Department of

Mathematics


Seminar Calendar
for Algebra Seminar events the year of Tuesday, December 3, 2019.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
    November 2019          December 2019           January 2020    
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
                 1  2    1  2  3  4  5  6  7             1  2  3  4
  3  4  5  6  7  8  9    8  9 10 11 12 13 14    5  6  7  8  9 10 11
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 17 18 19 20 21 22 23   22 23 24 25 26 27 28   19 20 21 22 23 24 25
 24 25 26 27 28 29 30   29 30 31               26 27 28 29 30 31   
                                                                   

Tuesday, April 2, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, April 2, 2019

Max-Min theorems for weak containment, square summable homoclinic points, and completely positive entropy

Ben Hayes (University of Virginia)

Abstract: I will present a max-min theorem for weak containment in the context of algebraic actions (i.e. actions of a discrete group by automorphisms of a compact group). Namely, given an algebraic action of $G$ on $X$, there is a maximal, closed $G$-invariant subgroup $Y$ of $X$ so that the action of $G$ on $Y$ is weakly contained in a Bernoulli shift. This subgroup is also the minimal subgroup so that any action weakly contained in a Bernoulli shift is "$G$$X/Y$-ergodic in the presence of $G$$X$" (this will be defined in the talk). Time permitting, I will discussion applications. These include showing that many algebraic actions are weakly contained in a Bernoulli shift, as well as applications to complete positive entropy of algebraic actions.

Friday, August 30, 2019

2:00 pm in 347 Altgeld Hall,Friday, August 30, 2019

Organizational Meeting

Chelsea Walton (UIUC)

Abstract: This will be a short organizational meeting to introduce the structure of the Algebra seminar. Here is the seminar website: https://faculty.math.illinois.edu/~notlaw/UIUC-Algebra.html. Half of the talk slots will be for (introductory +) research talks, and the other half of the slots will be dedicated to a reading group on Tensor Categories. The format of the reading group will be similar to that for the TQFT reading group in Spring 2019 (https://faculty.math.illinois.edu/~notlaw/teaching.html#past), except that the duration of talk slots will be longer.

Friday, September 6, 2019

2:00 pm in 347 Altgeld Hall,Friday, September 6, 2019

Quantum Symmetry in the context of co/representation categories

Chelsea Walton (UIUC)

Abstract: In the first 50 minutes, I will provide an introductory talk on quantum symmetry in the context of co/representation categories, which serves as one point of motivation for the reading group on Tensor Categories. I will then give a follow-up research talk in the second 50 minutes on joint work in progress with Elizabeth Wicks and Robert Won on quantum symmetry and weak Hopf algebras.

Friday, September 13, 2019

2:00 pm in 347 Altgeld Hall,Friday, September 13, 2019

Tensor Categories Reading Group

See seminar site

Abstract: See seminar site.

Friday, September 20, 2019

2:00 pm in 347 Altgeld Hall,Friday, September 20, 2019

Hopf Ore Extensions

Hongdi Huang (University of Waterloo (visiting UIUC for F19))

Abstract: Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism $\sigma$ and a $\sigma$-derivation $\delta$ of a Hopf $k$-algebra $R$ for when the skew polynomial extension $T=R[x, \sigma, \delta]$ of $R$ admits a Hopf algebra structure that is compatible with that of $R$. In fact, they gave a complete characterization of which $\sigma$ and $\delta$ can occur under the hypothesis that $\Delta(x)=a\otimes x +x\otimes b +v(x\otimes x) +w$, with $a, b\in R$ and $v, w\in R\otimes_k R$, where $\Delta: R\to R\otimes_k R$ is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that $\Delta(x)=\beta^{-1}\otimes x +x\otimes 1 +w$, with $\beta $ is a grouplike element in $R$ and $w\in R\otimes_k R,$ when $R\otimes_k R$ is a domain and $R$ is noetherian. In particular, this completely characterizes skew polynomial extensions of a Hopf algebra that admit a Hopf structure extending that of the ring of coefficients under these hypotheses. We show that the hypotheses hold for domains $R$ that are noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension.

Friday, September 27, 2019

2:00 pm in 347 Altgeld Hall,Friday, September 27, 2019

Tensor Categories Reading Group

See seminar site

Abstract: See seminar site.

Friday, October 4, 2019

2:00 pm in 347 Altgeld Hall,Friday, October 4, 2019

Tensor Categories Reading Group

See seminar site

Abstract: See seminar site.

Friday, October 11, 2019

2:00 pm in 347 Altgeld Hall,Friday, October 11, 2019

Tensor Categories Reading Group

See seminar site

Abstract: See seminar site.

Friday, October 18, 2019

2:00 pm in 347 Altgeld Hall,Friday, October 18, 2019

Brown-Goodearl conjecture for PI weak Hopf algebras

James Zhang (University of Washington, Seattle)

Abstract: Brown and Goodearl conjectured that every noetherian Hopf algebra is Artin-Schelter Gorenstein. This conjecture is known to be true for many cases, in particular, for affine polynomial identity Hopf algebras. Weak Hopf algebras are an important generalization of Hopf algebras, and the category of modules over a weak Hopf algebra has a monoidal structure. Let $W$ be a weak Hopf algebra that is a finitely generated module over its affine center. We prove that $W$ has finite self-injective dimension and is a direct sum of Artin-Schelter Gorenstein algebras. Therefore Brown-Goodearl conjecture holds in this special weak Hopf setting. We will also give some motivations and consequences of Brown-Goodearl conjecture. This is joint work with Dan Rogalski and Robert Won.

Friday, October 25, 2019

2:00 pm in 347 Altgeld Hall,Friday, October 25, 2019

Noncommutative algebra from a geometric point of view

Xingting Wang (Howard University)

Abstract: In this talk, I will discuss how to use algebro-geometric and Poisson geometric methods to study the representation theory of 3-dimensional Sklyanin algebras, which are noncommutative analogues of polynomial algebras of three variables. The fundamental tools we are employing in this work include the noncommutative projective algebraic geometry developed by Artin-Schelter-Tate-Van den Bergh in 1990s and the theory of Poisson order axiomatized by Brown and Gordon in 2002, which is based on De Concini-Kac-Priocesiís earlier work on the applications of Poisson geometry in the representation theory of quantum groups at roots of unity. This is joint work with Milen Yakimov and Chelsea Walton.

Friday, November 1, 2019

2:00 pm in 347 Altgeld Hall,Friday, November 1, 2019

Categorification and quantum symmetry

Colleen Delaney (Indiana University)

Abstract: One variation on the theme of ``quantum symmetry" is a categorical group action on a unitary modular tensor category, which can be interpreted physically as a global symmetry of a 2-dimensional topological quantum phase of matter. Much of our understanding of tensor category theory and hence topological phases comes from categorification: from generalizing theorems we have about rings to theorems about categories. For example, categorifying an easy theorem in commutative ring theory, the work of Etingof, Nikshych, and Ostrik established an equivalence between categorical G-actions on modular tensor categories (MTCs), and so-called G-crossed braided extensions of MTCs. Physicists Barkeshli, Bonderson, Cheng, and Wang then recognized that this correspondence can be understood as a tensor-categorical formulation of gauge coupling, wherein G-crossed braided extensions of MTCs give an algebraic theory of symmetry-enriched topological (SET) phases of matter. While the abstract theory of Etingof, Nikshych, and Ostrik is well understood, even constructing the de-categorified part of G-crossed braided extensions of MTCs, namely their fusion rings, is challenging problem in general. We will give a two-part talk, starting with an introduction to the algebraic theory of SET phases described above. In the second part of the talk we describe a topological phase-inspired approach to constructing the fusion rings of certain G-crossed extensions called permutation extensions and share work in progress with E. Samperton in constructing their categorifications.

Friday, November 8, 2019

2:00 pm in 347 Altgeld Hall,Friday, November 8, 2019

Cohomology for Hopf algebras and dualities

Cris Negron (University of North Carolina)

Abstract: In studies of finite-dimensional Hopf algebras one makes consistent use of a number of standard operations. The most fundamental of these operations include linear duality, (cocycle) deformation, and twisting by so-called Drinfeld twists. Many numerical invariants of Hopf algebras are known to be stable under these operations. However, one can see from examples, that the cohomology ring H*(A,k) for a finite-dimensional Hopf algebra A with trivial coefficients is not preserved under deformation or duality of A. In joint work with J. Plavnik we conjecture that, although the cohomology ring itself may vary, the Krull dimension of cohomology should be invariant under a general class of ``duality operations" which includes linear duality, deformation, and Drinfeld twisting. I this talk I will give the necessary definitions and examples, discuss the aforementioned conjecture, and provide some positive results obtained jointly with J. Plavnik.

Saturday, November 9, 2019

9:00 am in 245 Altgeld Hall,Saturday, November 9, 2019

Quasy-Con

Various regional speakers (various)

Abstract: "Quasy-Con" is an informal Quantum Symmetries Conference in the U.S. Midwest, which will be held on November 9-10, 2019. See site for more details: https://faculty.math.illinois.edu/~notlaw/QuaSy-Con2019.html.

Friday, November 15, 2019

2:00 pm in 347 Altgeld Hall,Friday, November 15, 2019

Tensor Categories Reading Group

See seminar site

Abstract: See seminar site.

Friday, November 22, 2019

2:00 pm in 347 Altgeld Hall,Friday, November 22, 2019

A category interpolating Yetter-Drinfeld modules over symmetric groups

Robert Laugwitz (University of Nottingham)

Abstract: P. Deligne introduced a remarkable tensor category Rep(St) interpolating the representation theory of symmetric groups, allowing for the natural number of permuted letters to be replaced by any complex number t. This category is defined using diagrams. We compute objects in the monoidal center of this category to obtain a ribbon category that interpolates the category of Yetter-Drinfeld modules over symmetric groups. As an application, interpolations of untwisted Dijkgraaf-Witten invariants. These are polynomial invariants of framed links. I will motivate Deligne's category Rep(St) and the monoidal center construction before describing how objects of the center or Rep(St) are obtained. This talk is based on joint work with Johannes Flake, RWTH Aachen University.

Friday, December 6, 2019

2:00 pm in 347 Altgeld Hall,Friday, December 6, 2019

Tensor Categories Reading Group

See seminar site

Abstract: See seminar site.

Friday, December 13, 2019

2:00 pm in 347 Altgeld Hall,Friday, December 13, 2019

Tensor Categories Reading Group

See seminar site

Abstract: See seminar site.