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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.
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.
Vasily Dolgushev, Temple University
I will talk about Kontsevich's graph complex and its link to the Lie algebra grt.
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.
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.
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".
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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