Expository Writings, Notes, Problem Solutions, etc.

The following are (expository) papers I wrote as an undergrad to satisfy various course and degree requirements.

• Fourier Analysis on the Torus – An introduction to Fourier analysis on the Torus, which includes Kolmogorov’s construction of an function whose Fourier series diverges almost everywhere and basic results on the Dirichlet and Fejer kernels. The paper was my final project for Math 212br (Spring ’12) with Professor A. Knowles.
• D.J. Newman PNT Proof – A detailed exposition of D.J. Newman’s extremely elegant proof of the prime number theorem. This was an expansion on my talk to my Math 229x class (Spring ’12) with Professor B. Mazur. It was also my submission to satisfy the Harvard math department’s expository paper requirement.
• Mathematical Methods in Wind Power Modeling – A survey of some mathematical methods in wind power estimation and modeling, with a focus on the use of the two-parameter Weibull distribution. This was my final project for the SPU 25 with Professor M. McElroy.
• Generalized Martingale ABRACADABRA Problem – Some basic results in the theory of discrete-time martingales, plus an application to the problem of determining the expected time until a fixed pattern occurs (informally, how long until a monkey types ABRACADABRA on a typewriter). The exposition is adapted from one of Professor S. Sternberg’s Math 212a problem sets.

I like to have proved at least once every non-trivial result I need to use in mathematical doings. What follows below is partial documentation of this idiosyncracy.

Classical Analysis

• Rademacher Differentiation Theorem
• Ergodic Theorems – Proofs of the von Neumann, mean, and pointwise ergodic theorems; proof of the strong Law of Large Numbers using the pointwise ergodic theorem.
• Banach Fixed Point Theorem and Applications – The Banach fixed point theorem and applications to matrix equations, integral equations, and a proof of the Picard-Lindelöf theorem.
• Spaces for
• Lp Norm and Duality Lemma

Functional Analysis

• Construction of Nonseparable Hilbert space 
• Jordan-von Neumann Theorem – Necessary and sufficient conditions for a (generalized) normed space to be an inner-product space.
• Hilbert-Schmidt Theory – Basics of Hilbert-Schmidt operators with focus on the theory.
• Hahn-Banach Theorem
• Uniform Convexity and Reflexivity – Radon-Riesz theorem, Goldstine’s theorem, Milman-Pettis theorem, McShane’s Lemma, and Maximum Principle.

Harmonic Analysis

• Hormander’s Staircase – Construction of Hörmander’s staircase used in the proof of the Malgrange-Ehrenpreis theorem.
• A Useful Fourier Transform – Computation of the Fourier transform of   for , where denotes the dimension of the underlying Euclidean space.
• Generalization of HL Maximal Inequality – Generalization of the Hardy-Littlewood Maximal Inequality to sigma-finite measure spaces where the underlying set is also a separable metric space.
• DJ Newman’s Proof of Fourier Uniqueness – A short, elegant proof of the injectivity of the Fourier transform as a map .

PDE

• Kelvin Transform

Complex Analysis

• Picard’s Little Theorem – Short proof of Picard’s little theorem using the universal cover of the twice punctured complex plane.
• Winding Numbers and Brouwer’s Fixed Point Theorem – An exposition of the winding number of a curve culminating with a proof of Brouwer’s well-known fixed point theorem in .

Measure Theory and Integration

• Category vs. Measure
• Lebesgue-Radon-Nikodym Theorem – A short introduction to signed measures, absolute variation, and an elegant proof of the Lebesgue-Radon-Nikodym Theorem.
• Construction of non-measurable set – Using the Axiom of Choice, we construct a set which is not in the Lebesgue -algebra.
• Countable Union of Sigma-Algebras – Sufficient conditions for the union of sigma-algebras to not be a sigma-algebra.
• Dynkin – Lemma – Some results on Dynkin systems, including the useful Dykin – lemma, uniqueness of measures, in particular the Lebesgue measure on .

Probability Theory

• Partial Converse to Borel-Cantelli Lemma
• Kolmogorov’s Inequality – Kolmogorov’s first inequality and a couple theorems on the a.s. convergence of random series.
• Laws of Large Numbers –  Proofs of the Weak and Strong Laws of Large Number.
• -Distribution and Applications

Analytic Number Theory

• Convergence of Dirichlet Series
• Zero-Free Region of Riemann Zeta Function
• Chebyshev Prime Number Estimates – Chebyshev’s prime number estimates, which can be viewed as a weaker version of the Prime Number Theorem.
• Other Proofs of PNT Handout – Two short proofs of the Prime Number Theorem: one is Fourier-analytic and is adapted from a proof originally given by Shikao Ikehara and the other is complex-analytic and is adapted from a proof originally given by D.J. Newman.
• Rational Approximation and Liouville’s Theorem
• Zeroes of Sums and Differences Gamma Functions

Linear Algebra

• Trace and Determinant – Coordinate-free approach to the trace and determinant of a linear operator on a finite-dimensional vector space.

General Topology

• Machado’s Theorem
• Chernoff’s Proof of Tychonoff Theorem
• Moore-Smith theorems – Proofs of the Moore-Smith theorem and the Arzela-Ascoli theorem for families of equicontinuous functions on complete, separable metric spaces into compact metric spaces.
• Compactness Equivalence and Application to Proper Maps – We define a topological space to be compact if and only if every open cover of contains a finite subcover. We prove a theorem which gives an equivalent criterion for a space to be compact in terms of the closedness of the projection map . We then use this theorem to prove that a proper map is closed and has compact fibers.

Applied Mathematics

• Compound Interest – A short note on the mathematics of continuously compounded interest.
• Monty Hall Problem
• Knaster-Kuratowski-Mazurkiewicz (KKM) Theorem – Proofs of KKM Theorem, Ky Fan minimax inequality, and mathematical existence of Nash equilibria.