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Department:

MATH

Course Number:

7339

Hours - Lecture:

3

Hours - Lab:

0

A comprehensive overview of advanced material in analysis. This is a Mother Course with 5 different subtitles; Recommended prerequisites may vary with each offering.

- Subtitle 1: Exponential Systems and Their Applications: A comprehensive course on nonharmonic Fourier series and related topics.
- Subtitle 2: Potential Theory and Applications: Potential theory in the complex plane, energy integrals, logarithmic and other potentials, solutions to certain PDEs, applications to polynomials and complex analysis.
- Subtitle 3: Singular Integral Operators: A comprehensive treatment of the theory of singular integral operators.
- Subtitle 4: Isoperimetric and Functional inequalities in convexity: Modern introduction to isoperimetric and functional inequalities, some of which involve convexity, using a variety of modern methods.
- Subtitle 5: Riemann Surfaces: Introduction to the theory of Riemann surfaces and theta functions.

For the categories below, match the bullet number with the subtitle number to find the corresponding information.

The course will be offered with subtitle 1 in Spring 2024.

Prerequisites:

- MATH 7337
- MATH 6337
- No prerequisite information available
- Recommended: MATH 6241, 6337, or 6307
- Math 6321

Course Text:

- Olevskii and Ulanovskii, “Functions with Disconnected Spectrum,” Young, “An Introduction to Nonharmonic Fourier Series,” and recent publications in the field.
- T. Ransford, “Potential Theory”.
- Unavailable
- Artstein–Avidan, Giannopoulos, and Milman, “Asymptotic Geometric Analysis”; Ledoux, “The Concentration of Measure Phenomenon”; Bakry, Gentil, and Ledoux; “Analysis and Geometry of Markov Diffusion Operators”; Lieb and Loss, “Analysis.”
- Farkas and Kra, “Riemann Surfaces” and and recent publications in the field.

Topic Outline:

1. Topics:

- Classical theory over a segment: Radius of completeness, Beurling and Maliavin theory, sampling and interpolation of band limited functions.
- Disconnected spectra: Relatively sparse complete systems of exponentials, Landau’s density theorem, relation to restricted invertibility and Kadison–Singer question, universal Sampling and interpolation, Matei and Meyer quasicrystals.
- Higher dimensions, convex sets.
- Bases of exponentials, Fuglede’s Conjecture, existence of Riesz bases.
- Weighted exponentials: Systems of translates, Gabor analysis.

2. Topics:

- Cauchy Principal Value Integrals.
- Harmonic functions.
- Subharmonic functions.
- Potential theory.
- The Dirichlet problem.
- Capacity.
- Applications.

3. Topics:

- Maximal functions: Covering lemmas and L p bounds.
- Singular Integral operators: L 2 theory, the Calder´on–Zygmund decomposition, and bounds in BMO and H1.
- Fourier transform and oscillatory integrals: The van der Corput Lemma, the role of curvature.
- Weighted theory: Duality theory, Muckenhoupt weights, sparse bounds
- T(1)-theorems: The Sawyer maximal theorem, positive dyadic operators, the David– Journ´e Theorem.
- Non-homogeneous Calder´on–Zygmund theory with applications to analytic capacity.
- Discrete singular integral operators and maximal functions.

4. Topics:

- Convexity. The Brunn-Minkowski inequality and its applications. The classical isoperimetric inequality. Johns position. Balls volume ratio and reverse isoperimetric inequality. Volume product and related inequalities.
- Log-concave measures, Borells theorem, Borells Lemma, moment estimates. Concentration of measure phenomenon and log-concavity.
- The Brascamp-Lieb inequality. Poincare inequality for strictly log-concave measures.
- Review of the theory of linear differential operators, Sobolev spaces, Ornstein–Uhlenbeck operator. The L2 method. The B-theorem of Cordero, Fradelizi and Maurey.
- Some elements of geometric measure theory
- Rearrangement inequalities, Schwartz symmetrization, Hardy-Littlewood principle. Energy minimization, Faber-Krahn inequality, Saint-Venant theorem, Talenti inequality.
- Gaussian isoperimetric inequality. Ehrhards symmetrization. Ehrhard inequality. Applications to concentration of measure. Nazarov-Ball reverse Gaussian isoperimetric inequality
- Modern methods in the calculus of variations, such as concentration compactness, compensated compactness, Ekeland principle. Variational proof of the isoperimetric inequality. Variational proof of the Gaussian Isoperimetric inequality.
- Log-Sobolev inequality, Youngs convolution inequality.

5. Topics:

- Topological aspects of Riemann surfaces.
- Differential forms and integration.
- Compact Riemann Surfaces: Holomorphic differentials, Riemann–Roch Theorem, Abel’s Theorem and Jacobi inversion problem.
- Theta functions and applications: Baker–Akhiezer function, solutions to nonlinear (soliton) PDEs (Korteweg–de Vries, Boussinesq, Kadomtsev–Petviashvili, Toda lattice, etc) in terms of theta functions, connections to commutative algebras of differential operators.