Poster Session

Kinematic Stratifications

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 and massive particles, with and without momentum conservation.

Evaluating Genus One String Amplitudes

The 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 Graphs

A 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.

The 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 cosmology

Scattering 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 Maps

Inspired 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 cones

Positivity 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 Gravity

The 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 lattice

The 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. Posch

Wilson Loop Correlators

An n-gluon scattering amplitude in N=4 SYM is equal to the expectation value of a light-like polygonal Wilson loop, <w>, 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, <w_1 w_2>, 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.</w_1></w>

Possibly: Vandermonde cells as positive geometries

We 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 covers

We 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 Amplitudes

We 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 Macaulay2

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). 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 Graph

The 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.