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

Session

Morning Session

22 Sept 2025, 09:30

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

1. Introduction and Welcome

22/09/2025, 09:30

2. Kinematic Stratifications

Bernd Sturmfels

22/09/2025, 10:00

This lecture discusses the stratification of regions in the space of real symmetric matrices. The points of these regions are Mandelstam matrices for momentum vectors in particle physics. The kinematic strata are indexed by signs and rank two matroids. Matroid strata of Lorentzian polynomials arise when all signs are nonnegative. We describe the posets of strata, for massless and massive...

3. The beta function in tropical phi^4 theory

Paul-Hermann Balduf

22/09/2025, 11:15

Tropical field theory is a limit of quantum field theory where the spacetime dimension and the propagator power simultaneously approach zero. This can equivalently be viewed as a specific limit of Mellin transforms of all Feynman integrals of the theory, where they become simple rational functions of a dimensional regulator. These tropicalized Feynman integrals retain much of the combinatorial...

4. Gravitational Waveforms as Twisted Period Integrals: Analytic 2PM Results via Fourier–Loop IBPs

Giacomo Brunello (Università degli Studi di Padova & IPhT-CEA/ Universitè Paris-Saclay & INFN-PD)

22/09/2025, 12:00

Gravitational waveforms generated by the scattering of two compact bodies can be expressed as Fourier transforms of five-point amplitudes in impact-parameter space (KMOC). In this talk I will combine the Fourier and loop integrations, treating scattering waveforms as twisted period integrals, and allowing scattering-amplitude techniques to be applied directly in frequency space. By...

8. Hypergeometric discriminants

Saiei Matsubara-Heo

23/09/2025, 10:00

Given a family of varieties, the Euler discriminant locus distinguishes points where the Euler characteristic differs from its generic value. In the context of Feynman integrals, this locus corresponds to the Landau variety. We introduce a Lagrangian cycle, which we call a hypergeometric discriminant. The shadow of the hypergeometric discriminant is the Euler discriminant. This approach...

9. The calculation of the 4-loop contribution to the electron g-2 and perspectives for the 5-loop calculation

Stefano Laporta

23/09/2025, 11:15

In this talk we describe the key points that made feasible the high-precision calculation of the 4-loop electron g-2: i.b.p. identities, high-precision solution of difference or differential equations and the analytical fit with PSLQ. We will take a first look to the 5-loop calculation, and discuss how the 4-loop approach could be extended to 5 loop.

10. Pfaffian Systems and Gröbner Bases

Nicolas Weiss

23/09/2025, 12:00

Differential equations for Feynman integrals appear both in the form of annihilating operators (as D-ideals) as well as in matrix form (as Pfaffian systems). In my talk I will relate both of them in the context of Weyl algebras and Gröbner bases. This is based on arXiv:2504.01362 which describes how to obtain Pfaffian systems from D-ideals. An implementation is provided in Macaulay2.

14. Euler stratifications of hypersurface families

Simon Telen

24/09/2025, 10:00

We stratify families of projective and very affine hypersurfaces according to their topological Euler characteristic. Our new algorithms compute all strata using algebro-geometric techniques. For very affine hypersurfaces, we investigate and exploit the relation to critical point computations. Euler stratifications are relevant in particle physics and algebraic statistics. They fully describe...

15. Annihilators and D-modules for Feynman integrals

Wojciech Fleiger

24/09/2025, 11:15

We present a novel algorithm for constructing differential operators with respect to external variables that annihilate Feynman-like integrals and give rise to the associated D-modules, based on Griffiths–Dwork reduction.
By leveraging the Macaulay matrix method, we derive corresponding relations among partial differential operators, including systems of
Pfaffian equations and Picard-Fuchs...

16. CALICO: parametric annihilators for loop integrals & special functions

Gaia Fontana

24/09/2025, 12:00

We elaborate on the method of parametric annihilators for deriving relations among integrals. Annihilators are differential operators that annihilate multi-valued integration kernels appearing in suitable integral representations of special functions and Feynman integrals. We describe a method for computing parametric annihilators based on efficient linear solvers and show how to use them to...

17. Integration in D-modules

Pierre Lairez

25/09/2025, 10:00

The aim of this talk is to explain the basics of the D-module formulation of Feynman integrals, that is how to compute with these integrals through the lens of systems of linear PDEs. I will particularly insist on the integration part: how to obtain relations on the integrals of functions, given a description of the functions with PDEs. I will hint at a few ideas developed with Hadrien Brochet...

18. Periods from mirror symmetry

Sergio Cacciatori

25/09/2025, 11:15

I will illustrate a method for computing period integrals of the mirror of a toric local Calabi-Yau manifold by analytic prolongation of a cohomology-valued hypergeometric function. 
With the aim to be as much as possible didactical, I will work with a very specific example.

19. Thimble decomposition and Wall Crossing Structure for Physical Integrals

Roberta Angius

25/09/2025, 12:00

A growing body of evidence suggests that the complexity of Feynman integrals is most naturally understood through geometry. Recent mathematical developments by Kontsevich and Soibelman [arXiv:2402.07343] have illuminated the role of exponential integrals as periods of twisted de Rham cocycles over Betti cycles, offering a structured approach to address this problem in a wide range of settings....

23. Genealogical Constraints on Feynman Integrals

Maria Polackova

26/09/2025, 09:30

Within the scattering amplitudes bootstrap program, I will present new hierarchical constraints on the symbol of the Feynman integral: genealogical constraints, which hold to all orders in dimensional regularisation. These constraints apply at any loop order, particle multiplicity, and for any configuration involving massive or massless virtual particles. I will also explain how one can find...

24. Combinatorial Feynman rules

Erik Panzer

26/09/2025, 10:00

The Feynman integral is a complicated analytic function in many variables, associated to a merely combinatorial object, namely a graph. In this talk I will focus on the highest order poles of the Feynman integral, and explain how they can be described entirely by combinatorics of the graph. In fact, we conjecture that the integral itself is fully determined by these residues, which turn out to...

25. From Critical Points to Syzygies for Feynman Integrals

Ben Page

26/09/2025, 11:15

In recent years it has become clear that the critical points of the "twist" play a crucial role in the study of relations between dimensionally regulated Feynman integrals. There is a great deal of understanding of these points both in the Lee-Pomeransky counting of master integrals and in the application of intersection theory to Feynman integrals. However, their role in the structure of...

26. Symmetry Invariant Twisted Cohomology

Cathrin Semper

26/09/2025, 12:00

It is well known that the number of master integrals for a family of Feynman integrals can be smaller than the dimension of the corresponding twisted cohomology group due to the presence of symmetries. This can lead to redundant computations of intersection numbers. We propose a basis choice such that only a subset of these intersection numbers needs to be computed. 
We show that in a basis...