Differential Galois Theory
Differential Galois theory is a Galois theory for differential equations and
mimicks the usual Galois theory for polynomials. The main difference is that the
splitting fields of differential equations (so-called Picard-Vessiot extensions)
are no longer algebraic extensions and the differential Galois groups are
therefore in general not finite groups. Just as usual Galois groups come with a
representation as the permuation groups of the zeros of a polynomial, the
differential Galois groups come with a representation as a linear algebraic
group acting on the vector space of solutions of a differential equation.
A central part of my research is devoted to
the inverse problem: Does every linear
algebraic group occur as the Galois group of some differential equation?
Of course, the answer to this question heavily depends on the differential field
under consideration. My thesis solves the case of a rational function field with
algebraically closed field of constants of characteristic zero. This had been
an open problem since Tretkoff and Tretkoff settled the so-called classical case
when the constants are the complex numbers in the late 70's.
Moreover, I am interested in differential Galois theory in positive characteristic
and relations between differential equations in characteristic p and p-adic
differential equations, as well as additional structures such as Frobenius modules.
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