Nicolas Kainz, M. Sc.

• Oct. 2017 - Sep. 2021: Bachelor of Science, Mathematics, Ulm University
• Oct. 2021 - Mar. 2023: Master of Science, Mathematics, Ulm University
• Since Apr. 2023: PhD student, Institute of Numerical Mathematics, Ulm University

• Elements of Topology (Summer term 2022)
• Numerical Methods for Ordinary Differential Equations (Summer term 2023)
• Modeling on Manifolds (Winter term 2023/2024)
• Numerical Methods for Ordinary Differential Equations (Summer term 2024)
• SAPS - Preparation course for the DAV maths entrance exam (Summer term 2024)
• Differential Topology and Mathematical Physics (Winter term 2024/2025)
• SAPS - Preparation course for the DAV maths entrance exam (Winter term 2024/2025)
• Differential Geometry and Mathematical Physics (Summer term 2025)
• Elements of Complex Analysis (Summer term 2025)
• SAPS - Preparation course for the DAV maths entrance exam (Summer term 2025)

I am interested in the dynamics of holomorphic vector fields and their associated flows, with a focus on the following aspects:

• Local-global relations influenced by the geometry of equilibria
• Homoclinic and heteroclinic orbits near multiple equilibria
• Qualitative behaviour of orbits near Infinity
• Quantitative characterization of separatrices in phase space

• Planar Analytic Dynamical Systems and their phase space structure. Master thesis, 03/2023, available here: master thesis
• Local geometry of Equilibria and a Poincare-Bendixson-type Theorem for Holomorphic Flows. Topology Proc. 65 (2025), pp. 99-116, e-published on November 7, 2024. Journal: https://topology.nipissingu.ca/tp/reprints/v65/ Preprint: https://doi.org/10.48550/arXiv.2402.07612
• Basins of Equilibria and geometry of Global Sectors in Holomorphic Flows. Article for Topology Proceedings, 10/2024, Preprint: https://doi.org/10.48550/arXiv.2410.04895
• Separatrix configurations in holomorphic flows. Article for Qualitative Theory of Dynamical Systems, 05/2025, Preprint: https://doi.org/10.48550/arXiv.2505.14594

• Global elliptic sectors in real time holomorphic flows. June 3, 2025, at the Summer School Topics in Complex Dynamics 2025: Connections to geometry and analysis, Barcelona (Spain)
• Elliptic sectors and heteroclinic regions in real time holomorphic flows. August 12, 2025, at the Summer Topology and Its Applications Conference, Mobile (South Alabama, USA)
• Basins, sectors and regions in holomorphic dynamical systems. August 26, 2025, at the Holomorphic Dynamics Conference, Oaxaca (Mexico)

I represent the interests of doctoral students at Ulm University in the following committees. If you have any concerns in this regard, please do not hesitate to contact me.

• Board of Promovierendenkonvent - Vorstand des Promovierendenkonvents (October 2023 - September 2025)
• Senate - Senat (October 2024 - September 2025)
• Faculty Council Mathematics and Economics - Fakultätsrat Mathematik und Wirtschaftswissenschaften (October 2023 - September 2025)
• Student Parliament - Studierendenparlament (October 2023 - September 2025)
• Working group for quality assurance funds - Arbeitskreis für Qualitätssicherungsmittel (April 2024 - September 2025)
• Budget Committee of the Senate - Haushaltsausschuss des Senats (October 2024 - September 2025)
• LGFG Scholarship Award Committee - Vergabekommission für LGFG-Stipendien (November 2023 - September 2025)