tackle complex theoretical concepts, relevant in various research and applied mathematics roles.