Conferences
Algebra & Geometry Day in Thessaloniki, Wednesday 18 December 2024, Aristotle University of Thessaloniki.
Derived Representation Theory and Triangulated Categories, Conference, Monday 23 June to Friday 27 June 2025, Aristotle University of Thessaloniki.
Research Seminars
Topics in Representation Theory, Summer Semester 2024.
Koszul Duality, Winter Semester 2024. The members of the HART project covered some standard references for Koszul Duality (for instance, the paper of Belinson, Ginzburg and Soergel:" Koszul duality patterns in representation theory").
Summer Semester 2025:
Monday 10/03/25, 13:00 via Zoom, Matt Booth (Imperial College London).
Title: Global Koszul duality
Abstract: Conilpotent Koszul duality, as formulated by Quillen, Hinich, Lefevre-Hasegawa, Positselski, and others, shows that the bar-cobar adjunction forms a Quillen equivalence between the model categories of augmented dg algebras and conilpotent dg coalgebras. This Quillen equivalence underlies the modern approach to derived noncommutative deformation theory: indeed conilpotent coalgebras are to be thought of as the coalgebras of distributions on noncommutative formal derived stacks, and the above equivalence manifests itself as the slogan that `noncommutative algebras control noncommutative deformation problems’ (the corresponding abelianised statement, that `Lie algebras control commutative deformation problems’, was originally conjectured by Deligne in the ’80s and is now known as the Lurie-Pridham correspondence).
A natural question to ask is: how can extend this to the nonconilpotent (or, geometrically, global) setting? The global setup means that we should also drop the augmentations – as geometrically, these correspond to points – and hence work with curved objects instead, since curvature arises when one wants to take the Koszul dual of a nonaugmented object. The relevant extended bar and cobar functors were examined by Anel and Joyal, and the module-comodule Koszul duality was recently worked out by Guan and Lazarev. The only thing left to do is to provide model structures on the categories of curved dg (co)algebras making bar and cobar into a Quillen equivalence. This was done by Andrey Lazarev and myself, using the new notion of MC equivalence, and in this stalk I’II explain the story.
Monday,17/03/25,12:00 in M2, two talks:
(i) David Martinez Carpena (Barcelona). Title: "Algebraic theories and sketches in higher category theory".
(ii) George Raptis (AUTh). Title: "The simplicial chain coalgebra of a space and Koszul duality".
Monday, 24/03/2025, 12:00 in M2, two talks:
(i) Odysseas Giatagantzidis (AUTh). Title: "Arrow reductions, quasi-uniform Loewy length algebras and the Finitistic Dimension Conjecture".
(ii) Panagiotis Kostas (AUTh). Title: "Regular triangulated categories".
Abstract: We propose a notion of regularity for triangulated categories. The latter requires the identification of certain (intrinsic) triangulated subcategories of a given triangulated category. Following the same idea we will also introduce finite global dimension, Gorensteinness and injective generation. The final aim of this talk is to explain how the above are related under the assumption that the category in question is compactly generated. This is based on joint work with Chrysostomos Psaroudakis and Jorge Vitória.
Monday, 31/03/2025, two talks:
(i) Nikita Müller (Universität Mainz). Title: Higher derivators and universal properties, 12:00 in room M2.
(ii) Manuel Rivera (Purdue University). Title: DG Koszul duality and path spaces, 18:00 via Zoom.
Abstract: DG Koszul duality may be understood as a homotopy theoretic duality theory between algebraic and coalgebraic objects. One of its versions says that, the classical bar and cobar functors establish an equivalence of homotopy theories between augmented dg algebras considered under quasi-isomorphisms and coaugmented conilpotent dg coalgebras considered under a suitable notion of weak equivalence. There are many different versions and tweaks that also fall under a general umbrella of DG Koszul duality statements. I will discuss some of these statements while establishing an analogy (and a precise relationship) with path spaces and classifying spaces in algebraic topology. This will allow us to understand DG Koszul duality from combinatorial, topological, and geometric perspectives.
Monday 07/04/2025,12:00 via Zoom, Leonid Positselski (Czech Academy of Sciences).
Title: Derived categories of the second kind and Koszul duality
Abstract: I will start with introducing curved DG-rings and the hat construction (the correspondence between curved DG-rings and acyclic DG-rings), and explaining that the DG-category of DG-modules over the double hat DG-ring is equivalent to the DG-category of
CDG-modules over the original CDG-ring. Then I will define the derived categories of the second kind as triangulated categories attached to an abelian DG-category. Finally, I will survey various versions of absolute and relative Koszul duality, including the case of DG-algebras and CDG-coalgebras over a field, the case of nonhomogeneous Koszul rings over base rings, and the case of nonhomogeneous Koszul semialgebras over coalgebras over a field, emphasizing the role of the inverse hat construction.
Monday 28/04/2025, 13:15 in M2, Andreas Hayash (AUTh).
Title: Quantum geometric Langlands and the fundamental local equivalence via factorization modules.
Monday 19/05/2025, 12:15 in M2,
Øyvind Solberg (NTNU. Title: Quantum Decomposing modules over finite fields.
Miltiadis Karakikes (NKUA). Title: Equivariant recollements and singular equivalences