Winter 2020 - UniversitÃ¤t Bonn

**MOTIVATION**

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.

**PROMISE**

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).

**WHERE TO SEE**

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

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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