IGAP Lecture on "Standard and less standard asymptotic methods"/ Don Zagier

[ICTP/math - Mabilo Koutou]

IGAP Lecture on "Standard and less standard asymptotic methods"

Starts 15 Feb 2022 16:00
Ends 23 Mar 2022 15:30
Central European Time

Online:
Join Zoom Meeting
https://unesco-org.zoom.us/j/91274263707

Meeting ID: 912 7426 3707
Passcode: 103118

Presence: ICTP Leonardo Da Vinci Building - Budinich Lecture Hall (only if
provided with a green pass)

Lecturer: Don Zagier (MPIM/ICTP)

Course Type: PhD Course
Academic Year: 2021-2022
Period: 15 February to 23 March 2022
Research Group: Geometry and Mathematical Physics
Duration: 20 h

Description:

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.

http://indico.ictp.it/event/9872/