ICTP SPECIAL LECTURE by DON B. ZAGIER - Thursday 18 May, 10:00 - 12:00 - Venue: ICTP Galileo Guest House - Fibonacci Lecture Room (hybrid seminar)

[ICTP Math Section]

*SPECIAL LECTURE ON *

*"Periods of modular forms, continued fractions, and generalizations"*


Thursday, 18 May, from 10:00 to 12:00 (Rome time)

_PRESENCE:_ ICTP Galileo Guest House - Fibonacci Lecture Room

_ONLINE: Please register in advance for this meeting:_
https://zoom.us/meeting/register/tJEpcOurrTsiHtAdPvhuDn-X8UskTVI7v4Gn
After registering, you will receive a confirmation email containing 
information about joining the meeting.

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Don B. Zagier (ICTP, Trieste/MPIM, Bonn)


*Abstract:* Modular forms are among the most beautiful and most fertile 
objects in all of number theory, with innumerable surprising 
properties.  One of these, going back to the work of Eichler and Shimura 
in the 1960s and of my  recently deceased colleague Manin in the 1970s, 
is that one can associate to every modular form a certain polynomial, 
its so-called period polynomial, without losing any information, thus in 
some sense reducing an a priori completely transcendental theory to the 
study of finite and algebraic objects.

The lecture will be divided into two parts, with a break in between to 
allow people to escape. In the first part I will explain briefly the 
definition and basic properties of modular forms and their period 
polynomials and will describe one or two particularly nice applications. 
The second part will address the question of reconstructing a cusp form 
from its period polynomial.  I will indicate a complete solution of this 
problem for the case of the full modular group relying on an elementary 
but not obvious lemma about the continued fraction expansion of real 
numbers, and will also explain how the attempt to generalize this 
statement to other modular groups led in a natural way to a conjecture 
about the dynamics of general Fuchsian groups that has been checked in 
many cases but is still open after more than 25 years.

/This will be a hybrid seminar. All are very welcome to join either 
online or in person/

/https://indico.ictp.it/event/10368/

-- 
Koutou Mabilo
ICTP Mathematics Group
Leonardo Da Vinci Building
Strada Costiera no. 11
34151 Trieste, Italy
Tel. no.: +39-040-2240455

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