This year's speaker for the Chow lectures is Nima Arkani-Hamed (IAS Princeton). Nima Arkani-Hamed is a theoretical physicist, mostly known for his groundbreaking contributions to the theory of scattering amplitudes. One of his key achievements is the introduction of the amplituhedron (with Jaroslav Trnka) and other geometric structures which simplify calculations for particle interactions in certain quantum field theories. Nima is Gopal Prasad Professor at the Institute for Advanced Studies in Princeton, New Jersey, USA. His work has been recognized by many awards, including the Gribov medal of the European Physical Society (2003), the Breakthrough Prize in Fundamental Physics (2012) and the Sakurai Prize of the American Physical Society (2021).
This lecture series is named after Wei-Liang Chow (October 1, 1911, Shanghai – August 10, 1995, Baltimore), who was a celebrated mathematician best known for his work in algebraic geometry, though he also made important contributions to other areas such as differential geometry, differential equations, and control theory. Within algebraic geometry he is known among other things for his work on intersection theory (Chow's moving lemma, Chow ring), for Chow's theorem on algebraicity of projective analytic varieties, and for many other achievements.
Chow was born in Shanghai, had his school education in the United States, and graduated from the University of Chicago in 1931. He obtained his PhD in Leipzig in 1936, where he worked with van der Waerden (who was a professor at the University of Leipzig), introducing in particular Chow coordinates. Subsequently, Chow was a professor in Nanjing, Princeton and at Johns Hopkins University were he worked until 1977.
The idea behind the Chow lectures is to give students the opportunity to enjoy lectures by internationally renowned experts on active fields of modern mathematics and to create a stimulating research environment.
WARNING: Please be wary if a company directly contacts you regarding payment for accommodation in Leipzig: this is fraud / phishing. The communication and organisation of this school is carried out exclusively by staff of the MPI MiS.
This activity is financially supported by 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.
Kinematic StratificationsVeronica Calvo CortesMPI MiS Leipzig, GermanyWe 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 and massive particles, with and without momentum conservation.Evaluating Genus One String AmplitudesEmiel ClaasenMax Planck Institute for Gravitational Physics (Potsdam), GermanyThe low-energy expansion of closed-string genus-one amplitudes introduces non-holomorphic modular forms for SL(2, Z) known as modular graph forms (MGFs). The expansion of the amplitude can be evaluated by integrating MGFs over the moduli space of the genus-one Riemann surface. A modern, more systematic approach to tackle this problem is by writing MGFs as equivariant iterated Eisenstein integrals, which are genus-one analogues of single-valued multiple polylogarithms. Their integrals over the moduli space result in among others multiple zeta values, logarithmic derivatives of zeta values and the Euler-Mascheroni constant. We compute the first seven orders in this expansion and conjecture an all-order schematic form of the amplitude.Tropical Principal Bundles on Metric GraphsArne KuhrsPaderborn University, GermanyA metric graph or tropical curve is the tropical analogue of a Riemann surface. There is a well-developed theory of line bundles and divisors on metric graphs, which is completely analogous to the algebraic theory.
Gross, Ulirsch and Zakharov defined tropical vector bundles on metric graphs as torsors over the group of tropical invertible matrices. Extending the GL_n case, we study the geometry of tropical bundles with other structure groups G on a metric graph. Using the root datum of reductive groups, we construct tropical analogues thereof and obtain natural descriptions as tropical matrix groups.
We generalize the description of tropical vector bundles on metric graphs via covers and line bundles to the setting of principal G-bundles. Furthermore, using Fratila’s description of the moduli space of semistable principal bundles on an elliptic curve, we develop a tropicalization procedure for semistable principal bundles on an Tate curve. This identifies the essential skeleton of the moduli space of semistable principal bundles on a Tate curve with the main component of the tropical moduli space of semistable principal bundles on its dual metric graph.
This is based on ongoing work with Andreas Gross, Martin Ulirsch and Dmitry Zakharov.On the V-number of binomial edge ideals: minimal cuts and cycle graphs.Emiliano LiwskiKU Leuven, BelgiumThe v-number of a graded ideal is an invariant recently introduced in the context of coding theory, particularly in the study of Reed–Muller-type codes. In this work, we study the localized v-numbers of a binomial edge ideal JG associated to a finite simple graph G. We introduce a new approach to compute these invariants, based on the analysis of transversals in families of subsets arising from dependencies in certain rank-two matroids. This reduces the computation of localized v-numbers to the determination of the radical of an explicit ideal and provides upper bounds for these invariants. Using this method, we explicitly compute the localized v-numbers of JG at the associated minimal primes corresponding to minimal cuts of G. Additionally, we determine the v-number of binomial edge ideals for cycle graphs.Matroid cosmologyFelix LotterMPI MiS Leipzig, GermanyScattering amplitudes, beginning with the amplituhedron, can be understood as 'volumes'. A key example, the Tr φ³ tree-level amplitudes, are volumes of associahedra, or equivalently Laplace transforms of their normal fans. These normal fans agree with the Bergman fan of the corresponding matroids. Motivated by this, recent work by Thomas Lam introduces a new class of 'amplitudes', associated to an arbitrary matroid.
It is natural to look for an analogous matroidal description of the cosmological wavefunction. In the case of Tr φ³, the wavefunction is governed by a family of polytopes, the Cosmohedra, obtained as certain 'blow-ups' of associahedra. We generalise this by proving that the Bergman fan of any matroid admits a cosmological refinement, recovering in a special case the normal fan of the n-point Cosmohedron.
This is based on joint work with Hadleigh Frost.Plabic Tangles and Cluster Promotion MapsMatteo ParisiMPI MPP München, GermanyInspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of `plabic tangles' that utilizes plabic graphs to define rational maps between products of Grassmannians called `promotions'. The central conjecture of our work is that promotion maps are quasi-cluster homomorphisms, which we prove for several classes of promotions. In order to define promotion maps, we utilize m-vector-relation configurations (m-VRCs) on plabic graphs. We relate m-VRCs to the degree (a.k.a `intersection number') of the amplituhedron map on positroid varieties and characterize all plabic trees with intersection number one and their VRCs. We also show that promotion maps admit an operad structure and, supported by the class of 4-mass box promotion, we point at new positivity properties for non-rational maps beyond cluster algebras. Promotion maps have important connections to the geometry and cluster structure of the amplituhedron and singularities of scattering amplitudes in planar N=4 super Yang-Mills theory.Geometry of effective field theory positivity conesPaula Naomi PilatusUniversity of Hamburg, GermanyPositivity bounds are theoretical constraints on the Wilson coefficients of an effective field theory. These bounds emerge from the requirement that a given effective field theory must be the low-energy limit of a relativistic quantum theory that satisfies the fundamental principles of unitarity, locality, and causality. The task of deriving these bounds can be reformulated as the geometric problem of finding the extremal representation of a closed convex cone $\mathcal C_W$. More precisely, in the presence of multiple particle flavors, the forward-limit positivity cone $\mathcal C_W$ consists of all positive semi-definite tensors in$W =\left\{ S \in \mathrm{Sym}^2 (\mathrm{Sym}^2\, V^*)\oplus \mathrm{Sym}^2 \left({\Lambda}^2 V^*\right) : \tau S = S \right\} \subset \mathrm{Sym}^2(V^*\otimes V^*)$,where $\tau$ denotes transposition in the second and fourth tensor factor and $V\cong\mathbb{R}^n$, where $n$ is the number of flavors. We solve this question up to three flavors, i.e. $n=3$, proving a full classification of all extremal elements in these cases. Based on arXiv:2508.18165.Renormalization of Gauge Theories and GravityDavid PrinzMax Planck Institute for Mathematics, Bonn, GermanyThe renormalization of gauge theories and, eventually, gravity is one of the biggest current challenges in mathematical physics. In this poster, I will describe recent progress in the Hopf algebraic approach to renormalization. Specifically, this amounts to the following two aspects:
1) Avoiding gauge and diffeomorphism anomalies, which translates to the validity of the corresponding Slavnov--Taylor identities: A theorem by van Suijlekom (2007), improved by myself (2022), states that they generate Hopf ideals.
2) Transversality of the formal Green's function, which can be implemented combinatorially via cancellation identities: In an ongoing project, I aim to implement these via a Feynman graph complex, similar to the one constructed by Kreimer et al. (2013).
Specifically, I aim to relate both aspects in said project by extending the renormalization Hopf algebra to a differential-graded renormalization Hopf module, as outlined in my doctoral thesis, cf. arXiv:2210.17510. Finally, I will address the application of these constructions to Quantum General Relativity, which is non-renormalizable by power counting: To overcome this issue, I will close with a recent working conjecture of mine for the appropriate UV-completion thereof.Tropical combinatorics of the 2D Toda latticeVincenzo RedaTrinity College Dublin, IrelandThe study of soliton solutions of the KP hierarchy is a classical topic. In seminal work of Kodama and Williams a tropical geometry approach to their study was introduced. More precisely, it was proved that the phase structure of smooth solutions of the KP hierarchy is described by a tropical curve that naturally arises from combinatorial decompositions of the Grassmannian. The 2D Toda lattice hierarchy extends the KP hierarchy. In this poster, we show that the tropical approach introduced by Kodama and Williams generalizes to the 2D Toda lattice answering a question posed by Kodama. This is joint work in progress with R. N. Betancourt, M. A. Hahn and V. PoschWilson Loop CorrelatorsMatthew RochfordUniversity of Southampton, United KingdomAn n-gluon scattering amplitude in N=4 SYM is equal to the expectation value of a light-like polygonal Wilson loop, , with n sides (0705.0303, 0707.1153, 0707.0243, 1009.2225, 1101.1329). Thus, expectation values of light-like polygonal Wilson loops satisfy the properties of gluon scattering amplitudes. Scattering amplitudes display many interesting mathematical properties, such as BCFW recursion relations and generalised unitarity. These properties allow amplitudes to be expressed in a vastly simpler way than the Feynman diagram expansion. Motivated by the scattering amplitude-Wilson loop duality, we investigated whether Wilson loop correlators, , satisfy similar mathematical properties. In this poster, I will introduce these observables in twistor space and discuss some of their mathematical properties, such as the chiral box expansion, with coefficients given by tree-level correlators, the Q-bar equation, which relates loop-level correlators to correlators of lower loop order, and a tree-level BCFW relation. I will then briefly detail future directions, which will involve using these correlators to compute other observables in N=4 SYM.Possibly: Vandermonde cells as positive geometriesSebastian SeemannKU Leuven, BelgiumWe study Vandermonde cells from the perspective of positive geometry, determining canoncial forms in the planar case and proving an obstruction to the existence of canoncial forms in the non-planar case.Moduli of tropical admissible cyclic coversPedro SouzaGoethe-Universität Frankfurt am Main, GermanyWe study the relationship between the skeleton $\overline{\Sigma}(\overline{\mathcal{H}}_{g,G,\xi})$ of the moduli space $\overline{\mathcal{H}}_{g,G,\xi}$ – parametrizing admissible $G$-covers of stable curves of genus $g$ with ramification profile $\xi$, where $G$ is a finite cyclic group – and the corresponding tropical moduli space $(\overline{\mathcal{H}}_{g,G,\xi})^{trop}$. This is part of the program started by Abramovich-Caporaso-Payne (and others) of comparing moduli spaces of algebraic geometry with their tropical counterparts using the theory of non-archimedean Berkovich spaces. Drawing on the application of techniques from the theory of smooth surfaces to study irreducibility of moduli spaces of covers, we show that the natural combinatorial objects produced from the each of the theories (non-archimedean and tropical) agree, when $G$ is a finite cyclic group.Positive Geometry for Stringy AmplitudesJonah StalknechtCharles University (Prague), Czech RepublicWe introduce a new positive geometry, the associahedral grid, which provides a geometric realization of the inverse string theory KLT kernel. It captures the full α′-dependence of stringified amplitudes for bi-adjoint scalar theory, pions in the NLSM, and their mixed ϕ/π amplitudes, reducing to the corresponding field theory amplitudes in the α′ → 0 limit. Our results demonstrate how positive geometries can be utilized beyond rational functions to capture stringy features of amplitudes. The kinematic δ-shift, recently proposed to relate field theory Tr(ϕ3) and NLSM pion amplitudes, naturally emerges as the leading contribution to the stringy geometry. We show how the connection between Tr(ϕ3) and NLSM can be geometrized using the associahedral grid.Connection Matrices in Macaulay2Nicolas Alexander WeissMPI MiS (Leipzig), GermanyDifferential equations for Feynman integrals appear both in the form of annihilating operators (as D-ideals) as well as in matrix form (as Pfaffian systems). Starting from a D-ideal of finite holonomic rank, one can compute a corresponding Pfaffian system by reduction to a basis (of standard monomials) via non-commutative Gröbner bases. We provide an implementation and tools for the computation and handling of Pfaffian systems in Macaulay2.The Scattering Correspondence of a GraphBailee ZacovicUniversity of Michigan (Ann Arbor), USAThe CHY scattering equations on the moduli space M_{0,n} play an important role at the interface of particle physics and algebraic statistics. We present the scattering correspondence when the Mandelstam parameters are restricted to a fixed graph G on n vertices. This leads us to associate a partial compactification M_G of M_{0,n}, which yields a combinatorial formula for the maximum-likelihood degree of the graph.
I present combinatorial operations on Feynman diagrams, described by cutting and pinching edges, that correspond to the algebraic coaction on the functions resulting from their integration. These operations correspond in turn to a coaction on generalized hypergeometric functions. I will comment on some implications for physical theories.
