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

22 Sept 2025, 08:30 → 26 Sept 2025, 16:15 Europe/Berlin

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

The interplay between advanced mathematical techniques and theoretical physics has become essential for deepening our understanding of quantum field theory, Feynman integrals, and scattering amplitudes.

Recent mathematical breakthroughs—ranging from twisted cohomology, intersection theory, and D-modules to tropical geometry and Euler integrals—have provided powerful frameworks for analyzing and computing these fundamental objects. At the same time, the intricate analytic structure of Feynman integrals provides concrete examples that both challenge and enrich these mathematical theories.

A primary goal of this meeting is to bring together mathematicians and theoretical physicists from diverse fields—including Differential and Algebraic Geometry, Topology, Number Theory, Feynman Calculus, and Scattering Amplitudes—to explore novel approaches for evaluating Feynman integrals and scattering amplitudes. The conference will also address the broader implications of these developments for collider physics, cosmology, string theory, statistics, and pure mathematics.

MathemAmplitudes 2025 will provide a collaborative environment to leading researchers and early-career scientists to foster interdisciplinary dialogue and identify new common research directions in understanding Euler-type integrals and their role in high-energy physics. This event builds on the successful tradition of the MathemAmplitudes conferences, previously held in Padova (2019 and 2023), and brings experts from mathematics and theoretical physics to explore new synergies between algebraic and geometric tools along with advanced computer algebra and the study of Feynman integrals and scattering amplitudes.

MAMP2025_Group Picture.JPG

Monday 22 September

09:30 → 10:45 Morning Session: Session-1

09:30 Introduction and Welcome

MAMP25_Info.pdf

10:00 Kinematic Stratifications

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 particles, with and without momentum conservation. This is joint work with Veronica Calvo and Hadleigh Frost.

Speaker: Bernd Sturmfels

SturmfelsSlides.pdf

10:45 → 11:15 Coffe-Break

11:15 → 12:30 Morning Session: Session-2

11:15 The beta function in tropical phi^4 theory

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 structure of UV subdivergences of the original theory, and there is strong numerical correlation between tropical and non-tropical theory. The special case of tropical subdivergence-free diagrams amounts to the Hepp bound (1908.09820), which has already found interesting applications for the study of Feynman integrals.

My talk is about the full amplitudes of tropical phi^4 theory at zero momentum transfer, including all Feynman integrals with subdivergences. These amplitudes can be computed efficiently from a combinatorial recurrence, which takes the form of a partial differential equation discovered by Michael Borinsky (2508.14263). Renormalization involves the usual combinatorics of subdiagram subtractions. By now, we know the exact renormalized perturbation series of tropical phi^4 theory to more than 200 loops, this allows to explicitly determine its asymptotic and non-perturbative structure, such as growth rates of the beta function or locations of singularities in the Borel plane. In particular, the data empirically answers the old questions of whether renormalons obstruct Borel-resummability of the beta function.

Speaker: Paul-Hermann Balduf

12:00 Gravitational Waveforms as Twisted Period Integrals: Analytic 2PM Results via Fourier–Loop IBPs

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 constructing the integrand via generalised unitarity and performing integration-by-parts with an exponential twist (Fourier–loop IBPs), combined Fourier–loop integrals can be decomposed into a compact basis of master integrals. This method yields the first fully analytic, velocity-exact two-body waveform at 2PM (one loop), together with its power spectrum. The framework extends naturally to spin and higher-PM orders.

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

Brunello_MathemAmplitudes.pdf

12:30 → 14:30 Lunch

14:30 → 15:45 Afternoon Session: Session-1

14:30 Differential equations for moving hyperplane arrangements

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.

Speaker: Anaelle Pfister

mamp2025_Pfister.pdf

15:00 Parametric cycles for Feynman integrals

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.

Speaker: Ruth Britto

Britto-MathemA-3.pdf

15:45 → 16:15 Coffe-Break

16:15 → 17:00 Afternoon Session: Session-2

16:15 Full Classification of Feynman Integral Geometries at Two Loops

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 order to find their corresponding integral geometry. We approach this both with maximal cuts performed in the Baikov representation, as well as using Picard-Fuchs operators.
We uncover a plethora of geometries, such as elliptic curves, hyperelliptic curves, and K3 manifolds, of which some are known, but many have not been investigated until now.

Speaker: Hjalte Frellesvig

mam25basic.pdf

Tuesday 23 September

10:00 → 10:45 Morning Session: Session-1

10:00 Hypergeometric discriminants

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 reveals new facets of the Euler discriminant and would be a key to analyzing its geometry. These include an interpretation in terms of a family of likelihood equations, as well as ongoing work on Fourier transform and projective duality. Time permitting, we will also mention some concrete open questions.

Speaker: Saiei Matsubara-Heo

mamp2025_MatsubaraHeo.pdf

10:45 → 11:15 Coffe-Break

11:15 → 12:30 Morning Session: Session-2

11:15 The calculation of the 4-loop contribution to the electron g-2 and perspectives for the 5-loop calculation

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.

Speaker: Stefano Laporta

laporta_mamp2025.pdf

12:00 Pfaffian Systems and Gröbner Bases

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.

Speaker: Nicolas Weiss

mamp2025_Weiss.pdf

12:30 → 14:30 Lunch

14:30 → 15:45 Afternoon Session: Session-1

14:30 Algebraic Structures in Multi-Scale Scattering Amplitudes

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 deep links to algebraic theories such as twisted cohomology and D-modules. I will survey several examples illustrating how these perspectives enrich our understanding of Feynman integrals and open new avenues for the analytic and numerical evaluation of scattering amplitudes in perturbative quantum field theory.

Speaker: Seva Chestnov

mamp2025_Chestnov.pdf

15:00 Homology for exponential integrals with multivalued potential

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 C. Wheeler (arXiv:2410.20973), and some ongoing work with J. E. Andersen and M. Kontsevich.

Speaker: Veronica Fantini

mamp2025_Fantini.pdf

15:45 → 16:15 Coffe-Break

16:15 → 17:00 Afternoon Session: Session-2

16:15 SPQR: A New Package for Polynomial Division and Elimination Theory

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.

Speaker: Giulio Crisanti

mamp2025_Crisanti.pdf

Wednesday 24 September

10:00 → 10:45 Morning Session: Session-1

10:00 Euler stratifications of hypersurface families

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 the dependence of the number of master integrals, respectively the maximum likelihood degree, on kinematic or model parameters. Joint work with Maximilian Wiesmann.

Speaker: Simon Telen

10:45 → 11:15 Coffe-Break

11:15 → 12:30 Morning Session: Session-2

11:15 Annihilators and D-modules for Feynman integrals

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 operators. For the studied examples, we observe that the holonomic rank of the D-modules coincides with the dimension of the corresponding de Rham co-homology groups. We also discuss some general structures of a D-ideal for the banana family.

Speaker: Wojciech Fleiger

mamp2025_Fleiger.pdf

12:00 CALICO: parametric annihilators for loop integrals & special functions

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 derive relations between a wide class of special functions. These include hypergeometric functions, Feynman integrals relevant to high-energy physics and duals of Feynman integrals. We finally present the public Mathematica package CALICO for computing parametric annihilators and its usage in several examples.

Speaker: Gaia Fontana

mamp2025_Fontana.pdf

12:30 → 14:30 Lunch

14:30 → 15:45 Free Session

15:45 → 16:15 Coffe-Break

Thursday 25 September

10:00 → 10:45 Morning Session: Session-1

10:00 Integration in D-modules

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 and Frédéric Chyzak to perform this operation faster [https://arxiv.org/abs/2504.12724].

Speaker: Pierre Lairez

mamp2025_lairez.pdf

10:45 → 11:15 Coffe-Break

11:15 → 12:30 Morning Session: Session-2

11:15 Periods from mirror symmetry

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.

Speaker: Sergio Cacciatori

mamp2025_Cacciatori.pdf

12:00 Thimble decomposition and Wall Crossing Structure for Physical Integrals

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. In this talk, I will first introduce the key tools underlying this structure and then apply them to show how families of physically relevant integrals, ranging from holomorphic exponentials to logarithmic multivalued functions, can be reformulated within this language. For holomorphic exponents, I will present an explicit decomposition of a family of integrals into thimble expansion together with a detailed analysis of the wall-crossing structure behind the analytic continuation in its relevant parameter. Finally, I will discuss the generalization to multivalued functions, which provides the appropriate framework for describing Feynman integrals in special representations. In this context, the thimble decomposition is expected to match the decomposition into Master Integrals, while the study of the wall-crossing structure yields a precise count of independent Master Integrals (or periods), circumventing complications arising from Stokes phenomena.

Speaker: Roberta Angius

mamp2025_Angius.pdf

12:30 → 14:30 Lunch

14:30 → 15:45 Afternoon Session: Session-1

14:30 Computational Algebraic Geometry for Feynman Integral Calculation

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 about syzygy IBP and lift differential equation with respect to atypical Feynman integrals, such as energy correlator integrals.

Speaker: Rourou Ma

mamp2025_Rourou.pdf

15:00 Computational strategies in modern multi-scale two-loop calculations

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 methods in this context, including finite fields, rational reconstruction, syzygies and differential equations.

Speaker: Tiziano Peraro

mamp2025_Peraro.pdf

15:45 → 16:15 Coffe-Break

16:15 → 17:00 Afternoon Session: Session-2

16:15 Canonical form of differential systems of displaced hyperplane arrangements from positive geometry

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, the connection admits a block structure that is tightly related to geometrical data of the arrangements, which we will illustrate on several concrete examples.

Speaker: Eric Pichon-Pharabod

mamp2025_Pichon-Pharabod.pdf

Friday 26 September

09:30 → 10:45 Morning Session: Session-1

09:30 Genealogical Constraints on Feynman Integrals

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 the hierarchical constraints without solving Landau equations explicitly, relying instead on minimal information about the integral’s kinematic singularities.

Speaker: Maria Polackova

MAMP2025_Polackova.pdf

10:00 Combinatorial Feynman rules

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 be products of the tropical limit of the integral (Hepp bound) and the diagonal coefficients of the integrand. This perspective opens up the possibility to prove identities of integrals purely by combinatorics.
I will illustrate these ideas concretely with periods in phi^4 theory, where by another miracle the combinatorics of spanning trees is related to circuit partitions, which allows for efficient recursive computations. In this context, I illustrate a conjecture that Euler-Mellin integrals (at integer Mellin arguments) are Apery-like limits of recurrence relations (similar to the construction in Apery’s proof of the irrationality of zeta(3)).

This talk covers past work with Karen Yeats (arXiv:2304.05299) and ongoing work with Francis Brown.

Speaker: Erik Panzer

MAMP2025_Panzer.pdf

10:45 → 11:15 Coffe-Break

11:15 → 12:30 Morning Session: Session-2

11:15 From Critical Points to Syzygies for Feynman Integrals

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 integration by parts relations remains less well understood.

In this talk, building on insights from intersection theory, we work in the syzygy formalism for relations between Feynman integrals and analyze its large-$\epsilon$ limit. Remarkably, we find that the critical locus of the twist naturally emerges in this setting. Moreover, the large-$\epsilon$ limit singles out a subset of syzygies, which we call "critical syzygies". We use techniques from commutative algebra to show that, when the critical locus is isolated and a multiplicity criterion is satisfied, critical syzygies generate all relations among Feynman integrals. This analysis provides a new perspective on the Lee-Pomeransky integral counting.

Finally, we discuss concrete applications at both one and two loops to demonstrate the relevance of critical syzygies to Feynman integral reduction in cutting-edge LHC scattering processes.

Speaker: Ben Page

mamp2025_Page.pdf

12:00 Symmetry Invariant Twisted Cohomology

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 choice that transforms under the irreducible representations of the symmetry group the intersection pairing is invariant. This leads to a block structure of the intersection pairing which makes it possible to compute only a subset of intersection numbers. An analogous structure occurs in the period and homology pairing. We demonstrate this property at the different mass configurations of the three-loop banana integral.

Speaker: Cathrin Semper

mamp2025_Semper.pdf

12:30 → 14:30 Lunch

14:30 → 15:45 Free Session

15:45 → 16:15 Coffe-Break