Starting February 15, Don Zagier will give a lecture course entitled "Standard and less standard asymptotic methods". This course will be given in collaboration with the IGAP (Institute of Geometry and Physics, a new joint venture between SISSA and ICTP in Trieste).

The course will be streamed from Trieste twice a week (Tu/Th 4-5:30 for the first four weeks and Mo/We 2-3:30 for the last two weeks).

Abstract:

In every branch of mathematics, one is sometimes confronted with the problem of evaluating an infinite sum numerically and trying to guess its exact value, or of recognizing the precise asymptotic law of  formation of a sequence of numbers ${A_n}$ of which one knows, for  instance, the first couple of hundred values.  The course will tell a number of ways to study both problems, some relatively standard (like the Euler-Maclaurin formula and its variants) and some much  less so, with lots of examples.   Here are three typical examples: 1. The slowly convergent sum  $\sum_{j=0}^\infty (\binom{j+4/3}{j})^{-4/3}$   arose in the work of a colleague.  Evaluate it to 250 decimal digits. 2.  Expand the infinite sum  $\sum_{n=0}^\infty (1-q)(1-q^2)...(1-q^n)$ as  $\sum A_n (1-q)^n$, with first coefficients 1, 1, 2, 5, 15, 53, ... Show numerically that  $A_n$  is asymptotic to  $n! * a * n^b * c$  for some real constants $a$, $b$ and $c$, evaluate all three to high precision, and recognize their exact values. 3.  The infinite series  $H(x) = \sum_{k=1}^\infty sin(x/k)/k$  converges for every complex number $x$.  Compute this series to high accuracy when $x$ is a large real number, so that the series is highly oscillatory.

The courses are scheduled as follows:

1 Tue 15-Feb 16.00 - 17-30
2 Thu 17-Feb 16.00 - 17-30

3 Tue 22-Feb 16.00 - 17-30
4 Thu 24-Feb 16.00 - 17-30

5 Tue 01-Mar 16.00 - 17-30
6 Thu 03-Mar 16.00 - 17-30

7 Tue 08-Mar 16.00 - 17-30
8 Thu 10-Mar 16.00 - 17-30

9 Mon 14-Mar 14.00 - 15.30
10 Wed 16-Mar 14.00 - 15.30

11 Mon 21-Mar 14.00 - 15.30
12 Wed 23-Mar 14.00 - 15.30


These will be hybrid courses. All are very welcome to join either online or in person (if provided with a green pass). Venue: Budinich Lecture Hall (ICTP Leonardo Da Vinci Building), for those wishing to attend in person. Zoom: https://unesco-org.zoom.us/j/91274263707

 

Time: Tuesdays, 4.30 - 6 pm
Place: MPIM Lecture Hall, Vivatsgasse 7
First lecture: on March 3, 2020, end ?

The Hasse Priciple is a fundamental topic in number theory. Anyway a complete proof is very seldom showed in undergraduate and graduate courses in number theory. We will decribe this classical result and prove it. After an overview of some long-established local-global principles, we will focus on the recent local-global question for divisibility in commutative algebraic group. In particular we will explain its relation with the Hasse principle for divisibility of elements of the Tate-Shavarevich group in the Weil-Châtelet group.  This latter problem arose as a generalization of a classical question considered by Cassels. We will describe the results achieved for the two problems during the last fifteen years and some questions that are still open.