Alternatively have a look at the program.

## Registration

## Modular q-difference modules

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

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

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

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

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

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

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 - ...

## Super-positivity for L-functions associated to modular forms

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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