# 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!

## General Information

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 enumerative 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.

## Recommended courses

I recommend that you attend some of the following courses before you start or while you are working on your project. Which one depends on the individual project.

## Some Projects

Here are some suggestions for projects. 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!

### Massey products in Galois cohomology and arithmetic geometry

Massey products are invariants which measure whether a differential graded algebra may contain more information than its cohomology. While there are many examples of non-vanishing Massey products in arithmetic, Hopkins and Wickelgren showed that all triple Massey products of degree one classes in the mod 2-Galois cohomology of global fields of character- istic different from 2 vanish, i.e., contain zero, whenever they are defined. Minac and Tan then showed the vanishing of mod p-triple Massey products for all fields. Moreover, they formulated the Massey vanishing conjecture stating that, for all fields k, all n ≥ 3 and all primes p, the n-fold Massey product of degree one classes in mod p-Galois cohomology should vanish whenever it is defined. The work of Hopkins–Wickelgren and Minac–Tan has inspired a lot of activity in recent years. In this project we will explain some of the existing results on the vanishing of Massey products in detail and will try to explore new cases for the vanishing in various cases which are relevant in arithmetic geometry.

### Examples of non-vanishing Massey products for complex algebraic varieties

The rational homotopy type of a projective smooth complex variety is formal and therefore all rational Massey products for such spaces vanish. In cohomology with finite coefficients, however, nontrivial Massey products exist. This was first deomnstrated by Ekedahl by constructing concrete examples of Massey products on smooth complex surfaces. More recently, Bleher, Chinburg and Gillibert discussed Massey products of degree one classes for complex algebraic curves and showed that they vanish. Over other fields, however, there are nonvanishing triple Massey products for elliptic curves. This project studies the examples of Ekedahl and explores the differences of algebraic dimensions one and two for the Massey products.

### Enumerative geometry and motivic homotopy

Enumerative geometry is a classical branch of algebraic geometry in which one tries to understand in how many different ways a geometric problem can be solved. A famous example is the computation of the number of lines on a smooth cubic surface in three-dimensional projective complex space by Salmon and Cayley. This number is 27. With roots reaching to the beginnings of geometry, it has now connections to many areas in mathematics, e.g. characteristic classes via Schubert calculus, moduli spaces, mathematical physics and equivariant K-theory.

In this project we aim to study an exciting new development, initiated by Kass and Wickelgren, who show that there are arithmetic refinements of many classical counts of geometric objects. The refinement is based on an important joint branch of algebraic geometry and topology: motivic homotopy theory. This project could consist of studying one of the very interesting and rich recent papers in this direction and attempts to perform own new calculations. Such calculations can in principal be done only based on a background on commutative algebra and quadratic forms and could be supported by writing computer programs in for example Python. But there are plenty of theoretical problems waiting to be explored. Learning about enumerative geometry and motivic homotopy is an excellent preparation for further adventures in modern theoretical 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.