Instructor: Florent Baudier
Office: Blocker 525J
Office hours: MW 3-4 p.m. by appointment
Lectures:
MW 5:45-7:00 p.m., Blocker 148
Textbook: no textbook is required but the following material will be useful for the course
- Hervé Queffelec and Daniel Li, Introduction to Banach Spaces: Analysis and Probability: Volume 1, Cambridge University Press, 2017.
- Hervé Queffelec and Daniel Li, Introduction to Banach Spaces: Analysis and Probability: Volume 2, Cambridge University Press, 2017.
- Michel Ledoux and Michel Talagrand, Probability in Banach Spaces, Springer, 2014.
- Jan van Neerven, Mark Christiaan Veraar, Tuomas Hytönen, and Lutz Weis, Analysis in Banach Spaces: Volume I: Martingales and Littlewood-Paley Theory, Springer, 2016.
- Jan van Neerven, Mark Christiaan Veraar, Tuomas Hytönen, and Lutz Weis, Analysis in Banach Spaces: Volume II: Probabilistic Methods and Operator Theory, Springer, 2017.
Schedule
| Date of Class |
Material covered |
| 01/12 |
lecture 1: the one about Khintchine-Kahane inequalities, probabilistic proof of Khintchine's inequalities, Borell's proof of Kahane's inequalities
|
| 01/14 |
lecture 2: the one about the 2-point hypercontractive inequality and isometric embeddings using Gaussian random variables.
|
| 01/19 |
MLK Day, class does not meet.
|
| 01/21 |
lecture 3: the one about p-stable random variables and isometric embeddings of \(L_p\) into \(L_q\) with \(1 \leqslant q \leqslant p \leqslant 2\) (finite representability and ultraproduct approach).
|
| 01/26 |
class cancelled due to inclement weather.
|
| 01/28 |
lecture 4: the one about Rosenthal's inequalities, symmetrically exchangeable random variables, and the Johnson-Maurey-Schechtman-Tzafriri inequalities (lower bound)
|
| 02/02 |
lecture 5: the one about the Johnson-Maurey-Schechtman-Tzafriri inequalities (upper bound) and the linear \(X_p\) inequalities (applications to embedding theory)
|
| 02/04 |
lecture 6: the one about the linear \(X_p\) inequalities and the optimal distortion to embed \(\ell_p^n\) into \(L_q\) for \(2 < p < q < \infty\).
|
| 02/09 |
lecture 7: the one about Rademacher type and cotype (definitions, basic properties, applications to embedding theory); introduction to Bochner integration and definition of the Bochner-Lebesgue spaces \(L_p(\Omega;X)\). |
| 02/11 |
lecture 8: the one about the type and cotype of \(L_p(\Omega;X)\) and the duality of type and cotype (Fourier analysis on the hypercube, K-convexity) |
| 02/16 |
class does not meet. |
| 02/18 |
lecture 9: This class will run 30' longer, Kwapien's theorem (Gaussian type and cotype, factorization of operators) |
| 02/23 |
lecture 10: the one about sequences of independent and centered random variables being unconditional basic sequences in \(L_p\) (scalar contraction principle), review of scalar-valued martingale theory, statement of Burkholder's inequality and application to the unconditionality of martingale differences |
| 02/25 |
lecture 11: the one about the proof of Burkholder's inequality |
| 03/02 |
lecture 12: the one about (scalar-valued) Doob's maximal inequalities, vector-valued integration and vector-valued conditional expectation (bounded extension of positive operator to Bochner-Lebesgue spaces), vector-valued contraction principle
|
| 03/04 |
lecture 13: the one about UMD spaces
|
| 03/16 |
lecture 14:
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| 03/18 |
lecture 15:
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| 03/23 |
lecture 16:
|
| 03/25 |
lecture 17:
|
| 03/30 |
lecture 18:
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| 04/01 |
lecture 19:
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| 04/06 |
lecture 20:
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| 04/08 |
lecture 21: |
| 04/13 |
lecture 22: |
| 04/15 |
lecture 23: |
| 04/20 |
lecture 24:
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| 04/22 |
lecture 25: |
| 04/27 |
lecture 26:
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