During the spring semester 2021 I arranged a seminar, mainly for the master students in mathematics at NTNU, where the willing students presented what they are doing on their master project. The reason was to further strengthen the social and academic bond between the students, as well as motivate and inform younger students what they can do during their own projects later in their studies. Because of COVID-19 we met digitally on a weekly basis. Below you can find information and slides from the different talks in chronological order.

Torgeir Aambø, 27.01.2021

Title: Formal dg-algebras
Abstract: A sad story one quickly learns in topology is that we can’t fully classify topological spaces by their cohomology. The cochain complex gives much better knowledge about the space we are working with, but it is often much harder to compute compared to the cohomology. Instead of topological spaces we can work more generally with differential graded algebras (dg-algebras). Since cohomology is easier to compute, a natural question to ask is; for which dg-algebras do we have “all information” if we know its cohomology? These dg-algebras are called formal, because we can “formally reconstruct them from their cohomology”. In this talk I will attempt to define properly the above situation, look at some properties of these dg-algebras, as well as how we can abstract an important part of the theory to so-called $A_{\infty}$-algebras.
Slides, Thesis

Kristoffer Brakstad, 03.02.2021

Title: Stable derivators and representation theory
Abstract: Triangulated categories have had a big success in many fields of mathematics. However, right from the beginning there where some problems with the axioms that became apparent. Many solutions were proposed to “fix” these issues and we will talk about one of them today; Derivators. We go into some depth about the motivation, and build up for the theory, before moving into more conceptual ideas. The plan is to give a good understanding of what a derivator is, how it enriches the theory of triangulated categories, and finally see some areas of representation theory where derivators has played a role in generalizing the results.
Slides, Thesis

Anders A. Andersen, 10.02.2021

Title: Differential geometry in abstract homotopy theory
Abstract: Classifying spaces of objects sometimes lie outside the category of those objects. An known example of this is the category of principal bundles with connection. But in 2013, M. Hopkins and D. Freed showed that there is a way to fix this issue, using abstract homotopy theory. This also gives a way to calculate all invariants of such objects. We will give an introduction to the language used, give a feel for the article, and then, if time allows it, look at what happens in the complex case.
Slides, Thesis

Katrine Johansen, 17.02.2021

Title: Post-quantum cryptography
Abstract: Quantum computers are getting better and better by the day. A big consequence of this is that our crypto systems will, eventually, no longer be secure and our information no longer be safe from the evil of the world. This is bad, doomsday seems to be upon us, oh no.

But fear not for all hope is not lost! There is multiple quantum safe systems being developed as you read this. And together we will try to understand how these new systems work. Let’s overcome the Sauron’s of our world!
Slides Thesis

Jacob F. Grevstad, 24.02.2021

Title: The finitistic dimension conjecture for finite dimensional algebras
Abstract: The finitistic dimension is an important homological invariant of the representation theory of a finite dimensional algebra. It’s a strengthening of the invariant of global dimension and measures in a sense how complicated projective resolutions in of $\Lambda$-modules can be.

The finitistic dimension conjecture, formulated around 1960, states that this invariant is always finite. We will give an overview of what work has been done on the conjecture up to present day, and for which families of algebras the conjecture is known to hold.
Slides, Thesis

Endre Rundsveen, 03.03.2021

Title: Multiparameter persistence
Abstract: Topological data analysis is a relatively new and promising tool for processing large amounts of data. During the development of this tool, a result from representation theory in Algebra was rediscovered, namely that vector space representations over a rectangular grid do not have a nice decomposition theory. This is and was a problem for data analysts who want to use topological data analysis, as these representations come up often and naturally in this context.

I will roughly try to convey the idea behind topological data analysis before I explain why representations over multidimensional grids show up. Towards the end I will review results from Bauer et. al (2020) which shows that in some special cases the decomposition will be a little less hopeless. Finally, I will address a still open question within cotorsion and torsion pairs in representation categories.
Slides, Thesis

Jørgen Juel, 17.03.2021

Title: Elliptic Curves, Isogenies and Quaternions
Abstract: Some post-quantum protocols rely on hardness assumptions related to finding certain isogenies between elliptic curves. In this talk I will explain what elliptic curves and isogenies are, give a high level overview of their cryptographic applications, and finally how we can use the structure of a specific quaternion algebra to break the security assumption. By the end of this talk you will have some pictures in mind when thinking of elliptic curves and elliptic curve cryptography, and hopefully be inspired to look further into how one can break the security and ruin the hard work of all the isogeny-cryptographers out there.
Thesis

Ole A. Berre, 24.03.2021

Title: Classifying exact structures on idempotent complete categories
Abstract: Given an additive category $\mathcal A$ we can impose different exact structure on $\mathcal A $ yielding different exact categories. In this talk will look at a way to classify all exact structures of an idempotent complete additive category through module categories. In the end we will see an example where we classify all exact structures on a quiver.
Slides, Thesis

Jacob Sjødin, 14.04.2021

Title: Computation on encrypted data through homomorphic encryption
Abstract: Fully homomorphic encryption allows us to compute arbitrary polynomials on encrypted data. This was a task long thought to be impossible, and It was only a decade ago that the first scheme was conceptualized. There are a variety of topics connected to homomorphic encryption, such as algebra, arithmetic, probability and many more. Even though we have schemes for computing polynomials over encrypted data, there are many nuances to this computation. We will explore these nuances in this talk, as well as look at a particular homomorphic encryption scheme.
Slides, Thesis

Elias K. Angelsen, 21.04.2021

Title: The K-theory and Morita Equivalence Classes of Noncommutative Tori
Abstract: When studying time-frequency analysis, one encounters unitary translation and modulation operators of great importance. These make up a framework for an operator algebraic approach to time-frequency analysis and can be studied through noncommutative tori, the universal $C^*$-algebras generated by two such operators. Noncommutative tori shows up in a lot of places, for example as dynamical systems in terms of rotation algebras, in representation theory as twisted group algebras and even in theoretical physics, as interesting arenas for Yang-Mills theory on noncommutative spaces.

We develop tools originating from ideas in algebra, such as Hilbert $C^*$-modules, which take us towards the operator algebraic formulation of Morita equivalence and we generalize topological K-theory to noncommutative scenarios and present powerful consequences, such as the classification theorem of AF-algebras. Towards the end, we attempt to apply the study of Morita equivalences and K-theory to noncommutative tori through the work of Rieffel and Pimsner-Voiculescu to give a classification of noncommutative tori.

The aim of this talk is giving an overview of some of the beautiful theory and fruitful techniques coming from collaboration between several branches of mathematics, and hence the focus will lie on developing the theory, but as most students have not (knowingly) touched a $C^*$-algebra before, we will certainly take the time to define the basics and state the celebrated results instead of rushing to the classification of noncommutative tori.
Slides, Thesis