Supervisor: Gunther Cornelissen

Second reader: Valentijn Karemaker

Abstract:

My thesis is about Hilbert's tenth problemand Diophantine maps. Hilbert's tenth problem asks for an algorithm that givena polynomial equation with coefficients in the integer decides whether or notthis equation has a solution over the integers. In 1970 it has been proven thatsuch an algorithm does not exist. But mathematicians did not stop here, sincewe can easily generalize this problem: Does there exist an algorithm that,given a system of polynomial equations with coefficients in some ring R_0, decideswhether or not it has a solution in some ring R that contains R_0? The maintool that is used to get negative answers for various rings is to reduce theproblem to the integers. I have formalized this type of reduction viaDiophantine maps, which I will introduce in my talk. 

Then we will move to Hilbert's tenthproblem over some noncommutative rings. In my thesis I have proven severalresults about Hilbert's tenth problem over the twisted polynomial rings andvariants thereof. I also looked at differential polynomials and its divisionring of fractions. In my talk I will introduce some of these rings and some ofthe results I have proven.

I will assume knowledge of (polynomial)rings and fields and some knowledge of logic.

Atiyah's Real K-Theory is a C2-spectrum with fixed points KO and underlying spectrum KU. We present a modern-style construction of it and give streamlined proofs of its basic properties. 

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