D-Modules and TDA

My research addresses 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 and can be investigated by means of D-modules. In this area, I focus on computational aspects and applications – among others, maximum likelihood estimation of statistical models.

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 datasets, from which one easily reads topological features. Behind the scenes, algebraic structure theorems keep the machinery running. My emphasis in this area lies on advancing the algebraic tools.

My current collaborators include:

  • Valeria Bertini (TU Chemnitz, Germany)
  • René Corbet (KTH Stockholm, Sweden)
  • Wojciech Chachólski (KTH Stockholm, Sweden)
  • Joscha Diehl (University of Greifswald, Germany)
  • Paul Görlach (TU Chemnitz, Germany)
  • Christian Lehn (TU Chemnitz, Germany)
  • Robin van der Veer (KU Leuven, Belgium)

Preprints and publications:

  1. Algebraic Analysis of the Hypergeometric Function 1F1 of a Matrix Argument (with Paul Görlach and Christian Lehn). Preprint arXiv:2005.06162, 2020. To appear in Beiträge zur Algebra und Geometrie.
  2. Algebraic Analysis of Rotation Data (with Michael F. Adamer, András C. Lőrincz, and Bernd Sturmfels). Preprint arXiv:1912.00396, 2019. To appear in Algebraic Statistics, issue 11:2.
  3. D-Modules and Holonomic Functions (with Bernd Sturmfels). Preprint arxiv:1910.01395, 2019. To be published in the proceedings of the MATH+ Fall School on Algebraic Geometry.
  4. Topological Computation of the Stokes Matrices of the Weighted Projective Line P(1,3). Preprint arXiv:1810.10954, 2018. An extended version Topological computation of Stokes matrices of some weighted projective lines is published in Manuscripta Mathematica, 2020. DOI: 10.1007/s00229-020-01193-3.