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Overview
Overview
Shift-cyclic functions form a rich class of functions in a variety of function spaces, with connections to many deep problems in mathematics such as the invariant subspace problem, dilation completeness problem, and even the Riemann hypothesis. The goal of this seminar is to hone in on the generalized Hardy spaces over the unit polydisk, and discuss the properties of their shift-cyclic functions. We include a wide range of topics, and cover the basics of function theory in several complex variables.
Shift-cyclic functions form a rich class of functions in a variety of function spaces, with connections to many deep problems in mathematics such as the invariant subspace problem, dilation completeness problem, and even the Riemann hypothesis. The goal of this seminar is to hone in on the generalized Hardy spaces over the unit polydisk, and discuss the properties of their shift-cyclic functions. We include a wide range of topics, and cover the basics of function theory in several complex variables.
Schedule
Schedule
The seminar will take place on Sundays every week from 9 - 10:30 AM (Israel Standard Time), starting December 31, 2023 for twelve weeks. The seminar will be held in a hybrid setting, meaning it will be hosted in-person at the Technion but will also be broadcasted via Zoom for those who cannot attend.
The seminar will take place on Sundays every week from 9 - 10:30 AM (Israel Standard Time), starting December 31, 2023 for twelve weeks. The seminar will be held in a hybrid setting, meaning it will be hosted in-person at the Technion but will also be broadcasted via Zoom for those who cannot attend.
For more details about the topics covered in the lectures, please read the attached PDF file.
For more details about the topics covered in the lectures, please read the attached PDF file.
Overview.pdfLecture notes and Speakers
Lecture notes and Speakers
- Motivation - December 31, 2023.
Notes: Dark background, bright background.
- The class of H^p(D) spaces - January 7, 2024.
Notes: Dark background, bright background.
- Canonical factorization theorem - January 14, 2024.
Speaker: Mikhail Mironov, Technion - Israel Institute of Technology.
Notes: Dark background, bright background
- Beurling's theorem - January 21, 2024.
Notes: Dark background, bright background.
- Function theory in D^n - January 28, 2024.
Notes: Dark background, bright background.
- The class of H^p(D^n) spaces - February 4, 2024.
Notes: Dark background, bright background.
- Factorization problem in H^p(D^n) - February 11, 2024.
Notes: Dark background, bright background.
- Shift-cyclicity vs. outerness - February 18, 2024.
Speaker: Mikhail Mironov, Technion - Israel Institute of Technology.
Notes: Dark background, bright background.
- Hardy space on the multidisk D^\infty_2 - February 25, 2024.
Notes: Dark background, bright background.
- The function theory junction - March 3, 2024.
Notes: Dark background, bright background.
- Algebraic properties of shift-cyclic functions - March 10, 2024.
Notes: Dark background, bright background.
- Cyclicity in complete Nevanlinna-Pick spaces - March 17, 2024.
Notes: Based on sections 2.3-2.5 from my thesis.
- Bonus lecture: Cyclicity preserving operators - March 24, 2024.
Notes: Based on my paper on CPO.
References
References
- "Theory of H^p spaces", by P. L. Duren (1970).
- "Function theory in polydiscs", by W. Rudin (1969).
- "Hardy spaces", by N. Nikolski (2019).
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