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Current contacts: Vasily Dolgushev, Ed Letzter, Martin Lorenz or Chelsea Walton
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.
Jonathan Beardsley, University of Washington
Given a fiber sequence of n-fold loop spaces X-->Y-->Z, and morphism of n-fold loop spaces Y-->BGL_1(R) for R an E_{n+1}-ring spectrum, we describe a method of producing a new morphism of (n-1)-fold loop spaces Z-->BGL_1(MX), where MX is the Thom spectrum associated to the composition X-->Y-->BGL_1(R). This new morphism has associated Thom spectrum MY, but constructed directly as an MX-module. In particular this induces a relative Thom isomorphism (i.e. a torsor structure) for MY over MX: MY \otimes_{MX} MY = MY \otimes Z. We will see a rough description of this construction as well as many examples. In many cases this torsor condition additionally satisfies a descent condition showing that the unit map MX-->MY is a Hopf-Galois extension of structured ring spectra. Moreover, the composition R-->MX-->MY describes an intermediate Hopf-Galois extension associated to thinking of X as a sub-bialgebra of Y. It seems likely that the methods described in this talk can be modified to apply to homotopy quotients of DGAs.
Johannes Flake, Rutgers University
Dirac cohomology has been employed successfully to analyze the representation theory of connected semisimple Lie groups and of degenerate affine Hecke algebras. We study a common generalization of these situations as suggested by Dan Barbasch and Siddhartha Sahi, certain PBW deformations satisfying an orthogonality condition, which we call Hopf-Hecke algebras. Besides the mentioned special cases, they also include infinitesimal Cherednik algebras as new examples. We will discuss a general result relating the Dirac cohomology with central characters, partial results on the classification of Hopf-Hecke algebras, and a concrete computation of the Dirac cohomology for infinitesimal Cherednik algebras of the general linear group. This is joint work with Siddhartha Sahi.
Chassidy Bozeman, Iowa State University
Zero forcing on a simple graph is an iterative coloring procedure that starts by initially coloring vertices white and blue and then repeatedly applies the following color change rule: if any vertex colored blue has exactly one white neighbor, then that neighbor is changed from white to blue. Any initial set of blue vertices that can color the entire graph blue is called a zero forcing set. The zero forcing number is the cardinality of a minimum zero forcing set. A well known result is that the zero forcing number of a simple graph is an upper bound for the maximum nullity of the graph (the largest possible nullity over all symmetric real matrices whose (ij)-th entry (for distinct i and j) is nonzero whenever {i,j} is an edge in G and is zero otherwise). A variant of zero forcing, known as power domination (motivated by the monitoring of the electric power grid system), uses the power color change rule that starts by initially coloring vertices white and blue and then applies the following rules: 1) In step 1, for any white vertex w that has a blue neighbor, change the color of w from white to blue. 2) For the remaining steps, apply the color change rule. Any initial set of blue vertices that can color the entire graph blue using the power color change rule is called a power dominating set. We present results on the power domination problem of a graph by considering the power dominating sets of minimum cardinality and the amount of steps necessary to color the entire graph blue.
Vasily Dolgushev, Temple University
I will introduce the concept of a star product and outline Fedosov's construction for star products on an arbitrary symplectic manifold. I will also state the classification theorem for star products on a symplectic manifold.
Lev Borisov, Rutgers University
The Grothendieck ring of complex algebraic varieties is defined as the space of formal sums $\sum_i a_i [X_i]$ of algebraic varieties with integer coefficients, subject to the relations $[X]=[X-Z]+[Z]$ for closed subvarieties $Z$ of $X$. I will talk about recent developments that show that the class of the affine line is a zero divisor in the Grothendieck ring.
Vasily Dolgushev, Temple University
Equivalence classes of star products on a symplectic manifold M can be described in terms of the second de Rham cohomology of M. I will review Fedosov's construction whose input is a series of closed two forms and whose output is a star product on a symplectic manifold.
Vasily Dolgushev, Temple University
To describe the equivalence classes of star products on an arbitrary Poisson manifold, we need some constructions related to differential graded Lie algebras. I am going to review these constructions in my talk.
Maitreeyee Kulkarni, Louisiana State University
Let G be a Lie group of type ADE and P be a parabolic subgroup. It is known that there exists a cluster structure on the coordinate ring of the partial flag variety G/P (see the work of Geiss, Leclerc, and Schroer). Since then there has been a great deal of activity towards categorifying these cluster algebras. Jensen, King, and Su gave a direct categorification of the cluster structure on the homogeneous coordinate ring for Grassmannians (that is, when G is of type A and P is a maximal parabolic subgroup). In this setting, Baur, King, and Marsh gave an interpretation of this categorification in terms of dimer models. In this talk, I will give an analog of dimer models for groups in other types by introducing a technique called “constructing cylinders over Dynkin diagrams”, which can (conjecturally) be used to generalize the result of Baur, King, and Marsh.
Lia Vas, University of the Sciences
I have been working in algebra and ring theory, in particular with rings of operators, involutive rings, Baer star-rings and Leavitt path algebras. These rings were introduced in order to simplify the study of sometimes rather cumbersome operator theory concepts. For example, a Baer star-ring is an algebraic analogue of an AW star-algebra and a Leavitt path algebra is an algebraic analogue of a graph C-star algebra. Such rings of operators can be studied without involving methods of operator theory. Thus algebraization of operator theory is a common thread between most of the topics of my interest. After some overview of the main ideas of such algebraization, I will focus on one common aspect of some of the rings of operators – the existence of a trace as a way to measure the size of subspaces/subalgebras. In particular, we adapt some desirable properties of a complex-valued trace on a C-star algebra to a larger class of algebras.
Edward Letzter, Temple University
History and background of results on finitely generated algebras of low (i.e., greater than zero but less than three) Gelfand-Kirillov dimension. Beginning with early results of Bergman, Small-Stafford-Warfield (and others), continuing through later results of Artin-Stafford, Bell, Small (and others), and concluding with recent work of Smoktunowicz and collaborators Bell, Lenagan, Small (and others).
Edward Letzter, Temple University
Continuation of Part I, surveying open questions, examples, and results in Gelfand-Kirillov dimension 2. As time permits, I'll discuss some broader features of the theory.
Martin Lorenz, Temple University
I will talk about rudiments of category theory: categories, functors, natural transformations, and adjoint functors. If time will permit, I will also talk about limits and colimits. Various examples will be given.
Vasily Dolgushev, Temple University
I will talk about adjoint functors, limits and colimits. I hope to give many examples. If time will permit, I will start talking about monoidal categories and monoidal functors.
Vasily Dolgushev, Temple University
I will talk about limits and colimits of functors. I will present various examples including profinite completions of groups. If time will permit, I will start talking about monoidal categories and monoidal functors.
Chelsea Walton, Temple University
I will give an introduction to monoidal categories.
Blake Farman, University of South Carolina
In their 1994 paper, Noncommutative Projective Schemes, Michael Artin and J.J. Zhang introduce a generalization of usual projective schemes to the setting of not necessarily commutative algebras over a commutative ring. Gonçalo Tabuada in 2005 endows the category of differential graded categories with the structure of a model category and in 2007 Toën shows that its homotopy category is symmetric monoidal closed. In this talk, we’ll give a brief overview of these results, adapting Artin and Zhang’s noncommutative projective schemes for the language of DG categories, and discuss a “geometric” description of this internal Hom for two noncommutative projective schemes. As an immediate application, we give a noncommutative projective derived Morita statement along the lines of Rickard and Orlov.
Zachary Cline, Temple University
Susan Montgomery and Hans-Juergen Schneider classified all non-trivial $n$-dimensional module algebras $A$ over the Taft algebras $H$ of dimension $n^2$, $n > 2.$ They further showed that each such module structure extends uniquely to make $A$ a module algebra over the Drinfeld double of $H$. We explore what it is about the Taft algebras that leads to this uniqueness, by examining Hopf algebras "close" to the Taft algebras in various directions, and their module algebras.
Abeer Kamal Al-Ahmadieh, Temple University
The groupoid ((Trees)) of labeled planar trees is a convenient tool for defining an operad and for working with operads. I will recall necessary rudiments of graph theory and introduce the groupoid ((Trees)). If time will permit, I will show that every collection (of vector spaces) allows us to define a functor from the groupoid ((Trees)) to the category of vector spaces.
Vasily Dolgushev, Temple University
Algebraic operad is a natural generalization of an associative algebra. Loosely speaking, it is a gadget with an infinite sequence of multiplications indexed by planar labeled trees. For every operad O, we can consider the category of O-algebras (or algebras over O). Thus associative algebras are algebras over a certain operads, Lie algebras are algebras over a certain operad and so on. As Tai-Danae Bradley wrote in her abstract for the graduate seminar, operads have a wide range of applications: deformation theory, algebraic topology, and mathematical physics. So if you want to learn even more about operads, please, come to my talk. I am going to use freely the language of planar labeled trees introduced by Abeer Al-Ahmadieh in her talk.
Vasily Dolgushev, Temple University
I will talk about solved and unsolved problems which involve operads and related structures. Some of these problems are related to the prounipotent version of the Grothendieck-Teichmueller group, while others are related to the profinite version of the Grothendieck-Teichmueller group.
Hsuan-Yi Liao, Penn State University
A Lie pair $(L,A)$ consists of a Lie algebroid $L$ together with a Lie subalgebroid $A$. A wide range of geometric situations can be described in terms of Lie pairs including complex manifolds, foliations, and manifolds equipped with Lie algebra actions. We establish the formality theorem for Lie pairs. As an application, we obtain Kontsevich-Duflo type theorems for Lie pairs. In this talk, I'll start with the case of $\mathfrak{g}$-manifolds, i.e., smooth manifolds equipped with Lie algebra actions. After that I'll explain formality theorem and Kontsevich-Duflo theorem for Lie pairs and other geometrical situations.
Lisa Carbone, Rutgers University
The Monster Lie algebra $m$, which admits an action of the Monster finite simple group $M$, was constructed by Borcherds as part of his program to solve the Conway-Norton conjecture about the representation theory of $M$. We associate the analog of a Lie group $G(m)$ to the Monster Lie algebra $m$. We give generators for large free subgroups and we describe relations in $G(m)$.
Julia Plavnik, Texas A&M University
The problem of classifying modular tensor categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter. These categories have also applications in different areas of mathematics like topological quantum field theory, von Neumann algebras, representation theory, and others.
In this talk, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular tensor categories, and we will also give some concrete examples to have a better understanding of their structures.
The idea of the talk is to give an overview of the current situation of the classification program for modular categories. We will explain some of the techniques that we found useful to push further the classification, with a focus on new constructions of modular tensor categories. If time allows, we will mention some results for the super-modular case.
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