Teaching / Previous Courses
Below are university mathematics courses that I have previously taught.
MTH5109: Geometry II: Knots and Surfaces
This is a secondyear undergraduate course on the differential geometry of curves and surfaces, along with a brief excursion to knot theory.
Approximate (combined) list of topics covered:
 Knots, knot diagrams, and the Reidemeister theorem
 Knot invariants: crossing number, chirality, tricolorability, Jones polynomial
 Curves and their parametrizations
 Curves: tangent lines, orientation
 Curves: arc length, path/line integrals
 Curves: curvature
 Plane curves: signed curvature, winding number
 Space curves: torsion, Frenet–Serret formulas
 Surfaces and their parametrizations
 Surfaces: tangent planes, first fundamental forms, unit normals
 Surfaces: orientability and orientation
 Surfaces: surface area, surface integrals
 Surfaces: first fundamental form, second fundamental forms
 Surfaces: principal, mean, and Gauss curvatures
 Curves on surfaces: normal and geodesic curvatures, Euler's theorem
 Geodesics: covariant derivatives, geodesic equations
 Some landmark results (distanceminimizing curves, theorema egregium, Gauss–Bonnet theorem)
M4P41: Analytic Methods in PDEs
 Position: Instructor
 Location: Imperial College London
 Term: Spring 2016
This is a fourthyear undergraduate course in partial differential equations (PDEs) and the mathematical methods used to study them.
Approximate list of topics covered:
 Review of ODE theory (existence, uniqueness, continuous dependence)
 Method of characteristics (firstorder scalar PDE)
 Analytic PDE, power series solutions (CauchyKovalevskaya and Holmgren theorems)
 Weak derivatives, weak solutions of PDE
 The Laplace and Poisson equations
 The heat equation
 The wave equation
 Mastery content: Existence and uniqueness for the cubic nonlinear Schrödinger equation.
The following supplementary notes covered a couple additional topics that were not fully covered in lecture.
TCC: Dispersive Equations
This is a graduatelevel partial differential equations (PDE) course, a collaborative effort with Jonathan BenArtzi, which introduced the following two areas of study:
 Kinetic theory (JBA): transport equations, classical theory of VlasovPoisson and VlasovMaxwell equations.
 Wave equations (AS): linear waves, classical theory of nonlinear wave equations.
The lectures were broadcast to universities in the TCC network: University of Bath, University of Oxford, University of Bristol, Imperial College London, University of Warwick.
Approximate list of topics covered, by week:
 (AS) Ordinary differential equations, connections to evolutionary PDE
 (JBA) PDE preliminaries (Fourier transforms, Sobolev spaces), linear transport equations (method of characteristics), introduction to kinetic theory
 (JBA) The VlasovPoisson system: local existence and uniqueness
 (JBA) The VlasovPoisson system: global existence
 (AS) Linear wave equations: physical and Fourier representation formulas, energy and dispersive estimates
 (JBA) The VlasovMaxwell system: conditional global existence
 (AS) Nonlinear wave equations: classical local existence and uniqueness
 (AS) Nonlinear wave equations: the vector field method, smalldata global and longtime existence
The full set of lecture notes for the course can be found on the official course website (created by Jonathan BenArtzi).
My contributions to the notes can be found below:
(Much thanks to Vaibhav Jena for proofreading and finding errors within the notes.)
MAT336: Elements of Analysis
 Position: Instructor, Course coordinator
 Location: University of Toronto
 Semester: Spring 2014
 Textbook: Real Analysis and Applications: Theory in Practice, by Kenneth. R. Davidson and Allan P. Donsig
This is a introductory course in real analysis.
The course covers the basic properties of real numbers, as well as a rigorous development of various calculus concepts.
Throughout, we also discuss some generalizations of these concepts to other useful settings.
The course is geared toward students without experience in fully rigorous mathematics courses.
Approximate list of topics covered, by week:
 Motivation (why real numbers are necessary), construction of real numbers (Dedekind cuts), defining properties of real numbers (least upper bound principle)
 Cardinality, limits (real numbers), monotone sequences
 Limits (metric spaces), Cauchy sequences, completeness
 Series, convergence tests, absolute and conditional convergence
 Discrete dynamical systems, fixed points, contraction mapping theorem
 Fractals (iterated function systems)
 Topology of metric spaces, limit and interior points, open and closed sets, continuity
 Compactness and connectedness, HeineBorel theorem, extreme and intermediate value theorems
 Differentiation (1d), mean value theorem, Taylor's theorem
 Inverse function theorem, implicit function theorem
 Riemann integration, fundamental theorem of calculus
 Pointwise and uniform convergence, spaces of continuous functions, existence and uniqueness for ordinary differential equations
Some exam questions and solutions from the course:
MAT244: Ordinary Differential Equations
 Position: Instructor
 Location: University of Toronto
 Semester: Spring 2014
 Textbook: Elementary Differential Equations and Boundary Value Problems (10th ed.), by William E. Boyce and Richard C. DiPrima
This is an introductory course on ordinary differential equations.
Approximate list of topics covered, by week:
 Firstorder equations: separation of variables, linear equations
 More firstorder equations: exact equations, general integrating factors
 Existence and uniqueness theory: linear equations and systems, nonlinear equations
 Secondorder homogeneous linear equations: equations with constant coefficients
 Secondorder nonhomogeneous linear equations: method of undetermined coefficients, variation of parameters
 Higherorder linear equations: equations with constant coefficients, methods of undetermined coefficients and variations of parameters
 Firstorder linear systems: basic theory, existence and uniqueness, connections with linear algebra
 Homogeneous firstorder linear systems: equations with constant coefficients, eigenvalue analysis
 More homogeneous firstorder linear systems: qualitative analysis, phase diagrams, stability
 Nonhomogeneous linear systems: methods of diagonalization, undetermined coefficients, and variation of parameters
 Nonlinear firstorder systems: autonomous systems, local linearizations, and stability analysis
 Series solutions of linear equations: power series, solutions near ordinary points
MAT334: Complex Variables
 Position: Instructor, Course coordinator
 Location: University of Toronto
 Semesters: Spring 2013, Fall 2012
 Textbook: Complex Variables (2nd ed.), by S. D. Fisher
This is an introductory course on the theory of functions of one complex variable.
Topics (approximate):
 Basics of complex numbers
 Functions and limits on the complex plane
 Line/contour integration
 Holomorphic functions, the CauchyRiemann equations
 Power series
 Cauchy's theorem, Cauchy's formula
 Analyticity, Morera's theorem, Liouville theorem
 Zeroes, isolated singularities, and Laurent series
 Residue theorem, and applications
 Argument principle, Rouche's theorem, and the maximum modulus property
 Linear fractional transformations
 Conformal transformations, Riemann mapping theorem
I am slowly compiling my pile of notes from this course into a coherent set of lecture notes.
The most recent version of this effort is below:
Here are some exam questions and solutions from previous versions of the course.
 Fall 2012, Midterm 1
 Fall 2012, Midterm 1 Solutions
 Fall 2012, Midterm 2
 Fall 2012, Midterm 2 Solutions
 Spring 2013, Practice Exam 1
 Spring 2013, Practice Exam 1 Solutions
 Spring 2013, Practice Exam 2
 Spring 2013, Practice Exam 2 Solutions
 Spring 2013, Midterm 1
 Spring 2013, Midterm 2
 Spring 2013, Midterm 2 Solutions
MAT235: Calculus for Life Sciences II
 Position: Instructor
 Location: University of Toronto
 Semesters: Fall 2011, Spring 2012
 Textbook: Calculus (7th ed.), by J. Stewart
This is a yearlong course on both differential and integral multivariable calculus.
Fall semester topics:
 Parametrizations and parametric curves
 Polar coordinates and curves
 Polar curves (continued), conic sections
 Vectors, dot products, cross products
 Lines and planes in 3D space
 (More) calculus of parametric curves
 Multivariable functions, multivariable limits
 Partial and directional derivatives
 Differentiability and differentials, linear approximations of functions
 Multivariable chain rules, gradients, calculus on level sets
 Extrema of functions, first and second derivative tests
 Lagrange multipliers
Spring semester topics:
 Double integrals, Fubini's theorem
 Double integrals in polar coordinates
 Applications of integrals, surface area of graphs
 Triple integrals, cylindrical coordinates
 Spherical coordinates, change of variables formula
 Vector fields, line integrals
 Line integrals and conservative fields
 Green's theorem
 Divergence and curl, parametrized surfaces
 Surface area, surface integrals
 Stokes' theorem
 Divergence theorem
