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Algebra Seminar

The Algebra Seminar provides graduate students, advanced undergraduate students, and faculty a forum to discuss and/or present Research, Problems, Applications, or Exposition in Algebra. For more information contact Arturo Magidin.

Fall 2023

For the Fall 2023 semester we will meet on Fridays, from 2:30 - 3:20, in Maxim Doucet 208. We may need to change the time and date on occasion to accommodate visitors.
For more information or connection details contact contact Arturo Magidin.

  • The 8 September talk by Professor Geline will be the first of a 4 lecture "mini course".
    Additional details about these talks will be posted below as they become available. The dates and locations of the four talks are:
    • Friday 8 September: Maxim Doucet 208, 2:30 - 3:20 (Algebra seminar regular time)
    • Tuesday 12 September: Maxim Doucet 307, 3:30 - 4:30 (Special meeting of the seminar)
    • Friday 15 September: Maxim Doucet 208, 2:30 - 3:20 (Algebra seminar regular time)
    • Tuesday 19 September: Maxim Doucet 307, 3:30 - 4:30 (Special meeting of the seminar)
  • 8 September 2023
    Rediscovering Brauer’s modular character theory
    Michael Geline (Northern Illinois University)
    Abstract: In this series of talks we will develop and examine the role of ordinary irreducible characters of finite groups in modular representation theory. The goal will be to break down an argument, due to Brauer and Feit, bounding the “heights” of irreducible characters into the most elementary pieces we can. It will be necessary to review the classical orthogonality relations, introduce the decomposition and Cartan matrices, and also to explain some less well-known things about units in centers of block algebras. Ultimately, the numerical result of Brauer and Feit is for us an illustration of how Archimedean and p-adic methods can interact.
  • 5 October 2023 (5:45 pm Thursday on Zoom)
    Classical Distributive Restriction Categories
    JS Lemay (Macquarie University, Sydney, Australia)
    Abstract: Restriction categories provide a categorical framework for working with partially defined maps. The canonical example of a restriction category is PAR, the category of sets and partial functions. In PAR, the Cartesian product × is not a categorical product. Instead the categorical product is given by A+B+A ×B, where + is the disjoint union. In this talk, I will give an introduction to restriction categories and also give a restriction category explanation of when A+B+A × B iproduct. The main result is that in a distributive restriction category, A+B+A × B is a product if and only if the restriction category is classical, that is, has joins and relative complements. We will also discuss on how this covers wide range of interesting examples of restriction categories, such as the opposite category of commutative rings and non-unital ring morphisms. This is joint work with Robin Cockett, based on our paper Classical Distributive Restriction Categories.
  • 20 October 2023 (on Zoom)
    On groups with few subgroups not contained in the Chermak—Delgado lattice
    William Cocke (Augusta University)
    Abstract: For a finite group G, the Chermak-Delgado lattice of G is a self-dual lattice of subgroups of G defined by the centralizer function. Following a standard line of research in group theory, Fasola and Tarnauceanu were able to classify groups with only one or two subgroups not contained in their respective Chermak-Delgado lattices. We extend on their work by classifying all groups with less than five subgroups not in the Chermak-Delgado lattice. In addition, we show that a group with less than five subgroups not in the Chermak-D​elgado lattice is nilpotent. Of particular interest to further research are a number of exclusion lemmas we produce, i.e., lemmas that show that groups with certain structural properties must have many subgroups not in the Chermak-Delgado lattice. This line of research has led to a major improvement of our results by An, Liu, and Qu.
  • 3 November 2023 (on Zoom)
    Dade's Conjecture and how to prove it
    Damiano Rossi (Loughborough University)
    Abstract: I will present some new results on Dade's Conjecture for finite reductive groups. If p is a prime number dividing the order of a finite group G, then Dade's Conjecture gives a formula to count the number of irreducible representations in a Brauer block of G and with a given p-defect in terms of the p-local structure of the group. This behavior is in accordance with a recurring phenomenon known as the local-global principle in representation theory of finite groups. When G is a finite reductive group arising as the set of rational points of a linear algebraic group and p is different from the defining characteristic, the p-local structure of G can be replaced by a geometric analog: the e-local structure. Using this observation I will explain how Dade's Conjecture can be explained within the framework of generalised Harish-Chandra theory.
  • 10 November 2023
    Contractibility of the orbit space of a fusion system after Steinberg
    Omar Dennaoui (UL Lafayette)
    Abstract: Recently, Steinberg proved that the orbit space of the p-subgroup complex of a finite group is contractible using Brown-Foreman discrete Morse theory. We extend Steinberg's argument to show the contractibility of the orbit space of the p-subgroup complex of a group that is pseudo finite at the prime p. We then apply the result to show that the orbit space of a saturated fusion system is contractible. This is joint work with Jonathon Villareal.
  • 17 November 2023 (on Zoom)
    Mackey functors and higher limits
    Antonio Díaz Ramos (Universidad de Málaga)
    Abstract: The concept of Mackey functor was introduced in the context of Representation Theory and of Equivariant Homotopy and Cohomology. In this talk, we will introduce several of the equivalent definitions of such functors, including a categorical version that does not resort to an ambient group. Then we will discuss some of their applications, including the vanishing of higher limits and the sharpness conjecture.

Spring 2023

For the Spring 2023 semester we will meet on Fridays, from 2:00 - 2:50, in Maxim Doucet 208. We may need to change the time and date on occasion to accommodate visitors.
For more information or connection details contact contact Arturo Magidin.

  • 3 May 2023 (Wednesday at 3:00 in room 208)
    Reflective hyperbolic lattices
    Daniel Allcock (University of Texas at Austin)
    Abstract: "Hyperbolic lattices" means bilinear forms of signature (n,1), the name reflecting the importance of hyperbolic space in their study. Reflective means that the isometry group is generated by reflections, up to finite index. The reflection groups arising this way are the hyperbolic version of finite and affine Weyl groups, and play a big role in Kac-Moody theory and the automorphism groups of algebraic surfaces. We have recently classified the reflective hyperbolic lattices of signature (3,1), corresponding to hyperbolic 3-space. I will explain this, and give an overview of the current state of the classification problem in higher dimensions.
  • 24 February 2023 (on zoom)
    Saturated fusion systems and rank two simple groups of Lie type
    Martin Van Beek (University of Manchester)
    Abstract: Saturated fusion systems capture and abstract conjugacy in $p$-subgroups of finite groups and have recently found application in finite group theory, representation theory and algebraic topology. A particularly interesting class of fusion systems is those supported on Sylow $p$-subgroups of rank $2$ simple groups of Lie type, as some of them have been shown to support families of \emph{exotic} systems. Motivated by this, for any prime $p$ and $S$ a $p$-group isomorphic to a Sylow $p$-subgroup of a rank $2$ simple group of Lie type in characteristic $p$, we determine all saturated fusion systems supported on $S$ up to isomorphism.
  • 10 March 2023 (on zoom)
    The fundamental group of the p-group complexes
    Kevin Piterman (Philipps-Universität Marburg)
    Abstract: For a finite group G and a prime p, let A_p(G) denote the poset of non-trivial elementary abelian p-subgroups of G. In 1993, M. Aschbacher characterized the simple connectivity of A_p(G) modulo a conjecture for which there is significant evidence. He also showed that this question is reduced basically to the study of almost simple groups. Based on these results, we proved that, in general, the fundamental group of A_p(G) is a non-abelian free group and that the possible failure of this property also arises from almost simple groups. Indeed, for G = A_10 and p = 3, pi_1(A_p(G)) is not a free group, and this is the smallest possible example. This is a work in collaboration with E.G. Minian.
  • 17 March 2023 (on zoom)
    How diverse can subgroups and quotient groups be?
    James Wilson (Colorado State University)
    Abstract: We'll start by constructing exponentially many groups where
    • no two are the same, but
    • each has the same set of subgroups (with multiplicity),
    • the same quotient groups,
    • the same automorphisms, and
    • the same character tables.
    This settles a question of Timothy Gowers and disprove a conjecture of Laci Babai. But more interestingly, the process also work in finite simple group and proves that the subgroups of finite simple groups are as diverse as possible.
    The moral: group isomorphism is just the outcome of a calculation, not the understanding of structure.
    Most of the talk is accessible to anyone who can multiply matrices with entries in a finite field.
  • 31 March 2023 (on zoom)
    Classification of Pointed Hopf Algebras in Characteristic 2
    Noel Bourne (Carnegie Mellon)
    Abstract. Over an algebraically closed field k of characteristic 0, Hopf algebras of prime power dimensions p and p^2 have been investigated and classified. Pointed Hopf algebras of prime dimensions p, and pq for two distinct primes p and q over an algebraically closed field k in characteristic 0 have also been classified. For an algebraically closed field k of characteristic 2, we will prove that a pointed Hopf algebra are isomorphic to the group algebra of odd prime dimension p. Also that the H is isomorphic to either a Taft algebra or a group algebra if p = q and H is isomorphic to a group algebra if p not equal q.

Fall 2022

For the Fall 2022 semester we will meet on Tuesdays, from 4:50 - 5:50, in Maxim Doucet 208. We may need to change the time and date on occasion to accommodate visitors.
For more information or connection details contact contact Arturo Magidin.

  • 20 September 2022
    An Introduction to Partial Groups
    Justin Lynd
    UL Lafayette
    Abstract: A partial group is a set X together with a multivariable product $x_1\cdots x_n$ which is only defined on a specified subset D of the set of words in X, satisfying some axioms that are similar to those for groups; in fact, if the set D is the set of all words in X, then we obtain a group. Partial groups are the main subject of study of the continuing IKAN ("I Know Almost Nothing") seminar. The talk will serve as both an introduction to anyone not familiar with partial groups, and a refresher for those who participated in the IKAN seminar last semester. If you didn't go to the IKAN last semester, but think you might to this semester, this talk should give you the necessary background.
  • 27 September 2022
    Extensions of partial groups and localities
    Philip Hackney
    Abstract: Discussion of the paper "Extensions of partial groups and localities" by A. Gonzalez.
  • 4 October 2022
    Chermak-Delgado-like lattices in groups
    Arturo Magidin
    Abstract: Given a finite group G, Chermak and Delgado considered the collection of all subgrups H of G for which
    m(H) = |H| |C_G(H)|
    is as large as possible (within G). These subgroups form a lattice with some interesting properties, and this lattice in turn can be used to establish the following theorem:
    THEOREM. Let G be a finite group. Then G has a normal (in fact, characteristic) abelian M with the property that for every abelian subgroup A of G, [G:M] .
    We have been investigating the corresponding construction when the centralizer subgroup is replaced by a different construction. We have determined conditions for the construction to yield a lattice, and have managed to establish analogues to many of the theorems from the Chermak-Delgado construction for this construction. I will describe the constructions and the results we have so far, and where we hope to take this investigation.
    Joint work with Luise-Charlotte Kappe, William Cocke, and Elizabeth Wilcox.
  • 25 October 2022
    How can we extend the statement: "Every ring can be imbedded in the endomorphism ring of an Abelian group," to non-Abelian groups; or who needs 2 distributive laws?
    Gary Birkenmeier
    Abstract: We will discuss the evolution, history, and some basic facts for algebraic systems motivated by the title.
  • 1 November 2022
    How can we extend the statement: "Every ring can be imbedded in the endomorphism ring of an Abelian group," to non-Abelian groups; or who needs 2 distributive laws? Part Two
    Gary Birkenmeier
    Abstract: We will discuss the evolution, history, and some basic facts for algebraic systems motivated by the title.
  • 8 November 2022
    Computational Methods for classifying $p$-groups
    David Burrell
    University of Florida
    Abstract: The problem of finding all the groups of a fixed order $n$ is an old one, initiated by an 1878 paper of Cayley. Significant progress has been made for small $n$ ($n \leq 20,000$) since his call to action. In this presentation we will discuss work related to a recent classification of the groups of order $3^9$ which shows that there are 5,937,876,645 such groups. Topics include the $p$-group generation algorithm, solving computational problems in group theory via parallelization, and the design of search data structures for computational results within computational group theory.
  • 15 November 2022
    Character Tables and Brauer’s Legacy
    Mandi Schaeffer Fry
    Department of Mathematics and Statistics
    Metropolitan State University of Denver
    Abstract: Widely known as the founder of modular representation theory, Richard Brauer set the stage for the study of the so-called local-global conjectures in character theory. In this talk, I'll discuss some of these conjectures and their implications for other character-theoretic properties and lingering questions of Brauer.
  • 22 November 2022
    How can we extend the statement: "Every ring can be imbedded in the endomorphism ring of an Abelian group," to non-Abelian groups; or who needs 2 distributive laws? Part Two
    Gary Birkenmeier
    Abstract: We will discuss the evolution, history, and some basic facts for algebraic systems motivated by the title.

Spring 2022

For the Spring 2022 semester we will usually meet 2:15 - 3:15 on Fridays. Some meetings will be on Zoom and others will be in-person in Maxim Doucet 208. We may need to change the time and date on occasion to accommodate visitors.
For more information or connection details contact contact Arturo Magidin.

  • 28 January 2022 (postponed from 21 Jan)
    More on the Benson-Solomon fusion systems
    Justin Lynd
    UL Lafayette
    Abstract: In this seminar Dr. Lynd will expand on his colloquium presentation of Thursday (20 January 2022).
    A fusion system can be thought of as a "finite group at a prime p". After some historical remarks around the Classification of the Finite Simple Groups, I will explain what a p-fusion system is. Most of the examples of fusion systems we know of come from finite groups, but there do exist many exotic examples, chiefly at odd primes. In fact, we know of just one infinite family of simple exotic fusion systems at the prime 2, the Benson-Solomon fusion systems. In the second part of the talk, I will explain some coincidences that allow the Benson-Solomon systems to exist and, time permitting, discuss some of what is known about them.
  • 11 February 2022
    An overview of the co-class conjectures: order out of chaos and general out of particular in the study of finite p-groups
    Arturo Magidin
    UL Lafayette
    Abstract: p-groups play an important role in the study of finite groups, and in a sense lie at the opposite end of finite simple groups, since every subgroup is contained in a proper normal subgroups. The study of p-groups, however, had often focused on special classes of groups (groups with large abelian subgroups, groups with certain invariant values, and so on), with ad hoc techniques that seemed unlikely to generalize.
    In 1980, Leedham-Green and Newman proposed a series of conjectures, known as the Co-class Conjectures, that sought to provide order out of this seeming chaos. The conjectures were proven by Leedham-Green and by Shalev in 1994.
    I will give a brief summary of some of the early attempts at studying p-groups, before giving an overview of what the Co-class conjectures say (and the necessary background and definitions to understand them) and how they provide some coarse-grained structure to the family of p-groups. If time permits, I will talk a bit about further developments.
  • 18 February 2022 (on Zoom)
    Towards the Amit-Ashurst Conjecture for Word Maps
    William Cocke
    Augusta University, August GA
    Abstract: The Amit-Ashurst Conjecture asks whether the probability distribution induced by a word map on a finite nilpotent group G has minimal non-zero value of at least 1/|G|. In this talk we will explore some of the history of this conjecture and some open questions related to it. Then we will discuss a recent result with Rachel Camina and Anitha Thillaisundaram showing that p-groups with a cyclic maximal subgroup satisfy this conjecture.
  • 4 March 2022
    A finite geometry problem arising from the character theory of finite groups
    Michael Geline
    Northern Illinois University
    A finite geometry problem arising from the character theory of finite groups Abstract: The local to global philosophy of finite group theory says that questions concerning a finite group G together with a prime p ought to have answers that are visible in the normalizers of nontrivial p-subgroups of G. One such question involves the (ordinary) irreducible characters of G and is known as Brauer's height zero conjecture. An elementary approach to this conjecture, involving p-adic representations, has given rise to a problem in finite geometry that I consider nice enough to be of independent interest. We will go through this problem and then discuss how exactly it arose.
  • 11 March 2022
    An Introduction to Sharp Permutation Groups
    Douglas Brozovic
    University of North Texas
  • 25 March 2022
    Limits and colimits, generators and relations of partial groups
    Omar Dennaoui
    UL Lafayette
    Abstract: This is an expository talk, mainly influenced by E. Salati's paper on the topic, I will briefly define what a partial group is and then will showcase what are the limits and colimits in the category Part or partial groups. I will then show that Part is (co)complete. Finally I will touch up on free partial groups (generators and relations).
  • 1 April 2022 (on zoom)
    An Introduction to NTRUEncrypt and the Math Problem The Keeps It Secure
    Elizabeth Wilcox
    SUNY-Oswego
    Abstract: NTRUEncrypt is a public key cryptosystem proposed back in the mid-90s by three number theorists. It's a secure, fast encryption algorithm, but not particularly well-known or often used -- yet, it's in the running as one of seven finalists in the National Institute of Standards and Technology call for post-quantum cryptography procedures. Like other public key cryptosystems, the mathematical underpinnings of NTRUEncrypt are in algebra but the procedure can be viewed through different mathematical lenses. In this talk, we'll learn about this cryptosystem and the mathematics, and see how some of the concepts from abstract algebra are used (but not successfully) to attack the system.

Fall 2021

During the Fall 2021 semester we will meet on Fridays from 1:00-2:00 in Maxim Doucet Hall, room 208. You can also join us via Zoom! Contact contact Justin Lynd for the Zoom info.

  • 17 September 2021
    An introduction to partial groups
    Justin Lynd, UL Lafayette
    Abstract: A partial group is a set together with a multivariable product which is only defined on a specified set of words in the underlying set. (A group is a partial group in which all products of words are defined.) This notion was introduced by Chermak in 2013 as the basic object in a framework for studying the $p$-local structure of a finite group G, i.e. how G conjugates its p-subgroups. I will explain briefly some of the history behind Chermak's work, which originates in topology. But mainly I plan give an introduction to partial groups, and explain several examples. This includes the motivational examples, the localities of finite groups.
  • 24 September 2021
    Generalizing the Chermak-Delgado lattice of a finite group: a very preliminary report
    Arturo Magidin, UL Lafayette
    Abstract: In 1989, Andrew Chermak and Alberto Delgado introduced a family of measures for subgroups of a finite group, connecting a subgroup with its centralizer. They then used the measure to obtain interesting results guaranteeing the existence of a characteristic abelian groups of bounded index.
    The normalized version of the measure leads to a collection of subgroups which form an interesting self-dual sublattice of the lattice of all subgroups, called the Chermak-Delgado lattice of the finite group.
    In joint work with William Cocke, L-C Kappe, and Elizabeth Wilcox, we are looking at possible generalizations of the measure that connects subgroups with certain "relative marginal subgroups." I will discuss what we are trying to do, what the obstacles are, and what questions these obstacles suggest we (and others) may want to address.
  • 1 October 2021
    Exotic and block-exotic fusion systems
    Patrick Serwene, Technische Universität Dresden (TU Dresden)
    Abstract: One of the main problems in the theory of fusion systems is the question whether a fusion system arises in the form of a finite group if and only if it arises in the form of a p-block of a finite group. There is a conjecture saying that a fusion system is induced by a group if and only if it is induced by a block. We present reduction theorems for this problem reducing it to blocks of quasisimple groups in certain cases. One of these reductions settles the conjecture for the family of Parker–Semeraro fusion systems. We discuss ongoing work concerning our strategy to prove the conjecture for some groups of Lie type.
  • 29 October 2021
    Endotrivial modules for cyclic p-groups and generalized quaternion groups via Galois descent
    Richard Wong, UCLA
    Abstract: One problem of interest in modular representation theory of finite groups is in computing the group of endo-trivial modules. This group was originally computed by Carlson-Thevenaz using the theory of support varieties. However, one can approach this problem through homotopy theory, where this group is known as the Picard group of the stable module category. Jointly with Jeroen van der Meer, we provide new, homotopical proofs of their results for the quaternion group of order 8, generalized quaternion groups, and cyclic p-groups using descent methods.
  • 5 November 2021
    Decreasing paths of triangles
    Leonel Robert, UL Lafayette
    Abstract: I will discuss a geometric problem on decreasing paths of triangles, as well as its higher dimensional generalizations (to tetrahedra, etc). The problem can be reformulated in terms of non-homogeneous Markov chains. In this context it is known as the "embedding problem". The problem can be approached using methods from control theory, but the farthest reaching results so far rely only on elementary methods from algebra and geometry.
  • 12 November 2021
    Two problems related to tilting modules for algebraic groups
    Paul Sobaje, Georgia Southern University
    Abstract: The modular representation theory of reductive algebraic groups (general linear groups being an example of such groups) has a number of longstanding open problems. Several of these problems have conjectured resolutions that involve special modules known as indecomposable tilting modules. In this talk we will look at how tilting modules relate to two problems in particular: 1) integrating representations from the Lie algebra to the algebraic group, and 2) finding a character formula for the irreducible representations. A good deal of background material will be provided throughout.
  • 19 November 2021
    Asymptotic Behavior of Tensor Powers of Modular Representations and Banach Algebras
    Peter Symonds,  University of Manchester
    Abstract: Let M be a finite dimensional modular representation of a finite group G. We consider the dimension of the non-projective part of a tensor power of M, and how this grows as the power increases. From this we obtain an invariant of M, which we investigate using tools from representation theory and from the theory of commutative Banach algebras. (Joint work with Dave Benson.)

Spring 2020

  • 11 February 2020
    The Chermak-Delgado Lattice and generalizations I: An introduction
    Arturo Magidin (UL Lafayette)
    In 1989, Andrew Chermak and Alberto Delgado introduced a "measure" for subgroups of a group $G$ being acted on by a group $H$: given a subgroup $A\leq G$, $m_r(A) = |A|^r|C_H(A)|$. The set of subgroups for which the measure achieves its largest value has interesting properties: for example, it forms a sublattice of the lattice of subgroups of $G$, is self-dual, and all its elements are equal to their double centralizers.
    L-C Kappe, Elizabeth Wilcox, William Cocke and are currently exploring some generalizations, replacing the centralizer with some other constructions. As a lead-up to discussing some of those results, I will discuss the properties of the measure and some of the results about the lattice.
  • 18 February 2020
    The Chermak-Delgado Lattice and generalizations II: A digression into words and margins
    Arturo Magidin (UL Lafayette)
    The center of a group and the commutator subgroup play roles that are "dual" to each other in a way that was formalized by Philip Hall in the 1940s, as part of a proposed scheme to classify finite $p$-groups. He also indicated how these notions could be generalized to other subgroups, including but not restricted to terms of the upper and lower central series. These are the marginal and verbal subgroups; the latter were well known at the time, but the former and the connection between them was new.
    One can view centralizer subgroups as a kind of "relative center"; the point of departure for our attempts at generalizing the Chermak-Delgado construction lattice is trying to take this view and apply it to the general setting of marginal and verbal subgroups, by defining a "relative marginal subgroup".
    To put this in context, I will give an introduction to verbal and marginal subgroups.
  • 3 March 2020
    The Chermak-Delgado Lattice and generalizations III
    Arturo Magidin (UL Lafayette)
  • 17 March 2020 (CANCELLED)
    Lower Finite Modules over Commutative Rings
    Luke Harmon (University of Colorado Colorado Springs)
    Abstract: A partially ordered set (P,lt) is lower finite provided P is infinite and for each x in P, there are but finitely many elements y in P such that y lt x. We will call a module M lower finite if the set of proper submodules of M, partially ordered by set-theoretic containment, is lower finite. We will use the (well-studied) class of Jonsson modules (along with other classical results) to classify the lower finite modules over a commutative ring with identity.

Fall 2019

During the Fall 2019 semester we will meet on Mondays from 4:00-5:00 in Maxim Doucet Hall, room 201.

  • 16 September 2019
    A Partial Order on Subsets of Baer Bimodules with Applications to C*-Modules
    Gary Birkenmeier
    Abstract: In a series of seminars, the concept of Baer (p, q)-sets is introduced. Using this notion, we define Rickart, Baer, quasi-Baer and pi-Baer (S, R)-bimodules, respectively. We show how these conditions relate to each other. We also develop new properties of the minus partial order \leq-, we extend the relation \leq- to (S, R)-bimodules, and use it to characterize the aforementioned Rickart, Baer, quasi-Baer, and pi-Baer (S, R)-bimodules. Note that \leq- generalizes the natural order on the projections of a C*-algebra. Moreover we specify subsets K of the power set of a (S, R)-bimodule for which \leq- is a partial order and for which \leq- determines a lattice. We analyze the relation \leq- by examining the associated Baer (p, q)- sets. Finally, we apply our results to C*-modules. Examples are provided to illustrate and delimit our results.
  • 23 September 2019
    Gary Birkenmeier
  • 30 September 2019
    Gary Birkenmeier
  • 7 October 2019
    Gary Birkenmeier
  • 14 October 2019
    Normal subgroups of invertibles and of unitaries in C^*-algebras
    Leonel Robert Gonzalez
    Abstract: I will talk about my work on the normal subgroups of the group of invertibles in the connected component of the identity in a C*-algebra.
  • 4 November 2019
    Algebraic properties of modular tensor categories
    Yilong Wang
    Louisiana State University
    Abstract: Modular tensor categories (MTC) are categorical generalizations of finite abelian groups with non-degenerate quadratic forms. They give rise to invariants of links and 3-manifolds that has similar "quantum" nature as the Jones polynomial. They also provide representations of SL(2, Z/NZ) and mapping class groups of higher genus surfaces which can be viewed as derived from (2+1)-topological quantum field theories (TQFTs) associated to the MTCs. MTCs are also indispensable ingredients in the setup of the mathematical foundations for topological quantum computation.
    In this talk, we will introduce the notion of modular tensor categories with an emphsis on its rich connection with representation theory, (2+1)-TQFTs and topological quantum computing. Then we will present our recent work on the algebraic properties of MTCs and the TQFTs associated to them.
  • 11 November 2019
    Principal Series Representations of GL(2) Over Finite Fields
    Regina Poderzay
    UL Lafayette
    Abstract: The goal of this talk is to construct the principal series representations of GL(2). To begin, we observe rudimentary results and examples using prerequisite knowledge from linear algebra and group theory. This is followed by inducing new representations from old ones. A key component in the classification of induced representations is Mackey’s theorem. At last is where we construct the principal series representations. This construction is motivated by the Bruhat decomposition of GL(2) into the Borel subgroup and is achieved by counting the conjugacy classes of GL(2,q).
  • 22 November 2019 (Friday, 3:00-4:00)
    Linearly Presented Toric Edge Ideals of Bipartite Graphs
    Jason McCullough
    Iowa State University
    Abstract: Let G = (V,E) be a finite simple graph and let k be a field. Denote by k[E] the polynomial ring with edges of G identified with variables and k[V] the polynomial ring with vertices of G identified with variables. The toric edge ring of G is the k-subalgebra of k[V] generated by v_i*v_j for all {i,j} in E. If we take the canonical surjection from k[E] to k[G], the kernel defines a (prime) toric ideal called the toric edge ideal I_G of G. In the case when G is a complete bipartite graph K_m,n, I_G is the defining ideal of the Segre embedding of P^(m-1) x P^(n-1) in P^(mn - 1). It is well-known that generators of I_G correspond to even closed walks in G. Hibi and Herzog previously characterized those bipartite graphs G such that I_G is generated by quadratic polynomials. We present a combinatorial description of those ideals whose ideals are linearly presented. This result has connections to polyomino ideals and Hibi rings that I will explain. This is joint work with Zach Greif.

Spring 2019

During the Spring 2019 semester we will meet on Fridays from 2:10-3:10 in Maxim Doucet Hall, room 210.

  • 1 February 2019
    The structure of a minimal counterexample to the Feit-Thompson theorem
    George Glauberman
    University of Chicago
    Abstract: This talk supplements my colloquium talk by describing in more detail the proof of the Feit-Thompson Theorem.
  • 15 February 2019
    Minimal bisets for fusion systems and blocks of finite group algebras, I
    Justin Lynd
    Abstract: I will discuss the background for ongoing work with M. Gelvin (Bilkent) in which we investigate the question: which finite groups are minimal bisets for their fusion systems? As it turns out, this question is closely related to the following question in the modular representation theory of finite groups: when k is an algebraically closed field of characteristic p, for which finite groups G is the group algebra kG indecomposable as a k-algebra?
    A saturated fusion system F over a finite p-group S is a finite category with objects the subgroups of S and with morphisms between subgroups which are "conjugation-like" group homomorphisms. Here, S plays the role of a Sylow p-subgroup of the category. In the first talk, I will give the definition of a saturated fusion system F. Although F need not be the localization of any finite group G, one can define an S-S biset associated with F that plays the same role as the S-S biset G, whenever F is the localization of G at p.
  • 8 March 2019
    Minimal bisets for fusion systems and blocks of finite group algebras, II
    Justin Lynd
  • 15 March 2019
    Minimal bisets for fusion systems and blocks of finite group algebras, III
    Justin Lynd
  • 22 March 2019
    Minimal bisets for fusion systems and blocks of finite group algebras, IV
    Justin Lynd
    Abstract: In the first three talks, we have defined the notion of a saturated fusion system \F on a finite p-group S, and we have associated to it an S-S biset X. When \F is the fusion system of a finite group G on its Sylow p-subgroup S, then one may take X = G. But such a characteristic biset is not uniquely determined by \F. In the remaining talks, we will look at the unique up to isomorphism minimal such biset under inclusion, and we will try to bring out the connection between this minimal biset, when \F = \F_S(G), and the blocks of the group algebra kG.
  • 25 April 2019
    Minimal bisets for fusion systems and blocks of finite group algebras, V
    Justin Lynd
    Abstract: In the first three talks, we have defined the notion of a saturated fusion system \F on a finite p-group S, and we have associated to it an S-S biset X. When \F is the fusion system of a finite group G on its Sylow p-subgroup S, then one may take X = G. But such a characteristic biset is not uniquely determined by \F. In the remaining talks, we will look at the unique up to isomorphism minimal such biset under inclusion, and we will try to bring out the connection between this minimal biset, when \F = \F_S(G), and the blocks of the group algebra kG.

Fall 2018

During the Fall 2018 semester we will meet on Fridays from 2:10-3:00 in Maxim Doucet Hall, room 201.

  • 14 September 2018
    Closure operators on subgroup and other lattices
    Arturo Magidin
     
    Abstract: I will talk about joint work with Martha Kilpack, in which we determined all finite lattices L for which the lattice of closure opeartors on L is isomorphic to the lattice of subgroups of a group; and also the same problem but starting from the lattice of subgroups of an infinite group.
  • 21 September 2018
    Closure operators on subgroup and other lattices II
    Arturo Magidin
     
    Abstract: I will continue the series of talks about joint work with Martha Kilpack, in which we determined all finite lattices L for which the lattice of closure operators on L is isomorphic to the lattice of subgroups of a group; and also the same problem but starting from the lattice of subgroups of an infinite group.
    I will present our main "exclusion result", which provides that certain lattices L do not have the desired property, as well as a reduction result that allows us to invoke inductive arguments. If time permits, we will complete the determination of all finite lattices L for which co(L) is isomorphic to sub(G) for some group G.
  • 28 September 2018
    Closure operators on subgroup and other lattices III
    Arturo Magidin
     
    Abstract: I will continue the series of talks about joint work with Martha Kilpack, in which we determined all finite lattices L for which the lattice of closure operators on L is isomorphic to the lattice of subgroups of a group; and also the same problem but starting from the lattice of subgroups of an infinite group.
    The next step is relate some downsets on the lattice of closure operators co(L) with the closure operators on certain subsets of L; this will allow us to show that if co(sub(G)) is isomorphic to a subgroup lattice, and N is a normal subgroup of G, then co(sub(G/N)) is also a subgroup lattice. This allows for inductive arguments on |G|, since we have already reduced the problem to groups whose order is of the form p^a*q^b.
  • 12 October 2018
    Closure operators on subgroup and other lattices IV
    Arturo Magidin
     
    Abstract: I will finish my series of talks about joint work with Martha Kilpack, in which we determined all finite lattices L for which the lattice of closure operators on L is isomorphic to the lattice of subgroups of a group; and also the same problem but starting from the lattice of subgroups of an infinite group.
    To finish off we will consider the case where G is an infinite group, leading to closure operators on an infinite lattice. We will discuss some technical issues that arise from this, and how the notion of "compact element" of a lattice and "algebraic closure operator" come in to cover the breach. Finally, we determine all infinite groups G for which the lattice of algebraic closure operators on Sub(G) form a subgroup lattice.
  • 19 October 2018
    π-Baer Rings
    Yeliz Kara
     
    Abstract: I will talk about joint work with Gary F. Birkenmeier and Adnan Tercan, in which we introduced and investigated the concept of π-Baer rings. I will present connections between the π-Baer condition and the related conditions such as the Baer and quasi-Baer conditions.
  • 19 October 2018
    π-Baer Rings II
    Yeliz Kara
     
    Abstract: I will complete the series of talks about joint work with Gary F. Birkenmeier and Adnan Tercan, in which we introduced and investigated the concept of π-Baer rings. I will explain the π-Baer ring results on polynomial rings and 2-by-2 generalized upper triangular matrix rings.
  • 2 November 2018
    Update on the Classification of Indecomposable Quasi-Frobenius Rings
    Gary Birkenmeier
     
    Abstract: This talk will review and update the nilary quasi-Frobenius rings. Recall that a ring R with 1 is "nilary" if AB = 0 implies that either A or B is nilpotent, for all ideals A and B of R. A ring is "quasi-Frobenius", denoted QF, if it is right selfinjective and right Noetherian.
  • 9 November 2018
    Update on the Classification of Indecomposable Quasi-Frobenius Rings II
    Gary Birkenmeier
     
    Abstract: This talk will review and update the nilary quasi-Frobenius rings. Recall that a ring R with 1 is "nilary" if AB = 0 implies that either A or B is nilpotent, for all ideals A and B of R. A ring is "quasi-Frobenius", denoted QF, if it is right selfinjective and right Noetherian.
  • 30 November 2018
    Update on the Classification of Indecomposable Quasi-Frobenius Rings II
    Gary Birkenmeier
     
    Abstract: This talk will review and update the nilary quasi-Frobenius rings. Recall that a ring R with 1 is "nilary" if AB = 0 implies that either A or B is nilpotent, for all ideals A and B of R. A ring is "quasi-Frobenius", denoted QF, if it is right selfinjective and right Noetherian.

Spring 2018

During the Spring 2018 semester we will meet on Tuesdays from 4:00-4:50 in Maxim Doucet Hall room 210.

  • 23 January 2018
    A Description of indecomposable QF-rings, part 1
    Gary Birkenmeier
     
    Abstract: In this series of talks, I will present a classification of indecomposable QF-rings in terms of the essentiality of the ideals generated by primitive idempotents.
  • 23 January 2018 (4:30-5:30 TIME CHANGE)
    A Description of indecomposable QF-rings, part 2
    Gary Birkenmeier
  • 13 February 2018
    No seminar this week
  • 20 February 2018
    No seminar this week
  • 27 February 2018
    no seminar this week
  • 6 March 2018
    No seminar this week
  • 13 March 2018
    No seminar this week
  • 27 March 2018
    A Description of indecomposable QF-rings, part 3
    Gary Birkenmeier
  • 3 April 2018
    no seminar this week (Spring Break)
  • 10 April 2018
    A Description of indecomposable QF-rings, part 4
    Gary Birkenmeier
  • 17 April 2018
    Support Varieties for Algebraic Groups and the Humphreys Conjecture
    William Hardesty
    Louisiana State University  
    Abstract: I will begin by introducing the notion of the support variety of a module over a finite group scheme. This will be followed by a brief overview of classical results and calculations for the case when the finite group scheme is the first Frobenius kernel of a reductive algebraic group G. In 1997, J. Humphreys conjectured that the support varieties of indecomposable tilting modules for G (a very important class of modules) is controlled by a combinatorial bijection, due to G. Lusztig, between nilpotent orbits and a certain collection of subsets of the affine Weyl group called "canonical cells". This later became known as the "Humphreys conjecture". I will discuss some recent developments concerning this conjecture, including its complete verification for G = GL(n) (appearing in my thesis) as well as some additional results in other types appearing in joint work with P. Achar and S. Riche.
  • 9 May 2018 (4:00-5:00 WEDNESDAY room 209)
    An application of algebra to topology
    George Glauberman
    University of Chicago
    Abstract: The theory of fusion systems is a new branch of mathematics with applications to finite group theory and algebraic topology. In particular, it is involved in a result in topology known as the Martino-Priddy Conjecture, which was proved recently by assuming the classification of finite simple groups. New research with J. Lynd has removed the classification from the proof by using group theory. I plan to describe these topics and some open problems.

Fall 2017

During the Fall 2017 semester we will meet on Tuesdays from 4:00-4:50 in Maxim Doucet Hall room 210.

  • 19 September 2017
    Some ideas and methods from the classification of the finite simple groups
    Justin Lynd
     
    Abstract: I intend to give a historical and high-level overview of the original program for the classification of the finite simple groups. Depending on one's point of view, this is a quest that began either in the late 1890s or in 1954, and it ended either in the early 1980s or in 2004. We will begin at the first beginning and end roughly at the first end. In the middle, the focus will be on the ideas behind the program, which partitions the collection of finite simple groups into three classes: the groups of low 2-rank (small), the groups of component type (odd), and the groups of characteristic 2-type (even). Inasmuch as there will be any details (unlikely, but this depends on one's definition of "details"), we will focus on the identification of groups in the small and odd cases.
  • 26 September 2017
    Some ideas and methods from the classification of the finite simple groups, part 2
    Justin Lynd
  • 10 October 2017
    Some ideas and methods from the classification of the finite simple groups, part 3
    Justin Lynd
  • 24 October 2017
    Some ideas and methods from the classification of the finite simple groups, part 4
    Justin Lynd
  • 31 October 2017
    Some ideas and methods from the classification of the finite simple groups, part 5
    Justin Lynd

Spring 2017

During the Spring 2017 semester we will meet on Fridays from noon-12:50 in Maxim Doucet Hall room 214.

  • 27 January 2017
    When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice?
    Arturo Magidin
     
    Abstract: (This is joint work with Martha Kilpack of Brigham Young University.)
    A lattice is a partially ordered set in which every pair of elements have a least upper bound and a greatest lower bound; it can also be viewed as an algebra with two binary operators satisfying certain identities. A typical example of a lattice is the lattice of all subgroups of a given group, ordered by inclusion (or more generally, of all substructures of a given structure).
    A closure operator on a partially ordered set P is a function f mapping P into P such that f satisfies three conditions:
    (i) f is increasing: x less than or equal to f(x) for all x in P;
    (ii) f is isotone: if x less than or equal to y, then f(x) less than or equal to f(y) for all x,y in P;
    (iii) f is idempotent: f(f(x)) = f(x).
    We can partially order all closure operators on P, by letting f less than or equal to g if and only if f(x) less than or equal to g(x) for all x in P. If P is a lattice, then this makes the set of all closure operators on P into a lattice.
    It is a theorem of Birkhoff that every complete lattice is the lattice of subalgebras of some (possibly infinitary algebra); and a theorem of Whitman that every lattice can be embedded as a sublattice of a subgroup lattice. This leads to the question of which lattices of closure operators are isomorphic to the subgroup lattice of a group.
    Previously, we had shown that if we look at the closure operators on a subgroup lattice Sub(G), then this resulting lattice, c.o.(Sub(G)) is itself a subgroup lattice if and only if G is cyclic of prime power order. We will extend the investigation first to the case of closure operators on an arbitrary finite lattice; and later, to the lattice of subgroups of an infinite group.
  • 3 February 2017
    When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 2)
    Arturo Magidin
  • 10 February 2017
    When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 3)
    Arturo Magidin
  • 17 February 2017
    When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 4)
    Arturo Magidin
  • 24 February 2017
    If G is an infinite group, when is the lattice of algebraic closure operators on Sub(G) isomorphic to the subgroup lattice of a group K?
    Arturo Magidin
  • 3 March 2017
    When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (last part)
    Arturo Magidin
  • 10 March 2017 (SPECIAL TIME 12:20)
    A description of indecomposable quasi-Frobenius rings
    Gary Birkenmeier
    Abstract In this talk, we characterize a nilary QF-ring R in terms of the essentiality of the ideals ReR where e is a primitive idempotent of R. Recall that a ring R is nilary if AB = 0 implies either A or B is nilpotent for all ideals A, B of R. Note that nilary rings are indecomposable rings. This is a preliminary report on joint research with Omar A. Al-Mallah andHafedh M. Al-Noghashi.
  • 24 March 2017
    A description of indecomposable quasi-Frobenius rings (part 2)
    Gary Birkenmeier
  • 31 March 2017
    A description of indecomposable quasi-Frobenius rings (part 3)
    Gary Birkenmeier
  • 7 April 2017
    A description of indecomposable quasi-Frobenius rings (part 4)
    Gary  Birkenmeier