# Seminars in Algebra

## Fall 2020

## December 15th, 2020, at 14:15, digital only on Zoom

**Title:** Morphism spectra in exact (\(\infty\)-)categories

**Speaker:** Erlend Due Børve

**Abstract:** We sketch the proof of a (rather old) result of Retakh: If \({\cal C}\) is an exact category in which \(A\) and
\(B\) are objects, then the extension categories \({\cal E}xt^n_{\cal C}(B,A)\) form an \(\Omega\)-spectrum. Time permitting, we will explain how this generalises to exact \(\infty\)-categories, as well as the connection to topological enhancements of extriangulated categories. This is a report on joint work in progress with Paul Trygsland.

Slides can be downloaded here

## December 8th, 2020, at 14:15, digital only on Zoom

**Title:** Classification of d-representation-finite trivial extensions of Dynkin type algebras

**Speaker:** Tor Kringeland

**Abstract:** Given a Dynkin quiver $Q$, we consider for which values $d\ge 2$ the trivial extension algebra $T(kQ)$ is $d$-representation-finite.

Download talk slides here.

## November 10th, 2020, at 14:15, digital only on Zoom

**Title:** Cryptographic voting and lattices

**Speaker:** Kristian Gjøsteen

**Abstract:** Cryptographic voting systems often need some way to prove that a given collection of ballots is the decryption of a collection of ciphertexts, without revealing which ciphertext decrypts to which ballots. How to do this is well-known with classical methods, but these are not quantum-safe. We explain how this can be done using simple linear algebra and lattice-based cryptography.

## November 3rd, 2020, at 14:15 in room S1

**Title:** Tau-exceptional sequences

**Speaker:** Aslak Bakke Buan

**Abstract:** I will review some classical results about exceptional sequences for hereditary algebras, and discuss a possible generalization of such to general finite dimensional algebras. This is (partly ongoing) joint work with Marsh. It builds on recent work by Igusa-Todorov on signed exceptional sequences and also on work by many, including Adachi-Iyama-Reiten and Jasso, on tau-tilting theory.

## October 27th, 2020, at 14:15 in room S1

**Title:** From Linear Algebra to Zero-Knowledge Proofs (and More)

**Speaker:** Jiaxin Pan

**Abstract:** In this talk, we will start with basic linear algebra facts and then, by combining them with computationally hard problems in number theory, we will survey some of the current advanced cryptography in a "simple" yet intuitive way. We will focus on zero-knowledge proof systems which is the goldmine of cryptography (and theoretical computer
science).

## October 20th, 2020, at 14:15, digitally on zoom (not in person)

**Title:** d-abelian categories are d-cluster tilting

**Speaker:** Sondre Kvamme

**Abstract:** In 2014 Jasso introduced d-abelian categories as an axiomatization of d-cluster tilting subcategories, and he showed that any projectively generated d-abelian category is d-cluster tilting. In this talk we will explain how any d-abelian category is d-cluster tilting, without the assumption on the projective objects. If time permits, we will also explain how the proof can be adapted to give axiomatization of more general subcategories of abelian categories.

This talk will be held digitally on Zoom (no in-person meeting). The Zoom meeting ID is **935 3915 4938**. You will need the password which will be emailed Tuesday morning (or just click the link there).

## October 13th, 2020, at 14:15 in room S1

**Title:** Support theory for triangulated categories in algebra and topology

**Speaker:** Drew Kenneth Heard

**Abstract:** We will survey the support theory of triangulated categories through the machinery of tensor-triangulated geometry. We will discuss the stratification theory of Benson—Iyengar—Krause for triangulated categories, the construction by Balmer of the spectrum of a tensor-triangulated category, and the relation between the two.

## October 6th, 2020, at 14:15 in room S1

**Title:** Representations which are realizable over Set

**Speaker:** Steffen Oppermann

**Abstract:** A representation of a quiver (poset, category) in the category of sets can be turned into a linear representation by turning any set into a free vector space. In my talk I will discuss the question which linear representations arise in this way. This is part of an ongoing discussion with Ulrich Bauer, Magnus Botnan, and Johan Steen.

## September 29th, 2020, at 14:15 in room S1

**Title:** Support varieties for finite tensor categories

**Speaker:** Petter Andreas Bergh

**Abstract:** Tensor categories appear naturally in several settings, such as group representations, representations of Hopf algebras (quantum groups), fusion categories and conformal field theory. This talk is a report on recent joint work with Julia Plavnik and Sarah Witherspoon, where we develop a theory of cohomological support varieties for finite tensor categories.