The Positive Geometry Seminar is an in-person seminar at MPI-MiS Leipzig. We feature talks by locals and visitors, with talks in or related to the emerging field of positive geometry, which straddles mathematics and physics. We focus on encouraging interactions between local researchers.
Visitors to Leipzig are most welcome to drop by. The seminar typically takes place Thursdays, 10h00, at MPI-MiS.
This activity is part of the ERC Synergy Grant UNIVERSE+ www.positive-geometry.com, funded by the European Union (ERC, UNIVERSE PLUS, 101118787). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
Explicit computations of scattering amplitudes have revealed a wealth of interesting structures such as: iterated integrals, their symbols and their Hopf algebra, momentum twistor geometry and cluster algebras, positive geometry and more. In this talk I will review some of these results and I will also describe a program to compute the iterated integral form from a detailed study of singularities (Landau equations) and the monodromies around branching hypersurfaces. This relies on older results of Leray, Thom and Pham.
A real function is said to be completely monotone (CM) in a region if the function and all its signed derivatives are positive at every point in that region. Complete monotonicity imposes infinitely many constraints on the function and its derivatives, and the space of CM functions in a region is convex.
In this talk, we will consider cases where this space is finite-dimensional - for instance, when the function’s Taylor coefficients satisfy a recursion relation or the function obeys a differential equation. Adopting a physics-inspired approach, we will discuss how recursion and positivity can be combined to formulate a convex optimisation problem that numerically constrains the values of the function throughout the CM region.
Our focus will be on scalar Feynman integrals, which are completely monotone in the Euclidean region under generic kinematics. These integrals are transcendental functions of the kinematic parameters and masses. However, when these parameters are rational, they evaluate to periods in the sense of Kontsevich and Zagier - that is, complex numbers given by integrals of algebraic differential forms over domains defined by polynomial inequalities with rational coefficients.
In our framework, the differential equations satisfied by the Feynman integrals provide the recursion structure, while complete monotonicity imposes positivity constraints. We will describe how to formulate the resulting numerical problem both as a linear programming problem (LPP) and as a semi-definite programming problem (SDPB), and present the bounds obtained.
Finally, time permitting, we will also discuss how this approach leads to rational approximations of certain transcendental functions.
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.
In high energy physics, correlators are what we use to connect theory to experiment. This lecture will outline how the 'master formulae' for scattering amplitudes and cosmological correlators are derived. This will include a discussion of the question 'what is a correlator', and an example of how symmetries can be used to constrain correlators.
This talk offers a guided journey through different spacetimes, introducing the geometric and algebraic structures that govern wave propagation in curved backgrounds. The path begins with the familiar Minkowski spacetime and leads to the near-horizon region of rotating black holes. Massless fields, such as electromagnetic or gravitational perturbations, around rotating black holes are described by the linear partial differential equation known as the Teukolsky equation, whose solutions are expressed in terms of Heun functions. For higher order perturbations, the self-interactions arising from non-linear terms require the systematic use of products, derivatives, and integrals of holonomic functions. I will argue that such higher-order analyses are essential for understanding gravitational turbulence in the vicinity of black holes. No prior knowledge of differential geometry or general relativity is assumed.
Many theories in high energy physics can be defined through an action, the principle of least action being what determines the dynamics of that theory. Actions must have symmetries that reflect the symmetries of the systems they are describing. Symmetries are very important in field theory. For example, they constrain the forms of correlators.
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.
In the 1850s, while trying to compute an integral from optics, George G. Stokes encountered a curious analytic phenomenon which many decades later turned out to be the key for a classification of linear complex differential equations. In this talk, we will first dive into the history of this discovery and its connection to the Riemann-Hilbert problem. At the end, we will see an explicit topological description of the Stokes phenomenon and some results about the behaviour under the Fourier transformation.
Copositive matrices and copositive polynomials are objects from optimization. In this talk, we connect these to the geometry of Feynman integrals. The integral is guaranteed to converge if its kinematic parameters lie in the copositive cone. We present methods for characterizing the copositive cone associated with a Feynman integral. In particular, we show how a modified version of Pólya’s classical theorem can be used to make containment in the copositive cone manifest.
Systems of linear PDEs can be encoded as $D$-ideals. When the ideal is holonomic, they can be expressed in terms of first-order matrix differential equations using connection matrices. In some cases, changes of basis (a gauge transform) allow to make the system more manageable, as, for example, with the $\epsilon$-factorized form in physics. To algorithmically compute connection matrices and such gauge transforms, one requires both Gröbner bases and the normal form algorithm in the rational Weyl algebra. We discuss the implementation of those in our new M2-package $\tt ConnectionMatrices$ and showcase its functionalities in examples.
This is joint work with: Paul Görlach, Joris Koefler, Anna-Laura Sattelberger, Mahrud Sayrafi, Hendrik Schroeder, and Francesca Zaffalon.
We study stratifications of regions in the space of symmetric matrices. Their points are Mandelstam matrices for momentum vectors in particle physics. Kinematic strata in these regions are indexed by signs and rank two matroids. Matroid strata of Lorentzian quadratic forms arise when all signs are non-negative. We characterize the posets of strata for massless with and without momentum conservation.
We study integrals on a punctured Riemann surface of genus g>1. The twisted cohomology groups associated to these integrals were studied by Watanabe. Here we study the corresponding twisted homology groups of these integrals, and bilinear pairings among homology and cohomology groups.
When does an ordinary differential equation (ODE) have a solution which is an algebraic function, ie a function which satisfies an (non-trivial) algebraic relation with its argument? This question was asked by Fuchs in 1875, and was investigated by many mathematicians such as Schwarz and Painlevé.
In a joint work with Daniel Litt, we formulate a conjectural answer to this question, even for non-linear ODEs, in terms of arithmetic, and we prove it for a large class of ODEs at special initial conditions. I’ll try to give examples throughout to illustrate what the question has to do with arithmetic or algebraic geometry.
The amplituhedron is a semi-algebraic set given as the image of the positive Grassmannian under a linear map subject to a choice of additional parameters. We define the limit amplituhedron as the limit of amplituhedra by sending one of the parameters to infinity. We derive its algebraic boundary in terms of Chow varieties and explain its geometric interpretation from the point of view of Positive Geometry in terms of secants of the rational normal curve. This discussion shows that the limit amplituhedron is a positive geometry with a unique differential form.
Joint work with Rainer Sinn.
In this talk I'll present the first steps towards a microlocal analysis of Feynman integrals. This approach aims to describe both the integrand and integral as distributions, this is dual to the algebraic D-module approach which studies the PDEs the integral satisfies rather than the integral itself. The goal of this talk is to introduce some of the tools needed to embark on this journey, culminating in the definition and explicit construction of the microlocal set of singularities (i.e. wave front set) for some of the ingredients of the Feynman integral.
String theory scattering amplitudes on a genus-one surface involve non-holomorphic modular forms in their low-energy expansion. These can be represented by iterated integrals over holomorphic Eisenstein series and have a close connection to multiple zeta values and other periods as studied in particular by Brown. I will explain the basic ideas and the role played by derivation algebras in this connection. Mainly based on 2403.14816 and 2406.05099.
We study scattering equations of hyperplane arrangements from the perspective of combinatorial commutative algebra and numerical algebraic geometry. We formulate the problem as linear equations on a reciprocal linear space and develop a degeneration-based homotopy algorithm for solving them. We investigate the Hilbert regularity of the corresponding homogeneous ideal and apply our methods to CHY scattering equations. This is a joint work with Viktoriia Borovik and Simon Telen.
In this talk, I will go over (in a way that is friendly for mathematicians) the calculation of the tree-level Yang-Mills NMHV scattering amplitude for six gluons (that is, of the simplest amplitude for which the amplituhderon makes sense). In particular, we will see that Feynman diagrams and computing the canonical form of the n =6, k=1, m=4 Amplituhedron give the same result for the amplitude. This is based on joint work with Shounak De, Marcus Spradlin and Anastasia Volovich.
There will be also a live stream: https://mpimis.zoom-x.de/j/68661777332?pwd=hV1uZZcscaVzFvpkt6kw9NDXh2RvU8.1
Cosmological correlators encode the statistical properties of the very early universe. They can be written as multi-dimensional Mellin transforms of specific rational functions. Recent progress has shown how systems of differential equations, that are used to efficiently solve these integrals, can be derived diagrammatically -- the so-called `kinematic flow'. The simplicity of these rules suggests an underlying mathematical structure. In this talk, I will present a few algebraic approaches to begin the exploration of this structure. I will discuss systematic ways of deriving shift-relations and how we can think of the cosmological integrals as restrictions of GKZ systems. Furthermore, I will explain a diagrammatic derivation of the partial fraction decomposition for the integrands and show why it can be useful when studying the integrals. Finally, I will highlight a few open questions about the cosmological integrals. This talk is based on the joint work arXiv:2410.14757 with Claudia Fevola, Anna-Laura Sattelberger and Guilherme L. Pimentel.
In this talk, I will present a method for constructing defining inequalities for the set of linear subspaces of fixed dimension that intersect a given polytope. This set can be described as a union of cells in the complement of a Schubert arrangement associated with the polytope, within the Grassmannian. In particular, I will provide a detailed description of the subspaces that intersect a cyclic polytope and explore its connections to the amplituhedron and positive geometries. Based on joint work with Sebastian Seemann.
The period matrix of a smooth complex projective variety encodes the isomorphism between its singular homology and its algebraic De Rham cohomology. Numerical approximations with sufficient precision of the entries of the period matrix allow to recover some algebraic invariants of the variety, such as the Néron-Severi group in the case of surfaces. Such approximations can be obtained from an effective description of the homology of the variety, which itself can be obtained from the monodromy representation associated to a generic fibration. We will describe these methods, and showcase implementations for the case of hypersurfaces and elliptic surfaces.
A recurring task in particle physics and statistics is to compute the complex critical points of a product of powers of affine-linear functions. The logarithmic discriminant characterizes exponents for which such a function has a degenerate critical point in the corresponding hyperplane arrangement complement. We study properties of this discriminant, exploiting its connection with the Hurwitz form of a reciprocal linear space.
This is joint work with Andreas Kretschmer (HU Berlin) and Simon Telen (MPI MiS).
In a recent paper with Kathlén Kohn, Ragni Piene, Kristian Ranestad, Felix Rydell, Boris Shapiro, Miruna-Stefana Sorea, and Simon Telen, we conjectured, based on numerical computation, that a general quartic plane curve is the adjoint of 864 heptagons in the plane. The adjoint curve of a polygon is the numerator of its canonical function appearing in the context of positive geometry. The goal of the talk is to explain the context of this result and give a proof of this number via intersection theory. This is joint work with Daniele Agostini, Daniel Plaumann, and Jannik Wesner.
N = 4 super Yang-Mills is the most symmetric of all four-dimensional quantum field theories. The simplest local operators in this theory are the so-called half-BPS operators. Correlators of these operators play a key role in the AdS/CFT correspondence and impact many key areas in current theoretical physics. In a certain limit, they relate to amplitudes through the so-called correlator/amplitude duality. On the other hand amplitudes in N=4 can be computed from the amplituhedron, a geometrical object whose boundaries encode the singularity structure of the amplitude. In this talk, I will review recent progress in building a geometrical analogue of the amplituhedron for correlators, named the correlahedron, by studying the correlator/amplitude duality. The talk is based on arXiv:2106.09372.
This talk will consist of two parts. In the first part I will review how to compute n-point scattering amplitudes in the $\phi^3$ theory from three different approaches. The first approach is using Feynman diagrams, which is the standard way to compute these observables in physics. Then I will explain how the tropical Grassmannian, TropG(2,n), is closely related to the space of the Feynman diagrams for this theory, and I will propose a formula as an integral over TropG(2,n) to compute first $\phi^3$ and then $\phi^p$ scattering amplitudes, for any p>2. The third approach is the CHY formula, which is an integral over the moduli space of n points on $CP^1$. In the second part of the talk, motivated by the isomorphism between the Grassmannians G(2,n) and G(n-2,n), I will explain how to generalize the CHY formula to an integral over the space of n points on higher dimensional projective spaces $CP^{k-1}$, thus providing a natural generalization of the notion of scattering amplitudes. I will also point out connections with higher dimensional tropical Grassmannians and will comment on some of the features of these new objects.
Scattering amplitudes in string theory are computed by integrals over moduli spaces of curves. At genus-zero, we can write the amplitudes of closed strings as a bilinear combination of amplitudes of open strings -- a fact physicists refer to as "the double copy."
I will recount the genus-zero story of the double copy with a view towards a genus-one realization.
