Seminars in Algebra (Spring 2025)

Thursday, March 13th, 2025, at 10:15 in room H1

Speaker: Jonathan Komada Eriksen (COSIC, KU Leuven)

Title: The use of isogenies in cryptography

Abstract: Elliptic curves were likely used by your computer just now, so that you could read this abstract (or rather, so that someone monitoring your internet traffic could NOT read it). In fact, their usefulness in cryptography can hardly be overstated; they are used in everything from key-exchanges and digital signatures to factoring algorithms and primality proving.

Post-quantum cryptography, i.e. cryptography which is not broken by quantum algorithms (yet), is not an exception. This talk will be an introduction to isogeny-based cryptography, where isogenies, which are certain maps between elliptic curves, are used to create new cryptographic primitives. I will introduce some of the main tools used to create these primitives, accompanied by many explicit, computational examples. Towards the end of the talk, we will also look at the main current trend in the field, where higher-dimensional abelian varieties (generalising elliptic curves) are currently causing a mini-revolution, despite their seemingly complicated computational nature.

Thursday, April 3rd, at 14:15–15:00 in room 656

Speaker: Sondre Kvamme (NTNU)

Title: The reconstruction problem for \(d\)-cluster tilting subcategories of exact categories

Abstract: We prove that any weakly idempotent complete \(d\)-exact category is equivalent to a \(d\)-cluster tilting subcategory of a weakly idempotent complete exact category. Furthermore, we show that the ambient exact category of a \(d\)-cluster tilting subcategory is unique up to exact equivalence, assuming it is weakly idempotent complete. We explain how this gives an answer to the reconstruction problem for \(d\)-cluster tilting subcategories of exact categories.

Thursday, April 10th, 2025, 10:15–11:00 in room H1

Speaker: Cristian Alonso Baeza Miranda (NTNU)

Title: Tropical Discrete Logarithm & Maximum Cycle Mean as function

Abstract: Tropical algebra are computationally cheaper operations that have some relatively new uses in cryptography. Here we describe the basic elements of tropical arithmetic, their extension to matrices and an interesting phenomenon that occurs in the function that encapsulates the tropical eigenvalue of a certain type of matrices.

Thursday, May 15th, 13:15–14:00 in room 656

Speaker: Håvard Terland (NTNU)

Title: A Transitive Braid Group Action

Abstract: Exceptional sequences over hereditary algebras enjoy nice combinatorial properties and are still studied, for example by Igusa and Sen who have recently investigated a connection to rooted trees. Of particular interest to us, there is a well-known transitive braid group action on exceptional sequences over hereditary algebras (Crawley-Boevey 93). For tau-exceptional sequences, a proposed generalization of exceptional sequences introduced by Buan and Marsh, a mutation generalizing the mutation of exceptional sequences was recently proposed by Buan, Marsh and Hanson. This mutation is neither transitive nor respects the braid group relations in general. We show that for a class of cyclic Nakayama algebras, however, the mutation does induce a transitive braid group action.

Tuesday, June 3rd, 13:15–14:00 in room F4

Speaker: Hipolito Treffinger (University of Buenos Aires)

Title: Scattering diagrams for Artin algebras

Abstract: In this talk we show that the construction of the scattering diagram for finite-dimensional algebras over an algebraically closed field due to Bridgeland can be interpreted in terms of the lattice of torsion classes of the algebra. As a consequence we obtain a notion of scattering diagram for every Artin algebra. Time permitting, we will discuss some applications of our construction to the study of picture groups.

Thursday, June 5th, 10:15–11:00 in room F4

Speaker: Jacob Grevstad (NTNU)

Title: Representation type, cohomological Mackey functors and non-algebraically closed fields

Abstract: In recent work by Clover May and myself we classify the representation type of the cohomological Mackey algebra for cyclic \( p \)-groups. We show that it’s finite for \( G=C_p \), tame for \( G=C_4 \) and wild in all other cases. In this talk I will talk a little bit about the development of representation type, which tools we use to prove this theorem, and some questions and conjectures about what the correct notions of representation type should be for fields that are not algebraically closed.