Julian Külshammer

Julian Külshammer

Associate professor working in representation theory

Supervision

Current PhD students

PhD projects in Sweden usually take about 5 years, including a 20% teaching load. Somewhere roughly in the middle, a licenciate thesis is handed in. PhD positions are usually advertised by the department in March of each year. Feel free to contact me via email if you are interested in writing a PhD thesis with me.

Main supervisor

Co-supervisor

Former PhD students

Former master students

Former bachelor students

A selection of possible theses topics

Bachelor theses

Below I list some topics, which I could imagine giving out as master thesis, but I am happy to discuss and find a topic which is more suited for you. Feel free to contact me.

Linear Algebra

\(\mathbb{F}_1\)-linear algebra

Prerequisites: Linear algebra II. Linear algebra III could be useful as well

Early on, one learns about vector spaces over the real numbers. In later courses, vector spaces over finite fields are also considered. However, a field generally has at least two elements, \(0\) and \(1\), which are assumed to be distinct. In some areas of modern mathematics, the concept of a “field with one element” has appeared. This is not an actual object, but more of a guiding principle. While in more advanced settings, it is not even clear what the definitions in this theory are, there is some agreement that a vector space over the “field with one element” is just a pointed sets. The goal of this thesis project is to explore analogues of classical concepts in Linear Algebra in the case of the “field with one element”.

Weyr canonical form

Prerequisites: Linear algebra III.

In Linear Algebra III an important topic is that of Jordan canonical form of matrices. An alternative to this normal form, that is much less popular is called Weyr canonical form. It differs from the Jordan canonical form by some permutation of rows/columns. Though very similar, the Weyr canonical form has some advantages over Jordan canonical form, for example the set of matrices which commute with a matrix in Weyr canonical form admits an easier description. The goal of this thesis project is to describe the Weyr canonical form and investigate some of the advantages over the more familiar Jordan canonical form.

Linear algebra over skew fields

Prerequisites: Linear algebra II, or better Linear algebra III.

In basic undergraduate courses, linear algebra is developed over a ground field (or even a specific field like the real or complex numbers). The goal of this thesis project is to investigate theorems that still hold when one generalises to skew fields (that is one drops the assumption that the scalars commute), and also to study counterexamples to theorems that no longer hold in this setup, e.g. the transpose of an invertible matrix is not necessarily invertible.

Number theory

Primes is in P

Prerequisites: Algebra I, and some familiarity with concepts of complexity theory

The famous P vs NP problem roughly asks whether for each problem, for which one can test whether something is a solution in polynomial time, one can also construct a solution in polynomial time. The goal of the thesis project is to discuss the result that one can test in polynomial time whether a given natural number is a prime number.

Linear codes

Prerequisites: Linear Algebra III, some familiarity with Elementary Number Theory might be useful

The goal of this thesis project is to summarize the theory of linear codes with the highlight of the Theorem of Tietäväinen and Van Lint that every non-trivial perfect code over a finite field must have the same parameters as one of the Hamming or Golay codes.

Eisenstein integers and Fermat’s last theorem

Prerequisites: Algebra II, some familiarity with Elementary Number Theory might be useful

Fermat’s last theorem deals with the famous equation \(x^n+y^n=z^n\). It was finally proven by Wiles and Taylor in the 90s, but many special cases were known before. The case of $n=4$ is covered in the lecture on Elementary Number Theory and uses the method of infinite descent. A bit harder is the case of $n=3$. One of the possible proofs uses the Eisenstein integers, the extension of the integers by the primitive third root of unity. The goal of this thesis is to outline the proof of this case, which was first noted by Euler using different methods.

Abstract algebra

Lüroth’s theorem

Prerequisites: Algebraic structures

For a field \(K\) and the field of rational functions \(K(X)\), Lüroth’s theorem states that any intermediate field is of the form \(K(f(X))\) for some rational function in \(X\). The goal of the thesis is to outline a proof of this theorem. If time and motivation permits, applications of this theorem in geometry can be discussed.

Category theory and homological algebra

Adjunctions and monads

Prerequisites: Algebra II, and some familiarity with basic category theory as in Modules and Homological algebra

An adjunction of functors between categories is a generalisation of an equivalence between categories. A monad is a functor from a category to itself together with a `multiplication’ similarly to that a monoid is a set together with a multiplication. The goal of the thesis is to discuss the correspondence between adjunctions and monads and provide examples of this correspondence.

The salamander lemma

Prerequisites: Algebra II, and some familiarity with basic homological algebra as in Modules and Homological algebra

A standard method of proof in homological algebra is called diagram chasing. Roughly speaking one looks at some commutative diagram diagram of \(R\)-modules and homomorphisms and proves a statement by considering an element in one of these \(R\)-modules and chasing it through the diagram using injectivity, surjectivity of certain maps and exactness properties. The salamander lemma is a very general such lemma which can be proved using diagram chase and which has as a corollary some of the more familiar statements in homological algebra which are usually proven using diagram chasing, e.g. the snake lemma.

Miscellaneous

Catalan combinatorics

Prerequisites: Algebra I, possibly Combinatorics

The Catalan numbers 1,1,2,5,14,42,132,… is probably the most ubiquitous sequence in mathematics. They count various types of objects, for example triangulations of convex polygons, or binary trees. The purpose of the thesis is to explore the properties of this sequence and the proof technique of a bijective proof.

Theorem proving in Lean

Prerequisites: Any proof-based course in mathematics

Recently, the possibility to do interactive theorem proving with the help of a computer has gained a lot of popularity. The goal of the thesis would be to explore this possibility with the favourite kind of mathematics you learned during your undergraduate by implementing it in Lean and writing about the process, difficulties you encountered, design decisions you took, and what you learned along the way.

Master theses

Below I list some topics, which I could imagine giving out as master thesis, but I am happy to discuss and find a topic which is more suited for you. Feel free to contact me.

Representation theory of finite dimensional algebras

Representation type of algebras over truncated polynomial rings

Prerequisites: Modules and Homological Algebra

Let \(kQ\) be the path algebra of a finite quiver. Gabriel’s theorem states that there are only finitely many indecomposable \(kQ\)-modules up to isomorphism if and only if the underlying graph of \(Q\) is a Dynkin quiver. The goal of this thesis project is to discuss this theorem and generalisations where \(k\) is replaced by \(k[x]/(x^n)\).

Representation type of higher Nakayama algebras

Prerequisites: Modules and Homological Algebra

Each finite dimensional module over a finite dimensional algebra has a composition series, i.e. a filtration by submodules such that the corresponding subquotients are simple. An algebra is called a Nakayama algebra if every finite dimensional module has a unique composition series. For a Nakayama algebra there are only finitely many indecomposable modules up to isomorphism. Together with Gustavo Jasso, a few years ago we introduced the concept of a higher Nakayama algebra. These higher Nakayama algebras do in general no longer have only finitely many indecomposable modules up to isomorphism. The goal of this thesis project is to investigate under which conditions there are only finitely many indecomposable modules up to isomorphism. The question has been partially answered in a preprint by Shen Li from 2023.

Fuller’s recursive formula for global dimension

Prerequisites: Modules and Homological Algebra

Each finite dimensional module over a finite dimensional algebra has a composition series, i.e. a filtration by submodules such that the corresponding subquotients are simple. An algebra is called a Nakayama algebra if every finite dimensional module has a unique composition series. The goal of this thesis is to review Fuller’s recursive formula for the global dimension for Nakayama algebra.

More exotic representation theory

\(\mathbb{F}_1\)-representation theory

Prerequisites: Modules and Homological Algebra

Early on, one learns about vector spaces over the real numbers. In later courses, vector spaces over finite fields are also considered. However, a field generally has at least two elements, \(0\) and \(1\), which are assumed to be distinct. In some areas of modern mathematics, the concept of a “field with one element” has appeared. This is not an actual object, but more of a guiding principle. While in more advanced settings, it is not even clear what the definitions in this theory are, there is some agreement that a vector space over the “field with one element” is just a pointed sets. The goal of this project is to review the theory of representation theory of quivers over \(\mathbb{F}_1\).

Representations of continuous quivers

Prerequisites: Modules and Homological Algebra

Crawley-Boevey and Igusa–Rock–Todorov recently discussed the representation theory of the real line, which can be seen as a type $\mathbb{A}$ continuous quiver. The goal of this thesis project is to review some of their work.

Advanced homological algebra

\(A_\infty\)-algebras with enough idempotents

Prerequisites: Modules and Homological Algebra

It is well-known that a small \(\Bbbk\)-linear category can equivalently be described by an algebra with enough idempotents. In 2016/17 Manuel Saorín wrote a paper generalising this to differential graded categories, i.e. categories with a self linear map \(d\) such that \(d\) satisfies the product rule, and \(d^2=0\) (this is called a differential). The goal of this thesis project is to generalise this result even further to \(A_\infty\)-categories and develop a notion of \(A_\infty\)-algebra with enough idempotents.