Felipe Gonçalves


You can schedule your oral exam for ANY day from July 9 - 12, from 10am to 12am and 2pm to 5pm. The exam will take place at my office. Please schedule the exam with 7 days in advance. To schedule your oral exam simply send me email with your preferred time. If no day of this week works for you, then there is second date for examination on 16.09.2019 (officially there is another in August 8, but that is a mistake, forget it). If that is the case send me an email with a reasonable explanation. The exam will be separated in two parts. Part 1 you will be asked about a specific topic agreed upon a priori by random selection during class. Part 2 you will be asked a couple of simple questions about something basic that was taught in class.
The topics are:

1) Finite dimensionality of M_k(Gamma_1) [Janina]
2) Functional Equation For Zeta via Theta functions[Nuno]
3) Zeta(3) is irrational [Andreas]
4) The Bieberbach Conjecture (de Branges Thm without the proving the needed positive sum) [Federica]
5) Positive Jacobi Sums (the positive sum from de Branges proof) [Max]
6) Sharp Hausdorff-Young Inequality (without the approximation argument, the K_N stuff) [Nuria]
7) The approximation argument in the proof of the Sharp Hausdorff-Young Inequality (the K_N stuff) [Leo]
8) Definition and properties of E2,E4,E6,eta,Delta [Parthiv]
9) Spherical Codes and Kissing Numbers (dim 8 and 24 solution) [Lyuchan]
10) All properties of Jacobi Polynomials (not the positivity part in de Branges Thm) [Dinesh]
11) All properties of Hermite Polynomials (probabilist ones) [Linus]
12) Spherical Harmonics (up to Funk-Hecke formula) [Yunseok]
13) Recurrence formula and the Christoffel-Darboux formula for orthogonal polynomials [Bjondina]

You should understand these assigned topics as one thing to focus, since you will be asked way more about it (say 70% of the time). You could even prepare a 20min board presentation of it, focusing on the ideas, but be aware that I will interrupt you several times. Then you will be asked about something else (but a basic question) just for me to make sure you not only know one thing about the class.

V5B7 - Advanced Topics in Analysis - Orthogonal Polynomials and Special Functions
Research Area C4, Universität Bonn
Mondays and Thursdays from 10 (c.t.) to 12 in Raum SR1.007, Endenicher Allee 60.

Course Description

The idea of the course is simple: 1) Study orthogonal polynomials and other special functions; 2) See how they are present in different areas of mathematics, by going through some awesome and unexpected proofs.

You may be wondering what a special functions is. What it is is not important, but what it does and how often it appears is what matters. This idea, however, is hard to define formally. My definition of a special function is very simple: A special function for me is whatever I say is special. You can easily deduce that a function, special for X, may not be special for Y. This does not implies that the definition is rubbish, since if you really want to know what X does you better learn his/her special functions.

Part 1
Initially we will follow closely Chihara - "An introduction to Orthogonal Polynomials", chapters 1 and 2. We then will mix parts of chapter 3, 4 and 5 of Chihara's book with parts of chapters 4, 5 and 8 of Szego - "Orthogonal Polynomials".
We then will explore unexpected applications:
0) Apery's Constant is irrational.
1) (Application of Hermite Polynomials) W. Beckner, Inequalities in Fourier Analysis, Annals of Mathematics Second Series, Vol. 102, No. 1 (1975), pp. 159-182.
2) (Application of Jacobi Polynomials) L. Branges, A proof of the Bieberbach conjecture, Acta Math. Vol 154, Number 1-2 (1985), 137-152.
3) (Application of Laguerre Polynomials) Something combinatorial (skipped) .

Part 2
We then move to Spherical Harmonics following closely Dai and Xu - "Approximation Theory and Harmonic Analysis on Spheres and Balls", chapter 1 and parts of chapter 2.
We then will explore unexpected applications:
Issue 3, (2015), 690-702. (skipped)
4) Delsarte's linear programming bounds for spherical codes ans kissing numbers.

Part 3
We will then move to modular and quasimodular functions following closely Zagier - "The 1-2-3 of Modular Forms" and parts of Diamond and Shurman - "A First Course in Modular Forms".
We then will explore (lightly) unexpected applications:
5) The partition function.
6) Functional equation for zeta.


Disclaimer: The hand notes below have several typos! Use your good sense to fix them when reading.

01/04 - Part1. We talked about Legendre and Charlier polynomials.

04/04 - Part2. We proved that Apery's constant is irrational. Magical day! Healthy snack this day!

08/04 - Canceled.

11/04 - Problems1. (Try to solve problems from 1.1 to 1.5 and the extra to attend the class. The first page on the pdf contains the needed definition of T_n for problem 1.1. João Ramos will supervise you during class, but students should present solutions to each other while João will fill the gaps when needed. Solutions will be posted here later: Solutions1)

15/04 - Part3. We reviewed Tchebycheff and Hermite polynomials present in the past Problems1 list. We talked about existence of OPS, the moment functional and positive definiteness. Problem for next class: Problems2 (hint: Pretend they are Legendre polynomials).

18/04 - Part4. We talked about least squares and the fundamental recurrence formula. Correction: B_n=(k_(n+1)~ - A_nk_n~)/k_n
Problem for next class: Without searching online, show that integrating the function (1-y)^nP_n(x)/(1-xy) in the box [0,1]X[0,1] gives a number of the form [rational +rational*zeta(2)], creating thus a linear approximation of zeta(2) that is "too good to be true" if zeta(2) was rational. P_n is the Legendre polynomial. Use the main lemma stated in the analogous argument for zeta(3). Since zeta(2)=Pi^2/6, this is yet another proof that Pi is irrational.

25/04 - Part4. Reproducing Kernel Spaces, Christoffel-Darboux Formula.

29/04 - Part5. Interpolation formulas, zeros locations and interlacing, Gauss quadrature.
FOR NEXT CLASS solve in Problems3 problems 5.4, 5.5 and 6.3. There are also some problems in the notes Part5 that you optionally may solve.

2/05 - Part6 Gauss quadrature. The Fourier transform and Hermite polynomials.

6/05 - Part6 Hermite polynomials and The sharp Hausdorff-Young inequality I.

7/05 (at SR 1.008 from 8-10am) - Part6 Hermite polynomials and The sharp Hausdorff-Young inequality II.

9/05 - Part6 Hermite polynomials and The sharp Hausdorff-Young inequality III.

13/05 - Canceled.

14/05 (at SR 1.008 from 8-10am) - Part6 Hermite polynomials and The sharp Hausdorff-Young inequality IV. Jacobi Polynomials.

16/05 - Part7 Jacobi polynomials. We talked about basic properties of Jacobi polynomials and ultraspherical polynomials.

20/05 - Part8 Positive Jacobi polynomial sums. We proved Theorem 3 from the paper "Positive Jacobi Sums II" of Askey and Gasper.

23/05 - Part9 The Proof of the Bieberbach Conjecture.

27/05 - Part10 Spherical Harmonics I. We atalked about basic properties of spherical harmonics. First chapter of "Approximation Theory and Harmonic Analysis on Spheres and Balls", by Dai and Xu.

30/05 - Canceled.

03/06 - Part11 Spherical Harmonics II - The Funk-Hecke Formula.

06/06 - Part12 Spherical Harmonics III - Kissing Problem, Spherical Codes and Delsarte's Linear programming bounds for codes.

10/06 - Canceled.

13/06 - Canceled.

17/06 - Part13 Basics of Modular Forms I (Following Zagier's first chapter of "The 1-2-3 of Modular Forms").

24/06 - Part13 Basics of Modular Forms II.

25/06 (at SR 0.011 from 8-10am) - Part14 Basics of Modular Forms III - Eisenstein Series.

27/06 - Part15 Basics of Modular Forms III - Theta Series.

01/07 - Part15 The Four Squares Thm and functional eq. for zeta.