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Courses I've taught

Notes that I've written on various topics:

Harmonic analysis

  1. Notes on the uniform uncertainty principle
  2. Talk on the uniform uncertainty principle
  3. Norms of trigonometric polynomials
  4. What is a Costas array?
  5. Lp norms of a sine sum
  6. The Dirac delta distribution and Green's functions
  7. The Poisson summation formula, the sampling theorem, and Dirac combs
  8. Lp norms of trigonometric polynomials
  9. L1 norms of products of sines
  10. Haar wavelets and multiresolution analysis
  11. The theorem of F. and M. Riesz
  12. The infinite-dimensional torus
  13. Tauber's theorem and Karamata's proof of the Hardy-Littlewood tauberian theorem
  14. The Fourier transform of holomorphic functions
  15. Locally compact abelian groups
  16. The Gelfand transform, positive linear functionals, and positive-definite functions
  17. The Wiener algebra and Wiener's lemma
  18. The Bernstein and Nikolsky inequalities for trigonometric polynomials
  19. Meager sets of periodic functions
  20. Zygmund's Fourier restriction theorem and Bernstein's inequality
  21. Bernstein's inequality and Nikolsky's inequality for R^d
  22. The Wiener-Pitt tauberian theorem
  23. Singular integral operators and the Riesz transform
  24. The Schwartz space and the Fourier transform
  25. The Hilbert transform on R

Asymptotic analysis

  1. Watson's lemma and Laplace's method
  2. Oscillatory integrals
  3. Chebyshev polynomials

Radial functions and harmonic analysis on the sphere

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

Number theory

  1. Notes on modular forms
  2. A note on Gilbreath's conjecture using PDE
  3. A proof of the pentagonal number theorem
  4. Newton's identities and the pentagonal number theorem
  5. The Polya-Vinogradov inequality
  6. Ramanujan's sum
  7. The Voronoi summation formula
  8. Nonholomorphic Eisenstein series, the Kronecker limit formula, and the hyperbolic Laplacian
  9. A series of secants
  10. Bernoulli polynomials
  11. Cyclotomic polynomials
  12. Lambert series in analytic number theory

p-adic numbers

  1. Hensel's lemma, valuations, and p-adic numbers
  2. The profinite completion of the integers, the p-adic integers, and Prufer p-groups
  3. The p-adic solenoid
  4. The Pontryagin duals of Q/Z and Q
  5. Explicit construction of the p-adic numbers
  6. Harmonic analysis on the p-adic numbers
  7. Valued fields
  8. p-adic test functions
  9. The adeles

Hamiltonian mechanics and dynamical systems

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

Linear algebra

  1. Alternating multilinear forms

Diophantine approximation and continued fractions

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

Partial differential equations

  1. Scaling for the nonlinear Schrodinger equation
  2. Proof by bootstrapping
  3. The nonlinear Schrodinger equation is Hamiltonian
  4. A derivation of the cubic nonlinear Schrodinger equation
  5. Orbital stability for the nonlinear Schrodinger equation
  6. The inhomogeneous heat equation on T
  7. The Euler equations in fluid mechanics
  8. The one-dimensional periodic Schrodinger equation

Spectral theory

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

Gaussian measures and Hermite polynomials

  1. Gaussian measures and Bochner's theorem
  2. Gaussian measures, Hermite polynomials, and the Ornstein-Uhlenbeck semigroup
  3. Hermite functions
  4. Gaussian Hilbert spaces
  5. Schwartz functions, Hermite functions, and the Hermite operator
  6. Stationary phase, Laplace's method, and the Fourier transform for Gaussian integrals
  7. The Segal-Bargmann transform and the Segal-Bargmann space
  8. The Heisenberg group and Hermite functions
  9. Gaussian integrals
  10. The Cameron-Martin theorem

The Laplace operator

  1. The Fredholm determinant
  2. Unbounded operators and the Friedrichs extension
  3. The heat kernel on R^n
  4. The heat kernel on the torus
  5. The functional determinant
  6. The Laplace operator is essentially self-adjoint

Probability and measure theory

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

Stochastic processes

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

Function spaces

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

Differential calculus

  1. Frechet and Gateaux derivatives
  2. Gradients and Hessians in Hilbert spaces
  3. The C-infinity Urysohn lemma
  4. Germs of smooth functions

Convex functions

  1. Semicontinuous functions and convexity
  2. Subdifferentials of convex functions


  1. Polish spaces and Baire spaces
  2. The Stone-Cech compactification of Tychonoff spaces
  3. Topological spaces and neighborhood filters
  4. The uniform metric on product spaces

Mathematics history

  1. Bibliography for the history of the Jacobian
  2. Notes on the history of Lioville's theorem
  3. The logarithmic integral
  4. log sin
  5. The Euler-Maclaurin summation formula
  6. What is a wave?
  7. Summable series and the Riemann rearrangement theorem
  8. Bibliography for the history of induction in mathematics
  9. Bibliography for the history of resonance
  10. The great year, calendars, and the incommensurability of celestial rotations
  11. Early instances of the martingale
  12. Gregory of Saint-Vincent and Zeno's paradoxes
  13. Book I of Euclid's Elements and application of areas
  14. Book IV of Euclid's Elements and ancient Greek mosaics
  15. Greek numbers
  16. Numbers and fractions in Greek papyri
  17. Approximating square roots in antiquity
  18. Greek music theory and Archytas's theorem
  19. The Euclidean algorithm and finite continued fractions
  20. Pell's equation
  21. Turning a rectangle into a square in the Sulbasutras

Quasi-mathematics history

  1. Zeno of Elea, locomotion, infinity, and time
  2. Plato's theory of forms and the axiom of foundation
  3. Denomination
  4. Latitude, intension, and remission
  5. Genus
  6. Amphibolia
  7. Ancient Greek weights and measures
  8. Minoan weights
  9. Ancient balance scales
  10. Vedic texts
  11. 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, 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.