This is the kick-off workshop for the ERC-synergy project with the same title which starts in June 2024. Lectures and discussions will explore novel mathematical structures in fundamental physics, ranging from elementary particles to the Big Bang, and revealing hidden principles beyond quantum mechanics and spacetime. We will focus on positive geometries, such as amplituhedron, and their recent applications to scattering amplitudes and cosmology. The program of the week will begin with plenary presentations by international experts, and it will conclude with shorter research talks, informal discussions, and intense working groups. To foster interactions with Leipzig University, Daniel Baumann (University of Amsterdam) will give the Physics Colloquium on Tuesday, and Nima Arkani-Hamed (IAS Princeton) will give the Felix-Klein Colloquium on Wednesday.

Limited travel funding can be provided for early-career participants such as postdoctoral researchers and PhD students. Applicants are expected to submit a brief academic CV and a motivation letter. The application for funding is included in the registration form.

The deadline for applying for funding is already closed.

I will review (and highlight open problems concerning) several aspects of the still rather mysterious appearance of positive geometries in quantum field theory scattering amplitudes, including connections between cluster algebras, symbol alphabets, and the amplituhedron.

We will review open Gromov-Witten theory, a mathematical theory of topological strings. We will argue that many of the main ingredients of positive geometries which appear in physics are also present in this theory. We will concentrate on an example which highlights these similarities - open r-spin theory.
The talk is based on joint works with S. Buryak, E. Clader; M. Gross, T. Kelly; Y. Zhao; and some recent progress.

The renormalization of gauge theories and, eventually, gravity is one of the biggest current challenges in mathematical physics. In my research, I am approaching this topic via the renormalization framework of Connes and Kreimer: In this setup, subdivergences are organized using the coproduct of a Hopf algebra of Feynman graphs and the renormalized Feynman rules are then constructed through an algebraic Birkhoff decomposition in the respective character group. When applied to (generalized) gauge theories, the obstructions for gauge anomalies — the so-called Slavnov—Taylor identities — form a Hopf ideal therein, as was originally shown by van Suijlekom and then refined by myself. In this talk, I first give a short introduction to the Connes—Kreimer renormalization framework and then explain its application to (generalized) gauge theories: Specifically, I will first discuss the case of Quantum Yang—Mills theory and then explain how this setup can be generalized to (effective) Quantum General Relativity. Finally, I will give an outlook how I aim to construct a Feynman graph cohomology that encodes the cancellation identities of the theory. Its compatibility with the renormalization operation then results in a differential-graded renormalization Hopf algebra, which is equivalent to the transversality of the theory. More information thereon can be found in my dissertation and the articles it is based upon, cf.arXiv:2210.17510 [hep-th].

A smooth complex quartic surface in the projective 3-space defines 22 complex numbers called periods. The integer relations between them describe the algebraic curves lying on this surface, in the form of the Néron-Severi group. I will explain how to compute numerically the periods with thousands of digits of precision and recover heuristically the lattice of integer relations. This provides an insight on the Néron-Severi group that purely algebraic methods fail to provide.
This is joint work with Eric Pichon-Pharabod (Inria), Emre Sertöz (Leiden University) and Pierre Vanhove (CEA).

It was recently appreciated how scattering amplitudes in a large class of theories can be described as a "curve integral", i.e. as an integral over the space of curves that can be drawn on a Riemann surface. At the heart of this formalism is a new description of Teichmuller space - and generalizations thereof - by polynomial equations to be solved over positive variables.
In this talk I will review these progresses, walking a path from purely combinatorial considerations about curves on surfaces, through tropical formulae for particle scattering amplitudes, and culminating in the positive description of Teichmuller space.
This talk is based on 2309.15913 and 2311.09284.

Physicists Arkani-Hamed and Trnka introduced the amplituhedron to better understand scattering amplitudes in N=4 super Yang-Mills theory. The amplituhedron is the image of the totally nonnegative Grassmannian under the "amplituhedron map". Examples of amplituhedra include cyclic polytopes, the totally nonnegative Grassmannian itself, and cyclic hyperplane arrangements. Of primary interest to physics are tilings of amplituhedra, which are roughly analogous to subdivisions of polytopes. This talk is a follow-up to Matteo's talk on BCFW tilings of m=4 amplituhedra. I will focus on the surprising connection between tilings of m=4 amplituhedra and the cluster algebra structure of the Grassmannian. No knowledge of cluster algebras will be assumed. All results mentioned in this talk are joint work with Even-Zohar, Lakrec, Parisi, Tessler and Williams.

I will present recent computer algebra tools from symbolic summation, symbolic integration and special functions that have been used non-trivially for the symbolic treatment of massive three loop Feynman integrals. All these techniques are illustrated by concrete examples that arose within a long term cooperation between RISC and the theory group of DESY Zeuthen (Deutsches Elektronen-Synchrotron).

I will give a general and gentle introduction to cosmological correlation functions — their usage as a tool to understand the beginning of the universe, and their connection to particle physics and scattering amplitudes. Then I will show in a toy model of cosmology a beautiful mathematical structure of differential equations that control the strength of the correlations as a function of distances in the sky. In this setup, time evolution emerges as a solution to these static differential equations.

Cosmology is famously an observational rather than an experimental science. No experimentalists were present in the early universe, and the birth and subsequent evolution of the universe cannot be repeated. Instead, we can only measure the spatial correlations between cosmological structures at late times. A central challenge of modern cosmology is to construct a consistent "history" of the universe that explains these correlations.
Recently, a new bootstrap approach was developed to understand this history using physical consistency conditions alone. In this colloquium, I will explain the basic idea behind this "cosmological bootstrap". I will also describe the search for new geometrical structures, called "positive geometries", which may underlie the theory of cosmological correlations. Finding such structures is one of the central aims of the ERC Synergy Grant project UNIVERSE+. I hope to make the talk accessible to a broad audience, and will not assume any background in cosmology or particle physics.

I will summarize recent progress in uncovering the analytic structure of scattering amplitudes. The overarching theme will be exploiting new intricate ways of analytically continuing time, extending beyond the Wick rotation. I will highlight a broad range of applications: from high-precision calculations in particle physics, through computations of gravitational waves, to formal topics in the scattering of strings.

Feynman integrals are Mellin integrals: they are the Mellin transform of the graph polynomial of the Feynman diagram taken to the power -d/2, if d denotes the dimension of Minkowski spacetime. It is important to construct relations among Feynman integrals in different dimensions of spacetime. One way to construct such relations is via integration by parts techniques. These IBP relations can be explained via cohomological theories. Also techniques from the theory of D-modules help to construct shift relations, namely by means of Bernstein-Sato operators and s-parametric annihilators, which I explain in this talk. This talk is based on the joint work arXiv:2208.08967 with D. Agostini, C. Fevola, and S. Telen, as well as on ongoing work.

Euler integral gains new attention after the introduction of the Lee-Pomeransky representation of Feynman integrals. As an integral is a transcendental object to study, it is natural to seek a system of equations that characterizes the integral. One can find such a system of differential, difference, and even difference-differential equations through twisted cohomology. We discuss what we can learn from it and what we need to develop.

I will start with an overview of the connection between scattering amplitudes, non-negative Grassmannian and on-shell diagrams (plabic graphs). I will show how the non-planar plabic graphs naturally appear and physics and how the explicit results for scattering amplitudes can associate them with subspaces in the Grassmannian G(k,n). In the talk I mostly focus on the G(3,6) case which corresponds to a particular six-gluon amplitude on the physics side.

The study of the singularities of Feynman integrals is a classical problem in scattering amplitudes. In this joint work with Sabastian Mizera and Simon Telen, we elaborate on the recent reformulation of Landau singularities of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations.
Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram.
I will illustrate some examples where our package implementation, PLD.jl, using numerical nonlinear algebra methods, allows us to compute components of the Landau singular locus that were previously out of reach.

The recent study of FRW correlators has revealed fascinating connections between positive geometries and modern on-shell methods as well as exhibiting novel mathematical structures. In particular, FRW correlators take the form of (degenerate) generalized Euler integrals. I will formulate the associated twisted cohomology governing these integrals then introduce the intersection pairing (an inner product on the space of FRW integrals) and the corresponding dual relative twisted cohomology. Then, I will advocate for the advantages of relative twisted cohomology; specifically how it manifests the unitarity structure of these correlators. Using this framework, I will explain how to predict the basis size of these integrals and describe several algorithms to compute their differential equations.

The theory of resurgence provides powerful tools to access the non-perturbative sectors of the factorially divergent asymptotic series that arise naturally as perturbative expansions in quantum theories. After introducing the basics of resurgence, I will review recent progress on the resurgent analysis of the strong and weak coupling limits of the spectral theory of toric Calabi-Yau threefolds, which is conjecturally dual to the topological string theory compactified on the same background. In the case of the local P^2 geometry, a remarkable analytic number-theoretic structure unfolds, revealing an exact and explicit strong-weak symmetry underpinning the resurgent properties of the perturbative expansions and paving the way for further insights. This talk is based on arXiv:2212.10606. A follow-up with V. Fantini will be available soon.

The multitude of integrals encountered in perturbative quantum field theory and string theory calls for systematic studies of function spaces that close under taking primitives. For integrations on the sphere — Riemann surfaces of genus zero — this is accomplished by multiple polylogarithms which had huge impact on Feynman integrals and string amplitudes. The more challenging iterated integrals on genus-one surfaces seen in various physics contexts were recently harnessed by means of elliptic polylogarithms. After reviewing the Brown-Levin formulation of elliptic polylogarithms, I will present a generalization to Riemann surfaces of arbitrary genus. The construction of higher-genus polylogarithms in this talk is based on a flat connection built from convolutions of Arakelov Green functions on the surface.

Tree-level scattering amplitudes of strings closely resemble Selberg integrals -- classical hypergeometric integrals. The twisted cohomology of Selberg integrals elucidates relations string amplitudes satisfy -- among them, the "double-copy" relations between scattering amplitudes of open & closed strings. Is there an elliptic analogue of Selberg integrals that, via twisted cohomology, tells us about the double-copy relations between one-loop string amplitudes?
Based on joint work with Rishabh Bhardwaj, Andrzej Pokraka and Lecheng Ren.

In this talk, I will reformulate the correlahedron as a loop-geometry over the positive space X_i X_j>0. This clarifies the connection between the geometric form and the O2222 correlator, while manifesting the partial non-renormalization theorem.
This reformulation allows us to invoke the novel idea of "chambers" to study the loop-geometry. We characterize the boundary of the chambers in each loop order and compute its loop-form. We have verified the construction up to L=3