LINEAR ALGEBRA SEMINAR

2014 - 2015

TUESDAYS 4:00 - 5:00 PARKER HALL 224

(PLEASE NOTE NEW ROOM)

**October 14**

**Title: Coordinatization using integral closures**

**Presenter:** Douglas A. Leonard

**Abstract:**Last talk we discussed blowups as a way of desingularizing a curve locally to coordinatize points. This talk we'll consider a global approach based on computing the integral closure of the associated quotient ring. As an example, the curve defined by z^3+zy+y^5=0 has two points P_i with (z(P_i),y(P_i))=(0,0). But =z^2/y is an integral integral element with (w(P_1),z(P_1),y(P_1))=(0,0,0) and (w(P_2),z(P_2),y(P_2))=(-1,0,0).

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**October 7**

**Title: Coordinatizing points on a curve**

**Presenter:** Douglas A. Leonard

**Abstract:** There are various local and/or global methods used to desingularize curves over algebraically closed fields. Some, such as integral closures, are purely algebraic; while others, such as blowups, are more geometric or topological in flavor. Since I use computer algebra systems to do my mathematics, I prefer the former. So I'll try to coordinatize curves such as those defined by:

1) y^2-x^3=0,

2) y^2-x^3-x^2=0,

3) y^25+y^6x^17+x^27=0,

4) y^8+y^6x+y^3x^3+y^2x^4+yx^6+x^9=0;

all with singularities at the origin, by desingularizing them somehow. (The first two are genus 0 curves, and at the level of examples on WIKI pages and in textbooks;

the latter two are more serious toy examples formulated by me to make a point.)

Audience participation is desired, as I'm trying to put this and similar non-traditional material into some sort of open-access book form on my website.

**September 30**

**NO SEMINAR**

**September 23**

**NO SEMINAR**

**September 16
**

**Abstract:** In this study, we present a single formula for the entries of the rth (r∈ℕ) power of a certain real circulant matrix of odd and even order, in terms of the Chebyshev polynomials of the first and second kind. In addition, we give two Maple 13 procedures along with some numerical examples in order to verify our calculation.

**Presenter:** Ahmet Oteles

**September 9**

**Title: Primitive Decompositions of Elements of a Free Metabelian Lie algebra of Rank Two**

**Abstract: ** We give a primitive decomposition of any element of a free metabelian Lie algebra and we determine the primitive length of an element.

**Presenter: ** Ela Aydin

**Title: A Single Formula for Integer Powers of Certain Real Circulant Matrix of Odd and Even Order**

**Abstract: ** In this study, we present a single formula for the entries of the rth (r∈ℕ) power of a certain real circulant matrix of odd and even order, in terms of the Chebyshev polynomials of the first and second kind. In addition, we give two Maple 13 procedures along with some numerical examples in order to verify our calculation.

**Presenter: ** Ahmet Oteles

**September 2**

**Title: Some Open Problems in Matrix Theory II**

**Abstract: ** We will discuss some open problems in Matrix Theory. The topics are the limiting theorem of Aluthge iteration and the explicit form of the limit for the 2x2 case.

**Presenter: ** Tin-Yau Tam

**August 26**

**Title: Some Open Problems in Matrix Theory I**

**Abstract: ** We will discuss some open problems in Matrix Theory in two talks. The first talk will cover Marcus-de Oliveira Conjecture and the explicit limit form of the Aluthge iteration of a 2x2 matrix if we have enough time.

**Presenter: ** Tin-Yau Tam