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Combinatorics/Algebra Seminar
Fall 2019 Archive:
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December 2
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Myra Cohen (Computer Science): Learning to build covering arrays with hyper-heuristic search.
Abstract:
A covering array CA(N;t,k,v) is an N x k array on v symbols where each N x t sub-array contains all ordered t-sets at least once. Covering arrays and their generalizations have been used extensively in software testing to define samples of tests or configurations that can systematically and efficiently detect interactions between different inputs or configuration options of a system. Today there are a large number of techniques and tools to build covering arrays. Some of the algorithms are optimized for constrained or unconstrained variants of the problem or work best on specific symbol sizes or strengths. Yet, no one strategy fits all. In this talk I will discuss some of the core algorithms used to find covering arrays and present a hyper-heuristic algorithm that learns strategies for building covering arrays. Our experiments show that a generalist approach is competitive with the best-known search algorithms, sometimes even improving on bounds. We also present some evidence that our algorithm's strong generic performance is the result of unsupervised learning.
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November 18
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Oyeyemi Oyebola (Federal University of Agriculture, Abeokuta, Nigeria): Extra polyloops and their applications to biological inheritance.
Abstract:
We introduce non-associative algebraic hyperstructures known as polyloops, and examine a special class of polyloops, extra polyloops, which generalize the binary algebraic structures known as extra loops. We investigate and characterize the inter-relationships among extra polyloops of types 1, 2, and 3. The corresponding extra identities and hyper-algebraic properties are applied to measure the significant level of nonassociativity in weakly associative structures
derived from biological inheritance.
This is joint work with Temitope G. Jaiyeola (Obafemi Awolowo University).
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November 11
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Jason McCullough: Monomial and binomial ideals.
Abstract:
This talk is largely an advertisement for the Math 610 EGR course I am teaching in the spring semester. As such, I won't be giving detailed proofs but rather introducing topics from algebraic geometry, commutative algebra, and combinatorics and how they are interrelated. The main computational tools will be free resolutions and Groebner bases of graded ideals in a polynomial ring. I will discuss the relationships between resolutions of an ideal and its initial ideal, polarization of monomial ideals, and the Stanley-Reisner correspondence between square-free monomial ideals and simplicial homology. If time allows I'll mention some results from combinatorial commutative algebra and how one can compute these in Macaulay2. The talk should be accessible to any graduate student interested in algebra, geometry or combinatorics.
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November 4
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Jonathan Smith: Jordan algebras and projective spaces.
Abstract:
Jordan algebras are commutative but nonassociative (linear) algebras where the multiplications by x and x2 commute. In this expository talk, we discuss their connections with quantum mechanics, projective geometry, and Hopf fibrations; and the classification of formally real Jordan algebras.
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October 28
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Jonathan Smith: Real, complex, and quaternion representations.
Abstract:
This talk will mainly be an expository account of the defining skewfield for ordinary group representations, and the Frobenius-Schur indicator. If time permits, we will discuss potential extensions to quasigroup and association scheme characters.
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October 14, 21
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Sung-Yell Song:
Group-case association schemes and their classification.
Click here for PDF abstract
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September 30, October 7
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Alex Nowak:
Distributive Mendelsohn triple systems and the Eisenstein integers.
Abstract:
Mendelsohn triple systems (MTS) constitute a class of block designs
which generalize Steiner triple systems.
Quasigroups obeying the semisymmetric law x(yx)=y
and the idempotence law xx=x provide an algebraic account for MTS.
Further imposing x(yz)=(xy)(xz),
left self-distributivity,
on MTS multiplications narrows our focus to linear quasigroups,
which are quasigroups whose multiplications can be expressed entirely
in terms of abelian group and commutative Moufang loop automorphisms.
In the first half of this two-part talk, we will describe the connection
between self-distributivity and linearity in quasigroups.
We then show how the structure of primes in the Eisenstein integers,
Z[X]/(X2-X+1),
gives us a direct product decomposition for distributive MTS.
We will then classify, up to isomorphism,
all distributive MTS of order coprime with 3,
and enumerate these isomorphism classes.
In week two, we will discuss progress on, and complications related to,
the case of distributive MTS with point set having order 3km.
The theory of congruence subgroups of SL2(Z)
provides a good language in which to articulate these complications.
We will also draw connections
between the previous week's discussion and orthogonality of Latin squares.
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September 23
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Anna
Romanowska (Warsaw University of Technology):
Semilattice sums of algebras and Mal'tsev products of varieties.
Abstract:
Let Q and R be (quasi)varieties of algebras of the same type.
Assume additionally that algebras in R are idempotent.
Then the Mal'tsev product Q o R of Q and R
consists of algebras A with a congruence relation r
such that the quotient A r is in R
and each congruence class a r is in Q.
(The congruence classes are subalgebras of A.)
Each algebra A in Q o R decomposes into a
disjoint union of its subalgebras a r.
It is known that the Mal'tsev product of two varieties of similar algebras is a quasivariety.
We consider the question of when this quasivariety is a variety.
Let V be a variety of algebras with no nullary operations,
with at least one non-unary operation,
and satisfying any (strongly irregular) identitity
of the form x * y = x.
Let S be the variety of algebras of the same type as V,
equivalent to the variety of semilattices.
We were able to prove that in this case
the Mal'tsev product V o S is a variety,
and to show how to find an equational basis for the product
if an equational basis for V is known.
Members of the product V o S
are called semilattice sums of V-algebras.
We also have an example showing that if V is a regular variety,
then the Mal'tsev product may not be a variety.
The results are from a paper (under preparation) by C. Bergman, T. Penza and A. Romanowska.
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September 16
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Durga Paudyal (Ames Laboratory):
Symmetry and topology of rare-earth magnetic materials.
Abstract:
The increasing technological importance of rare-earth based magnetic materials
is attracting the attention of scientists from all over the world.
Although a lot of work has been done on the experimental side,
a true theoretical understanding of the
symmetry, topology, and magnetism of these materials is lagging behind.
We offer a survey of the systematic theoretical research on rare-earth magnets
performed over the years using density functional theory.
We also present suitable finite-temperature models for comparison
with the properties obtained from precise experiments.
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September 9
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Jonathan Smith: Quasigroup homotopy, semisymmetry, and Mendelsohn triple systems.
Abstract:
Quasigroups are non-associative groups, whose multiplication table is a Latin square.
While quasigroup homomorphisms are important,
a more general notion of quasigroup homotopy, defined by a triple of maps,
is often more relevant in geometric and physical applications.
For example, the spin group of dimension 8 is the group of invertible homotopies
from the quasigroup of non-zero octonions to itself.
A quasigroup is semisymmetric if it satisfies the identity
y(xy)=x. A semisymmetrization functor,
defining a semisymmetric quasigroup structure on the direct cube of any quasigroup,
reduces a homotopy between quasigroups
to a homomorphism between their semisymmetrizations.
Idempotent semisymmetric quasigroups correspond to Mendelsohn triple systems,
combinatorial designs in which a set of vertices is decomposed into 3-cycles,
such that any ordered pair of distinct points lies in a unique 3-cycle.
Recently, Krapež and Petrić introduced a more "economical" semisymmetrization,
defined on the direct square of a quasigroup.
Since the construction of Krapež and Petrić
actually produces an idempotent quasigroup,
we describe it as a Mendelsohnization. For isotopes of abelian groups,
we investigate the relation between semisymmetrization and Mendelsohnization.
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