LINEAR ALGEBRA SEMINAR

2014 - 2015

MONDAYS 4:00 - 5:00 PARKER HALL 224

(PLEASE NOTE NEW ROOM)

**2015**

Speaker: Zach Sarver

Title: Kostant's pre-order and other tools

Abstract: Kostant's pre-order on semisimple Lie groups is a useful tool that has mysteriously not appeared in textbooks. This talk is an introduction to Kostant's pre-order and the tools needed to define it.

April 6

Speaker: Dr. Luke Oeding

Title: Equations for the Fifth Secant Variety of Segre Products of Projective Spaces

Abstract: We describe a computational proof that the fifth secant variety of the Segre product of five copies of the projective line is a codimension 2 complete intersection of equations of degree 6 and 16. Our computations rely on pseudo-randomness, and nu- merical accuracy, so parts of our proof are only valid ``with high probability.'' This is joint work with Steven Sam (UC Berkeley).

March 30

Speaker: Sima Ahsani

Title: "Two famous conjectures: Permanent on Top Conjecture and Permanent Dominance Conjecture"

Abstract: I plan to talk about the Schur Power Matrix and some of its open problems and conjectures. In addition, I will give the definition and basic properties of the permanent of an n-square matrix. After that, I will talk about the historical background and current status of the two famous conjectures: Permanent on Top Conjecture and Permanent Dominance Conjecture. Some examples will be given in my talk.

March 16

Speaker: Huajun Huang

Title: A survey of the QR decomposition

Abstract: I will give a brief survey of basic properties of the QR decomposition, and present several research results of the QR decomposition derived from these properties. These results include the asymptotic behaviors of the QR decomposition of powers and the QR iteration, and the majorization relationship between the diagonal component of the QR decomposition and the counterparts of Jordan decomposition, SVD, Cholesky decomposition, etc. These results cover some joint works with Tin-Yau Tam, Randall R. Holmes, and Sangjib Kim.

March 9

Speaker: Luke Oeding

Title: Homotopy techniques for tensor decomposition and perfect identifiability

Abstract: Given a tensor (or hyper-matrix), we would like to express it in the simplest possible way as the sum of the smallest number of decomposable (or rank-1) tensors. While there are many algorithms that attempt to accomplish this task, it is known to be a very difficult problem. Moreover, such a decomposition may not be unique. When a generic tensor of a given format has a unique decomposition, we say that tensors of that format are "generically identifiable."

We propose a new method to find tensor decompositions via homotopy continuation. This technique allows us to find all decompositions of a given tensor (at least for relatively small tensors). Our experiments yielded a surprise - we found two new tensor formats, (3,4,5) and (2,2,2,3), where the generic tensor has a unique decomposition. Using techniques from algebraic geometry, we prove that these cases are indeed "generically identifiable".

This is joint work with J. Hauenstein, G. Ottaviani and A. Sommese.

March 2

Speaker: Deyu Wu

Title: Spectral Theory of Unbounded Block Operator Matrices and Applications

Abstract: Block operator matrices are matrices the entries of which are linear operators between Banach or Hilbert spaces. They arise in various areas of mathematics and mechanics, and they are paid considerable attention for their theoretical research value and practical applications. In this talk, we will introduce the definition of spectrum of block operator matrix and illustrate some examples of applications. In the end, some basic research methods of block operator matrices are introduced to characterize the spectral properties.

February 23

Speaker: Hanyu Li

Title: New rigorous perturbation bounds for the LU and QR factorizations

Abstract: In this talk, we will introduce some new rigorous perturbation bounds for the LU and QR factorizations with normwise or componentwise perturbations in the given matrix, where the componentwise perturbations have the form of backward error resulting from the standard factorization algorithms. Moreover, some new sharper first-order perturbation bounds including two optimal ones for the LU factorization are also presented.

February 16

Presenter: Ming-Cheng Tsai

Title: Completely positive interpolations of compact, trace-class and Schatten-p class operators.

Abstract: In this talk, we extend Li and Poon's results on interpolation problems for matrices. We give characterizations of the existence of a completely positive linear map between compact (or Schatten-p class) operators sending a particular operator A to another B. It is shown that such a completely positive map exists if and only if a positive one does. Moreover, we show that such a completely positive map exists as above if and only if a multiple of the numerical range of A contains the numerical range of B.

February 9

Presenter: Dr. Luke Oeding

Title: Staircase flattenings and border ranks of monomials (part II)

Abstract: We will explain the method of flattening and its variants that allow one to compute bounds on the rank and border rank of tensors. I’ll show how a recent variant of the construction allows us to obtain new bounds for the border ranks of monomials, which can be considered as a special class of symmetric tensors.

February 2

Presenter: Dr. Luke Oeding

Title: Staircase flattenings and border ranks of monomials

Presenter: Dr. Luke Oeding

Abstract: We will explain the method of flattening and its variants that allow one to compute bounds on the rank and border rank of tensors. I’ll show how a recent variant of the construction allows us to obtain new bounds for the border ranks of monomials, which can be considered as a special class of symmetric tensors.

**2014**

November 18

Title: Derivations of the Lie algebra of dominated upper triangular matrices

Presenter: Prakash Ghimire

Abstract: Let M_n be the general linear Lie algebra consisting of all (n * n) matrices over a characteristic-zero field F, and M_L the subalgebra of M_n consisting of all matrices corresponding to a dominated upper triangular ladder L. Then any derivations of M_L can be expressed as the sum of ad(X), where X is a block upper triangular matrix in M_n, and a linear transformation mapping [M_L, M_L] to zero, and M_L to Z(M_L)∩M_L, where Z(M_L) is the center of M_L in M_n.

November 11

Title: The Category Hask, with Examples

Presenter: Zachary Sarver

Abstract: The Haskell programming language is a computer language which strives for mathematical elegance in its design. As such, it is a language growing in popularity for mathematical software. Most of the constructions in Haskell are in fact category theoretic constructions in the category Hask. This talk examines the categorical nature of the language. The existence of finite coproducts in Hask is proved, and detailed examples of functors and natural transformations are given.

For slide show, click here

November 4

Title: Endomorphism Rings of Bimodules

Presenter: Ulrich Albrecht

Abstract: Let M be an R-R-bimodule over a semi-prime right and left Goldie ring R. We investigate how non-singularity conditions on M when viewed as a right R-module are related to such conditions on M when viewed as a left module. In particular, an R-R-bimodule M, which is non-singular as a right and left R-module, has the right essentiality property if the right submodule IM is essential in M for all essential right ideals I of R, and investigate several questions related to this property.

For slide show, click here

October 28

Title: Covering ideals of morphisms

Presenter: Furuzan Ozbek

Abstract: A significant result of cotorsion theory proven by Eklof & Trlifaj is that if (F, C) is cogenerated by a set, then it is complete. Recently the cotorsion pairs of ideals (I,J ), where I, J are subfunctors of Hom_R, have been of interest. In this talk we will look at a few results motivated by Eklof & Trlifaj argument for an ideal I when it is generated by a set. Moreover, we will show how identifying an ideal I with a certain class of objects in A_2 (category of all representations of 2-quiver by modules) can help us to obtain sufficient conditions for I to be a covering ideal.

October 21

Title: Applications of Multilinear Algebra to World Wide Web Search

Presenter: Daniel Brice (Tuskeegee University)

Abstract: Linear algebra, specifically matrix decomposition, plays a crucial role in modern Web search engines. The methods typically employ decomposing the adjacency matrix of the World Wide Web, or alternatively, decomposing a certain heuristically-chosen subgraph of the World Wide Web. Some recent work in applied Web search algorithms extends and refines the conventional approaches by representing the hyperlink-structure of the Web as a multi-dimensional array and framing the problem in the context of tensor decompositions. In this talk, we will examine the multilinear algebra that provides the bases for the TOPHITS, CubeSVD, and TripleRank search algorithms.

For slide show, click here

For paper, click here

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).

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

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 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