Algebraic and Topological Data Analysis

I am enthusiastic about linking different fields of mathematics and about developing algebraic techniques to solve problems arising in the sciences. My research centers around applied algebraic analysis and topological data analysis.

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-modules. In this area, I focus on computational aspects and applications – among others, the maximum likelihood estimation of discrete statistical models for the inference of data. At my (virtual) office door, you can have a look at my poster about Algebraic Analysis and Applications.

Tological data analysis extracts information from data by topological methods. It has concrete applications in the life sciences and machine learning, for instance. One main tool is persistent homology. It associates a filtered simplicial complex to the data. From the barcodes of its homology modules, the persistence modules, one reads topological features. Behind the scenes, algebraic invariants keep the machinery running. My emphasis in this area lies on the development of stable invariants of multipersistence modules.

Mathematics reveals structures of data. They are not always as easy to detect as this sheep which is trying to hide behind a sheaf of grass.

Preprints and publications

  1. The Shift-Dimension of Multipersistence Modules (with Wojciech Chachólski and René Corbet). In preparation.
  2. 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. Available “online first” here. Submitted version also available at arXiv:2103.16300.
  3. Combinatorial Differential Algebra of xp (with Rida Ait El Manssour). Preprint arXiv:2102.03182, 2021. Submitted.
    Supplementary material: my painting Geometrische Gemeinheit 8 was inspired by the regular triangulation T2,2 (cf. Figure 1 in the article).
  4. Maximum Likelihood Estimation from a Tropical and a Bernstein-Sato Perspective (with Robin van der Veer). Preprint arxiv:2101.03570, 2021. Submitted.
  5. Algebraic Analysis of the Hypergeometric Function 1F1 of a Matrix Argument (with Paul Görlachand 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.
  6. 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.
  7. D-Modules and Holonomic Functions (with Bernd Sturmfels). Preprint arxiv:1910.01395, 2019. To appear in the volume Varieties, polyhedra, computation of EMS Series of Congress Reports.
  8. Topological computation of Stokes matrices of some weighted projective lines. Manuscripta mathematica, 164(3):327-347, 2021. DOI: 10.1007/s00229-020-01193-3.