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Abstracts for Conference "Modular forms are everywhere"

Alternatively have a look at the program.

Registration

Posted in
Datum: 
Mon, 2017-05-22 09:00 - 09:30

Modular q-difference modules

Posted in
Speaker: 
Maxim Kontsevich
Zugehörigkeit: 
Institut des Hautes Scientifiques
Datum: 
Mon, 2017-05-22 09:30 - 10:20
Location: 
MPIM Lecture Hall

I will propose a definition of modular families of q-difference
modules via a q-version of Riemann-Hilbert correspondence.

Holomorphic modular forms and cocycles

Posted in
Speaker: 
Roelof Bruggeman
Zugehörigkeit: 
Universiteit Utrecht
Datum: 
Mon, 2017-05-22 11:00 - 11:50
Location: 
MPIM Lecture Hall

I'll speak about joint work with YoungJu Choie and Nikos Diamantis on
the cocycles that one can attach to holomorphic modular forms. Knopp
has shown that there is a generalization of the classical
Eichler-Shimura theory tocusp forms of real weight. We consider a map
to cohomology from the space of holomorphic functions with modular
transformation behavior (without any growth condition at the cusps).
For weights that are not integers at least two the results differ
considerably from the classical Eichler-Shimura theory, and are

A meromorphic extension of the 3D index

Posted in
Speaker: 
Stavros Garoufalidis
Zugehörigkeit: 
Georgia Institute of Technology, Atlanta
Datum: 
Mon, 2017-05-22 14:00 - 14:50
Location: 
MPIM Lecture Hall

The 3D-index of Dimofte-Gaiotto-Gukov is a collection of q-series with integer coefficients which is defined for 1-efficient ideal triangulations, and gives topological invariants of hyperbolic manifolds, in particular counts the number of genus 2 incompressible and Heegaard surfaces. We give an extension of the 3Dindex to a meromorphic function defined for all ideal triangulations, and invariant under all Pachner moves. Joint work with Rinat Kashaev.

A class of non-holomorphic modular forms

Posted in
Speaker: 
Francis Brown
Zugehörigkeit: 
IHES/All Souls College, University of Oxford
Datum: 
Mon, 2017-05-22 15:00 - 15:50
Location: 
MPIM Lecture Hall

I will define an elementary theory of non-holomorphic modular forms and
describe some of its basic properties.
Within this family, there exists a class of functions which correspond
to certain mixed motives.
They are constructed out of single-valued iterated integrals of holomorphic
modular forms, and are closely related to a problem in string theory.

Fourier coefficients and singular moduli of modular functions

Posted in
Speaker: 
Masanobu Kaneko
Zugehörigkeit: 
Kyushu University
Datum: 
Die, 2017-05-23 09:00 - 09:50
Location: 
MPIM Lecture Hall

The generating function of traces of singular moduli of the modular j-invariant
becomes a modular form of weight 3/2.  This is Don's celebrated discovery, inspired
by a work of R. Borcherds. Using this modular form, one can obtain a formula for
the Fourier coefficients of the modular j-invariant in terms of singular moduli.
In this talk, I shall review these works, and introduce recent developments
regarding an application of the formula (due to R. Murty and K. Sampath)
as well as generalizations (due to T. Matsusaka).

Motivic supercongruences

Posted in
Speaker: 
Fernando Rodriguez Villegas
Zugehörigkeit: 
ICTP, Triest
Datum: 
Die, 2017-05-23 10:30 - 11:20
Location: 
MPIM Lecture Hall

 Certain congruences between truncated hypergeometric polynomials and unit roots
of their associated motives appear to hold to higher powers of primes than expected.
We will discuss how this phenomenon, generally known as supercongruences, is tied
to Hodge theory and is more widespread than previously thought. This is joint work
with D. Roberts.

Modular forms defining gothic cathedrals

Posted in
Speaker: 
Martin Möller
Zugehörigkeit: 
Universität Frankfurt
Datum: 
Die, 2017-05-23 11:30 - 12:20
Location: 
MPIM Lecture Hall

Flat surfaces with the floorplan of gothic cathedrals define an exceptional series of
Teichmüller curves. We give an overview of the ways to define Teichmüller curves
using Hilbert modular forms of non-parallel weight and the flat surfaces invariants
that can be computed from this viewpoint.

Polylogarithms - regulators - quantum modular polylogarithms - ...

Posted in
Speaker: 
Alexander Goncharov
Zugehörigkeit: 
Yale University
Datum: 
Die, 2017-05-23 14:00 - 14:50
Location: 
MPIM Lecture Hall

Super-positivity for L-functions associated to modular forms

Posted in
Speaker: 
Dorian Goldfeld
Zugehörigkeit: 
Columbia University (New York)
Datum: 
Die, 2017-05-23 15:00 - 15:50
Location: 
MPIM Lecture Hall

Zhiwei Yun and Wei Zhang introduced the notion of "super-positivity of self-dual L-functions" which specifies that all derivatives of the completed L-function (including Gamma factors and power of the conductor) at the central value s=1/2 should be non-negative. They proved that the Riemann hypothesis implies super-positivity for self dual cuspidal automorphic L-functions on GL(n).

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