MAST 837
Selected Topics in Analysis
Fall 2026 · Convex and Non‑Linear Analysis
Topics
- Convex sets, separation theorems and polar sets
- Ellipsoids and John’s theorem
- Convex functions, differentiability criteria and inequalities
- Conjugate functions
- Brunn–Minkowski theorems
- Polarity and duality
- Symmetrizations and rearrangements
- Loomis–Whitney inequalities, projections and sections
- Centers of mass
- Selected topics in advanced convexity
Readings
- Reference (not explicitly required): C. P. Niculescu and L.-E. Persson. Convex Functions and Their Applications: A Contemporary Approach. Springer, 2018.
- Reference (not explicitly required): Boyd and Vandenberghe. Convex Optimization. Cambridge University Press, 2004.
- Reference (not explicitly required): D. Hug and W. Weil. Lectures in Convex Geometry. Springer, 2020.
- Reference (not explicitly required): Peter M. Gruber. Convex and Discrete Geometry. Springer, 2007.
Fall 2026 · Measure Theory
Topics
- Outer measure, Lebesgue measure and positive measures
- Lebesgue integration and convergence theorems
- Differentiation and maximal functions
- Lebesgue Lp spaces
- Product measures
- Real and complex measures
- Radon–Nikodym theorem
Readings
- Listed textbook (requirement not stated): S. Axler. Measure, Integration & Real Analysis. Springer, 2020. https://measure.axler.net/
- Other textbook (requirement not stated): E. M. Stein and R. Shakarchi. Real Analysis, Measure Theory and Integration. Princeton University Press, 2005.
- Other textbook (requirement not stated): E. M. Stein and R. Shakarchi. Functional Analysis. Princeton University Press, Macmillan, 2011.
- Other textbook (requirement not stated): G. B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2013.
- Other textbook (requirement not stated): H. L. Royden. Real Analysis, 3rd or 4th edition.
Fall 2026 · Real Analysis
Topics
- Sets, functions, relations and countability
- Construction and topology of the real numbers
- Metric spaces, normed linear spaces and point-set topology
- Compactness, Heine–Borel and Bolzano–Weierstrass properties
- Baire category theorem
- Continuous functions, connectedness and intermediate values
- Function spaces and uniform convergence
- Arzelà–Ascoli theorem
- Stone–Weierstrass approximation
- Fixed-point theorems and applications
Readings
- Reference (not explicitly required): H. L. Royden. Real Analysis, 2nd, 3rd or 4th edition. MacMillan.
- Reference (not explicitly required): W. Rudin. Principles of Mathematical Analysis.
Winter 2026 · MATH 494
Readings
- Ingrid Daubechies, Ten Lectures on Wavelets.
- H. Dym and H. P. McKean, Fourier Series and Integrals.
- G. B. Folland, Fourier Analysis and Its Applications.
- G. B. Folland, Real Analysis: Modern Techniques and Their Applications.
- T. W. Korner, Fourier Analysis.
- W. Rudin, Real and Complex Analysis.
- W. Rudin, Functional Analysis.
- E. M. Stein and R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis.
- Yves Meyer, Wavelets and Operators.
Winter 2025 · Topics in Pure and Applied Mathematics
Topics
- Hilbert and Banach spaces
- Dual spaces and bounded linear operators
- Adjoints
- Hahn–Banach and Baire category theorems
- Banach–Steinhaus, open mapping and closed graph theorems
- Compact operators and the spectral theorem
- Fredholm alternative
- Weak and weak-star topologies
- Distributions and Sobolev spaces
Readings
- Listed textbook (requirement not stated): Barbara D. MacCluer. Elementary Functional Analysis. Springer, 2009. Chapters 1–4.
- Reference (not explicitly required): H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.
- Reference (not explicitly required): Avner Friedman. Foundations of Modern Analysis. Dover, 2003.
- Reference (not explicitly required): Elias M. Stein and Rami Shakarchi. Functional Analysis: Introduction to Further Topics in Analysis. Princeton, 2011.
Fall 2024 · MATH 467
Readings
- Measure, Integration & Real Analysis, by S. Axler, Springer, 2020.
- Real Analysis, Measure Theory and Integration, by E.M. Stein and R. Shakarchi, Princeton University Press, 2005.
- Functional Analysis, by E.M. Stein and R. Shakarchi, Princeton University Press, Macmillan, 2011.
- Real Analysis: Modern Techniques and Their Applications, by G.B. Folland, Wiley, 2013.
- Real Analysis, by H.L. Royden (3rd or 4th Edition).
Winter 2024 · MATH 494
Readings
- Ingrid Daubechies, Ten Lectures on Wavelets.
- H. Dym and H. P. McKean, Fourier Series and Integrals.
- G. B. Folland, Fourier Analysis and Its Applications.
- G. B. Folland, Real Analysis: Modern Techniques and Their Applications.
- T. W. Korner, Fourier Analysis.
- W. Rudin, Real and Complex Analysis.
- W. Rudin, Functional Analysis.
- E. M. Stein and R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis.
- Yves Meyer, Wavelets and Operators.
Fall 2023 · MATH 494
Readings
- M. S. Birman, M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space.
Fall 2022 · Complex Analysis II
Topics
- Holomorphic functions and Cauchy’s theorem
- Open mapping and maximum modulus principles
- Meromorphic functions, Laurent series and residues
- Argument principle
- Riemann mapping theorem and analytic continuation
- Riemann surfaces
- Periodic meromorphic functions on tori
- Weierstrass and Jacobi elliptic functions
- Time permitting: higher-genus algebraic curves, holomorphic differentials, Abel’s theorem, theta functions and Jacobi inversion
Readings
- L. V. Ahlfors. Complex Analysis (supplementary).
- H. M. Farkas and I. Kra. Riemann Surfaces, 2nd edition. Springer, 1992 (supplementary).
- Elias M. Stein and Rami Shakarchi. Complex Analysis. Princeton University Press, 2003.
Fall 2022 · MATH 494
Readings
- Elementary Functional Analysis by Barbara D. MacCluer, Springer.
- Functional Analysis, Sobolev Spaces and Partial Differential Equations by H. Brezis, Springer, 2011.
- Foundations of Modern Analysis by Avner Friedman, Dover, 2003.
- Functional Analysis: Introduction to Further Topics in Analysis by Elias M. Stein & Rami Shakarchi, Princeton, 2011.
Winter 2022 · MATH 467
Readings
- Measure, Integration & Real Analysis, by S. Axler, Springer, 2020.
- Real Analysis, by H.L. Royden (3rd or 4th Edition).
- Real Analysis, Measure Theory and Integration, by E.M. Stein and R. Shakarchi, Princeton University Press, 2005.
- Functional Analysis, by E.M. Stein and R. Shakarchi, Princeton University Press, Macmillan, 2011.
Fall 2021 · MATH 478
Readings
- Convex Optimization by Boyd and Vandenberghe.
Fall 2021 · MATH 494
Readings
- H. Dym and H. P. McKean, Fourier Series and Integrals.
- G. B. Folland, Fourier Analysis and Its Applications.
- G. B. Folland, Real Analysis: Modern Techniques and Their Applications.
- T. W. Korner, Fourier Analysis.
- W. Rudin, Real and Complex Analysis.
- W. Rudin, Functional Analysis.
- E. M. Stein and R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis.
- P. Koosis, An Introduction to Hp Spaces.
- E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces.
- E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals.
- E. M. Stein, Singular Integrals and Differentiability Properties of Functions.
- C. Pereyra and L. Ward, Harmonic Analysis from Fourier to Wavelets.
- A. Zygmund, Trigonometric Series.
Winter 2021 · MATH 494
Readings
- A Mathematical Introduction to Compressive Sensing by Simon Foucart and Holger Rauhut. Springer, 2013.
- Sparse and Redundant Representations by Michael Elad, Springer, 2010.
- B. Adcock and A. C. Hansen, Compressive Imaging: Structure, Sampling, Learning, Cambridge University Press (listed as forthcoming).
- Statistical Learning with Sparsity: The Lasso and Generalizations by Trevor Hastie, Robert Tibshirani, and Martin Wainwright Taylor and Francis Group, 2015.
Sources
- Course catalogue ↗2026–2027
- Syllabus or course outline ↗Fall 2026
- Syllabus or course outline ↗Fall 2026
- Syllabus or course outline ↗Fall 2026
- Historical course outline ↗Winter 2026
- Syllabus or course outline ↗Winter 2025
- Historical course outline ↗Fall 2024
- Historical course outline ↗Fall 2024
- Historical course outline ↗Winter 2024
- Historical course outline ↗Fall 2023
- Syllabus or course outline ↗Fall 2022
- Historical course outline ↗Fall 2022
- Historical course outline ↗Winter 2022
- Historical course outline ↗Fall 2021
- Historical course outline ↗Fall 2021
- Historical course outline ↗Winter 2021
- Fall 2026 outline. Course header explicitly lists MATH 479, MAST 661, MAST 837. These are optional sources students “may consult”; author/publisher spelling in the second entry has been lightly normalized.
- Fall 2026 outline. Course header explicitly lists MATH 467, MAST 669, MAST 837. The “Other Textbooks” list is not labelled required; publisher wording is preserved as printed.
- Fall 2026 outline. Course header explicitly lists MATH 464, MAST 661, MAST 837. Books are listed as reference textbooks; no required status is stated.
- Historical source. Generic Textbook/References headings remain requirement-unspecified. Bibliographic editions and years are retained as printed; source inconsistencies may remain. Automated extraction with conservative filtering; distinct bibliography strings screened for false positives; entire syllabus not manually reviewed.
- Winter 2025 outline. Course header explicitly lists MAST 662, MATH 494, MAST 837.
- Historical source. Generic Textbook/References headings remain requirement-unspecified. Bibliographic editions and years are retained as printed; source inconsistencies may remain. Automated extraction with conservative filtering; distinct bibliography strings screened for false positives; entire syllabus not manually reviewed.
- Historical source. Generic Textbook/References headings remain requirement-unspecified. Bibliographic editions and years are retained as printed; source inconsistencies may remain. Automated extraction with conservative filtering; distinct bibliography strings screened for false positives; entire syllabus not manually reviewed.
- Historical source. Generic Textbook/References headings remain requirement-unspecified. Bibliographic editions and years are retained as printed; source inconsistencies may remain. Automated extraction with conservative filtering; distinct bibliography strings screened for false positives; entire syllabus not manually reviewed.
- Historical source. Generic Textbook/References headings remain requirement-unspecified. Bibliographic editions and years are retained as printed; source inconsistencies may remain. Automated extraction with conservative filtering; distinct bibliography strings screened for false positives; entire syllabus not manually reviewed.
- Historical Fall 2022 outline. Main text is not explicitly marked required; the other two books are recommended supplementary texts.
- Historical source. Generic Textbook/References headings remain requirement-unspecified. Bibliographic editions and years are retained as printed; source inconsistencies may remain. Automated extraction with conservative filtering; distinct bibliography strings screened for false positives; entire syllabus not manually reviewed.
- Historical source. Generic Textbook/References headings remain requirement-unspecified. Bibliographic editions and years are retained as printed; source inconsistencies may remain. Automated extraction with conservative filtering; distinct bibliography strings screened for false positives; entire syllabus not manually reviewed.
- Historical source. Generic Textbook/References headings remain requirement-unspecified. Bibliographic editions and years are retained as printed; source inconsistencies may remain. Automated extraction with conservative filtering; distinct bibliography strings screened for false positives; entire syllabus not manually reviewed.
- Historical source. Generic Textbook/References headings remain requirement-unspecified. Bibliographic editions and years are retained as printed; source inconsistencies may remain. Automated extraction with conservative filtering; distinct bibliography strings screened for false positives; entire syllabus not manually reviewed.
- Historical source. Generic Textbook/References headings remain requirement-unspecified. Bibliographic editions and years are retained as printed; source inconsistencies may remain. Automated extraction with conservative filtering; distinct bibliography strings screened for false positives; entire syllabus not manually reviewed.