# Teaching

Since 2007, I've taught in university classrooms in various capacities ranging from teaching assistant to instructor of record. I've instructed classes in EPFL, Peking University, Princeton University, and University of Cambridge. Below you will find a summary of what I've been teaching since 2015, when I started my tenure track position at Michigan State University.

In addition to regularly schedule courses, I also direct independent student research projects and reading courses. For more about those, please see the student info page.

## Currently Teaching

Spring 2022

#### Stability of Matter

Summer topics course based on the book of Lieb and Seiringer.

#### Partial Differential Equations I

First part of the year-long qualifying sequence course for partial differential equations. The course focused on the basic properties of the Laplace/Poisson equations, the linear heat equation, the linear wave equation, and first order (possibly nonlinear) scalar equations.

#### Honors Intro to Analysis

A first semester honors course in real analysis. Summary of topics:

• Review of set theory and properties of the naturals, the integers, and the rationals.
• Intro to order theory; totally ordered sets; Dedekind completeness; and construction of reals.
• Nets and convergences in the reals.
• Metric spaces; nets and convergences in them; Cauchy completeness.
• Subnets. Infinite sums.
• Continuity of single variable real functions.
• Differentiability of single variable real functions.
• Integration theory for single variable real functions: Riemann integrals; indefinite integration and the fundamental theorems; Henstock integrals; Stieltjes integrals.

Notes and course material available

#### Analysis I

A first semester course in real analysis. I intend to cover:

• The real number as an ordered complete field; with emphasis on existence of supremum and infimum. (No Dedekind cut construction, just the axiomatization of the basic properties)
• The Archimedean property and its implications
• Limits of sequences in the reals: emphasis on limsup and liminf, emphasis on subsequences
• Basic aspects of the ordered topology: open versus closed intervals, Bolzano-Weierstrass
• Continuity of functions of one variable; intermediate value theorem
• Differentiability of functions of one variable; mean value theorem

#### Calculus II

Calculus 2 with Labs; the course material now broadly deployed for all MTH133 students. I also served as the Course Supervisor this semester.

#### Calculus II

Sixth and final pilot of the Calculus 2 Labs project.

Co-instructing with Mark Iwen

#### Calculus II

Fifth pilot of the Calculus 2 Labs project.

Co-instructed with Mark Iwen

#### Calculus II

Fourth pilot of the Calculus 2 Labs project.

Co-instructed with Mark Iwen

#### Calculus II

Third pilot of the Calculus 2 Labs project.

Co-instructed with Mark Iwen

#### Calculus II

Second pilot of the Calculus 2 Labs project.

Co-instructed with Mark Iwen and Rachael Lund

#### Calculus II

Initial pilot of the Calculus 2 Labs project.

Co-instructed with Mark Iwen and Rachael Lund

#### Real Analysis II

A second semester course in real analysis. Most enrolled students were studying math education (targeting secondary education) or actuarial science.

In my course I covered

• Rigorous definition of Riemann integration via Darboux sums.
• Error estimates for Darboux sums.
• Mean value theorem, continuity of the primitive, fundamental theorem of calculus.
• Real vector spaces, norms, open and closed sets.
• Metric-space topology, convergence, compactness.
• Banach fixed point theorem.
• Multivariable functions, linear mappings, operator norms.
• Continuity and uniform continuity, properties of continuous functions, curves and paths.
• Differentiability of curves, directional derivatives of multivariate functions, Fréchet differentiability, strong differentiability, relationship between different notions of differentiability.
• Geometry of level sets and critical points.
• Implicit and inverse function theorems.

I ran the course in a flipped style with worksheets and exercises that the students are supposed to complete before arriving to class, and discussions during class time.

Course material for this class is available upon request.

#### Transitions

Transitions is an "introduction to proofs" class. Content covered include

• prepositional logic
• mathematical reasoning skills (proof by induction, contradiction, contrapositive)
• basic number theory
• basic definitions in linear algebra and real analysis

# Course Development

After arriving at Michigan State University, I've been involved in the development of some new courses and in the revamping of curricular material for some existing courses.

## Dispersive and Wave Equations

An Introduction

Coincidental with my arrival at MSU, the PDE group decided to revamp our graduate curriculum. In addition to now providing a two-semester introductory sequence of courses (the "qualifying sequence"), we also established four regularly scheduled higher-level courses in partial differential equations. One of which, MTH943, has as its subject matter dispersive and wave equations, which I was charged with developing.

I've set my goal for this course to be an introductory course on dispersive and hyperbolic partial differential equations and their a priori estimates. At the end of the course, students will understand the physical origins of dispersion and how it manifests in qualitative and quantitative behaviors of solutions to dispersive PDEs. A rough list of topics that I originally hoped to include are

• A first look at dispersion via the Vlasov equation (linear transport in classical phase space); both the representation formula and vector-field method approaches to dispersive estimates will be introduced.
• Brief introduction to the quantum phase space; just enough Fourier theory will be introduced.
• Model equations: Schroedinger, Airy, wave, Klein-Gordon.
• Dispersive estimates via oscillatory integration ("representation formula").
• Dispersive estimates via the vector field method.
• Basic interpolation theory, $L^p$ decay.
• Introduction to Littlewood-Paley decompositions, the $TT^*$ method and Strichartz estimates.
• Nonlinear applications.
• Local smoothing and local energy decay.
• Multilinear estimates and wave-Sobolev ($H^{s,\delta}$) or Bourgain ($X^{s,b}$) spaces.

This list turns out to be too ambitious for a one semester class, especially one where Fourier theory has to be introduced from scratch. Though I still do hope to, at some point in time, complete my notes to my original specification. In the mean time, you can find a draft of my lecture notes below.

The notes are very unpolished right now and are heavily laden with typos. Use them at your own risk. (Though if you found any mistakes, do let me know!)