Disputation: Håkon Andreas Kolderup
Doctoral candidate Håkon Andreas Kolderup at the Department of Mathematics, Faculty of Mathematics and Natural Sciences, is defending the thesis "Geometric and arithmetic properties of motivic cohomology theories" for the degree of Philosophiae Doctor.
Håkon Andreas Kolderup
The University of Oslo is closed. The PhD defence and trial lecture will therefore be fully digital and streamed directly using Zoom. The host of the session will moderate the technicalities while the chair of the defence will moderate the disputation.
Ex auditorio questions: the chair of the defence will invite the audience to ask ex auditorio questions either written or oral. This can be requested by clicking 'Participants -> Raise hand'.
The meeting opens for participation just before 13.15 pm, and closes for new participants approximately 15 minutes after the defense has begun.
Submit the request to get access to the thesis.
19th June, 10.15, zoom
The Artin comparison theorem between singular and etale cohomology with finite coefficients
Join the trial lecture (deactivated)
The meeting opens for participation just before 10.15 pm, and closes for new participants approximately 15 minutes after the trial lecture has begun.
Main research findings
The theme of my thesis lies at a borderland between algebraic topology and algebraic geometry called motivic homotopy theory. I have studied so-called motivic cohomology theories, which are tools used to distinguish certain mathematical shapes from one another.
In algebraic topology the primary goal is to classify geometric shapes, much like the goal of particle physics is to classify the fundamental particles. A way of attacking this problem is to construct suitable invariants, which can be used to distinguish different shapes. For an early example of an invariant we can go back to 1758. In this year, the famous Swiss mathematician Leonhard Euler proved that the alternating sum of the number of vertices, edges and faces of a polygon always equals 2. This sum, now known as the Euler characteristic, is an important example of an invariant of a geometric object. Later it was discovered that the Euler characteristic is intimately linked with so-called cohomology groups, which are indispensable tools in the algebraic topologist’s toolkit.
However, a geometric object often arises as the set of solutions to an equation. These shapes, known as varieties, are the main objects of study in algebraic geometry. Varieties are equipped with a richer structure coming from the combination of arithmetic and geometric information. The classical cohomology groups are not sufficient to capture this extra information. Therefore new tools were needed, and these tools now go under the name motivic cohomology theories, or motives.
In my thesis I have shown how certain geometric data, known as correspondences, give rise to a parametrized family of motives. I have focused in particular on so-called Milnor-Witt motives, which can in a certain sense be thought of as the initial motivic cohomology theory. I have also studied how motives can be viewed as modules over appropriate ring spectra. Finally, I have made some explicit computations in Milnor-Witt motivic cohomology highlighting how motivic cohomology theories capture arithmetic information contained in varieties.