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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 Gary F. Birkenmeier.

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