
Combinatorics/Algebra Seminar
The seminar meets Mondays at 4:10 pm CDT in 401 Carver Hall.
(But note the exceptional times and locations!)
If you wish to invite a guest or give a talk, please visit the
Post Office.

April 18 (Thursday, 4pm in 294 Carver Hall)

Connor Depies: New approaches to the octonions and their maximal orders (Ph.D. thesis defense, public part).

April 9 (Tuesday, 2pm in 401 Carver Hall)

Mitchell Ashburn: Vectormatrix algebras and formed monotopies of formed Lie algebras (Ph.D. thesis defense, public part).

April 1

Joe Miller: Constructing nonisomorphic real projective planes using algebraically defined graphs.
Abstract:
Projective planes are often defined using three axioms which say: (i) any two points lie on a unique line, (ii) any two lines intersect at a unique point, and (iii) there exist four points, no three of which lie on the same line. This presentation will show how the standard real projective plane RP^{2} can be understood in this way. We will also show that an affine subset of RP^{2} can be described as a graph where adjacency between two vertices occurs if and only if a polynomial equation is satisfied. We give these graphs the insightful name of algebraically defined graphs. We ask ourselves the question "Are there any nonisomorphic algebraically defined graphs?", discuss why it's challenging to answer, and provide two nonisomorphic such graphs.

March 27 (Wednesday, 2pm in 294 Carver Hall)

Justin Stevenson: Character quasigroups: A generalization of Pontryagin duality (Ph.D. thesis defense, public part).

February 26, March 4, 18

SungYell Song: Strongly regular graphs that give rise to real equiangular tight frames.
Abstract:
A set of M lines passing through the origin in Ndimensional Euclidean space ( M > N > 1 ) is (real) equiangular if the cosine of the angle between any two of these lines is a fixed constant c. A set of real equiangular lines is called a (real) equiangular tight frame (ETF) if the N x M matrix F, where the columns are equiangular unit vectors and the rows are mutually orthogonal, all with the same length r, gives the matrix product of F with its transpose as the scalar matrix with diagonal entries r = M / N.
Real ETFs are obtained from various different mathematical contexts including such combinatorial objects as strongly regular graphs, regular twographs, and primitive rank3 association schemes. We will describe a few examples of ETFs that are obtained from strongly regular graphs, and discuss their relationship to regular twographs and primitive association schemes of class two. Our aim is to study some combinatorial characteristics of graphs, designs and association schemes that give rise to real ETFs.
(We will recall terms and notions, such as ETF, strongly regular graph, Seidel switching class, BoseMesner algebra, regular twograph, and 2class primitive association schemes, as needed if time permits.)

February 19

GeeChoon Lau (Universiti Teknologi MARA Johor, Malaysia): Constructions of local antimagic 3colorable tripartite graphs of fixed odd size
by labeling matrices.
Abstract:
An edge labeling of a (connected) graph G = (V,E) is a local antimagic labeling if it
is a bijection f : E → {1, . . . , E} such that for any pair of adjacent vertices x and y,
f^{+}(x) is distinct from f^{+}(y), where the induced vertex label f^{+}(x) is the sum of all the
f(e), with e ranging over all the
edges incident to x. If x is assigned the color f^{+}(x), then f^{+} is a proper vertex coloring of G
induced by f. If c(f) is the number of distinct induced vertex colors, we say G is c(f)
colorable. The local antimagic chromatic number of G, denoted by χ_{la}(G), is min{c(f)} over
all local antimagic labelings f of G.
In this talk, we first discuss some basic known results
that motivate us to construct matrices of size (3m + 2) × (2k + 1) with integers from 1 to
(3m + 2)(2k + 1), for m = 1, 3 and k ≥ 1, that meet certain properties. These matrices are
then used to construct various families of tripartite graphs of size (3m+2)(2k+1) that admit
a local antimagic 3coloring. By merging certain vertices, new local antimagic 3colorable
tripartite graphs are then obtained. Open problems are also introduced.
This is joint work with Wai Chee Shiub (The Chinese University of Hong Kong), K. Premalathac (Kalasalingam Academy of Research and Education, India), and M. Nalliahd (Vellore Institute of Technology, India).

February 12

Mitchell Ashburn: Formed monotopies of orthogonal Lie algebras.
Abstract:
From an anticommutative algebra with distinguished bilinear form, we construct a vectormatrix algebra. The conditions for two vectormatrix algebras to be isomorphic can be reduced to a special type of isotopy on the underlying anticommutative algebras. In this talk we will consider the family of Lie algebras so(n) and show which, if any, other Lie algebras generate isomorphic vectormatrix algebras as them by examining these special isotopies.

February 5

Jonathan Smith: Linear quasigroups and quaternions.
Abstract:
Linear quasigroups over a commutative, unital ring S are equivalent to representations of the homogeneous algebra
𝕳_{S} =
S
<
T_{i}

1+
T_{i}
T_{i1}
T_{i2}
>_{i = 1,2,3}
in invertible, noncommuting indeterminates T_{i} and their inverses. The indices on the indeterminates are integers modulo 3. The quotient
H_{S} =
S
<
T_{i}

1+
T_{i}
T_{i1}
T_{i2}
,
1+
T_{i}^{2}
>_{i = 1,2,3}
of the homogeneous algebra is a quaternion algebra over the ring S. Modules over this quaternion algebra are equivalent to linear quasigroups satisfying a set of quasigroup identities, which are permuted under triality. Natural examples of such quasigroups come from classical and quantum logic.

January 29

Jonathan Smith: Linear quasigroups and quasigroup triality.
Abstract:
Linear quasigroups are equivalent to representations of the free group on two generators. However, the usual expression of this equivalence does not fit well with the full S_{3} or triality symmetry of the language of quasigroups. In this talk, we will examine a new homogeneous algebra which does provide a representation of linear quasigroups that is naturally invariant under the triality symmetry.

January 22

Jonathan Smith: Linear quasigroups and group representations.
Abstract:
Linearization is a standard trick in universal algebra. A universal algebra (A,F) is linear if the underlying set A is a module over a commutative, unital ring S, and each basic operation f (x_{1},...,x_{n}) is a linear combination x_{1}X_{1}+...+x_{n}X_{n} of the variables x_{1},...,x_{n} acted upon by successive Sendomorphisms of A. The identities of algebras in a variety lead to relations on the various Sendomorphisms. In this talk, we show how linear quasigroups are equivalent to representations of twogenerated groups.
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