Number | Title | ECTS | Hours | Lecturers |
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**401-3129-67L** | **Differential Galois Theory** | 4 credits | 2V | P. S. Jossen |

Abstract | Algebraic theory of linear differential equations, Picard-Vessiot theory, Differential Galois groups, local theory of differential equations, the Frobenius method, Newton polygons, Connections and local systems, Riemann-Hilbert correspondence on ℙ¹. |

Learning objective | We introduce differential Galois theory and mathematical concepts surrounding it. We formulate and prove an important case of the Riemann-Hilbert correspondence. |

Content | We study linear differential equations from an algebraic perspective, introducing differential rings, fields and differential modules (so-called Picard-Vessiot theory), and very soon the Galois group of a differential equation. We relate then the algebraic theory with the analytic theory, which leads us to the classical Riemann-Hilbert correspondence. In particular we will prove that differential equations on the complex projective line ℙ¹ with regular singularities in a finite set S correspond to representations of the fundamental group of ℙ¹∖S. If time permits, we have a look at differential equations in positive characteristic. |

Literature | M. van der Put and M. F. Singer, Galois theory of linear differential equations, Grundlehren der Math. Wiss. Vol 328, Springer 2003 |

**401-3680-67L** | **Persistent Homology and Topological Data Analysis** Number of participants limited to 8. | 4 credits | 2S | P. S. Jossen |

Abstract | We study the fundamental tools of topological data analysis: Persistent homology, persistence modules and barcodes. Our goal is to read and understand parts of the paper "Principal Component Analysis of Persistent Homology..." by Vanessa Robins and Kate Turner (ArXiV 1507.01454v1). |

Learning objective | To get familiar with the basic concepts of topological data analysis and see some applications thereof. |

Literature | Herbert Edelsbrunner and John L. Harer: Computational Topology, An Introduction. AMS 2010 |

Prerequisites / Notice | Participants are supposed to be familiar with singular homology. |

**401-5110-00L** | **Number Theory Seminar** | 0 credits | 1K | Ö. Imamoglu,
P. S. Jossen,
E. Kowalski,
P. D. Nelson,
R. Pink,
G. Wüstholz |

Abstract | Research colloquium |

Learning objective | |