The Department of Mathematics at Western Michigan University presents an algebra seminar.
Day and time: Wednesdays from 11 a.m.-noon
Place: 6625 Everett Tower
The purpose of the Algebra Seminar is to reflect the research and scholarly interest of the Algebra Faculty: Clifton E. Ealy Jr – Quasi-groups, Groups, and related structures; Gene Freudenburg – Algebraic Geometry; Terrell Hodge – Algebraic groups and modular representation theory; John Martino – Group theory and classifying spaces of finite and compact groups; Annegret Paul–Representation of Lie groups; David Richter–Lie Algebras and matroids; and Jay Wood–coding theory over rings. The Algebra Seminar may be registered for as MATH 6930. The prerequisites for the Seminar as a course is graduate standing.
An Introduction to Linear Finite State Moore Machines with Output presented by Akshat Gulgulia in the Alavi Commons at 4 p.m.
Abstract: In this talk, given a finite state Moore Machine, M = (S , l , F , G s0 , O , T) with output, will introduce Linear Sequential Machines, LSM’s. In this context we will discuss the serial and parallel decomposition of LSM’s and controllability and connectivity of LSM’s as time allows.
Introduction to UFDs: Part 1 presented by Gene Freudenburg, Ph.D., Department of Mathematics, Western Michigan University
Abstract: The theme of the seminar this semester is: Unique Factorization Domains "with a View Toward Algebraic Geometry".
The first two lectures will lay out many of the main algebraic preliminaries for our investigations.
Retracts of polynomial rings and the Abhyankar-Sathaye conjecture presented by Takanori Nagamine, Ph.D., Niigata University (Japan)
Abstract: Let B = k[x1,...,xn] be the polynomial ring in n ≥ 1 variables over a field k. A polynomial f in B is called a coordinate if there are g2,...,gn in B such that k[f, g2,...,gn] = B. The statement of Abhyankar-Sathaye conjecture is the following:
If B/f B ≈k k[y1,...,yn-1], then f is a coordinate.
In this talk, we will discuss this conjecture in terms of retracts of polynomial rings. In particular, we consider the case where n = 3 and k = C.
Closed polynomials over an integral domain: Part 2 presented by Hideo Kojima, Ph.D., Niigata University (Japan)
Abstract: In the second part, we will discuss closed polynomials over integral domains. Let R[X] be a polynomial ring in n variables over an integral domain R and let f in R[X]be a non-constant polynomial such that Q(R)[f] ∩ R[X] = R[f]. We give necessary and sufficient conditions for f to be closed in R[X], which generalize some results given in the first part. We also introduce some results of T. Nagamine and discuss some open questions.
Closed polynomials over an integral domain: Part 1 presented by Hideo Kojima, Ph.D., Niigata University (Japan)
Abstract: Let R[X] be a polynomial ring in n variables over an integral domain R. A non-constant polynomial f in R[X] is said to be closed if R[f] is integrally closed in R[X]. When R is a filed, we have several characterizations for a non-constant polynomial in R[X] to be closed. In these two talks, we will discuss closed polynomials over integral domains. First of all, we recall some results on closed polynomials over a field. Then we give some results on closed polynomials over a UFD.
Let R be a UFD, and let M(R, n) be the set of all subalgebras of the form R[g], where g in R[X] is a non-constant polynomial. The main result of this talk is that, for a non-constant polynomial F in R[X], R[f] is a maximal element of M(R, n) if and only if f is closed and Q(R)[f] ∩ R[X] = R[f], where Q(R) is the quotient field of R. Moreover, we give that, in the case where the characteristic of R equals zero, R[f] is a maximal element of M(R, n) if and only if there exists an R-derivation on R[X] whose kernel equals R[f].