Prosjekt- og masteroppgaver for Gereon Quick
Jeg skal beskrive mine forslag av prosjekter og oppgaver på engelsk. Det er ingen krav til at oppgaven må forfattes på engelsk. Vi kan diskutere alt på norsk, og oppgaven kan også skrives på norsk. Men literaturen vi vil bruke, er vanligvis skrevet på engelsk. Derfor kan det gi mening å begynne å tenke på matematikk på engelsk. Du bestemmer!
In my own research I am mostly interested in algebraic topology and algebraic geometry and their interactions.
Algebraic geometers try to understand the solutions of polynomial equations with coefficients over different fields, for example the complex numbers, the rational numbers, or a finite field, using geometric ideas and concepts. This is directly related to many classical problems in number theory and geometry.
One origin for algebraic topology was the attempt in the 19th and beginning 20th century to understand complicated functions, that arise for example in physics, using geomtry and algebra. Algebraic topologists have developed many powerful tools that now play an essential role in many areas of mathematics, and many fundamental mathematical problems can be formulated and solved using algebraic topology. Since the 1990s algebraic topology and algebraic geometry overlap in motivic homotopy theory.
I recommend that you attend some of the following courses before you start your project. Which one depends on the individual project.
Here are some suggestions for projects. More information will follow soon. There are a lot of other exciting projects we can work out together according to your interests and background.
If you are interested in working on a project with me, I am looking forward to hearing from you. Just contact me!
Chern-Weil theory and abstract homotopy
Chern-Weil theory is a beautiful story that connects some of the main players in algebraic topology and differential geometry, such as characteristic classes, vector and principal G-bundles, Lie groups and Lie algebras, differentical forms, connections, curvature etc. A key role is played by the Chern-Weil homomorphism which does roughly the following. For a given Lie group G and its Lie algebra g, it describes the characteristic classes of a principal G-bundle as certain complex valued functions on g which are invariant under the natural action of G. In particular, it expresses an elegant and deep connection between fundamental geometric and topological concepts.
Freed and Hopkins have interpreted the Chern-Weil theory in the context of abstract homotopy theory. The goal of this project is to understand this modern point of view of a classical theory and to compute the invariants involved for suitable examples.
Smooth structures on spheres and manifolds
Given a topological manifold M, can we equip M with a differentiable structure? And if yes, how many different structures are there? For example, are there any differentiable structures on the n-dimensional sphere other than the standard one? These questions are at the heart of differential topology and have inspired many mathematicians to create beautiful theories. In the 1950s, Milnor showed that there are indeed such "exotic" differentiable structures on the 7-sphere. Kervaire on the other hand showed that not every topological manifold is also differentiable. Kervaire and Milnor then joined forces and studied the differentiable structures on n-spheres for all dimensions. They were able to determine all these structures, except for a possile factor of 2 in certain dimensions. This factor is connected to the so-called Kervaire invariant and the question in which dimensions there can be manifolds with Kervaire invariant one. To determine these dimensions became known as the Kervaire-Invariant-One-Problem. It took in fact several decades unitl Hill, Hopkins and Ravenel solved this famous problem (almost completely) in their seminal work in 2009. Their solution relies on groundbreaking new techniques in equivariant stable homotopy theory.
The goal of this project is to study some favourite part, depending on one's personal interest, of this story in detail and to learn the important techniques and ideas that are highly influential in many areas of mathematics.
Algebraic vector bundles on surfaces and 3-folds
Let X be a complex affine algebraic variety (the set of zeroes of a system of polynomial equations with 3 variables and complex number coefficients). Such an X is in particular also a nice complex manifold. If E is a topological complex vector bundle on X (considered as a nice manifold), then E has associated Chern classes in the singular cohomology of X. Then one can ask a fundamental question: Under which conditions on the Chern classes of E is E algebraic (i.e. we can construct E by just using polynomial equations)? In dimensions 2 and 3, it is known that a certain "algebraicity" condition on the first two Chern classes suffices. In dimension 4, Asok, Fasel and Hopkins have very recently shown that this condition is not enough to make sure that E is algebraic.
The goal of this project is to understand the situation in dimension 2 and to study concrete examples. Depending on the candidate's prior knowledge, one can then go further and have a look at the obstructions constructed by Asok-Fasel-Hopkins.