Hilbert Spaces of Entire Functions
Winter 2020 - Universität Bonn

One of the greatest class I ever did was a course based on de Branges theory of analytic function spaces. The theory is beautiful and written entirely in de Branges book "Hilbert Spaces of Entire Functions", which I am very fond of. De Branges became famous for his remarkable solution of the Bieberbach conjecture (which I presented in my Special Functions class). Perhaps he acquired a bad fame for his subsequent failed attempt to solve the Riemann Hypothesis (based on the things we will learn in this class), which I will maybe present if we have time. Do not be mistaken, he is an outstanding mathematician. But if you are curious, he still has an approach available on his website.

His book is well known (in the related mathematical circles) to be very hard, but this is what I most like about it. For me, the fame comes entirely on the problem sets of the book, not on the theorems he proves, which are actually very nice. These problems are designed to complement the theory and therefore crucial to be solved, as he quotes problems in the proof of theorems very often. Really, around 50% of these problems should be lemmas. I struggled a lot with his problems, but in the end the journey made me mentally strong, and now I tend to face new mathematical adventures with more confidence.

I promise you that after doing this course you will definitely become a more skilled and confident mathematician. When I did this class, it was like doing measure theory following Rudin's book and solving every single problem; pretty hard. The level of confidence and pride you get after accomplishing something like this is enormous. I am sure you will feel empowered after this class. My goal in teaching is to make you think in different ways about math and create the mindset of "I can learn and work on anything". The best tool for that is constant struggle (reading hard papers or books).

MATERIAL (send me an email to join the mailing list)
The course will be based almost entirely on de Branges book "Hilbert Spaces of Entire Functions" plus some other source of material for Hardy and Paley-Wiener spaces (and perhaps one or two applications). I should cover chapters 1--24 of de Branges book (each chapter has one theorem). De Branges book in free on his webpage. His preface on the book introduces well the theory. As usual I will make some hand written notes that will be available here. These are mainly to guide me in the lectures, but I will make them available anyway (so watch out for typos).


We should have around 28 classes streamed live via zoom every Tuesday and Friday at 10:15am. These will later be posted on my youtube page.
Meeting ID: 992 3546 0364
Password: debranges

I will hold office hours on zoom every Thursday 2:30-4:00pm for any sort of questions.
Meeting ID: 992 3546 0364
Password: debranges

Every Tuesday we will have a new list of problems (selected from the book) to be solved. For each list you will have 2 weeks to solve.

Set 1: Problems 1,2,3,4,5 (due Nov. 10, midnight)

Set 2: Problems 6,7,8,9 (due Nov. 17, midnight)

Set 3: Problems 11,12,13,16,17,19,24 (due Nov. 24, midnight)

Set 4: Problems 26,27,28,29,30,31 (due Dec. 1, midnight)

Set 5: Problems 33,34,35,36,37,38,39 (due Dec. 15, midnight)

Set 6: [Extra one] and Problems 50,51,52,53 (due Jan. 8, midnight)

Set 7 (final): Problems 54 to 64 (read chapter 24) (due Feb. 2, midnight)

I will grade them using my tablet and send it back to you. To be admitted in the final oral exam you will need an average of at least ?%. (first I have to see how students perform on the first few lists to decide a cutoff). The oral exam should not be hard. Probably a series of theoretical questions with quick answers. The point is: the hard part should be the problems not the oral exam.

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