Which new structures in data can algebra detect? How can algebra advance the toolbox of data analysis and can we exploit computations for insights of theoretical nature?
My research centers around algebraic analysis, applied algebraic geometry, and topological data analysis, and addresses the questions stated above. I love to link different fields of mathematics and to develop algebraic techniques to solve problems arising in the sciences.
Algebraic analysis investigates systems of linear partial differential equations by algebraic methods. It elegantly combines methods from algebraic geometry, algebraic topology, category theory, and complex analysis. Many special functions in the sciences are encoded by a holonomic annihilating ideal in the Weyl algebra D and can be investigated by means of D-ideals, such as hypergeometric functions, some probability densities, Feynman integrals, and many more. In this area, I focus on computational aspects and applications – among others, the maximum likelihood estimation of discrete statistical models for the statistical inference of data. At my (virtual) office door, you can have a look at my poster about Algebraic Analysis and Applications. In contrast to linear algebra, nonlinear algebra is currently still underrepresented in applications. Yet, it provides deep structural insights and computational techniques, some of which are showcased here.
Topological data analysis extracts information from data by topological methods. One main tool is persistent homology. Behind the scenes, algebraic invariants keep the machinery running. The one-parameter case is algebraically well-understood and established in applications. The multiparameter case is algebraically intricate. My emphasis in this area lies on the development of stable invariants of multipersistence modules.
Together with MPI MiS, KTH, EPFL, MIT, and the University of Oxford, we run the international consortium AlToGeLiS: Algebra, Topology, and Geometry in the Life Sciences. Find out more or subscribe to our mailing list on www.altogelis.com!
Preprints and publications
Vector Spaces of Generalized Euler Integrals (with Daniele Agostini, Claudia Fevola, and Simon Telen). Preprint arXiv:2208.08967, 2022. Submitted.
Supplementary material: Jupyter notebooks with code in Julia, Macaulay2, and Singular.
Bayesian Integrals on Toric Varieties (with Michael Borinksy, Bernd Sturmfels, and Simon Telen). Preprint arXiv:2204.06414, 2022. Submitted.
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). Preprint arXiv:2112.06509, 2021. Submitted.
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, 2021. DOI:10.3934/naco.2021045. 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. DOI:10.1016/j.jsc.2022.04.010. 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). Preprint arXiv:2101.03570, 2021. International Mathematics Research Notices, 2022. DOI:10.1093/imrn/rnac016.
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. DOI:10.1007/s13366-020-00546-z. Final version also available at arXiv:2005.06162.
Topological computation of Stokes matrices of some weighted projective lines. Manuscripta mathematica, 164(3):327-347, 2021. DOI: 10.1007/s00229-020-01193-3.