My email address is my first name dot my last name at gmail.com.

Courses I've taught

- UTM MAT 232 Winter 2017. Multivariable calculus I.
- UTSC MAT B44 Fall 2016. Differential Equations I
- St. George Mat 237 Summer 2014
- St George Mat 188 Fall 2013. Linear algebra. Practice questions.
- St George Mat 244 Summer 2013. Introduction to ordinary differential equations.

Notes that I've written on various topics:

Harmonic analysis

- Notes on the uniform uncertainty principle
- Talk on the uniform uncertainty principle
- Norms of trigonometric polynomials
- What is a Costas array?
- Lp norms of a sine sum
- The Dirac delta distribution and Green's functions
- The Poisson summation formula, the sampling theorem, and Dirac combs
- Lp norms of trigonometric polynomials
- L1 norms of products of sines
- Haar wavelets and multiresolution analysis
- The theorem of F. and M. Riesz
- The infinite-dimensional torus
- Tauber's theorem and Karamata's proof of the Hardy-Littlewood tauberian theorem
- The Fourier transform of holomorphic functions
- Locally compact abelian groups
- The Gelfand transform, positive linear functionals, and positive-definite functions
- The Wiener algebra and Wiener's lemma
- The Bernstein and Nikolsky inequalities for trigonometric polynomials
- Meager sets of periodic functions
- Zygmund's Fourier restriction theorem and Bernstein's inequality
- Bernstein's inequality and Nikolsky's inequality for R^d
- The Wiener-Pitt tauberian theorem
- Singular integral operators and the Riesz transform
- The Schwartz space and the Fourier transform
- The Hilbert transform on R

Asymptotic analysis

Radial functions and harmonic analysis on the sphere

- Positive definite functions, completely monotone functions, the Bernstein-Widder theorem, and Schoenberg's theorem
- Hausdorff measure
- The Fourier transform of spherical surface measure and radial functions
- Harmonic polynomials and the spherical Laplacian
- The Schrodinger kernel, spherical surface measure, Fourier restriction, and the Strichartz inequality
- The cross-polytope, the ball, and the cube

Number theory

- Notes on modular forms
- A note on Gilbreath's conjecture using PDE
- A proof of the pentagonal number theorem
- Newton's identities and the pentagonal number theorem
- The Polya-Vinogradov inequality
- Ramanujan's sum
- The Voronoi summation formula
- Nonholomorphic Eisenstein series, the Kronecker limit formula, and the hyperbolic Laplacian
- A series of secants
- Bernoulli polynomials
- Cyclotomic polynomials
- Lambert series in analytic number theory

p-adic numbers

- Hensel's lemma, valuations, and p-adic numbers
- The profinite completion of the integers, the p-adic integers, and Prufer p-groups
- The p-adic solenoid
- The Pontryagin duals of Q/Z and Q
- Explicit construction of the p-adic numbers
- Harmonic analysis on the p-adic numbers
- Valued fields
- p-adic test functions
- The adeles

Hamiltonian mechanics and dynamical systems

- The Poincare-Dulac normal form for formal vector fields
- Arnold's theorem on the analytic linearization of analytic circle diffeomorphisms
- Liouville's theorem, symplectic geometry, Gibbs measures and equivariant cohomology
- Gibbs measures and the Ising model
- Denjoy's theorem
- Notes on the KAM theorem
- What is the domain of the solution of an ODE?
- Complexification, complex structures, and linear ordinary differential equations
- Hamiltonian flows, cotangent lifts, and momentum maps
- The Hamilton-Jacobi equation
- The Legendre transform
- The Gottschalk-Hedlund theorem, cocycles, and small divisors
- Weak symplectic forms and differential calculus in Banach spaces
- The left shift map and expanding endomorphisms of the circle

Diophantine approximation and continued fractions

- Estimating a product of sines using Diophantine approximation
- Kronecker's theorem
- Vinogradov's estimate for exponential sums over primes
- The Gauss map
- Diophantine vectors
- Diophantine numbers
- Measure theory and Perron-Frobenius operators for continued fractions
- The inclusion map from the integers to the reals and universal properties of the floor and ceiling functions
- Functions of bounded variation and a theorem of Khinchin

Partial differential equations

- Scaling for the nonlinear Schrodinger equation
- Proof by bootstrapping
- The nonlinear Schrodinger equation is Hamiltonian
- A derivation of the cubic nonlinear Schrodinger equation
- Orbital stability for the nonlinear Schrodinger equation
- The inhomogeneous heat equation on T
- The Euler equations in fluid mechanics
- The one-dimensional periodic Schrodinger equation

Spectral theory

- The principal axis theorem and Sylvester's law of inertia
- Hilbert-Schmidt operators and tensor products of Hilbert spaces
- Categorical tensor products of Hilbert spaces
- Self-adjoint linear operators on a finite dimensional complex vector space
- The spectrum of a self-adjoint operator is a compact subset of R
- Abstract Fourier series and Parseval's identity
- Projection-valued measures and spectral integrals
- Decomposition of the spectrum of a bounded linear operator
- Gelfand-Pettis integrals and weak holomorphy
- The spectra of the unilateral shift and its adjoint
- Trace class operators and Hilbert-Schmidt operators
- Compact operators on Banach spaces
- Unordered sums in Hilbert spaces
- The singular value decomposition of compact operators on Hilbert spaces
- Banach algebras
- Unbounded operators in a Hilbert space and the Trotter product formula
- Unbounded operators, resolvents, the Friedrichs extension, and the Laplacian on L2(Td)
- Laguerre polynomials and Perron-Frobenius operators
- Integral operators
- Spectral theory, Volterra integral operators and the Sturm-Liouville theorem

Gaussian measures and Hermite polynomials

- Gaussian measures and Bochner's theorem
- Gaussian measures, Hermite polynomials, and the Ornstein-Uhlenbeck semigroup
- Hermite functions
- Gaussian Hilbert spaces
- Schwartz functions, Hermite functions, and the Hermite operator
- Stationary phase, Laplace's method, and the Fourier transform for Gaussian integrals
- The Segal-Bargmann transform and the Segal-Bargmann space
- The Heisenberg group and Hermite functions
- Gaussian integrals
- The Cameron-Martin theorem

The Laplace operator

- The Fredholm determinant
- Unbounded operators and the Friedrichs extension
- The heat kernel on R^n
- The heat kernel on the torus
- The functional determinant
- The Laplace operator is essentially self-adjoint

Probability and measure theory

- Rademacher functions
- Total variation, absolute continuity, and the Borel sigma-algebra of C(I)
- The Banach algebra of functions of bounded variation and the pointwise Helly selection theorem
- L^0, convergence in measure, equi-integrability, the Vitali convergence theorem, and the de la Vallee-Poussin criterion
- The symmetric difference metric
- The Dunford-Pettis theorem
- The Glivenko-Cantelli theorem
- Regulated functions and the regulated integral
- Infinite product measures
- Levy's inequality, Rademacher sums, and Kahane's inequality
- Martingales, Levy's continuity theorem, and the martingale central limit theorem
- The weak and strong laws of large numbers
- The Lindeberg central limit theorem
- Subgaussian random variables, Hoeffding's inequality, and Cramer's large deviation theorem
- Khinchin's inequality and Etemadi's inequality
- Varadhan's lemma for large deviations
- The Berry-Esseen theorem
- The law of the iterated logarithm
- Orthonormal bases for product measures
- Vitali coverings on the real line
- Functions of bounded variation and differentiability
- Random trigonometric polynomials

Stochastic processes

- The narrow topology on the set of Borel probability measures on a metrizable space
- The Kolmogorov extension theorem
- The Bochner-Minlos theorem
- Finite-dimensional distributions of stochastic processes
- Markov kernels, convolution semigroups, and projective families of probability measures
- The Kolmogorov continuity theorem, Holder continuity, and the Kolmogorov-Chentsov theorem
- Convolution semigroups, canonical processes, and Brownian motion
- Jointly measurable and progressively measurable stochastic processes
- Donsker's theorem

Function spaces

- Projective limits of topological vector spaces
- The weak topology of locally convex topological vector spaces and the weak-* topology of their duals
- Fatou's theorem, Bergman spaces, and Hardy spaces on the circle
- The Frechet space of holomorphic functions on the unit disc
- C^k spaces and spaces of test functions
- C[0,1]: the Faber-Schauder basis, the Riesz representation theorem, and the Borel sigma-algebra
- Test functions, distributions, and Sobolev's lemma
- Sobolev spaces in one dimension and absolutely continuous functions
- Real reproducing kernel Hilbert spaces

Differential calculus

- Frechet and Gateaux derivatives
- Gradients and Hessians in Hilbert spaces
- The C-infinity Urysohn lemma
- Germs of smooth functions

Convex functions

Topology

- Polish spaces and Baire spaces
- The Stone-Cech compactification of Tychonoff spaces
- Topological spaces and neighborhood filters
- The uniform metric on product spaces

Mathematics history

- Bibliography for the history of the Jacobian
- Notes on the history of Lioville's theorem
- The logarithmic integral
- log sin
- The Euler-Maclaurin summation formula
- What is a wave?
- Summable series and the Riemann rearrangement theorem
- Bibliography for the history of induction in mathematics
- Bibliography for the history of resonance
- The great year, calendars, and the incommensurability of celestial rotations
- Early instances of the martingale
- Gregory of Saint-Vincent and Zeno's paradoxes
- Book I of Euclid's Elements and application of areas
- Book IV of Euclid's Elements and ancient Greek mosaics
- Greek numbers
- Numbers and fractions in Greek papyri
- Approximating square roots in antiquity
- Greek music theory and Archytas's theorem
- The Euclidean algorithm and finite continued fractions
- Pell's equation
- Turning a rectangle into a square in the Sulbasutras

Quasi-mathematics history

- Zeno of Elea, locomotion, infinity, and time
- Plato's theory of forms and the axiom of foundation
- Denomination
- Latitude, intension, and remission
- Genus
- Amphibolia
- Ancient Greek weights and measures
- Minoan weights
- Ancient balance scales
- Vedic texts
- Nicole Oresme

Please write me if you find any of them interesting and would like to talk with me about them, or if you find any mistakes.

I have also translated a number of Euler's papers from the Latin. They're posted at arxiv.org, under author name Euler. My translated of Euler's "De summis serierum reciprocarum" ("On the sums of series of reciprocals"), in which Euler first works out a formula for the sum of the squares of the reciprocals of the natural numbers (namely zeta(2)=pi squared divided by 6), is included in Stephen Hawking's "God Created the Integers", new edition.