I will present some results from the joint work with Ralf Meyer [Isomorphism and stable isomorphism in “real” and “quaternionic” K-theory, New York J. Math. 31 (2025) 690–700]. 

In this talk, we speak about the Burau representation faithfulness of the four-string braid group B4, which is a well-known open problem. It is equivalent to the faithfulness of the Jones representation, which itself is related to the Jones hypothesis stating that the Jones polynomial detects the unknot. I will review the Garside and dual Garside structures on the braid group and their role in the Lawrence-Krammer-Bigelow representation. Finally, I will discuss the current progress on the Burau representation faithfulness problem in collaboration with my co-authors.

A classical result in relative homological algebra asserts that a precovering class of modules closed under direct limit is always covering. Enochs' conjecture asks if the converse of this result is true. In the talk we will give a conection between Enochs' conjecture and the existence of certain relative almost free modules, suggesting that Enochs' conjecture may be independent of ZFC.

Using methods of stable homotopy theory, the category of symmetric quasi-coherent sheaves associated with non-commutative graded algebras with extra symmetries is introduced and studied in this paper. It is shown to be a closed symmetric monoidal Grothendieck category with invertible generators. It is proven that the category of quasi-coherent sheaves on a projective scheme is recovered out of symmetric quasi-coherent sheaves.

Two elements a, b in a ring R form a right coprime pair if aR + bR = R. In this talk, we will define right strongly exchange rings in terms of descending chains of right coprime pairs, and we will show that they are semiregular. We will also demonstrate that the class of right strongly exchange rings contains left injective, left pure-injective, left cotorsion, local, or left continuous rings. This allows for a unified study of all these classes of rings in terms of the behavior of descending chains of right coprime pairs. We will also extend this concept to coprime summable families of endomorphisms of a module and provide a general criterion to determine when a module satisfies the full exchange property. In particular, we will use this criterion to prove that flat cotorsion modules enjoy this property. (Joint work with M. Cortés-Izurdiaga and Ashish Srivastava).

For several decades by now a very general kind of duality has been available, embedding the opposite of the category of all algebras from an equational variety V into the classifying topos of V.

The celebrated Birkhoff theorem tells us that the equational class of all algebras which satisfy the same identities as those from some class of algebras K coincides with the closure of K under Homomorphic images of Subalgebras of Products of algebras from K.

Under the above duality embedding, these operators correspond - in the same order - to taking subobjects of quotients of coproducts of objects in a certain category of sheaves. When such sheaves can be realized as sheaves of sections of local homeomorphisms over a space X, succession of the latter operations turns out to correspond to another well-known procedure: gluing a local homeomorphism over a space X from a bunch of copies of open subsets of X.

In our work we are trying to employ this correspondence to the study of varieties of Heyting algebras. In the talk, we will present some of our findings, along with some other interesting examples of entirely different kinds. Some questions that remain open to us will be mentioned.

In his 1986 paper, Willems introduced the notion of the jet completeness property, which can be interpreted as a local-global principle. We refer to this as Willems’ General Principle.  In addition to this, we introduce a related property, which we term Willems’ Special Principle.  We demonstrate that these two principles, when combined with the natural assumptions of linearity and invariance under differentiation, uniquely characterize linear differential systems among all continuous dynamical systems.

I will introduce the bicategory of C*-algebras and correspondences and briefly review its usefulness as a setting to study Morita equivalence and generalisations of group actions. Then I will explain that the bicategory of proper correspondences is a simplicial localisation of the category of C*-algebras at corner embeddings.

In this talk, we give a characterization of those quantaloids (categories enriched in the symmetric monoidal closed category of sup-lattices) that are equivalent to modular categories over quantales. Based on this characterization, necessary and sucient conditions are derived for two quantales to be Morita-equivalent, i. e. have equivalent module categories. As an application, it is shown that the category of internal sup-lattices in a Grothendieck topos is equivalent to the module category over a suitable chosen ordinary quantale. 

We present a survey of techniques from relative homological algebra as applied to the study of Kasparov's bivariant  K-theory. We will focus on developing the framework necessary to derive universal coefficient theorems (UCTs) within this context and showcase results when one considers separable C*-algebras with finite group actions. This exposition is based on joint work with Ralf Meyer. 

The classical Farrell-Tate cohomology measures the failure of duality in group (co)homology. Brown in 70s gave a general method for computing the p-local part of the Farrell-Tate cohomology. Using Brown’s methods Farrell-Tate cohomology has been computed for various arithmetic groups, mapping class groups and Out(F_n)-s, outer automorphism groups of Free groups. Later Klein introduced generalised Farrell-Tate cohomology with coefficients in an arbitrary spectrum. In this project we investigate the Farrell-Tate K-theory of Out(F_n). We will show that for any discrete group with finite classifying space for proper actions, the p-adic Farrell-Tate K-theory is rational. Then using Lück’s Chern character, we will give a general formula for the p-adic Farrell-Tate K-theory in terms of centralisers. In particular, we apply this formula to Out(F_{p+1}) which has curious p-torsion behaviour: It has exactly one conjugacy class of a p-torsion element which does not come from Aut(F_{p+1}). Computing the rational cohomology of the centraliser of this element allows us to fully compute the p-adic Farrell-Tate K-theory of Out(F_{p+1}). As a consequence we show for example that the 11-adic Farrell-Tate K-theory of Out(F_{12}) is non-trivial, thus detecting a non-trivial class in odd K-theory of Out(F_{12}) without using any computer calculations.

This is joint work with Naomi Andrew.

We describe a relationship between the following two notions that were claimed to be unrelated:

1) a Frobenius functor, i.e., a functor that is simultaneously left and right adjoint,

2) a Frobenius monoidal functor, i.e., a lax and oplax monoidal functor whose structural morphisms satisfy compatibilities similar to those of a Frobenius algebra.

For this, we identify general conditions, formulated using the projection formula morphisms, for a functor that is simultaneously left and right adjoint to a strong monoidal functor to be a Frobenius monoidal functor.

Moreover, we identify stronger conditions for the adjoint functor to extend to a braided Frobenius monoidal functor on Drinfeld centers.

This is joint work with Johannes Flake (Universität Bonn) and Robert Laugwitz (University of Nottingham). 

Link to the corresponding paper.

If R is a ring we consider those additive functors on the category of R-modules which commute with direct products and directed colimits; such functors have nice presentations in terms of matrices with entries in R and they form a locally coherent Grothendieck category.  Evaluation at an R-module M is finite-type localisation of the functor category and the localised functor category acts on the definable category of modules generated by M.  These finite-type localisations are in natural bijection with the closed sets of a topological space - the Ziegler spectrum - which was originally defined in the model theory of modules.  I will describe and illustrate these ideas and also indicate how they extend beyond categories of modules, for example to categories of sheaves and to triangulated categories.

In a paper with Anand Pillay (Notre Dame University) in the Journal of Algebra 670, (2025), we gave a model-theoretic characterization—in terms of descending chain conditions on certain first-order formulas—of the following algebraic phenomenon. Given any class of finitely presented modules, A, over a ring R, and C = lim add A, the class of all direct limits of finite direct sums of modules from A. When is every countably generated module from C a direct summand of a(n arbitrary) direct sum of modules from A? I will present two new applications of this result, one with A, the class of cyclically presented modules, and the other with A, the class of cyclic finitely presented modules. Both lead to interesting natural classes of first-order formulas, which have played a role in the model theory of the corresponding modules. 

We will see how the telescope conjecture for compactly generated triangulated categories can be seen as a particular case of a problem raised by Miller in 1975. This naturally leads to bilocalisations of module categories. In this talk we will show the connection between the local finite presetability of a bilocalizing subcategory of a module category connects with the mentioned telescope conjecture.

Using a two cocycle deformation of a double biproduct of quasi-Hopf algebras we will construct a new class of quasi-Hopf algebras similar to the Drinfeld-Jimbo quantum groups.

Extending the classical notion of cohomological Mackey functors of finite groups to profinite groups yields an abelian group category which can be used for many problems in the theory of profinite groups. We will illustrate how this approach can be also used for studying finitely generated pro-p groups with more than 1 ending the sense of A. Korenev. The talk gives a summary of an ongoing collaboration with Pavel Zalesskii.