Which new structures can algebraic analysis and algebraic geometry detect in problems in the sciences? Which insights can nonlinear algebra provide for data analysis – and can we exploit computations for results of theoretical nature?
My research centers around algebraic analysis and applied algebraic geometry and addresses the questions stated above. I love to link different fields of mathematics. By doing so, I develop algebraic techniques to tackle problems arising in high energy physics, cosmology, algebraic statistics, and the theory machine learning.
Links: arXiv, Google Scholar, ORCİD
Algebraic analysis investigates systems of linear partial differential equations by algebraic methods. It combines methods from algebraic geometry, algebraic topology, category theory, and complex analysis. The main actor is the Weyl algebra, denoted D. It is a non-commutative ring which gathers linear differential operators with polynomial coefficients. Left D-ideals encode systems of linear PDEs. Many special functions in the sciences – such as hypergeometric functions, some probability densities, Feynman integrals, polylogarithms, and many more – are encoded by their holonomic annihilating ideal in the Weyl algebra and can be investigated via this D-ideal. In this area, I focus on applications – among others, the study of scattering processes of particles, or maximum likelihood estimation of discrete statistical models. At my virtual office door, you can have a look at a poster about algebraic analysis and applications.
In contrast to linear algebra, nonlinear algebra is still underrepresented in applications. Yet, the world is highly non-linear. Nonlinear algebra, in which algebraic geometry is playing a key role, provides deep structural insights and new computational techniques, such as for the evaluation of Bayesian integrals, and many more. Algebraic geometry is also useful to tackle fundamental questions in learning theory – for instance, for a thorough study of the geometry of function spaces of neural networks. Among others, this helps for the design of equivariant linear neural networks, or to locate the critical points of the loss when training a network.
Also behind the scenes of topological data analysis, which infers information from data by topological methods, algebraic invariants keep the machinery running. One main tool is persistent homology. While the one-parameter case is algebraically well-understood and established in applications, the multiparameter case is algebraically intricate.
Together with MPI MiS, KTH, EPFL, MIT, the University of Oxford, and newly the University of Tromsø, we run the international consortium AlToGeLiS: Algebra, Topology, and Geometry in the Life Sciences. Find out more on www.altogelis.com!
Publications and preprints
Border Bases in the Rational Weyl Algebra (with Carlos Rodriguez). Preprint arXiv:2510.23411. Supplementary material: Mathematica notebooks
Connection Matrices in Macaulay2 (with Paul Görlach, Joris Koefler, Mahrud Sayrafi, Hendrik Schroeder, Nicolas Weiss, and Francesca Zaffalon). Preprint arXiv:2504.01362. A demo is available at the MathRepo.
Algebraic and Positive Geometry of the Universe: from Particles to Galaxies (with Claudia Fevola). Notices of the American Mathematical Society, 72(8):808-817, 2025. Accepted version also available at arXiv:2502.13582.
Differential Equations for Moving Hyperplane Arrangements (with Anaëlle Pfister). Le Matematiche, 80(1):409-429, 2025. Special volume on Positive Geometry. Accepted version also available at arXiv:2412.09479. Supplementary material: computations in Macaulay2 and Singular.
Algebraic Approaches to Cosmological Integrals (with Claudia Fevola, Guilherme L. Pimentel, and Tom Westerdijk). Le Matematiche, 80(1):303-324, 2025. Special volume on Positive Geometry. Accepted version also available at arXiv:2410.14757. Supplementary material: computations in Singular and Mathematica.
Geometry of Linear Neural Networks: Equivariance und Invariance under Permutation Groups (with Kathlén Kohn and Vahid Shahverdi). SIAM Journal on Matrix Analysis and Applications, 46(2):1378-1415, 2025. Accepted version also available at arXiv:2309.13736. Supplementary material: implementation for MNIST in Python.
D-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals (with Johannes Henn, Lizzie Pratt, and Simone Zoia). Letters in Mathematical Physics, 114(28), 2024. Accepted version also available at arXiv:2303.11105. Supplementary material: implementation in Sage.
D-Algebraic Functions (with Rida Ait El Manssour and Bertrand Teguia Tabuguia). Journal of Symbolic Computation, 128(102377), 2025. Accepted version also available at arXiv:2301.02512. Supplementary material: implementations in Macaulay2 and Maple.
Vector Spaces of Generalized Euler Integrals (with Daniele Agostini, Claudia Fevola, Simon Telen, and an appendix by Saiei-Jaeyeong Matsubara-Heo). Communications in Number Theory and Physics, 18(2):327-370, 2024. Accepted version also available at arXiv:2208.08967. Supplementary material: Jupyter notebooks with code in Julia, Macaulay2, and Singular.
Bayesian Integrals on Toric Varieties (with Michael Borinsky, Bernd Sturmfels, and Simon Telen). SIAM Journal on Applied Algebra and Geometry, 7(1):77-103, 2023. Accepted version also available at arXiv:2204.06414. Supplementary material: Julia code, Geometrische Gemeinheit 9 (inspired by a toric sector decomposition of the pentagon).
The Shift-Dimension of Multipersistence Modules (with Wojciech Chachólski and René Corbet). Journal of Applied and Computational Topology, 8(3):643-667, 2024. Final version also available at arXiv:2112.06509. Supplementary material: C++ implementation of our algorithm to compute the shift-dimension of interval modules in the 2-parameter case.
Nonlinear Algebra and Applications (with Paul Breiding, Türkü Ö. Çelik, Timothy F. Duff, Alexander Heaton, Aida Maraj, Lorenzo Venturello, and Oğuzhan Yürük). Numerical Algebra, Control and Optimization, 13(1):81-116, 2023. Submitted version also available at arXiv:2103.16300.
Combinatorial Differential Algebra of xp (with Rida Ait El Manssour). Journal of Symbolic Computation, 114:193-208, 2023. Accepted version also available at arXiv:2102.03182. Supplementary material: my painting Geometrische Gemeinheit 8 was inspired by the regular triangulation T2,2 (cf. Figure 1 in the article).
Maximum Likelihood Estimation from a Tropical and a Bernstein-Sato Perspective (with Robin van der Veer). International Mathematics Research Notices, 2023(6):5263-5292, 2023. Preprint also available at arXiv:2101.03570.
Algebraic Analysis of the Hypergeometric Function 1F1 of a Matrix Argument (with Paul Görlach and Christian Lehn). Beiträge zur Algebra und Geometrie, 62:397-427, 2021. Final version also available at arXiv:2005.06162.
Algebraic Analysis of Rotation Data (with Michael F. Adamer, András C. Lőrincz, and Bernd Sturmfels). Algebraic Statistics, 11(2):189-211, 2020. Submitted version also available at arXiv:1912.00396.
D-Modules and Holonomic Functions (with Bernd Sturmfels). In Varieties, polyhedra, computation. Volume 22 of EMS Series of Congress Reports, pp. 251-293, EMS Press, Berlin, 2025. Preprint also available at arxiv:1910.01395.
Topological computation of Stokes matrices of some weighted projective lines. Manuscripta mathematica, 164(3):327-347, 2021.
Anna-Laura Sattelberger 