MathemAmplitudes: Co-homology and Combinatorics of GKZ systems, Euler-Mellin-Feynman Integrals and Scattering Amplitudes

Session

Afternoon Session

22 Sept 2025, 14:30

02.430 (Mainz Institute for Theoretical Physics, Johannes Gutenberg University)

5. Differential equations for moving hyperplane arrangements

Anaelle Pfister

22/09/2025, 14:30

Inspired by the cosmological correlators, we construct explicitly an annihilating D-ideal for Mellin integrals of products of hyperplanes, each raised to an individual power. This ideal has an easy combinatorial description and is conjectured to be the D-ideal derived from the restricted GKZ system.

6. Parametric cycles for Feynman integrals

Ruth Britto

22/09/2025, 15:00

I will discuss integration contours for integrals over Feynman parameters that reveal discontinuities, with dimensional regularization. I explore their variation in kinematic phase space and explain how they can identify sequential discontinuities.

7. Full Classification of Feynman Integral Geometries at Two Loops

Hjalte Frellesvig

22/09/2025, 16:15

In this talk I will discuss ongoing work on the complete classification of the Feynman Integral geometries that can appear in two-loop corrections to scattering amplitudes in any quantum field theory, including the standard model. We systematically categorize all graphs that may contribute in four dimensions, finding 79 in total. We then investigate them for generic mass configurations, in...

11. Algebraic Structures in Multi-Scale Scattering Amplitudes

Seva Chestnov

23/09/2025, 14:30

The demand for high-precision predictions for scattering processes at modern colliders drives the study of multi-particle amplitudes and their basic constituents, Feynman integrals. The increasing complexity of these integrals in multi-scale processes naturally calls for methods from computational algebraic geometry. At the same time, their interpretation as twisted period integrals reveals...

12. Homology for exponential integrals with multivalued potential

Veronica Fantini

23/09/2025, 15:00

Motivated by resuming perturbative invariants of hyperbolic knots in 3-manifolds, we define an algorithm to construct exponential integrals whose asymptotic expansion recovers the original divergent series. In a nutshell, this amounts to defining a rapid decay homology for the dilogarithm. In this talk, I will explain the main ideas of our construction, which are based on a joint project with...

13. SPQR: A New Package for Polynomial Division and Elimination Theory

Giulio Crisanti

23/09/2025, 16:15

Using recently developed techniques from high energy physics, we describe a new program to perform polynomial division over finite fields, without the need to reconstruct intermediate Groebner bases. We showcase how these tools can be applied to a broad class of problems, ranging from constraint equation solving to Landau Analysis.

20. Computational Algebraic Geometry for Feynman Integral Calculation

Rourou Ma

25/09/2025, 14:30

There are many powerful tools in computational algebraic geometry that can help us simplify the calculation of Feynman integrals, such as syzygy equation for integral by parts(IBP), lift equation for differential equations. I will introduce an IBP package "NeatIBP" which is based on syzygy method and Gaussian elimination in a local ring for singularity free IBP. I will also give an example...

21. Computational strategies in modern multi-scale two-loop calculations

Tiziano Peraro

25/09/2025, 15:00

I will report on recent progress in computational methods for modern multi-loop calculations.  I will focus on computational strategies used in recent results for multi-scale loop integrals, such as two-loop six-point integrals and two-loop integrals contributing to top pair production plus a W boson at hadron colliders.  I will describe efficient implementations and applications of modern...

22. Canonical form of differential systems of displaced hyperplane arrangements from positive geometry

Eric Pichon-Pharabod

25/09/2025, 16:15

We consider affine arrangements of hyperplanes with a displacement parameter. Such systems notably appear in the study of scattering amplitudes in cosmology. The twisted cohomology of such a system inherits a differential connection. We show that in a certain choice of basis of the cohomology that comes from positive geometry, this connection is automatically in canonical form. Furthermore,...