Matematik Bölüm Seminerleri : “Differential Galois Theory”

Konu : “Differential Galois Theory

Konuşmacı : Ahmet Berkay KEBECİ

Tarih: 19.12.2018

Saat: 14:00

Yer: Matematik Bölümü D-II

ÖzetAbstract:  Galois Theory is a powerful tool to study the roots of a polynomial. In this sense, the dierential Galois theory is the analogue of Galois theory for linear dierential equations. In this talk, we will construct the notion of a dierential eld and Picard-Vessiot extension of a linear dierential equation as the analogue of a eld and the splitting eld of a polynomial, respectively. Then we dene the dierential Galois group and we see that it has a linear algebraic group structure. Using those, we have a Galois correspondence for algebraic subgroups of the dierential Galois group similar to the correspondence in the Galois theory. Moreover, we nd a characterization for Liouvillian functions corresponding to the solvability of G0 , the identity component of dierential Galois group G. This is the analogue of the characterization of solvability by radicals of a polynomial equation in Galois theory. As a corollay we find that identity component of the dierential Galois group of an elementary function is abelian. Using this tool we can prove that  | e-x2 cannot be expressed in terms of elementary functions.

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