Applied Math Colloquium

I will describe some recent work with R. Kobayashi of Hokkaido and J. Warren of NIST where an orientation field is introduced into phase field calculations of solidification and impingement of a polycrystal from its melt. It will be shown how rotational invariance of the homogeneous part of the free energy functional requires a non-analytic form of the gradient energy for localized and stable grain boundaries.

Computational results will illustrate the utility of the orientation field; they include effects such as colliding dendrites and simultaneous grain boundary evolution in a second phase during the growth of the second phase.

Quantum effects are seldom evident in today's electronic devices since the quantum states of many millions of atoms are averaged together blurring their discreteness. But in quantum computing the foundations of quantum mechanics are finding direct and visible applications in information processing. The unreasonable effectiveness of quantum computing is founded on coherent quantum superposition or entanglement which allows a large number of calculations to be performed simultaneously. This coherence is lost as a quantum system interacts with its environment and an important challenge today is to devise means of preserving it.

A quantum error correcting code is a way of encoding quantum states into qubits so that error or decoherence in a small number of individual qubits has little or no effect on the encoded data. This talk will describe a beautiful group theoretic framework that simplifies the presentation of known quantum error correcting codes and greatly facilitates the construction of new examples.

Joint work with Eric Rains, Peter Shor, and Neil Sloane

Starting with the classical Strassen algorithm, we will give in this talk an overview of the algebraic and combinatorial methods that lead to the fastest known matrix multiplication algorithms

I shall summarize the results of calculations of the eigenfunctions and eigenfrequencies of normal modes of oscillation trapped within the inner region of their accretion disk by the strong-field gravitational properties of a black hole (or a compact, weakly-magnetized neutron star). The focus will be on the most robust class: the analogue of internal gravity modes in stars. Their frequency which corrresponds to the lowest mode numbers depends almost entirely upon only the mass and angular momentum of the black hole. Such a feature may have been detected in the X-ray power spectra of two galactic 'microquasars', allowing the angular momentum of the black hole to be determined in one case.

The basic stability radius problem can be defined as follows. Let A be a stable matrix with all its eigenvalues in the stability region S (typically the open left half plane or the open unit disc). The stability radius of A is the norm of smallest (complex) perturbation D that causes at least one eigenvalue of A+D to become unstable, i.e.

$r_c = min\{||D||_2, D\in C^{nxn}, \exists\lambda(A+D \notin S)\}$

This problem is well studied by several authors and there exists an analytic formulation for rc as well as a quadratically convergent algorithm to compute it.

In this talk we discuss several variants and extensions of this standard problem to the generalized eigenvalue problem, to polynomial matrices, to the periodic eigenvalue problem and to structured matrices. We also talk about the use of different norms as well as structured perturbations and link this problem to the concept of pseudospectra.

This is ongoing work with Y. Genin, C. Lawrence, Y. Nesterov, C. Oara, J. Sreedhar, A. Tits and V. Vermaut.