# Omar Leon Sanchez - Existence of PPV-extensions over differentially large fields.

Dates: | 6 November 2019 |
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Times: | 15:00 - 16:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Current University students |

Speaker: | Omar Leon Sanchez |

Join us for this seminar, part of the Logic Seminar Series. Note the unusual room: University Place 3.213

In a 2016 paper, Crespo, Hajto, and van der Put proved that if a differential field (K,d) is real (or p-adic) and its constant subfield C_K is real closed (respectively p-adically closed) then any homogeneous linear differential equation over K has a Picard-Vessiot extension which is real (respectively p-adic). Turns out that this type of existence result holds as long as C_K is existentially closed in K (as fields) and the first Galois cohomology of C_K is finite for any linear algebraic group (so, by a classical result of Serre, it suffices for C_K to be bounded as a field). This was observed in a paper of Kamensky and Pillay, who provided a model-theoretic proof (other proofs use the Tannakian formalism). In this talk I will give an overview of the model-theoretic proof and explain how one can attempt to extend the argument to show existence of Parameterized Picard-Vessiot (PPV) extensions; there are some subtleties of course that have to do with differential Galois cohomology and so at this point we assume differential largeness. All definitions will be explained. This is joint (but separate) work with J. Nagloo, A. Pillay, and M. Tressl.

### Speaker

Omar Leon Sanchez

Organisation: University of Manchester