LINEAR ALGEBRA SEMINAR

Tuesday 4:00-5:00 p.m. Parker 224 NEW ROOM!!!!

**April 15, 2014**

Speaker: **Ted Kilgore**

Title: **Lagrange interpolation, the Bernstein-Erdos Conjectures, and an n x (n-1) sign matrix**

**Abstract:** Lagrange interpolation is, among other things, a bounded linear projection operator. The quality of approximation obtained by interpolation is related to the operator norm, and that norm in turn depends on the placement of the nodes of interpolation. An old conjecture of Bernstein, later expanded by Erdos, proposes to characterize the placement of the nodes which will minimize the operator norm.

The method by which these conjectures were affirmatively resolved will be outlined, and a portion of the proof will be presented. In the proof, one must investigate the non-singularity properties of a matrix whose entries are based upon interlacing polynomials. The proof is then completed by investigation of a resulting sign pattern.

**April 8, 2014 ** **Part II**

**April 1, 2014 Part I**

Speaker: **Liping Wang**

Title: **Kazhdan-Lusztig coefficients for Affine Weyl groups**

**Abstract:** Click here

**March 18, 2014**

Speaker: **Dr. T.-Y. Tam**

Title: **Recent matrix asymptotic results and their generalizations**

Part III

**March 4, 2014**

Speaker: **Dr. Frank Uhlig**

Title: **The Matrix Symmetrizer Problem**

**Abstract:** Click here

**February 18, 2014**

**and**

**February 11, 2014**

Speaker: **Dr. T.Y. Tam**

Title: **Recent matrix asymptotic results and their generalizations**

**Abstract:** We will discuss some recent asymptotic results in matrix space and their extensions in Lie group, namely, (1) Beurling-Gelfand-Yamamoto's theorem and the generalization of Huang and Tam; (2) QR and Iwasawa asymptotic results of Huang and Tam, (3) Antezana-Pujals-Stojanoff convergence theorem on Aluthge iteration and the generalization of Tam and Thompson; (4) Rutishauer's LR algorithm and the generalization of Thompson and Tam; (5) Francis-Kublanovskaya's QR algorithm and the generalization of Holmes, Huang and Tam. These results are related to several important matrix decompositions, namely, SVD, QR, Gelfand-Naimark, Jordan and their counterparts namely, Cartan, Iwasawa, Bruhat, complete multiplicative Jordan decomposition.

**January 28 Canceled **

**AND**

**February 4 CANCELED**

** **

**November 19, 2013**

Speaker: **Douglas Leonard**

Title:** Syzygy**

**Abstract:** Please click here

**November 12, 2013**

Speaker: **Luke Oeding**

Title:** ** **Principal minors, exclusive minors and the tangential variety**

**Abstract:** The tangential variety of the Segre product contains tensors that have high rank, but can be approximated by tensors of rank 2. I will explain an unexpected connection between the tangential variety and the variety of principal minors of symmetric matrices. This connection motivated the definition of "exclusive minors.” I will discuss exclusive minors and their symmetry, explain the connection to principal minors, and show how they were used to solve a conjecture of Landsberg and Weyman on the defining equations of the tangential variety.

**November 5, 2014**

Speaker: **Peter Nylen**

Title:** ** **On the norm of zero one $3$ dimensional matrices**

**Abstract:** It is quite trivial to show that a zero one matrix $A=[a_{ij}]$ satisfies $||A|| \le 1$, where $|| \cdot ||$ is the spectral norm, if and only if $A$ is a sub permutation matrix, i.e., has no more than one $1$ in each row and column. The spectral norm inequality may be expressed as $\sum a_{ij} x_i y_j \le ||x|| ||y||$. This latter expression has a natural extension to $3$ dimensional matrices $[a_{ijk}]$

In this talk I will describe a necessary condition for the $3$ dimensional inequality to hold which is a reasonable analogue of the necessary and sufficient condition for the $2$ dimensional inequality. It is not known if this condition is also sufficient. I will also describe a sufficient condition for the $3$ dimensional inequality, and discuss why this sufficient condition is not necessary.

The contents of this talk are from the 1993 LAA paper authored by C.R. Johnson and the speaker, titled "The Sprinkling Problem."

**October 29, 2013**

Speaker: **Daniel Brice**

Title: **Zero product determined algebras III - Upper Triangular Ladders and Further Examples**

**Abstract:** We provide example and non-examples of zero product determined algebras. There include the free associative and free commutative algebras, as well as others. We define a class of matrix algebras that generalizes block upper triangular matrices and show that such algebras under suitable conditions are zero product determined.

**October 22, 2013**

Speaker: **Daniel Brice will continue his talk**.

Title:** ** **Zero Product Determined Algebras II - Homomorphic Images, Examples**

**Abstract:** Let $K$ be a commutative ring with identity.

A $K$-algebra $A$ is said to be \emph{zero product determined} if for every $K$-bilinear $\varphi$ having the property that $\varphi(a_1,a_2) = 0$ whenever $a_1a_2=0$ there is a $K$-linear $\tilde{\varphi} : A^2 \longrightarrow \mathop{\mathrm{Cod}} \varphi$ such that $\varphi(a_1, a_2) = \tilde{\varphi}(a_1 a_2)$ for all $a_1, a_2 \in A$.

We complete the proof that the tensor product of two zero product determined algebras is zero product determined. We continue by examining conditions under which the homomorphic image of a zero product determined algebra is itself zero product determined, as well as use previous results and direct computation to provide several new examples of zero product determined algebras.

**October 15, 2013**

Speaker: **Daniel Brice**

Title: **Zero Product Determined Algebras I - Direct Sums and Tensor Products**

**Abstract:** Let $K$ be a commutative ring with identity. A $K$-algebra $A$ is said to be zero product determined if for every $K$-bilinear $\phi: A\times A\to B$ having the property that $\phi(a_1,a_2) = 0$ whenever $a_1a_2=0$ there is a $K$-linear $\mu : A^2 \to Im \phi$ such that $\phi(a_1, a_2) = \mu(a_1 a_2)$ for all $a_1, a_2 \in A$.

We provide a necessary and sufficient condition for an algebra $A$ to be zero product determined and use the condition to show that the direct sum of arbitrarily many algebras is zero product determined if and only if each component algebra is zero product determined and that the tensor product of two zero product determined algebras is zero product determined in case $K$ is a field or in case the algebra multiplications are surjective.

**October 8, 2013**

**Professor Thomas Pate** will continue his talk

Title: **Some Unsolved Problems Involving Multilinear Algebra and Norms of Multilinear Functions**

**Abstract:** There are many interesting problems in involving norms of multilinear functions that are partially solved. We shall present some conjectures which could lead to solutions of some of these problems.

**October 1, 2013**

Speaker: **Professor Thomas Pate **

Title: **Some Unsolved Problems Involving Multilinear Algebra and Norms of Multilinear Functions
**

**Abstract:** There are many interesting problems in involving norms of multilinear functions that are partially solved. We shall present some conjectures which could lead to solutions of some of these problems.

**September 24, 2013**

***Luke notes that the lecture notes on secant varieties are now available on the arxiv: http://arxiv.org/abs/1309.4145

Speaker: ** Luke Oeding**

Title: **Toward a salmon conjecture**

**Abstract:** Methods from numerical algebraic geometry are applied in combination with techniques from classical representation theory to show that the variety of 3 × 3 × 4 tensors of border rank 4 is cut out by polynomials of degree 6 and 9. Combined with results of Landsberg and Manivel, this furnishes a computational solution of an open problem in algebraic statistics, namely, the set-theoretic version of Allman’s salmon conjecture for 4 × 4 × 4 tensors of border rank 4. A proof without numerical computation was given recently by Friedland and Gross.

**September 17, 2013**

Speaker: **Luke Oeding
** Title:

**Abstract:** This lecture is intended to be basic covering background material related to tensors, representation theory and geometry.

Topics I hope to include: Representation Theory of S_n and SL_n(C), Schur's Lemma, Schur-Weyl duality, Tensor products of vector spaces, Polynomials on tensors, Some classical algebraic varieties related to tensors and their ideals, Open questions.

**September 10, 2013**

Speaker: ** Luke Oeding
** Title:

**Abstract:** A principal minor of a matrix is the determinant of a submatrix centered about the main diagonal. A basic linear algebra question asks to what extent is it possible to prescribe the principal minors of a matrix. The algebraic problem is to find defining equations for the associated algebraic variety. In particular, knowing such equations would provide a test for whether a given list of numbers can be the principal minors of a matrix.

I will explain a solution to this problem in the case of symmetric matrices, answering a conjecture of Holtz and Sturmfels. Then I will explain the notion of "exclusive rank" and how it relates principal minors to classically studied algebraic varieties such as the Segre variety and its tangential variety, answering a conjecture of Landsberg and Weyman. Finally I will discuss new work in the case that the principal minors of equal size are required to have the same value.

PDF available here