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.

Topological Data Analysis analyzes the shape of data by topological methods. It has concrete applications in neurosciences and machine learning, for instance. One main tool is persistent homology. It associates persistence modules and so called barcodes to data, from which one easily reads topological features. Behind the scenes, algebraic invariants keep the machinery running. My emphasis in this area lies on the development of stable invariants for multipersistence modules.

Preprints and publications

  1. Stable Invariants for Multipersistence Modules arising from Hierarchical Stabilization (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). Preprint arXiv:2103.16300, 2021. Invited expository article by Numerical Algebra, Control and Optimization. Submitted.
  3. Combinatorial Differential Algebra of xp (with Rida Ait El Manssour). Preprint arXiv:2102.03182, 2021. Submitted to the special issue of the Journal of Symbolic Computation on the occasion of MEGA 2021.
    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 to International Mathematics Research Notices.
  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 be published 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.