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

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 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.

Gabriel’s structure theorem

Prerequisites: Modules and Homological Algebra

Gabriel’s structure theorem is one of the reasons why quivers are such a powerful tool within representation theory. It states that every finite-dimensional algebra over an algebraically closed field is Morita equivalent to the quotient of a path algebra of a finite quiver by an admissible ideal of relations. The goal of this thesis is to explore this theorem and its variants for non-algebraically closed fields and/or finite length categories.

Quasi-hereditary structures on hereditary algebras

Prerequisites: Modules and Homological Algebra, possibly Lie algebras

The prototypical example of a hereditary algebra is the path algebra of a finite acyclic quiver. It is hereditary in the sense that submodules of projective modules inherit the property of being projective. The class of quasi-hereditary algebras generalises the class of hereditary algebras, in particular the acyclicity of the quiver, encoded by a total order on the set of simple modules. This class of algebras is important in Lie theory as it comes with a class of standard modules resembling the properties of Verma modules. Hereditary algebras are precisely the algebras which are quasi-hereditary with respect to every total order. However, different total orders can yield the same standard modules. The goal of this project is to explore recent work of Flores, Kimura, and Rognerud as well as work of Rodriguez Rasmussen counting quasi-hereditary structures giving rise to different standard modules.

More exotic representation theory

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.

\(A_\infty\)-structures on Ext-algebras of simples

Prerequisites: Modules and Homological Algebra

The notion of an \(A_\infty\)-algebra is a generalisation of the notion of a differential graded algebra (that is a graded algebra together with a self linear map \(d\)such that \(d\) satisfies the proeduct rule, and \(d^2=0\)). According to a result of Kadeishvili, one can put an \(A_\infty\)-structure on the Ext-algebra of the direct sum of all simple modules. According to work of Keller–Lefevre-Hasegawa this is related to the quiver and relations of an algebra via a concept called \(A_\infty\)-Koszul duality, a theory that is not only important in algebra but also in symplectic geometry. The goal of this thesis is to explore this relationship in examples.