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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.
Martin Lorenz, Temple University
This is the first in a series of three lectures that will be accessible to students, including everybody currently taking Math 8011. I will start by briefly reviewing some material that was covered in Math 8011 this semester: the definition of the braid groups (by generators and relations), the visualization of braids (by braid diagrams), and some standard homomorphisms. Then I will dig a little deeper into the structure of the braid groups using the book by Kassel and Turaev as a reference.
Bach Nguyen and Vasily Dolgushev, Temple University
This meeting will be devoted to some exercises on quivers and cluster algebras. Some of the exercises come from https://arxiv.org/abs/1608.05735. A list of exercises will be distributed on October 26.
He Wang, University of Nevada, Reno
Fix a connected graded commutative algebra $H$ over a field of characteristic zero. To study the set of rational homotopy types with cohomology $H$, Halperin and Stasheff constructed a filtered model for each commutative differential graded algebra $A$ with cohomology $H$ by perturbing the differential of the bigraded model of $H$, while Kadeishvili studied the C-infinity structures on $H$ transferred from $A$.
Motivated by the work of Schlessinger and Stasheff, we construct an explicit L-infinity quasi-isomorphism between the following two (filtered) differential graded Lie algebras: the derivations of the bigraded model of $H$, and the Harrison cochain complex of $H$. Passing to their Deligne-Getzler-Hinich infinity-groupoids, we produce a homotopy equivalence of the corresponding simplicial sets. In particular, on the level of $\pi_0$, we obtain a bijection from the moduli space of filtered models with cohomology $H$ to the moduli space of C-infinity structures on $H$. This talk is based on joint work with Chris Rogers.
Vasily Dolgushev, Temple University
The talk will be devoted to open questions of the theory of cluster algebras: the positivity conjecture, its strong version and the conjecture on cluster monomials. I will follow closely the paper by Lauren Williams "Cluster algebras: an introduction".
Bach Nguyen, Temple University
Cluster algebras of geometric type will be defined and some examples will be given.
Bach Nguyen, Temple University In this talk we will continue to discuss another motivating example involving the Grassmannian Gr(2,n+3), which also relate to the two examples covered in the first talk. Then we will move on to talk about total positivity. If time permit, we will start the definition of cluster algebras of geometric type.
Valentijn Karemaker, University of Pennsylvania
A (genus 0) Belyi map is a finite map from the projective line to itself, branched exactly at 0, 1, and infinity. Such maps can be described combinatorially by their generating systems. Assuming further that 0, 1, and infinity are both fixed points and the unique ramification points above 0, 1, and infinity respectively yields dynamical Belyi maps, since the resulting maps can be iterated and will therefore exhibit dynamical behaviour. In this talk, we will discuss several results on the dynamics, reductions, and monodromy of dynamical Belyi maps, and the interplay between these. (This is joint work with J. Anderson, I. Bouw, O. Ejder, N. Girgin, and M. Manes.)
Bach Nguyen, Temple University
Cluster algebras was invented by Fomin and Zelevinsky around 2000 to study total positivity and canonical bases in Lie theory. Since then they have been applied to study many different subjects in mathematics such as commutative and noncommutative algebraic geometry, number theory, (quiver) representation theory, and mathematical physics.
This talk is a part of series of talks desired to give an inviting introduction to the theory of cluster algebras. We will discuss many motivating examples of cluster algebras and study properties and classification of cluster algebras. We also will cover some interesting connection between cluster algebras with number theory, Lie theory, dynamical system, and Poisson geometry if time permits.
Neal Livesay, The University of California, Riverside
The problem of classifying singular differential operators has a long and rich pedigree. An algebro-geometric variant of this problem involves the construction of moduli spaces of irregular singular connections on vector bundles (over the Riemann sphere $\mathrm{P}^1$). Locally (i.e., around a singularity), a selection of a basis for the vector bundle induces a matrix form for the connection. The study of matrices associated to connections is analogous to the study of matrices associated to linear maps, and is amenable to representation-theoretic tools. I will discuss recent work in this direction by D. Sage and N. Livesay. No prior knowledge of connections will be assumed in this talk.
Vasily Dolgushev, Temple University
The story about infinite dimensional Galois extensions of fields is related to the classification of etale algebras over a field. I will give the definition of an etale algebra over a field F and then talk about the classification of etale algebras over F in terms of the absolute Galois group of F.
Vasily Dolgushev, Temple University
We will talk about the fundamental theorem of infinite dimensional Galois theory and its consequences.
Vasily Dolgushev, Temple University
We will introduce profinite groups and show that the Galois group of an infinite Galois extension is a profinite group. Then we will talk about the fundamental theorem of infinite Galois theory.
Vasily Dolgushev, Temple University
We will show that, for every Galois extension, the Galois group G is compact (Hausdorff) and totally disconnected. We will also show that G is a projective limit of finite groups with the discrete topology. If time will permit, we will talk about the fundamental theorem of infinite Galois theory.
Vasily Dolgushev, Temple University
I will introduce infinite Galois extensions and show that the automorphism group of an infinite Galois extension is naturally a topological group (with the Krull topology). If time will permit, I will talk about the fundamental theorem of infinite Galois theory.
Vasily Dolgushev, Temple University
This series of lectures will be devoted to infinite Galois extensions and I plan to follow Chapter 7 of Milne's book "Fields and Galois Theory". In this talk, I will give a brief reminder of topological groups and introduce the Krull topology on the Galois group.
Angela Gibney, Rutgers University
In this talk I will give a tour of recent results and open problems about vector bundles on the moduli space of curves constructed from the representation theory of affine Lie algebras. I will discuss how these questions fit into the context of some of the open problems about the birational geometry of the moduli space.
Benjamin Collas, The University of Bayreuth
Following Grothendieck's "Esquisse d'un Programme", the moduli spaces of curves present remarkable arithmetic-geometry properties which translate to an elegant study of the absolute Galois group of rationals. This program results in the construction of Grothendieck-Teichmueller groups that express how the topological combinatoric of the compactification of spaces encaptures their arithmetic. In this duality, the arithmetic side is expressed through the deformation of curves and the notion of tangential structure, while the topological side recently found an elegant expression in terms of homotopy of the little 2-discs operad by Fresse, Horel et al. The goal of this talk is to present how these two sides intersect each other in the study of the absolute Galois group of rationals. We will thoroughly present both aspects in some recent work for genus 0 curves, and explain how it indicates some promising research lines in higher genus.
Martin Lorenz, Temple University
This time, the focus will be on affine algebraic groups, with some outlook/problems for "quantum groups" at the end. Again, the talk will be largely self-contained inasmuch as no details from the first two talks will be assumed. I will remind you of the general Nullstellensatz/Dixmier-Moeglin picture from the second lecture at the beginning of the talk.
Martin Lorenz, Temple University
Continuing with the theme of the first lecture, I will present some known results on group algebras and speculate on possible extensions to more general classes of Hopf algebras. Thus, much of the second lecture will again be concerned with groups and group algebras. It will be possible to understand this lecture even if you missed the first one.
Martin Lorenz, Temple University
This series of talks will be concerned with actions of Hopf algebras on other algebras ("quantum invariant theory"). I will present a few observations and then proceed to discuss some speculations and open questions. In the first talk, I plan to focus on "local finiteness."
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