Analysis

These are notes that I have compiled during my self-study for Sobolev Spaces. At the end, I have included several exercises, largely taken from Partial Differential Equations by Lawrence Evans. I am sure there may be errors in here, in which case feel free to email me.

This is an outline for the Rutgers Grad Analysis seminar talk that I delivered on 4/25/2025.

Abstract: In this talk, we will define energy minimizing maps and introduce key examples. Then, we will derive the associated variational formulae. A central theme of the talk will be the monotonicity formula: we will explore its significance and highlight its appearance in related geometric contexts such as minimal surfaces, mean curvature flow, and Ricci flow. We will then prove the monotonicity formula for energy minimizing maps. Finally, we will define the density function, introduce tangent maps, and discuss the role of monotone quantities in the study of regularity and singularities.

These are a collection of exercises in minimal surfaces, most of which are taken from “A Course in Minimal Surfaces” by Colding and Minicozzi. I am still adding to this document, so please feel free to contact me if you find any errors.

This is the outline for a talk delivered on Feburary 21st, 2025 at the Rutgers student operated Grad Analysis Seminar.

The following is a short presentation on Hardy Spaces for MATH507 - Functional Analysis with Professor Maxime Van de Moortel. The presentation includes a brief introduction to H^p spaces and a proof that H^p is a Banach space for all p from 1 to infinity.

The following exercises are an assortment of homework problems from MATH 502 - Theory of Functions of a Real Variable II with Professor Dennis Kriventsov at Rutgers University. Topics include Radon Measures, weak convergence, Haar measures, Fourier analysis, PDEs, distributions, and probability theory.