The seminar series will feature presentations and discussions on various topics relevant to the ERC UNIVERSE+ Synergy Project. Each session will offer valuable insights and opportunities for interaction, allowing us to keep the momentum going and delve deeper into our research areas. The Seminars will be held via Zoom bi-weekly, Wednesdays at 4:00 PM CET.
The organizers are Viktoriia Borovik (Max Planck Institute for Mathematics in the Sciences, Leipzig), Hadleigh Frost (Institute for Advanced Study), Andrzej Pokraka (University of Amsterdam), and Cristian Vergu (Max Planck Institute for Physics, Garching).
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.
I will discuss different approaches to and results about the question in the title. The main focus is on the case m=2. My goal is to discuss the strengths and technical challenges in analogy to the case of polytopes, where different approaches, based on triangulations, adjoints, or mixed Hodge theory, all work out nicely. For amplituhedra, each approach has a generalisation, yet none have lead to a proof yet.
Tropical Grassmannians Trop Gr(k,n) were introduced by Speyer and Sturmfels, and in the case of 2-planes, the tropical Grassmannian can be identified with the space of phylogenetic trees. Besides the applications to mathematical biology, in recent years the tropical Grassmannian has seen new connections to scattering amplitudes in physics.
Despite its importance, the construction of a combinatorial model for higher tropical Grassmannians has remained an open problem for many years. In this work, we propose such a combinatorial model, and discuss connections with tropical linear spaces, buildings, webs, noncrossing tableaux, and matroid decompositions.
This is based on joint work with Nick Early.
In this talk we will introduce border bases - a generalization of Gröbner bases - for a noncommutative ring of differential operators. In the context of Feynman integrals, these border bases will consist of differential operators that annihilate a given Feynman integral. We can obtain such border bases from the 1st order linear PDEs Feynman integrals satisfy (i.e. their differential equations). This method returns an ideal of annihilating differential operators that is "big enough," i.e. of finite holonomic rank.
Loop amplitudes are central to precision collider phenomenology, yet their computation remains a major challenge due to the complexity of intermediate steps. A powerful modern strategy is to bypass this complexity through numerical evaluations over finite fields, followed by the reconstruction of the amplitudes’ analytic form from a set of numerical samples using number-theoretic and algebro-geometric techniques.In this talk, I will present the recent re-computation of the two-loop leading-colour amplitudes for the production of a heavy electroweak vector boson, $V=\{W^\pm,Z,\gamma^*\}$, in association with two light jets at hadron colliders ($pp\to Vjj$) ([arXiv:2503.10595](https://arxiv.org/abs/2503.10595)), including leptonic decays of the electroweak boson ($V\to \ell\bar{\ell}$). Compared to the previous state-of-the-art result ([arXiv:2110.07541](https://arxiv.org/abs/2110.07541)), the new approach achieves a three-order-of-magnitude reduction in size (1.4 GB to 1.9 MB), while also lowering the number of required numerical samples from one million to fifty thousand. To illustrate the applicability of these techniques in settings with more massive external legs, I will also briefly discuss the one-loop processes $pp \to HHH$ and $pp \to t\bar t H$.I will outline the core ingredients of this computation: a basis-change algorithm in the vector space of rational functions based on correlations among multivariate residues; numerical sampling in number fields with non-Archimedean metrics (finite fields and $p$-adic numbers); the use of redundant spinor-helicity variables organised via polynomial quotient rings; and the role of primary decompositions in identifying allowed multivariate partial fraction decompositions. These methods not only yield efficient and stable results ready for phenomenological applications, but also expose structural features that we expect to generalise to more complex multi-loop amplitudes.
In this talk I will discuss a geometric organisation of the differential equations for the correlation functions of conformally coupled scalars in power-law cosmologies. To do this, I will introduce a basis of functions inspired by a decomposition of the correlators into time-ordered components and explore how this is related to the arrangement of hyperplanes in energy space. Expressing these functions as graph tubings allows us to deduce the differential equations from just two straightforward rules—activation and merger. The merger rule can alternatively be understood as the collapse of time ordered propagators, helping to demystifying the origin of the kinematic flow.
The adjoint polynomial of a polytope is the numerator of its canonical form. A mathematically (and, it seems, physically) interesting question is whether adjoints have determinantal representations, that is, can be written as determinants of matrices of linear forms. In this talk I will report on this question. In particular, I will discuss when this is true (and when this is false), present constructive methods to obtain determinantal representations of adjoints and some counterexamples. Our attention will be mostly focused on two- and three-dimensional polytopes as well as on ABHY associahedra. This is joint work in progress with Clemens Brüser and Mario Kummer.
Novel representation of scattering amplitudes reveals a rich relations in particle/string theories. In this talk, I will first introduce a universal expansion for one-loop gluon amplitudes, which holds for general gauge theories with gluons/fermions/scalars in the loops. These amplitudes are expressed as linear combinations of scalar-loop integrands weighted by gauge-invariant building blocks. Then, I will show how tree-level open superstring amplitudes can be constructed from sum of stringy Tr$(\phi^3)$ amplitudes with specific kinematic shifts. Upon taking $n$ scaffolding residues, this leads to a new formula of the $n$-gluon superstring amplitude, which is manifestly symmetric in $n-1$ legs, as a gauge-invariant combination of mixed bosonic string amplitudes with gluons and scalars.
Multiple polylogarithms and their elliptic counterparts provide powerful tools for the algorithmic evaluation of large classes of integrals on the sphere and the torus which greatly expanded our computational reach for Feynman integrals and string amplitudes. This talk is dedicated to generalizations of multiple polylogarithms to higher-genus Riemann surfaces and the associated function spaces that close under integration on the surface. I will describe two closely related constructions of higher-genus polylogarithms, one via meromorphic integration kernels, the other via modular integration kernels, and highlight their parallels with the Kronecker-Eisenstein kernels of the elliptic case. Recent work has produced families of algebraic and differential relations among these higher-genus kernels which take the same form in their meromorphic and modular formulation. I will comment on possible applications of these results.
Motivated by the recent discovery of hidden zeros in particle and string amplitudes, we characterize zeros of individual graphs contributing to the cosmological wavefunction. We construct polytopal realizations of the relevant graph associahedra and show that the cosmological zeros have natural geometric and physical interpretations as well as demonstrate that the wavefunction splits near these zeros. We also establish an equivalence between a family of flat-space wavefunction coefficients and $Tr[\phi^3]$ amplitudes, consequently rediscovering their hidden zeros.
High-energy physics can be encoded into low-energy effective field theories (EFTs) as an infinite set of Wilson coefficients. Requiring that the high-energy theory respects the fundamental physical principles of unitarity, locality, and causality bounds the values of these coefficients for 2->2 scattering amplitudes to live inside of a positive geometry in the space of Wilson coefficients known as the “EFT-hedron.” As constraints on an increasing number of Wilson coefficients are considered, the EFT-hedron has several surprising features that I will describe in my talk. I will additionally explain how requiring that Wilson coefficients live inside the EFT-hedron can place new constraints on what physics is allowed at high energies.
Multi-loop Feynman integrals for collider physics are known to contain intricate geometries and to evaluate to complicated transcendental numbers and functions. After an introduction to how those arise in Quantum Field Theory, I will investigate Feynman integrals contributing to the emission of gravitational waves in classical gravity. I will provide a full classification of the contributing Feynman integrals up to fifth order in the post-Minkowskian expansion, identifying new geometries that lead to new transcendental functions. Moreover, I will demonstrate how the corresponding differential equations can be solved by bringing them into canonical form.
Associating a “canonical form” associated to a semi-algebraic region in the complex points of an algebraic variety, whose poles lie along its boundary, has recently risen in popularity. In this talk I will introduce the notion of "canonical function" associated to two such regions. In brief, it is a rational function which has *poles* along the boundary of the first region, and *zeros* along the boundary of the second region. Associated to this data is a canonical Picard-Fuchs recurrence relation, periods and mysterious symmetry groups. Examples constructed in this way provide rare mathematical gems: for instance Apéry’s famous approximations to zeta(2), zeta(3), surprising connections with modular forms, and much more besides.
The purpose of this talk is to review recent progress in Landau analysis, which aims to predict the singularity structure of Feynman integrals before explicitly evaluating them. In the first part, I will discuss the advantages and disadvantages of formulating this problem in both Schwinger parameter space and momentum (Baikov) space. In the second part, I will explain how the latter approach extends the powerful unitary-based method of arXiv:2406.05241 beyond two-particle cut-reducible graphs. To demonstrate its efficiency, I will present new results for multi-loop and multi-scale Feynman integrals, derived using an automatized Mathematica implementation in preparation.
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.
Fabian Schmidt's research area is cosmology, with a main focus on theory, numerics, the analysis of quasilinear and nonlinear large-scale structures, and how they can inform us about gravity, dark energy, and the physics of inflation.
The UNIVERSE+ Online Seminar Series is designed to foster dialogue and collaboration among project partners and those interested in positive geometry.
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.
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.
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.
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.
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.
