The aim of this talk is to give an overview of the beautiful mathematical relationships between matrices, moments, orthogonal polynomials, quadrature rules and Krylov subspace methods. The underlying goal is to obtain efficient numerical methods for estimating quantities of the form $I[f]=${\bf u}^T f(A){\bf v}$, where ${\bf u}$ and ${\bf v}$ are given vectors, $A$ is a symmetric nonsingular matrix, and $f$ is a smooth function.

An obvious application is the computation of some elements of the matrix $f(A)$ when all of $f(A)$ is not required. Computation of quadratic forms can yield error estimates in methods for solving systems of linear equations. Bilinear or quadratic forms also arise naturally for the computation of parameters in some numerical methods for solving least squares or total least squares problems, and also in Tikhonov regularization for solving ill-posed problems. Furthermore, computation of bilinear forms is also useful in spectral methods for the numerical solution of PDE.

The main idea is to write $I[f]$ as a RiemannStieltjes integral and then to apply Gaussian quadrature rules to approximate the integral. The nodes and weights of these quadrature rules are given by the eigenvalues and eigenvectors of tridiagonal matrices whose nonzero coefficients describe the three-term recurrences satisfied by the orthogonal polynomials associated with the measure of the RiemannStieltjes integral. Beautifully, these orthogonal polynomials can be generated by the Lanczos algorithm.

Results pertaining to orthogonal polynomials and quadrature rules may not be so well known in the matrix computation community, and conversely, researchers working with orthogonal polynomials and quadrature rules may not be very familiar with their applications to matrix computations. We will see that it can be very fruitful to mix techniques coming from different areas. The resulting algorithms can also be of interest to scientists and engineers who are solving problems in which computation of bilinear forms arises naturally.

Heterogeneity among cells, a universal phenomenon in the noisy biological world, has stimulated much interest in systems biology. Traditional bulk measurements, however, are insufficient to understand such important phenomenon. Researchers have realized that many such complex biological systems need higher-resolution information than just ensemble average. Single cell approaches, instead, can reveal important information about how these complex systems function. We have studied two systems using a single live-cell genealogical tracking method. In one, we found that stochastic switching between different gene expression states in budding yeast is heritable. This striking behavior only became evident using genealogical information from growing colonies. Our model based on burst induced correlation can explain the bulk of our results. In the other system investigated, we have provided clear evidence for circadian gating in cyanobacteria, in which circadian rhythms regulate the timing of cell divisions. Simultaneous measurement of both circadian and cell cycle dynamics in individual cells reveals the direct relationships between these two biological clocks. We fit the data to a simple model to determine when cell cycle progression slows as a function of circadian and cell cycle phases. We infer that cell cycle progression in cyanobacteria slows during a specific circadian interval but is uniform across cell cycle phases.

Zeta functions are central topics in number theory and arithmetic algebraic geometry. They arise in many different forms. They can be viewed as generating functions for counting "points"
(or "solutions") of polynomial equations, thus contain deep arithmetic and geometric information of the equations. In this lecture, we will explain various zeta functions from this point of view, including the Riemann zeta function, the Hasse-Weil zeta function, and zeta
functions over finite fields.