Titles and abstracts

Yuri Bahturin: Kurosh' Problem for Lie algebras

Abstract: This a short survey of some results on finitely generated infinite-dimensional (restricted) Lie algebras in which the adjoint derivation of each element has finite order.

Dave Benson: NilCoxeter algebras, their representations, and cohomology.

Abstract: The nilCoxeter algebra of a Coxeter group is the algebra obtained by replacing the relations in the group algebra saying that the distinguished generators square to one, with the relations saying they square to zero. The finite types give rise to local finite dimensional algebras, and I shall describe their cohomology. The affine types are examples of prime affine polynomial identity algebras, which gives us interesting information on the representation theory. Hyperbolic types seem harder to approach, but there are some interesting conjectures.

Alexander Berglund: Poincaré duality homomorphisms and graph complexes

Abstract: I will introduce certain operations, governed by Kontsevich's Lie graph complex, that can be associated to a Poincaré duality homomorphism of commutative dg algebras. These operations have bearing on seemingly disparate problems in homological algebra and differential topology: on one hand, the problem of finding certain types of multiplicative structures on semifree resolutions and, on the other hand, the problem of promoting Poincaré duality fibrations to smooth manifold bundles.

Benjamin Briggs: Hidden Koszul duality patterns in commutative algebra

Abstract: Golod rings and complete intersection rings live on opposite ends of the homological spectrum (and so, in the philosophy of homological commutative algebra, they define very different kinds of singularities). However, there are some suspiciously similar regularity properties in free resolutions over both of these types of rings (and others). I'll talk about how some of this can be explained by the fact that both Golod and complete intersection rings are secretly Koszul, in a sense that I will explain. This involves a relative version of Koszul duality being developed in joint work with James Cameron, Janina Letz, Josh Pollitz, and Keller VandeBogert. I'll also talk about some applications, such as constructing free resolutions over these rings in a way that generalises constructions in the literature due to Priddy, Shamash, Eisenbud, Iyengar, and Burke.

Lars W. Christensen: Rational Poincaré series and Bass series

Abstract: A considerable amount of homological data about a commutative Noetherian local ring is encoded in its Poincar\'e and Bass series, which are generating functions for sequences of ranks of (co)homology modules. For rings "small enough" they are known to rational functions, and I will provide a status update on a project to describe exactly which rational functions can be realized in this way.

Rostislav Grigorchuk: Self-similar torsion groups and their applications

Abstract. In the first part of the talk I will shortly describe a history of events around the Burnside problem, Golod-Shafarevich inequality, Golod groups and Golod-Shafarevich groups and the relation of all this to the growth, amenability, random walks and ergodic theory.

Then I will focus on the idea of self-similarity and contraction in group theory, recall my construction of torsion p-groups and associated pro-p-groups and list a dozen of solved and unsolved problems around it. Branch groups naturally will appear as a representative of the class of just-infinite groups and representative of the class of finitely constrained groups.

Finally symbolic systems, including the Morse system will be related to the discussed groups and some results connecting the group theory with modern dynamics will be presented at the end of the talk.

Farshid Hajir: A historical survey of the Golod-Shafarevich theorem and its consequences: From Evgenii Golod to Nigel Boston

Abstract: In the early 1960s, Shafarevich articulated the conjecture that a pro-p group whose minimal presentation has “few” relations as compared to its number of generators must be infinite. Soon thereafter, Golod and Shafarevich formulated and proved a very explicit version of this conjecture in their 1964 paper “On Class Field Towers.” This fundamental theorem of group theory had its origins in the investigation of an intriguing problem in algebraic number theory. In this historical talk, I will sketch the influence of the Golod-Shafarevich theorem in algebraic number theory in the ensuing decades, especially with the view to illustrating the group theory - number theory synergies displayed prominently in the contributions of Nigel Boston.

Vladimir Hinich: Lagrangian equivalence relations

Abstract: Given an equivalence relation R on a complex vector space V, one defines the ring of invariants as the subring of polynomial functions on V that have equal values at equivalent points. We are interested when the maximal ideals of the ring of invariants describes the equivalence classes of R.

The main application is to the description of the centers of finite dimensional kac-Moody superalgebras that are known to be invariant subrings corresponding to certain interesting equivalence relations.

This is a joint work with M. Gorelik and V. Serganova.

Natalya Iyudu: Golod-Shafarevich estimate for potential algebras’

Andrei Jaikin: Embeddings of group rings into artinian rings and their applications

Abstract: I will describe several instances of embeddings of group rings into Artinian rings and show how these embeddings can be used to study the corresponding groups.

Steffen Kionke: Hereditarily just-infinite torsion groups

Abstract: I will present a new method for the construction of finitely generated, residually finite, infinite torsion groups. In contrast to known related constructions, a profinite perspective makes it possible to control finite quotients and normal subgroups of these torsion groups. I will explain, how this can be used to give the first examples of residually finite, hereditarily just-infinite groups with positive first L2-Betti-number. These groups have polynomial normal subgroup growth, which answers a question of Barnea and Schlage-Puchta.

Claudia Miller: Eagon resolution from a bar resolution beyond the Golod case

Abstract: We will recall some history of the Eagon resolution, involving Massey operations in the Golod case and some mysterious lifts in general, and clarify it via a certain isomorphism of bar resolutions with roots in work of Avramov. This is joint work with Ben Briggs, Hamid Rahmati, and Zheng Yang.

Volodymyr Nekrashevych: Simple torsion groups of intermediate growth

Abstract: We will discuss a method of constructing amenable finitely generated infinite torsion groups from actions of the infinite dihedral group on a Cantor set. Many of the examples will have intermediate growth. We will discuss the methods of finding growth estimates for these groups and properties of random walks on them.

Taras Panov: Polyhedral products, Golod rings and moment-angle complexes

Julia Pevtsova: Finite generation of cohomology: From Golod to van der Kallen

Abstract: The question of finite generation of the cohomology ring for a finite group/finite group scheme/finite dimensional Hopf algebra/finite tensor category goes back to the pioneering work of Golod and Venkov over 60 years ago but still has no shortage of open problems and unsettled conjectures. I’ll try to give (an incomplete) historical overview of the subject and describe where it stands now.

Dmitri Piontkovski: Non-commutative geometry and coherent rings

Abstract: In noncommutative projective geometry, finitely generated graded coherent algebras play the role of coordinate rings for “noncommutative projective schemes”. In this talk, we explore noncommutative schemes arising from non-Noetherian coherent rings, illustrating the theory with key examples such as those associated with regular and monomial algebras. Related conjectures and recent results concerning asymptotic derived invariants of such noncommutative schemes will also be discussed. My research on noncommutative rings began under the supervision of Golod, and I will also discuss some of his earlier ideas that underlie the results presented.

Eduard Schesler: Finite quotients of torsion groups and how to modify them

Abtract: Since the formulation of the Burnside problem, the search for constructions of finitely generated infinite torsion groups has been a constant theme in group theory. Among the rich sources of residually finite torsion groups are Golod–Shafarevich groups, branch groups, and groups arising as limits of hyperbolic groups. Assuming that hyperbolic groups are residually finite, Olshanskii and Osin showed that there exist finitely generated infinite residually finite torsion groups G that are residually finite-simple, meaning that every non-trivial element of G has a non-trivial image in a finite simple quotient of G. Since such groups do not arise from the previously known constructions, this raised the question of whether such groups can exist at all. In this talk, I will show that in fact every infinite finitely generated residually finite torsion group can be modified to yield a torsion group that is residually finite-simple.

Liana Sega: Poincare series over rings defined by general forms

Ryo Takahashi: On the dominance of Golod local rings

Abstract: We call a local ring dominant if the residue field is finitely built from any nonzero object in the singularity category in the sense of Buchweitz and Orlov. In this talk, I will speak about the question asking whether a Golod local ring is dominant. All the new things presented in this talk come from my ongoing joint work with Toshinori Kobayashi.

Mark Walker: On homotopy Lie algebras and deformations

Abstract: An embedded deformation of a local ring R refers to the realization of R as the quotient of another local ring by a regular element in the square of its maximal ideal. It is not difficult to see that an embedded deformation of R leads to the existence of a central element of degree two in its homotopy Lie algebra. A question of Avramov from 1989 asks whether every such central element arises in this manner. I will report on joint work with Ben Briggs, Eloísa Grifo, and Josh Pollitz in which, among other things, we produce examples showing that the answer to Avramov's question is "no". We also give counter-examples to some conjectures on related topics. Our work raises as many questions as it settles, however, and I will pose some of them.

Volkmar Welker: Golod property of quotients of polynomial rings and its applications in combinatorics

Abstract: The Golod property of the quotient of a polynomial ring $k[x_1,..,x_n]$ by a homogeneous ideal is defined by extremal behavior of the free resolution of the field $k$ over the quotient, or equivalently by the vanishing of Massey operations. It has been shown that certain algebraic and combinatorial properties of the ideal imply the Golod property. This has been particularly fruitful in the case of Stanley–Reisner ideals, where moment-angle complexes provide an additional strong link to geometry and topology. In this talk, we will survey these developments and connections.