# Abstracts for Motivic structures on quantum cohomology and pencils of Calabi-Yau motives. Final reports

Alternatively have a look at the program.

## Talks tba. (Motivic structures on quantum cohomology and pencils of Calabi-Yau motives)

Subjects: Fano manifolds and their mirror dual pencils, maximally mutable Laurent polynomials, the first and second structure connections, gamma conjectures and the gamma class, drops in the images of monodromies in Landau-Ginzburg models, congruence Galois reps and congruence sheaves, Bloch-Kato classes and numerator/denominator of L-values, GSp_4 - Galois representations and paramodular non-lifts, hypergeometric families and weighted projective spaces, hypergeometric motives and their L-functions, extensions of Picard-Fuchs equations and subcritical L-valu

## Talks tba. (Motivic structures on quantum cohomology and pencils of Calabi-Yau motives)

Subjects: Fano manifolds and their mirror dual pencils, maximally mutable Laurent polynomials, the first and second structure connections, gamma conjectures and the gamma class, drops in the images of monodromies in Landau-Ginzburg models, congruence Galois reps and congruence sheaves, Bloch-Kato classes and numerator/denominator of L-values, GSp_4 - Galois representations and paramodular non-lifts, hypergeometric families and weighted projective spaces, hypergeometric motives and their L-functions, extensions of Picard-Fuchs equations and subcritical L-valu

## Reducibility of Galois representations and critical L-values: another mod 691 congruence

I will start with Ramanujan's famous mod 691 congruence, after a

brief discussion of rational torsion on elliptic curves, and the

Birch-Swinnerton-Dyer conjecture. Then I will look at an experimental mod

691 congruence of Bergstroem, Faber and van der Geer, involving the Hecke

eigenvalues of a genus 3 Siegel cusp form. This, like Ramanujan's congruence

and Harder's conjecture, is an example of an ``Eisenstein'' congruence. I

will then explain the coincidence of the moduli, visiting the Bloch-Kato

conjecture and Selmer groups along the way."

## Amplitude in QFT and string theory, and the relation between them.

String theory amplitudes are expressed as integral over a moduli

space of Riemann surface. The QFT is the degeneration limit of this.

I'll discuss how limitied Hodge structures are useful to bridge

between what we know in string theory and QFT.

## Borcherds Products Everywhere Theorem

This is a report on my joint results with Cris Poor and David Yuen about

Borcherds Products on groups that are simultaneously orthogonal and symplectic,

the paramodular groups of degree two and the elementary divisors (1,t).

This work began as an attempt to make Siegel paramodular cusp forms

that are simultaneously Borcherds Products and additive Jacobi lifts

(or Gritsenko lifts). We prove the Borcherds Products Everywhere

Theorem, that constructs holomorphic Borcherds Products from all

Jacobi forms that are theta blocks without theta denominator.

## New approach to Calabi-Yau operators

Pencils of Calabi-Yau threefolds give rise to local systems and differential operators with very strong arithmetical properties. One interesting source of examples come from regularised quantum cohomology D-modules of Fano varieties. Although no classification is known, one may hope to characerise and construct such operators from first principles. I will describe joint work in progress with V. Golyshev and A.