Past Courses

  • 2023
  • 2022
    • Summer: five-lecture minicourse on "Introduction to the mathematics of X-ray imaging: X-ray transforms" (JSS - MA2 - University of Jyväskylä). Lecture notes
    • Spring: Introduction to Numerical Methods (Math 148/L - UCSC)
    • Winter: Analysis II (graduate) (Math 205 - UCSC).

      A core course on measure theory.

    • Winter: Real Analysis (Math 105A - UCSC).
  • 2021
    • Spring: Introduction to Numerical Methods (Math 148/L - UCSC)
    • Winter: Analysis II (graduate) (Math 205 - UCSC).

      A core course on measure theory.

    • Winter: Advanced Analysis (graduate) (Math 216 - UCSC).

      X-ray transforms on Riemannian manifolds. Geometric preliminaries: geometry of the double tangent bundle, geodesic flow, integration by parts, Santalo's formula, convexity, non-trapping, the surface case, isothermal coordinates. The X-ray transform on surfaces: forward mapping properties, surjectivity, non-injectivity, fiber-harmonic decomposition and formulation of injectivity problems in the framework of transport equations. Conjugate points, simplicity, and the injectivity of the X-ray transform on functions and 1-forms via energy identities. The case of higher-dimensional manifolds and higher-order tensors (tensor tomography, via holomorphic integrating factors in 2D). Pestov-Uhlmann inversion formulas in two dimensions.
      Introduction to spectral theory. Unbounded operators, spectrum, spectral theorem, operators with compact resolvent, essential spectrum and its stability under relatively compact perturbations. Introduction to potential scattering and scattering resonances on the real line.

  • 2020
    • Fall: Inverse Problems and Integral Geometry (graduate) (Math 264 - UCSC).

      Generalities on inverse problems: scales of smoothness, injectivity, stability, inversion, singular value decomposition, regularization. Three prototypes of integral geometry: the Funk transform on the sphere, the Radon transform on the plane, the Radon transform on the disk. Several methods inverting the Radon transform on the plane: inversion based on the Fourier Slice Theorem; diagonalization through circular harmonics and Abel transforms; complexification of transport equations and a Riemann-Hilbert problem.

    • Spring: Introduction to Proofs and Problem-Solving (Math 100 - UCSC)
    • Spring: Introduction to Numerical Methods (Math 148/L - UCSC)
    • Winter: Analysis II (graduate) (Math 205 - UCSC).
    • Winter: Nonlinear Functional Analysis (graduate) (Math 219 - UCSC).

      Focus on pseudodifferential operators and the Nash-Moser theorem, using the book by Alinhac and Gérard of the same name, and the article by Richard Hamilton on the Nash-Moser theorem.

  • 2019
    • Winter: Inverse Problems and Integral Geometry (graduate) (Math 264 - UCSC).

      We review the various techniques available to address the following question: on a two-dimensional Riemannian manifold, what can be reconstructed of a function (or a tensor field) from knowledge of its integrals over all geodesics ? We first look at the Euclidean case and all the methods that work there: Fourier-slice based methods; diagonalization through circular harmonics; Helgason support theorem; Pestov identities; Singular Value Decomposition. We then move on to simple surfaces and revisit the reconstruction of functions, and the tensor tomography problems. In the end, we study some non-linear generlizations: inverse problems for connections and Higgs fields, and boundary rigidity.

    • Winter: Honors Calculus II (Math 20B - UCSC).
  • 2018
    • Fall: Real Analysis (Math 105A - UCSC).
    • Fall: Honors Calculus I (Math 20A - UCSC).
    • Winter: Real Analysis (Math 105A - UCSC).
    • Winter: Analysis II (graduate) (Math 205 - UCSC).
  • 2017
    • Fall: Calculus with Apps (Math 11A - UCSC).

      The first quarter of the calculus sequence.

    • Spring: Complex Analysis (graduate) (Math 207A - UCSC)
    • Winter: Real Analysis (Math 105A - UCSC).

      Chapters 1 through 6 of Strichartz' "The way of Analysis".

    • Winter: Partial Differential Equations I (graduate) (Math 213A - UCSC).

      Laplace, heat and wave equations, studied using various techniques: classical solutions, potential theory, distributions and Fourier analysis, stationary phase, weak formulations, L2 Sobolev spaces and other Hilbert space methods.

  • 2016
    • Winter: Introductory Differential Equations (Math 216, two sections, U. Michigan) - (Course website). Additional material specific to the sections I taught available here.
  • 2015
    • Winter: Complex Analysis II (Math 428 A, U. Washington) - Course website.
      Draft of lecture notes here. (topics similar to Winter '14)
    • Winter: Introductory Real Analysis I (Math 327 B, U. Washington) - Course website.

      Sequences of real numbers, limits, point set theory, uniform continuity.

  • 2014
    • Summer: Introductory Real Analysis II (Math 328 B, U. Washington) - Course website

      Power series, continuity, uniform continuity, theory of integration, improper integrals, the Gamma function, Stirling's formula. Emphasis on proofs.

    • Summer: Seminar in Analysis (Math 530 C, U. Washington)

      Preparatory problem-solving sessions for graduate students toward taking the "Linear Analysis" Preliminary Exam. Topics: distributions, linear algebra, spectral theory, Fourier analysis, numerical methods for ODEs.

    • Summer: Special Lecture on math and origami, taught at SIMUW 2014 and at the 2014 discovery seminar on "Left brain, right brain: creative thinking in the sciences and in the arts.". Slides (Warning: 47MB file).
    • Winter: Complex Analysis II (Math 428 A, U. Washington) - Course website

      Some applications of residues, Rouché's theorem, conformal mappings, linear fractional transformations, Riemann mapping theorem, complex dynamics of rational maps.

  • 2013
    • Autumn: Introductory Real Analysis I (Math 327 A, U. Washington) - Course website

      sequences of real numbers, limits, point set theory, uniform continuity.

    • Autumn: Complex Analysis I (Math 427 A, U. Washington) - Course website

      Making our way from the algebra of complex numbers to residue theory, omitting no proof.

    • Summer: Computer Labs at MSRI Summer School on "The mathematics of seismic imaging".
    • Summer: Computer Labs at UW RTG IPDE Summer School - Detail of sessions

      Implementation and inversion of the thermoacoustic tomography problem. Illustration of the principle of propagation of singularities for the wave equation. Generalized X-ray transforms.

  • 2012
    • Autumn: Linear Analysis (Math 309 B&C, U. Washington) - Course website

      Systems of ODE's, solutions to PDE's using Fourier series and separation of variables. Emphasis on problem-solving.

  • 2011
    • Summer '11: Computer Labs at UW RTG IPDE Summer School - Detail of sessions

      The fast fourier transform. Implementation of the Radon tranform and attenuated Radon transform, and their inversion. Partial data problems. Generalization to other families of integration curves.