Combinatorics/Algebra Seminar

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: Vector-matrix algebras and formed monotopies of formed Lie algebras (Ph.D. thesis defense, public part).

April 1

Joe Miller: Constructing non-isomorphic 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 RP² can be understood in this way. We will also show that an affine subset of RP² 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 non-isomorphic 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

Sung-Yell Song: Strongly regular graphs that give rise to real equiangular tight frames.

Abstract:

A set of M lines passing through the origin in N-dimensional 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 two-graphs, and primitive rank-3 association schemes. We will describe a few examples of ETFs that are obtained from strongly regular graphs, and discuss their relationship to regular two-graphs 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, Bose-Mesner algebra, regular two-graph, and 2-class primitive association schemes, as needed if time permits.)

February 19

Gee-Choon Lau (Universiti Teknologi MARA Johor, Malaysia): Constructions of local antimagic 3-colorable 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 3-coloring. By merging certain vertices, new local antimagic 3-colorable 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 vector-matrix algebra. The conditions for two vector-matrix 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 vector-matrix 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_i-1 T_i-2  >_i = 1,2,3

in invertible, non-commuting 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_i-1 T_i-2  ,  1+ T_i²  >_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 S₃ 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 (x₁,...,x_n)  is a linear combination  x₁X₁+...+x_nX_n  of the variables  x₁,...,x_n  acted upon by successive  S-endomorphisms of  A. The identities of algebras in a variety lead to relations on the various  S-endomorphisms. In this talk, we show how linear quasigroups are equivalent to representations of two-generated groups.

Archive of earlier seminars

Back to the Mathematics Institute

Back to Main Street