Arick Shao 邵崇哲

Below are university mathematics courses that I have previously taught.

## MTH5109: Geometry II: Knots and Surfaces

This is a second-year undergraduate course on the differential geometry of curves and surfaces, along with a brief excursion to knot theory.

Approximate (combined) list of topics covered:

1. Knots, knot diagrams, and the Reidemeister theorem
2. Knot invariants: crossing number, chirality, tricolorability, Jones polynomial
3. Curves and their parametrizations
4. Curves: tangent lines, orientation
5. Curves: arc length, path/line integrals
6. Curves: curvature
7. Plane curves: signed curvature, winding number
8. Space curves: torsion, Frenet–Serret formulas
9. Surfaces and their parametrizations
10. Surfaces: tangent planes, first fundamental forms, unit normals
11. Surfaces: orientability and orientation
12. Surfaces: surface area, surface integrals
13. Surfaces: first fundamental form, second fundamental forms
14. Surfaces: principal, mean, and Gauss curvatures
15. Curves on surfaces: normal and geodesic curvatures, Euler's theorem
16. Geodesics: covariant derivatives, geodesic equations
17. Some landmark results (distance-minimizing curves, theorema egregium, Gauss–Bonnet theorem)

## M4P41: Analytic Methods in PDEs

• Position: Instructor
• Location: Imperial College London
• Term: Spring 2016

This is a fourth-year undergraduate course in partial differential equations (PDEs) and the mathematical methods used to study them.

Approximate list of topics covered:

1. Review of ODE theory (existence, uniqueness, continuous dependence)
2. Method of characteristics (first-order scalar PDE)
3. Analytic PDE, power series solutions (Cauchy-Kovalevskaya and Holmgren theorems)
4. Weak derivatives, weak solutions of PDE
5. The Laplace and Poisson equations
6. The heat equation
7. The wave equation
8. 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 graduate-level partial differential equations (PDE) course, a collaborative effort with Jonathan Ben-Artzi, which introduced the following two areas of study:

• Kinetic theory (JBA): transport equations, classical theory of Vlasov-Poisson and Vlasov-Maxwell 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:

1. (AS) Ordinary differential equations, connections to evolutionary PDE
2. (JBA) PDE preliminaries (Fourier transforms, Sobolev spaces), linear transport equations (method of characteristics), introduction to kinetic theory
3. (JBA) The Vlasov-Poisson system: local existence and uniqueness
4. (JBA) The Vlasov-Poisson system: global existence
5. (AS) Linear wave equations: physical and Fourier representation formulas, energy and dispersive estimates
6. (JBA) The Vlasov-Maxwell system: conditional global existence
7. (AS) Nonlinear wave equations: classical local existence and uniqueness
8. (AS) Nonlinear wave equations: the vector field method, small-data global and long-time existence

The full set of lecture notes for the course can be found on the official course website (created by Jonathan Ben-Artzi). 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:

1. Motivation (why real numbers are necessary), construction of real numbers (Dedekind cuts), defining properties of real numbers (least upper bound principle)
2. Cardinality, limits (real numbers), monotone sequences
3. Limits (metric spaces), Cauchy sequences, completeness
4. Series, convergence tests, absolute and conditional convergence
5. Discrete dynamical systems, fixed points, contraction mapping theorem
6. Fractals (iterated function systems)
7. Topology of metric spaces, limit and interior points, open and closed sets, continuity
8. Compactness and connectedness, Heine-Borel theorem, extreme and intermediate value theorems
9. Differentiation (1-d), mean value theorem, Taylor's theorem
10. Inverse function theorem, implicit function theorem
11. Riemann integration, fundamental theorem of calculus
12. 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:

1. First-order equations: separation of variables, linear equations
2. More first-order equations: exact equations, general integrating factors
3. Existence and uniqueness theory: linear equations and systems, nonlinear equations
4. Second-order homogeneous linear equations: equations with constant coefficients
5. Second-order nonhomogeneous linear equations: method of undetermined coefficients, variation of parameters
6. Higher-order linear equations: equations with constant coefficients, methods of undetermined coefficients and variations of parameters
7. First-order linear systems: basic theory, existence and uniqueness, connections with linear algebra
8. Homogeneous first-order linear systems: equations with constant coefficients, eigenvalue analysis
9. More homogeneous first-order linear systems: qualitative analysis, phase diagrams, stability
10. Nonhomogeneous linear systems: methods of diagonalization, undetermined coefficients, and variation of parameters
11. Nonlinear first-order systems: autonomous systems, local linearizations, and stability analysis
12. 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):

1. Basics of complex numbers
2. Functions and limits on the complex plane
3. Line/contour integration
4. Holomorphic functions, the Cauchy-Riemann equations
5. Power series
6. Cauchy's theorem, Cauchy's formula
7. Analyticity, Morera's theorem, Liouville theorem
8. Zeroes, isolated singularities, and Laurent series
9. Residue theorem, and applications
10. Argument principle, Rouche's theorem, and the maximum modulus property
11. Linear fractional transformations
12. 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.

## 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 year-long course on both differential and integral multivariable calculus.

Fall semester topics:

1. Parametrizations and parametric curves
2. Polar coordinates and curves
3. Polar curves (continued), conic sections
4. Vectors, dot products, cross products
5. Lines and planes in 3-D space
6. (More) calculus of parametric curves
7. Multivariable functions, multivariable limits
8. Partial and directional derivatives
9. Differentiability and differentials, linear approximations of functions
10. Multivariable chain rules, gradients, calculus on level sets
11. Extrema of functions, first and second derivative tests
12. Lagrange multipliers

Spring semester topics:

1. Double integrals, Fubini's theorem
2. Double integrals in polar coordinates
3. Applications of integrals, surface area of graphs
4. Triple integrals, cylindrical coordinates
5. Spherical coordinates, change of variables formula
6. Vector fields, line integrals
7. Line integrals and conservative fields
8. Green's theorem
9. Divergence and curl, parametrized surfaces
10. Surface area, surface integrals
11. Stokes' theorem
12. Divergence theorem