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

Fall 2022

Honors Intro to Analysis

2022 FallMichigan State UniversityMTH327H
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; tosets, poset, proset, and directed sets. Max/min/sup/inf in posets.
  • Axiomatic definition of real numbers; basic properties.
  • Nets and convergences in the reals.
  • Subnets and continuity of real functions.
  • Infinite sums.
  • Normed linear spaces: completion, convergence, topology, compactness.
  • (Time permitting) continuity of functions between normed linear spaces.

Previously Taught Graduate Courses

PDE II

2022 SpringMichigan State UniversityMTH849

Dispersive and Hyperbolic PDEs

2021 SpringMichigan State UniversityMTH943

Dispersive and wave equations

2019 SpringMichigan State UniversityMTH943

Stability of Matter

2018 SummerMichigan State UniversityMTH890 sec. 102
Summer topics course based on the book of Lieb and Seiringer.

Partial Differential Equations I

2017 FallMichigan State UniversityMTH847
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.

Dispersive equations

2017 SpringMichigan State UniversityMTH943

Previously Taught Undergraduate Courses

Analysis II

2022 SpringMichigan State UniversityMTH421 sec. 1

Analysis I

2021 FallMichigan State UniversityMTH320 sec. 2

Honors Intro to Analysis

2020 FallMichigan State UniversityMTH327H

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

2020 FallMichigan State UniversityMTH320 sec. 2
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

2020 SpringMichigan State UniversityMTH133 sec. 15--28, 38
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

2019 SpringMichigan State UniversityMTH133 sec. 22--28

Sixth and final pilot of the Calculus 2 Labs project.

Co-instructing with Mark Iwen

Calculus II

2018 FallMichigan State UniversityMTH133 sec. 16--20 & 23

Fifth pilot of the Calculus 2 Labs project.

Co-instructed with Mark Iwen

Calculus II

2018 SpringMichigan State UniversityMTH133 sec. 40 & 41

Fourth pilot of the Calculus 2 Labs project.

Co-instructed with Mark Iwen

Calculus II

2017 FallMichigan State UniversityMTH133 sec. 40 & 41

Third pilot of the Calculus 2 Labs project.

Co-instructed with Mark Iwen

Calculus II

2017 SpringMichigan State UniversityMTH133 sec. 37

Second pilot of the Calculus 2 Labs project.

Co-instructed with Mark Iwen and Rachael Lund

Calculus II

2016 FallMichigan State UniversityMTH133 sec. 62

Initial pilot of the Calculus 2 Labs project.

Co-instructed with Mark Iwen and Rachael Lund

Real Analysis II

2015 FallMichigan State UniversityMTH421

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

2015 FallMichigan State UniversityMTH299
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!)

Link to Draft Lecture Notes

Calculus II with Matlab Labs

Together with my favorite co-conspirator, Mark Iwen, I've been engaged in developing enrichment material showcasing applications of calculus concepts in the broader STEAM disciplines for Calculus II (MTH133). We ran 6 semesters of pilots (Fall 2016 - Spring 2019), where we replaced traditional recitations with computer-based lab activities.

These labs are built around Matlab-based simulations. The main goal is for students to interpret the observed results using calculus concepts, thereby applying the curricular material to non-textbook scenarios. Secondary goals for students include becoming generally acquainted with modelling concepts, as well as basic techniques in numerical simulations.

The first four semesters were mostly spent on development and re-development of course material, tested with smaller cohorts of students (~30 per semester in AY16-17, ~70 per semester in AY17-18). Rachael Lund also worked with us during this initial period, sharing with us her considerable expertise and experience teaching Calculus II at Michigan State.

In AY18-19 we ramped it up to a full-size large lecture (150+ students) and, through a generous grant by MathWorks, worked with Andrew Krause and Ryan Maccombs to study the impact of the lab activities on student learning, both in terms of traditional assessment metrics as well as broader cognitive, aspirational, and affective domain changes.

The project itself has grown considerably, more-so than can be summarized here. Please refer to its dedicated project page for more information.