# Trimester Seminar Series

**December 14, 2022 (CEST)**

3:00 - 4:00pm Yuqing Shi (Utrecht)

Title: Spectral Methods in Unstable Homotopy Theory

Abstract: Unstable homotopy theory concerns classifying topological spaces up to weak homotopy equivalences. Analogous to the chromatic structure of spectra, one can decompose the homotopy type of a topological space using v_h-periodicities for every natural number h, working p-locally for a fixed prime number p. The h = 0 case is about rational homotopy theory, where Quillen shows that we can understand the rational homotopy types of topological spaces via Lie algebras in rational spectra. In this talk I will briefly introduce the localisation of p-local topological spaces with respect to v_h periodic equivalences, for positive natural number h. This is also known as the unstable monochromatic layer of height h, which relates to its stable counterpart via the Bousfield--Kuhn functor. Up to v_h-periodic equivalences, Heuts shows that topological spaces are spectral Lie algebras in spectra, generalising naturally the Lie algebraic model for rational homotopy theory. Under this correspondence one can consider the Bousfield--Kuhn functor as the forgetful functor forgetting Lie algebra structures. I will talk about the universal property of the Bousfield--Kuhn functor, which relates to the costabilisation of v_h-periodic homotopy types.

4:30-5:30pm Josh Pollitz (Salt Lake city)

Title: Cohomological support varieties in local algebra

Abstract: Cohomological support varieties have been invaluable as a geometric approach to study the representation theory of algebras. They were imported to commutative algebra by Avramov in ‘89, and further developed by Avramov and Buchweitz in 2000 during their investigations of cohomology modules over complete intersection rings. In the past twenty years this theory has been further extended and applied by Avramov-Iyengar, Burke-Walker, Jorgensen, Stevenson, and myself, as well as many others. In this talk, I’ll highlight some recent applications of the theory; notably, I will present a collaboration with Briggs and Grifo where we establish bounds for the Krull dimensions of these geometric objects that have a number of applications in local algebra.

**November 30, 2022 (CEST)**

3:00 - 4:00pm Severin Barmeier (Köln)

Title: Deformations and spectra in noncommutative algebraic geometry

Abstract: One compelling construction of noncommutative analogues of commutative varieties is through deformation of the Abelian category Qcoh(X) of a commutative scheme X developed by Lowen and Van den Bergh. Such deformations will be "close to" commutative varieties, but can be viewed as incorporating "quantum effects" as they are often obtained by quantizing algebraic Poisson structures. I will present joint work with Zhengfang Wang in which we give a combinatorial description of the deformation theory of Qcoh(X) for any separated Noetherian scheme X which also provides tools for overcoming the technical hurdle of passing from formal to strict deformations.

Given the explicit nature of these strict deformations, one may obtain a "geometric" picture of noncommutative varieties through Rosenberg's construction of the spectrum of an Abelian category which I hope to sketch in some simple examples. These considerations also suggest revisiting the definition of the structure sheaf of the Rosenberg spectrum in the noncommutative case.

4:30 - 5:30pm Aurélien Djament (Lille)

Title: Structure results for generic representations of infinite general linear groups

Abstract: Functors from finitely-generated A-modules to K-vector spaces form a nice abelian category which can be thought as generic representation theory of general linear groups over the ring A with coefficients in the field K. The structure and cohomology of these functors, in particular the polynomial (following Eilenberg and Maclane) ones, have deep interplay with algebraic topology, classical representation theory, cohomology of groups and K-theory. In this talk, which relies on joint works with A. Touzé and C. Vespa, I will present some recent structure theorems for generic representations of general linear groups over an arbitrary ring, which are meaningful in the case of infinite rings, for which the subject was very few studied. These results include a classification of simple functors with finite-dimensional values and generalisations of finite presentation, in particuliar for polynomial functors.

November 23, 2022 (CEST)

4:30 - 5:30pm Neil P Strickland (Sheffield)

Title: Global rational representation theory(joint with Luca Pol)

Abstract: Let U be the category of finite groups and conjugacy classes of surjective homomorphisms, or some reasonable subcategory of that. Let A be the category of contravariant functors from U to rational vector spaces (which is equivalent to a certain category of globally equivariant spectra with rational homotopy groups). The category A has some unusual properties: there is a good theory of duality but finitely generated projective objects are not strongly dualisable, all projective objects are injective but not vice-versa, and so on. This makes it difficult to analyse the Balmer spectrum of the associated derived category, but we will explain some progress towards that goal.

**November 18, 2022 (CEST)**

11:00 - 12:00pm Marc Stephan (Bielefeld)

Title: $\infty$-categories starter pack

Abstract:I will provide an introduction to the theory of quasi-categories as developed by Joyal and Lurie. The goal is to make talks that start with "Let $C$ be a stable $\infty$-category" more accessible.

Part II: stable $\infty$-categories and the triangulation of their homotopy categories.

The talk is targeted to researchers who have encountered the definition of simplicial sets and maybe some ideas about $\infty$-categories.

**November 16, 2022 (CEST)**

3:00 - 4:00pm Karin Erdmann (Oxford)

Title: Finite generation of Hochschild cohomology

Abstract: The Hochschild cohomology of a selfinjective algebra is often not finitely generated. But if it satisfies suitable finite generation properties (known as Fg) then modules have support varieties, similar to the ones via group cohomology. We show that many Brauer graph algebras satisfy Fg, and quite likely any generalizations of dihedral or semidihedral type algebras.

4:30-5:40pm Gong show, grand finale:

4:30-4:40 Pablo Ocal

4:45-4:55 Angel Toledo

5:00-5:10 Severin Barmeier

5:15-5:25 Jack Davies

**November 15, 2022 (CEST)**

4:30 - 5:30pm Marc Stephan (Bielefeld)

Title: $\infty$-categories starter pack

Abstract:I will provide an introduction to the theory of quasi-categories as developed by Joyal and Lurie. The goal is to make talks that start with "Let $C$ be a stable $\infty$-category" more accessible.

Part I: $\infty$-categories, functors, and natural transformations.

The talk is targeted to researchers who have encountered the definition of simplicial sets and maybe some ideas about $\infty$-categories.

**November 9, 2022 (CEST)**

4:30 - 5:30pm Jon Carlson (University of Georgia)

Title: Modules: Idempotent and Otherwise

Abstract: Suppose that G is a finite group scheme and that k is a field of characteristic p > 0. We concentrate on the important case that kG is the group algebra of an elementary abelian p-group or commutative restricted Lie algebra. We give an explicit description of the structure of certain idempotent modules that are associated to closed points in the spectrum ProjH(G, k) and define localizing subcategories of the stable category of all kG-modules. The endomorphism rings of these modules can be computed and provide something of a local support with some interesting properties.

**November 7, 2022 (CEST)**

11:00 - 12:00pm Jesper Grodal (University of Copenhagen) Part I

**November 9, 2022 (CEST)**

11:00 - 12:00pm Jesper Grodal (University of Copenhagen) Part II

**November 11, 2022 (CEST)**

11:15 - 12:15pm Jesper Grodal (University of Copenhagen) Part III

Title: Smith theory in algebra, geometry and topology

Abstract: Smith theory is the study of what can be said about the fixed-points X^P from knowing X, for X an object with an action of a finite p-group P. In my 3 informal lectures I will discuss ways this can be made precise (e.g., what does it mean to "know"?), and give occurrences of Smith theory in algebra, geometry, and topology. Given the broad nature of the TP, the Monday lecture will mainly be overview and elementary, and parts probably boring to even non-experts; hence our aim is to make different parts boring to slightly different people, to hopefully avoid a complete waste of time. Wednesday and Friday I'll survey some theorems with Smith theory in action, with the exact content depending on how Monday goes down. I plan to keep the talks self-contained (whatever that means), but still hope to make some sort of bridge to Geordie Williamson's and Nick Kuhn's talks earlier in the TP.

**November 2, 2022 (CEST)**

4:30 - 5:30pm Rosanna Laking (University of Verona)

Title: T-structures with Grothendieck hearts in the derived category of a finite-dimensional algebra

Abstract: In this talk I will discuss a particularly nice class of t-structures arising from the representation theory of finite-dimensional algebras called (pure-injective) cosilting t-structures. In a compactly generated triangulated category they can be characterised as the non-degenerate smashing t-structures with Grothendieck hearts. One remarkable aspect of such t-structures is that, in some settings, it is possible to classif (at least subsets of) them. We will give an overview of the structural results that make them amenable to classification and describe one such classification theorem obtained in joint work with Karin Baur.

**October 19, 2022 (CEST)**

3:00 - 4:00pm Jan Stovicek (Charles University)

Title: The singularity category of a finite dimensional algebra and finitistic dimensions

Abstract: I will explain how the finiteness of the big finitistic dimension of a finite dimensional algebra (which can be viewed as a non-commutative homological analog of the Krull dimension of a commutative noetherian ring) is related to the singularity category of the algebra. This is related to recent work of Rickard on injective generation of the derived category as well as to the delooping level of an algebra studied by Gélinas.

4:30 - 5:30pm Joseph Ayoub (Zürich)

Title: Motivic Galois group

Abstract: I'll give an introduction to the theory of motives with emphasis on the motivic Galois group. In particular, I'll explain how the motivic Galois group arises naturally as the group of autoequivalences of categories of local systems on algebraic varieties.

**October 14, 2022 (CEST)**

11:00-12:00** **Amnon Neeman (ANU)

Title: Approximability and metrics in triangulated categories Part3 -> Slide (PDF)

Gong show – last episode

3:00 - 3:10pm Jan Stovicek

3:15 - 3:25pm Shaul Barkan

3-30 - 3:40pm Teresa Conde

3:45 - 3:55pm Alexandra Zvonareva

Coffee Break

4:30 - 4:40pm Paolo Tomasini

4:45 - 4:55pm Luca Pol

**October 12, 2022 (CEST)**

11:00 - 12:00** **Amnon Neeman

Title: Approximability and metrics in triangulated categories Part2 -> Slide (PDF)

Gong shows continued...

3:00 - 3:10pm Bernhard Keller

3:15 - 3:25pm Chiara Sava

3:30 - 3:40pm Sil Linskens

3:45 - 3:55pm Juan Omar Gomez

Coffee Break

4:30 - 5:30pm Mike Hill

Title: Equivariant approaches to chromatic homotopy

Abstract: I'll talk about how developments in Equivariant homotopy theory over the last decade have resulted in new ways to study and understand classical questions in chromatic homotopy. This is a preliminary report on work with various subsets of Beaudry, Bobkova, Lawson, Shi, Stojanoska, & Zeng.

**October 10, 2021 (CEST)**

11:00-12:00** **Amnon Neeman (ANU)

Title: Approximability and metrics in triangulated categories Part1-> Slide (PDF)

**October 7, 2022 (CEST)**

Gong shows continued...

3:00 - 3:15pm Peter Symonds

3:15 - 3:25pm Venkata Sai Narayana Bavisetty

3:30 - 3:40pm Jordan Williamson

3:45 - 3:55pm Christian Carrick

**October 6, 2022 (CEST)**

3:00 - 4:00pm Geordie Williamson

Title: Smith-Treumann theory, geometric representation theory and

dreams of K-theory

Abstract: Topologists have long known that Z/pZ can be seen as a "discrete circle" when coefficients are mod p. I'll first explain this circle of ideas ("Smith theory"), and try to motivate Smith-Treumann theory, which is a version of Smith theory for constructible sheaves. I'll then sketch how to use Smith-Treumann theory to establish deep facts in the representation theory of algebraic groups via the geometry of the affine Grassmannian. (This was the subject of a recent week-long Arbeitsgemeinschaft at Oberwolfach, so I certainly won't be able to cover everything in a small amount of time! I'll just give the key ideas.) Topologists tell me that Smith theory is also powerful for understanding K-theory and other generalized cohomology theories. If time and interest allows, I'll try to outline some ver preliminary calculations of what happens when one puts KU-modules on the Satake stack. (New results in the first part are joint with Simon Riche, whilst the dreamy bit is based on computations with Ben Elias.)

**October 5, 2022 (CEST)**

First half: Gong shows continued

3:00 - 3:10pm John Greenlees

3:15 - 3:25pm Yuqing Shi

3:30 - 3:40pm Anish Chedalavada

3:45 - 3:55pm Shai Keidar

Second half:

4:30 - 5:30pm Sarah Petersen

Title: Ravenel-Wilson Hopf ring methods in $C_2$-equivariant homotopy theory and the $H\underline{\mathbb{F}}_2$-homology of $C_2$-equivariant Eilenberg-MacLane spaces.

Abstract: This talk describes an extension of Ravenl-Wilson Hopf ring techniques to $C_2$-equivariant homotopy theory. Our main application and motivation for introducing these methods is a computation of the $RO(C_2)$-graded homology of $C_2$-equivariant Eilenberg-MacLane spaces. The result we obtain for $C_2$-equivariant Eilenberg-MacLane spaces associated to the constant Mackey functor $\underline{\mathbb{F}}_2$ gives a $C_2$-equivariant analogue of the classical computation due to Serre at the prime 2. We also investigate a twisted bar spectral sequence computing the homology of these equivariant Eilenberg-MacLane spaces.

**September 16, 2022 (CEST)**

Gong shows

3:00 - 3:10pm Jeremiah Heller

3:15 - 3:25pm Sarah Petersen

3:30 - 3:40pm Collin Litterell

3:45 - 3:55pm Guchuan Li

Coffee break

4:30 - 4:40pm Elizabeth Tatum

4:45 - 4:55pm Scott Balchin

**September 14, 2022 (CEST)**

Gong shows

3:00 - 3:10pm Vesna Stojanovska

3:15 - 3:25pm Merlin Crist

3:30 - 3:40pm Henning Krause

3:45 - 3:55pm Rudradip Biswas

Coffee break

4:30 - 4:40pm Janina Letz

4:45 - 4:55pm George Raptis