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BHZ & beyond: Exploring new
"heights"
Mandi Schaeffer Fry (University of Denver) Friday 1:30
Featured Talk
In recent joint work
with Gabriel Navarro, Gunter Malle, and Pham Huu
Tiep, we completed the proof of Brauer’s long-standing Height Zero
Conjecture (BHZ), one of the first "local-global" conjectures in the
representation theory of finite groups. This now-theorem says that
every character in a p-block of a finite group has height zero if
and only if the corresponding defect groups are abelian. In the first
part of this talk, I will introduce the BHZ and some of its
history. Then, I will move on to the primary goal, which is to discuss
several extensions of the BHZ, including the Eaton-Moretó conjecture,
a Galois version, and a normal version. This will include joint work
with Gunter Malle, Alex Moretó,
and Noelia Rizo, along with various
combinations of the four of us.
Poisson structure on
quantum groups
Bach Nguyen (Xavier University of Louisiana) Friday
2:30
Quantum groups are noncommutative algebras coming from
deformations of the universal enveloping algebra of the Lie algebras. They
are intimately connected to the underlying Lie algebra and its Lie group,
especially when the quantum group is specialized at a root of unity. In
the remarkable work of De Concini and Procesi in the early 1990's, they
showed that the center of the quantum group at a root of unity admits a
canonical Poisson bracket which is related to the standard Poisson
structure on the Lie group. Using this Poisson structure on the center, De
Concini and Procesi proved many representation theoretic results which
play a crucial role in the modular representation theory of the Lie group.
However, such a picture is missing for the Yangians, another kind of
deformation of the Lie algebra. In this talk, we will review the quantum
group picture and also show how it can be carried over to the Yangians.
This is a joint work with Prasad Senesi.
On symmetric
representations of SL(2,Z)
Siu-Hung Ng (Louisiana State University)
Friday 3:30
The linear representations of SL(2,Z)
arising from
modular tensor
categories are symmetric
and have congruence kernel. Conversely, one may also reconstruct
modular data from finite-dimensional symmetric, congruence
representations of SL(2,Z).
In this talk,
I will introduce the
notions of symmetric and symmetrizable representations of
SL(2,Z) and
demonstrate some
unsymmetrizable and symmetrizable examples. By investigating a
(Z/2Z)-symmetry of some
Weil representations at prime power levels, all finite-dimensional
congruence representations of SL(2,Z)
are proved to be
symmetrizable. This talk is based on a joint work with Yilong Wang and
Samuel Wilson
Contractibility of the orbit space
of a saturated fusion system after
Steinberg
Omar Dennaoui (UL Lafayette) Friday 4:00
Recently, Steinberg used discrete Morse
theory to give a new proof of
a theorem of Symonds that the orbit space of the poset of nontrivial
p-subgroups of a finite group is contractible. We extend Steinberg’s
argument in two ways, covering more general versions of the theorem
that were already known. In particular, following a strategy of
Libman, we give a discrete Morse theoretic argument for the
contractibility of the orbit space of a saturated fusion system.
MV-frames
Jean Nganou (University of Houston-Downtown) Friday 4:30
MV-frames, which are complete MV-algebras are investigated from
frame-theoretic point of view. We completely characterize algebraic
MV-frames as well as regular MV-frames. In addition, we consider
nuclei on MV-frames in general and on MV-frames of ideals of
Łukasiewicz rings in particular.
Leibniz homology and some
non-relativistic invariants
Guy Biyogmam (Georgia College & State University) Saturday
11:00
Non-relativistic invariants play an important role in non-relativistic
quantum field theories. In this talk, we will discuss how Lie
algebra and Leibniz algebra homology calculated on various non
semisimple Lie algebras of theoretical and mathematical physics
captures several non-relativistic invariants.
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Algebraic groups and Lie algebras in
positive characteristic
Cornelius Pillen (University of South Alabama) Saturday 8:00
Featured Talk
Let G be a simple algebraic group over an algebraically closed field
k. In the classical case, the characteristic zero case, the finite
dimensional rational representations of G are just lifts of
representations of the associated Lie algebra. In positive
characteristic the restricted enveloping algebra u(g) and the
Frobenius kernels still approximate G, but they provide a picture
that is far from complete. In the positive characteristic case certain
classes of representations of the Lie algebra can still be lifted to
the algebraic group. But there are situations where the lifting fails
depending on the characteristic of k.
In this talk we delve into the rich history of questions on
lifting. Our discussion will start with the seminal paper by Humphreys
and Verma from 1972. We will give partial answers to questions and
conjectures due to Humphreys, Verma, Jantzen and Donkin.
Lie algebras whose derivation
algebras are simple
Jörg Feldvoss (University of South Alabama), Saturday
9:00
It is well known that a finite-dimensional Lie algebra over a field of
characteristic zero
is simple exactly when its derivation algebra is simple. In this talk
we characterize those
Lie algebras of arbitrary dimension over any field that have a simple
derivation algebra.
As an application we classify the Lie algebras that have a complete
simple derivation
algebra and are either finite-dimensional over an algebraically closed
field of prime
characteristic p>3 or Z-graded of finite growth over an
algebraically closed field of characteristic zero.
This is joint work with Salvatore Siciliano from the Università
del Salento.
Maximal hyperplane sections of
Schubert varieties with applications to coding theory
Sudhir Ghorpade (Indian Institute of Technology Bombay) Saturday
10:30
Schubert varieties are classical objects whose study goes back to the
work in the 19th century by Grassmann and Schubert. We focus on
Schubert variety in a Grassmannian and consider its nondegenerate
embedding in a projective space (induced by the Plucker embedding). We
then consider hyperplanes of this projective space and their
intersections with the given Schubert variety. The question that we
are interested in is the following:
What are the maximal hyperplane sections and how many of the
hyperplane sections are maximal?
Of course, the word "maximal" can be interpreted in several ways. A
precise interpretation that we follow is to fix a finite field
Fq and regard a hyperplane section as maximal if it has
the maximum possible number of Fq-rational points. The
above question then amounts to asking (i) what is the maximum possible
number of Fq-rational points on a hyperplane section of a
Schubert variety in Grassmannian? and (ii) can we characterize the
Fq-rational hyperplanes for which the corresponding
section of the Schubert variety has the maximum
possible number of Fq-rational points, and further, can we
enumerate the number of such maximal hyperplanes?
Questions such as these arise naturally in coding theory (albeit, with
a different nomenclature), but can also be of interest in algebraic
geometry (over finite fields). Answers to both these questions are
known due to Nogin (1996) when the Schubert variety is the full
Grassmannian. A conjectural answer to (i) was proposed in 2000 and
this conjecture was eventually settled in the affirmative in
2008. More recently, a conjectural answer to (ii) has been proposed in
2018 and the general case is still open although several partial
results
are known. We shall outline these developments and discuss some recent
progress.
This talk is partly based on a joint work with Prasant Singh and with
Mrinmoy Datta and Avijit Panja.
On Borel adjoint orbits in maximal
nilpotent subalgebras of Type C
Meaza Bogale (Hampton University) Saturday 10:30
Let B, H, and U
be the set of
Borel, Cartan, and maximum unipotent subgroup of symplectic group
Sp2n,
respectively. We describe the adjoint orbits of a Borel
subgroup on a maximal nilpotent algebra of type C. We consider the
adjoint action of Borel subgroup of the symplectic Lie group Sp2n
on the maximum nilpotent subalgebra $\mathfrak{n}$ of the Lie algebra
sp2n to study adjoint orbits in the type C case. Our
goals are to describe elementary adjoint actions in n in
terms of the positive root system, give a redefined version of
Belitskii's algorithm and use this algorithm to describe the
corresponding canonical forms on the symmetrized lattice of positive
roots. (Joint work with Dr. Huajun Huang)
Rigid automorphisms of the linking systems of finite groups of
Lie type
Jonathon Villareal (UL Lafayette) Saturday 11:30
Let L be a centric linking system associated with a
saturated fusion system on a finite p-group S. An automorphism of
L$ is said to be rigid if it restricts to the identity on
the fusion system. An inner rigid automorphism is conjugation by some
element of the center of S. If L is the centric linking
system of a finite group G, then rigid automorphisms of
L are closely related to automorphisms of G that
centralize S. When the prime is odd, it is known that all rigid
automorphisms are inner, but this is not true for the prime 2. We
determine which known quasisimple linking systems at the prime 2 have
a noninner rigid automorphism. Based on previous results, this
involves handling the case of the linking system at the prime 2 of a
finite simple group of Lie type in odd characteristic. These have no
noninner rigid automorphisms with two families of exceptions: the
2-dimensional linear groups and even dimensional orthogonal groups for
quadratic forms of nonsquare discriminant.
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Basic extensions of modules of
infinite dimension
Sergio Lopez-Permouth (Ohio University) Saturday 2:00
Featured Talk
For any algebra A, over a field F, an infinite
dimensional module M, with
a basis B (as an F-vector space) is naturally related to
the vector space
FB, the product of copies of F indexed
by B. The module structure of
F(B), isomorphic to that of M, may be extended
to FB when
B satisfies a
summability condition we refer to as being amenable. The module thus
obtained is said to be a basic module, or a basic extension of M.
In this talk, we present the state-of-the-art in the ongoing process of
answering natural questions that arise in this context and reflects the
work of the speakers with many collaborators who will be acknowledged
during the presentation.
Square-difference factor absorbing
ideals of a commutative ring
Ayman Badawi (American University of Sharjah) Saturday 3:00
Let R be a commutative ring with 1≠0. A proper
ideal I of
R is a square-difference factor absorbing ideal (sdf-absorbing
ideal) of R if whenever a2 - b2 is
in I for 0≠a,b in R,
then a + b in I or a - b
in I. In this paper, we introduce and
investigate sdf-absorbing ideals.
The complete Koszul Structure
Theorem for trivariate monomial ideals
Jared Painter (University of North Alabama) Saturday 4:00
We will outline the complete set of Koszul structures for R=S/I,
where S=k[x,y,z] and I is a monomial ideal. As
minimal free
resolutions of such ideals correspond geometrically to planar graphs,
it is known that the specific Koszul algebra structure can be
determined from this planar graph. When I is minimally generated
by n monomials and is contained in m2 where
m is the homogeneous maximal ideal of S, we will show
how to determine the Koszul structure from simply looking at the
minimal generating set for I. Our outcomes from this exploration
yield Koszul structures of T, B, and
H(p,q), The as we will also ignore the simple case when R
is a complete intersection. For the most part, we do not restrict our
description of these Koszul structures by type (the rank of the last
free module in the minimal free resolution of R), but we will show
that R has max type and has Koszul structure of T if and
only if I is a generic monomial ideal.
The ideal topology on an integral
domain
Simplice Tchamna (Georgia College) Satruday 4:30
The ideal topology on an integral domain R is the linear topology
which has a fundamental system of neighborhoods of 0 the nonzero
ideal of R. We present properties of the ideal topology on a
Noetherian local domain. We give conditions under which the completion
in the ideal topology is Noetherian.
Prime ideals of B[x]
Naufil Sakran (Tulane University) Saturday 5:00
We give a complete classification of the prime ideals in the tropical
semiring B[x] where B is the Boolean algebra
{0,1}. We show that the set of prime ideals in B[x] can
be partitioned into classes indexed by integers and there are at most
three types of prime ideals in every class.
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Counting the number of fixed-point
subgroups of some finite groups
Gary Walls (Southeastern Louisiana University) Sunday 8:30
Let be a group and suppose that α∈Aut(G).
We
let CG(α)={x∈G |
xα=x} and we let
Acent(G)=CG(α) | α
∈ Aut(G)}.
In this talk we determine |Acent(G)| for various
classes of groups. In particular, we determine
|Acent(G)| when G
is a finite group and G/Z(G) is isomorphic
to D2n where D2n
is the dihedral group of order 2n
with n odd.
Partial monoid of embedded
surfaces in 3-Manifolds
Birch Bryant (University of North Alabama) Sunday 9:00
In 1961, Wolfgang Haken introduced a cut and paste operation on closed
surfaces embedded in a 3-manifold, M. Given two embedded, transverse
surfaces F, G ⊂ M, this operation, the Haken sum denoted F⊕
G, is dependent on a choice of orientation around each curve
γ∈ F ∩ G. Given a triangulation T of M, if
F,G are transverse to the 2-skeleton T(2), a condition
can be imposed on F,G so that a natural orientation can be chosen,
making ⊕ a well-defined binary operation. With these
restrictions, the Haken sum, defines a finitely generated commutative
partial monoid on the set of embedded surfaces transverse to T(2). We will discuss these partial monoids, and their extension
to a commutative monoid.
Centers and generalized centers of
zero-symmetric sandwich
nearrings without identity
G. Alan Cannon (Southeastern Louisiana University) Sunday
10:00
Let (G,+) be a finite group written additively with identity 0,
but not necessarily abelian, and let X be a finite, nonempty set.
Let φ G →X be a fixed function
with φ(0)=x0
Then M0(X,G) = {f: X → G |f(x0) = 0} is a
right zero-symmetric nearring under pointwise addition and
multiplication defined by f1*f2 =
f1∘ φ ∘ f2 for
all f1,f2 ∈ M0(X,G). For |G| ≥ 2 and |X| ≥ 2, we
characterize when a zero-symmetric sandwich
nearring M0(X, G) has a
multiplicative identity and, in that situation, determine those
functions with multiplicative inverses. We find the center
of M0(X,
G) and also find generalized centers in certain cases
when M0(X,
G) does not have an identity.
Simplicity of full centralizer
nearrings
Kent Neuerburg (Southeastern Louisiana University) Sunday
10:30
Let G be a group written additively with identity 0, but not
necessarily abelian. For a semigroup S of endomorphisms
of G, the set of functions
MS(G) = {f : G → G | f(0) = 0 and f
∘ s = s ∘ f for all s ∈ S} forms a right nearring
under function addition and composition, called the centralizer
nearring determined by G and S. Every nearring with identity is
isomorphic to an MS(G) for some G
and S. Thus centralizer
nearrings are integral in nearring theory. In this paper, we study of
the simplicity of centralizer nearrings, but restrict our attention to
the full centralizer nearrings, i.e., the centralizer nearrings where
S is one of I =
Inn G, A = Aut G,
or E = End G.
Horospherical Schubert
varieties
Mahir Bilen Can (Tulane University) Sunday 11:30
Horospherical varieties are an important class of spherical varieties
that generalize toric varieties and exhibit rich geometric and
combinatorial structures. In this talk, we first discuss the role of
horospherical varieties in representation theory. Then, we focus on
Schubert varieties with a horospherical structure, highlighting their
remarkable properties.