# Algebra Seminar 2021

Current contacts: Vasily Dolgushev, Ed Letzter or Martin Lorenz.

The Seminar usually takes place on Mondays at 1:30 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract.

• Monday February 1, 2021 at 13:30, the Zoom link is shared with the department
The Lie algebra grt and Kontsevich's graph complex

Vasily Dolgushev, Temple University

This is the first talk in the mini-series of (online) meetings on the Grothendieck-Teichmueller Lie algebra grt and Kontsevich's graph complex. I will give an introduction to the topic and an overview of the goals.

• Monday February 8, 2021 at 13:30, The Zoom link is shared with the department
The Lie algebra grt and Kontsevich's graph complex II

Vasily Dolgushev, Temple University

I will talk about Kontsevich's graph complex and its link to the Lie algebra grt.

• Monday February 15, 2021 at 13:30, The Zoom link is shared with the department
Kontsevich's graph complex versus the Poisson cochain complex

Vasily Dolgushev, Temple University

The space PV of polyvector fields carries a natural Lie bracket. It is called the Schouten bracket or the Nijenhuis-Schouten bracket. I will show that there is a homomorphism of differential graded (dg) Lie algebras from Kontsevich's graph complex GC to the Chevalley-Eilenberg complex of the Lie algebra PV. If time permits, I will show that, using a Poisson structure alpha, we can construct a (degree 2) chain map from GC to the Poisson cochain complex of alpha.

• Monday February 22, 2021 at 13:30, The Zoom link is shared with the department
Kontsevich's graph complex versus the Poisson cochain complex II

Vasily Dolgushev, Temple University

I will finish talking about the link between Kontsevich's graph complex and the Poisson cochain complex. Then we will discuss future plans of this activity.

• Monday March 8, 2021 at 13:30, The Zoom link was shared with the department
Introductory Magma session

Vasily Dolgushev, Temple University

We will start with the basic Magma datatypes: integers, rationals, sequences, sets, tuples, and do simple programming exercises using Magma. If time permits, we will work with permutation groups, rings and matrices using Magma. You will benefit from this session more if you have an access to the HPC server "compute".

• Monday March 15, 2021 at 13:30, The Zoom link will be shared with the department
An A-infinity analog of the Goldman-Millson Theorem in characteristic p

Patricia Milham, University of Nevada, Reno

Over fields of characteristic zero, the relationship between L-infinity algebras and formal deformation problems is established via the simplicial Maurer-Cartan functor. This functor assigns to each L-infinity algebra L a Kan simplicial set whose vertices are the solutions to the Maurer-Cartan equation in L. However, this equation is not well-defined over fields of positive characteristic, so the Maurer-Cartan simplicial functor cannot be used to study deformation problems in characteristic p. As a step towards resolving this, de Kleijn and Wierstra showed that there is an A-infinity analog to the Maurer-Cartan simplicial functor which can be used to study non-symmetric deformation problems in characteristic p.

In this talk, I will describe work in progress that establishes an A-infinity analog of the Goldman-Millson theorem in characteristic p, as conjectured by by de Kleijn and Wierstra. In particular, I prove that the simplicial Maurer-Cartan functor for A-infinity algebras sends filtration preserving A-infinity quasi-isomorphisms between complete A-infinity algebras to weak equivalences of the corresponding Maurer-Cartan simplicial sets.

• Monday March 22, 2021 at 13:30, The Zoom link was shared with the department
Genus zero modular operad & Grothendieck-Teichmueller group's avatar

Noemie C. Combe, Max Planck Institute Leipzig

In this talk, we develop the geometry of canonical stratifications of the spaces $\bar M_{0,n}$ and prepare ground for studying the action of the Galois group or the field of rational numbers upon strata. We introduce a categorical framework for the description of symmetries of genus zero modular operad. This description merges the techniques of recent "persistence homology" studies and the classical formalism of groupoids. We provide a new avatar of profinite Grothendieck-Teichmueller group acting upon this operad, but seemingly not related with representations of the Galois group of all algebraic numbers.

• Monday March 29, 2021 at 13:30, the Zoom link is shared with the department
The Merkulov-Willwacher graph complex, moduli spaces of curves and Lie bialgebras

Alexey Kalugin, University of Luxembourg

In 80's Mumford, Penner, Harer, and Thurston related the cohomology of a moduli stack of curves with marked points to the cohomology of a certain combinatorial cochain complex (the Kontsevich-Penner ribbon graph complex). In 2015 Merkulov-Willwacher introduced a version of the Kontsevich-Penner ribbon graph complex (the Merkulov-Willwacher graph complex) which has roots in the deformation theory of Lie bialgebras. In my talk, I am going to compute the cohomology of this graph complex and explain its relation to the recent work of Chan-Galatius-Payne. If time permits I will also explain a (conjectural) relation to the formality problem of the Goldman-Turaev Lie bialgebra studied by Alekseev-Kawazumi-Kuno-Naef.

• Monday April 5, 2021 at 13:30, The Zoom link was shared with the department
What is an operad and where to find it?

Xingting Wang, Howard University

Coined in J. Peter May’s The Geometry of Iterated Loop Spaces, an operad is an abstraction of a family of composable functions of n variables for various n, useful for the “bookkeeping” and applications of such families. We will take an adventure in the history of operad with the help of trees and Schur functors. Examples of operads and algebras over them will be discussed in details.

• Monday April 12, 2021 at 13:30, The Zoom link was shared with the department
Examples of operads and where to look for them

Vasily Dolgushev, Temple University

This is a continuation of Xingting's talk about operads. I will introduce the endomorphism operad (of a set) and define an algebra over an operad. I will describe the operads that govern magmas, monoids and commutative monoids. If time permits, I will also talk about operads in "more sophisticated" symmetric monoidal categories: topological spaces, vector spaces and groupoids.

• Monday April 19, 2021 at 13:30, The Zoom link was shared with the department

Xingting Wang, Howard University

We will take another journey to the world of vector spaces, where commutative algebras, associative algebras, Lie algebras and other types of algebras are all governed under the rule of operads constructed from their free algebras.

• Monday April 26, 2021 at 13:15, The Zoom link was shared with the department
Poincaré-Birkhoff-Witt-type theorems for algebras over operads

Pedro Tamaroff, Trinity College Dublin

In joint work with Vladimir Dotsenko https://arxiv.org/abs/1804.06485, we developed a framework to state and prove PBW-type theorems about universal enveloping algebras of various algebraic structures. We used it to recover the classical PBW theorem for universal envelopes of Lie algebras, to prove new PBW-type theorems for other types of algebras, answering a question of J.-L. Loday, and to deduce some PBW-type theorems are unattainable in other situations. In this talk, I will survey the results of https://arxiv.org/abs/1804.06485 and explain the role Groëbner bases for operads (as developed in Dotsenko--Khoroshkin https://arxiv.org/abs/0812.4069) play in our work. Previous knowledge of Groëbner bases is not assumed: we will introduce them along the way.

• Monday August 30, 2021 at 14:00, Wachman 617
Algebra Seminar: Discussion of the plans for the fall semester

• Monday September 13, 2021 at 13:30, Wachman 617
Growth of groups and algebras, I

Martin Lorenz, Temple University

This series of three talks will deal with "growth" of groups and ofalgebras. Despite its elementary combinatorially flavored definition,the concept of growth has played in important role in algebra andother areas; in fact, for groups, its origins lie in geometry and themain theorems have been contributed by geometers. Certain theoremsabout groups become “easy” when viewed in the context ofalgebras. The talks aim to explore the potential and the currentlimitations of this approach.

• Monday September 20, 2021 at 13:30, Wachman 617
Growth of groups and algebras, II

Martin Lorenz, Temple University

After providing some more group-theoretical background, I will focus on“representable” algebras in this talk. By definition, these are algebras that can be embedded into matrix algebras over some commutative algebra. Despite the seemingly elementary nature of this class of algebras, there are quite a few mysteries remaining to be resolved.

• Monday September 27, 2021 at 13:30, Wachman 617
Growth of groups and algebras, III

Martin Lorenz, Temple University

The focus in this talk, the last in the series, will remain on affine representable algebras, that is, finitely generated algebras that can be embedded into matrix algebras over some commutative algebra. In particular, I plan to give an outline of the proof that the Gelfand-Kirillov dimension of such an algebra is always an integer. Several open problems will also be formulated.

• Monday October 4, 2021 at 13:30, Wachman 617
For the Grothendieck-Teichmueller group to GT-shadows

Vasily Dolgushev, Temple University The Grothendieck-Teichmueller group GT introduced by V. Drinfeld in 1990 connects topology to number theory in fascinating way. GT receives an injective homomorphism from the absolute Galois group G_Q of rational numbers, it acts on Grothendieck's child's drawings and this action is compatible with that of G_Q. I will start this series of talks with defining what I call the gentle version of GT. In the subsequent talks, we will introduce the groupoid of GT-shadows and explain its link to (the gentle version of) GT.

• Monday October 11, 2021 at 13:30, Wachman 617
Linking congruences of modular forms to solvable extensions of the rational numbers

Carl Wang-Erickson, University of Pittsburgh

Going back to Ribet's converse to Herbrand's theorem, there is a relationship between two phenomena: congruences, modulo a prime p, of Hecke eigensystems between Eisenstein series and cusp forms; and Galois extensions of the rational numbers with metabelian Galois group. In many cases, this relationship is enough to establish a much more expansive relationship between "all" congruences and "all" solvable Galois extensions: an "R=T" theorem. We will describe situations where studying metabelian extensions is not enough to deduce R=T, but studying three-step solvable (meta-metabelian?) extensions is enough. We will give explicit examples of both of these two types of situations. This includes joint works with Preston Wake (Michigan State University) and with Catherine Hsu (Swarthmore College).

• Monday October 18, 2021 at 13:30, Wachman 617
For the Grothendieck-Teichmueller group to GT-shadows II

Vasily Dolgushev, Temple University

I will describe the link between the absolute Galois group of rationals and the gentle version of Grothendieck-Teichmueller group. Then I will start explaining the construction of the groupoid of GT-shadows.

• Monday October 25, 2021 at 13:30, Wachman 617
For the Grothendieck-Teichmueller group to GT-shadows III

Vasily Dolgushev, Temple University

GT-shadows form a groupoid GTSh whose objects are certain finite index normal subgroups of Artin braid group B_3 on 3 strands. In my talk, I will introduce this groupoid and describe its link to (the gentle version of) the Grothendieck-Teichmueller group.

• Monday November 1, 2021 at 13:30, Wachman 617
Formal exponential maps and the Atiyah class of dg manifolds

Seokbong Seol, Penn State University

Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. The infinite jets of these classical exponential maps are related to Poincaré-Birkhoff-Witt isomorphism and the complete symbols of differential operators. We will investigate the question on how to extend these maps to dg manifolds. As an application, we will show there is an L-infinity structure on the space of vector fields in connection with the Atiyah class of a dg manifold. In a special case, it is related to Kapranov’s L-infinity structure on the Dolbeault complex of a Kähler manifold. This is a joint work with Mathieu Stiénon and Ping Xu.

• Monday November 15, 2021 at 13:30, Wachman 617
TBA

Ningchuan Zhang, University of Pennsylvania

TBA