Publications: Max Planck Institute of Molecular Cell Biology and Genetics

^* joint first author ^# joint corresponding author

2026

Michael Borinski, Chiara Meroni, Maximilian Wiesmann
Asymptotic number of edge-colored regular graphs.
ArXiv, Art. No. arXiv:2601.18994 (2026)
Open Access

We prove a formula for the asymptotic number of edge-colored regular graphs with a prescribed set of allowed vertex-incidence structures. The formula depends on specific critical points of a polynomial encoding the vertex-incidences. As an application, we compute the expected number of proper -edge-colorings of a large random -regular graph.

Paul Breiding, John Cobb, Aviva K. Englander, Nayda Farnsworth, Jonathan D. Hauenstein, Oskar Henrikson, David K. Johnson, Jordy Lopez Garcia, Deepak Mundayur
Elimination Without Eliminating: Computing Complements of Real Hypersurfaces Using Pseudo-Witness Sets.
ArXiv, Art. No. arXiv:2601.04383 (2026)
Open Access

Many hypersurfaces in algebraic geometry, such as discriminants, arise as the projection of another variety. The real complement of such a hypersurface partitions its ambient space into open regions. In this paper, we propose a new method for computing these regions. Existing methods for computing regions require the explicit equation of the hypersurface as input. However, computing this equation by elimination can be computationally demanding or even infeasible. Our approach instead derives from univariate interpolation by computing the intersection of the hypersurface with a line. Such an intersection can be done using so-called pseudo-witness sets without computing a defining equation for the hypersurface - we perform elimination without actually eliminating. We implement our approach in a forthcoming Julia package and demonstrate, on several examples, that the resulting algorithm accurately recovers all regions of the real complement of a hypersurface.

Emma Cardwell, Aida Maraj, Alvaro Ribot
Toric multivariate Gaussian models from symmetries in a tree.
Advances in Applied Mathematics , 174 Art. No. 103024 (2026)
DOI

Given a rooted tree T on n non-root leaves with colored and zeroed nodes, we construct a linear space L-T of n x n symmetric matrices with constraints determined by the combinatorics of the tree. When LT represents the covariance matrices of a Gaussian model, it provides natural generalizations of Brownian motion tree (BMT) models in phylogenetics and a step toward a more accurate model for phylogenetic networks with symmetries for species hybridization. When L-T represents a space of concentration matrices of a Gaussian model, it gives certain colored Gaussian graphical models, which we refer to as BMT derived models. We investigate conditions under which the reciprocal variety L-T(-1) is toric. Relying on the birational isomorphism of the inverse matrix map, we show that if the BMT derived graph of T is vertex-regular and a block graph, under the derived Laplacian transformation, which we introduce, L-T(-1) is the vanishing locus of a toric ideal. This ideal is given by the sum of the toric ideal of the Gaussian graphical model on the block graph, the toric ideal of the original BMT model, and binomial linear conditions coming from vertex-regularity. To this end, we provide monomial parametrizations for these toric models realized through paths among leaves in T.

Maximilian Wiesmann
Lee-Yang phenomena in edge-coloured graph counting.
ArXiv, Art. No. arXiv:2601.02525 (2026)
Open Access

We study the accumulation of zeros of a polynomial arising from the enumeration of edge-coloured graphs along certain limit curves. The polynomial is a variant of an edge-chromatic polynomial, which specialises to the partition function of the ferromagnetic Ising model on a random regular graph. We call this accumulation behaviour a Lee-Yang phenomenon in analogy with the Lee-Yang theorem. The limiting loci are semialgebraic and arise from anti-Stokes curves of an exponential integral.

2025

Kexin Wang, Aida Maraj, Anna Seigal
Contrastive independent component analysis for salient patterns and dimensionality reduction.
Proc Natl Acad Sci U.S.A., 122(50) Art. No. e2425119122 (2025)
Open Access DOI

In recent years, there has been growing interest in jointly analyzing a foreground dataset, representing an experimental group, and a background dataset, representing a control group. The goal of such contrastive investigations is to identify salient features in the experimental group relative to the control. Independent component analysis (ICA) is a powerful tool for learning independent patterns in a dataset. We generalize it to contrastive ICA (cICA). For this purpose, we devise a linear algebra-based tensor decomposition algorithm, which is more expressive but just as efficient and identifiable as other linear algebra-based algorithms. We establish the identifiability of cICA and demonstrate its performance in finding patterns and visualizing data, using synthetic, semisynthetic, and real-world datasets, comparing the approach to existing methods.

Taylor Brysiewicz, Aida Maraj
Lawrence lifts, matroids, and maximum likelihood degrees.
Algebr Stat, 16(2) 217-242 (2025)
DOI

We express the maximum likelihood (ML) degrees of a family of toric varieties in terms of Möbius invariants of matroids. The family of interest are those parametrized by monomial maps given by Lawrence lifts of totally unimodular matrices with even circuits. Specifying these matrices to be vertex-edge incidence matrices of bipartite graphs gives the ML degrees of some hierarchical models and three dimensional quasi-independence models. Included in this list are the no-three-way interaction models with one binary random variable, for which we give closed formulae.

Jane Ivy Coons, Shelby Cox, Aida Maraj, Ikenna Nometa
ML degrees of Brownian motion tree models: Star trees and root invariance.
J Symb Comput, 132 Art. No. 102482 (2025)
DOI

A Brownian motion tree (BMT) model is a Gaussian model whose associated set of covariance matrices is linearly constrained according to common ancestry in a phylogenetic tree. We study the complexity of inferring the maximum likelihood (ML) estimator for a BMT model by computing its ML-degree. Our main result is that the ML-degree of the BMT model on a star tree with n + 1 leaves is 2n+1-2n-3, which was previously conjectured by Am & eacute;ndola and Zwiernik. We also prove that the ML-degree of a BMT model is independent of the choice of the root. The proofs rely on the toric geometry of concentration matrices in a BMT model. Toward this end, we produce a combinatorial formula for the determinant of the concentration matrix of a BMT model, which generalizes the Cayley-Pr & uuml;fer theorem to complete graphs with weights given by a tree. (c) 2025 Published by Elsevier Ltd.

Amer Goel, Aida Maraj, Alvaro Ribot
Halfspace Representations of Path Polytopes of Trees.
ArXiv, Art. No. arXiv:2502.21204 (2025)
Open Access

Given a tree , its path polytope is the convex hull of the edge indicator vectors for the paths between any two distinct leaves in . These polytopes arise naturally in polyhedral geometry and applications, such as phylogenetics, tropical geometry, and algebraic statistics. We provide a minimal halfspace representation of these polytopes. The construction is made inductively using toric fiber products.

2024

Toric Multivariate Gaussian Models from Symmetries in a Tree with Emma Cardwell and Alvaro Ribot, (arxiv.org:407.02357)

Contrastive Independent Component Analysis with Kexin Wang and Anna Seigal, (https://arxiv.org/abs/2412.00895)

Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance with Jane Ivy Coons, Shelby Cox, Ikenna Nometa (https://arxiv.org/abs/2402.10322)

2023

Lawrence Lifts, Matroids, and Maximum Likelihood Degrees with Taylor Brysiewicz (arXiv:2310.13064)

Symmetry Lie Algebras of Varieties with Applications to Algebraic Statistics with Arpan Pal (arXiv:2309.10741)

2022

Shift Invariant Algebras, Segre Products and Regular Languages with Uwe Nagel, Journal of Algebra, Volume 631, 236-266, 2023 (arXiv:2204.07849)

2021

Symmetrically Colored Gaussian Graphical Models with Toric Vanishing Ideals with Jane Ivy Coons, Pratik Misra, and Miruna-Stefana Sorea, SIAM Journal of Applied Algebra and Geometry 7(1), 2023 (arXiv:2111.14817)

Staged Tree Models with Toric Structure with Christiane Gorgen and Lisa Nicklasson, Journal of Symbolic Computation 113, 242-268, 2022 (arXiv:2107.04516).

Nonlinear Algebra and Applications with Paul Breiding, Türkü Özlüm Çelik, Timothy Duff, Alexander Heaton, Anna-Laura Sattelberger, Lorenzo Venturello, Oğuzhan Yürük, Numerical Algebra, Control and Optimization, 2021 (arXiv:2103.16300).

2020

Reciprocal Maximum Likelihood Degrees of Brownian Motion Tree Models with Tobias Boege, Jane Ivy Coons, Christopher Eur, and Frank Rottger, Le Matematiche 76 (2), 383-398, 2021, special issue on Linear Spaces of Symmetric Matrices (arXiv:2009.11849).

Generalized Cut Polytopes for Binary Hierarchical Models with Jane Ivy Coons, Joseph Cummings, and Ben Hollering, Algebraic Statistics Vol. 14 (2023), No. 1, 17–36, 2023 (arXiv:2008.00043).

Algebraic and Geometric Properties of Hierarchical Models, Ph.D. Thesis, https://doi.org/10.13023/etd.2020.232.

2019

Equivariant Hilbert Series for Hierarchical Models with Uwe Nagel, Algebraic Statistics 12(1), 21–42, 2021 (arXiv:1909.13026).