Marvin Plogmann

Abstract: An interesting class of A_infinity-algebras, naturally arising in representation theory, are the Ext-algebras of simple modules for a finite-dimensional algebra. But how are these distinguished among general A_infinity-algebras?
We will see that there is a bijective correspondence, given by Koszul duality, between finite-dimensional algebras and coconnective, augmented, locally finite, homologically smooth A_infinity-algebras that are generated in degrees 0 and 1 in a precise, coherent sense.
I will discuss dg-realisation functors, as well as conditions for them to be an equivalence and how to use those to obtain the above correspondence and generalise it to proper connective dg-algebras.

This talk is based on arXiv:2603.03122[math.RT].

Abstract: An abelian length category can often be recovered from the derived endomorphism algebra of its simple objects. This leads to the following recognition problem: which dg algebras arise in this way?
In this talk, I will describe a characterization for module categories and certain abelian length categories. The key condition is that the associated simple-minded collection is 0-complicial, in the sense of Lurie.

This talk is based on arXiv:2603.03122[math.RT].

Abstract: By a theorem of Keller, an abelian length category with finitely many simple objects can be recovered from the derived endomorphism algebra of its simple objects. This leads to the following question: Up to quasi-isomorphism, which differential graded algebras arise in this way?

In this talk, I will explain a characterization for module categories and for abelian length categories with finitely many simple objects. The key condition is that the associated simple-minded collection is 0-complicial in the sense of Lurie. If time permits, I will also explain the role of d-complicial simple-minded collections, for d a natural number, in Koszul duality for proper connective differential graded algebras.

This talk is based on arXiv:2603.03122[math.RT].

Abstract: For a finite-dimensional algebra A, the category mod(A) can be recovered inside the bounded derived category of A as the full subcategory of objects whose cohomology is concentrated in degree 0. A natural enlargement is obtained by considering, for a fixed n>0, those objects whose cohomology is concentrated in degrees 1-n,...,0. This subcategory is called the n-extended heart of the standard t-structure. 

For n=1, one recovers the abelian category mod(A), while for n>1 the extended heart is no longer abelian. Nevertheless, it retains many features reminiscent of module categories; for example, it admits Auslander--Reiten sequences. I will explain why extended hearts are useful for studying finite-dimensional algebras and their derived analogues.

This talk is based on arXiv:2505.16560[math.RT] and (parts of) arXiv:2603.03122[math.RT].