\documentstyle{article}
\tt
%\pagestyle{plain}
%\vspace*{0.1in}
%\samepage
\pagenumbering{arabic} \setlength{\textwidth}{5.71in}
\setlength{\textheight}{8.5in} \setlength{\parskip}{0.07in} \voffset
= -.36in \hoffset = -0.5in
\renewcommand{\baselinestretch}{1.3}
\begin{document}

%\begin{sloppypar}

\begin{center}
{\large \bf Total Positivity, Bidiagonal Decompositions, and 
Variation-Diminishing
Properties}
\end{center}
\begin{center}
Plamen Koev\\
{\small Department of Mathematics, Massachusetts Institute of Technology,\\
77 Massachusetts Ave., Cambridge, MA 02139, U.S.A.\\
plamen@math.mit.edu}
\end{center}

\begin{abstract} In this talk I will concentrate on the properties on the 
totally positive matrices, i.e., the matrices with all minors positive. 
Famous examples include the Pascal, Hilbert, and Vandermonde matrices (as 
long as the nodes of the latter are positive and increasing). These 
matrices possess very elegant structure going well beyond the positivity 
of the minors. For example:
\begin{itemize}
\item despite the general nonsymmetry, all 
eigenvalues are real and positive;
\item the multiplication by a totally 
positive matrix does not increase the number of sign changes in a vector;
\item the eigenvector corresponding to the $j$th largest eigenvalue has 
exactly $j-1$ changes of sign.
\end{itemize}

How can one establish these properties systematically?

Answer: By considering the bidiagonal decompositions of the totally 
positive matrices. 

This (old) idea has become particularly prominent recently and has allowed 
for an easy and straightforward insight into many theoretical properties 
of these matrices.

I will survey the key ideas behind these bidiagonal decompositions and 
will demonstrate how easy it is to establish classical results such as the 
positivity of the eigenvalues or the variation-diminishing properties. I 
will conclude with some new results, including a full characterization, in 
the language of bidiagonal decompositions, of the eigenvector matrices of 
totally positive matrices. \end{abstract} \end{document}
