Departmental Colloquium

Nonlinear optimization problems represent the bed-rock of numerous real-life applications, such as data assimilation for weather prediction, radiation therapy treatment planning, optimal design of energy transmission networks, and many more. The solution of these problems usually involves iteratively constructing easier-to-solve local models of the function to be optimized, with the optimizer of the model taken as an estimate of the sought-after solution. Linear or quadratic models are usually employed locally in this context; however, these approximations are often unsatisfactory either because they are unbounded in the presence of nonconvexity and hence cannot be meaningfully optimized, or they are accurate representations of the function only in a small neighbourhood, yielding only small or no iterative improvements. Hence such models require some form of regularization to improve algorithm performance and avoid failure; traditionally, linesearchand trust-region techniques have been employed for this purpose and represent the state-of-the-art. Here, a new class of methods for nonlinear nonconvex unconstrained problems will be presented that approximately globally minimize a quadratic model of the objective regularized by a cubic term, inspired by earlier regularization approaches of Nesterov (2007) and Griewank (1982). An overestimation property of functions with Lipschitz-continuous Hessians underlies and justifies the model construction in the work to be presented. Preliminary numerical experiments show our methods to perform better than a trust-region implementation, while our convergence and complexity results show it to be at least as reliable as the latter approach. Extensions to problems with simple constraints and a simplified application to the subclass of nonlinear least-squares problems will also be presented. This is joint work with Nick Gould (Rutherford Appleton Laboratory, UK), Philippe Toint (University of Namur, Belgium), and partly, also with Stefania Bellavia and Benedetta Morini (University of Florence, Italy). February 5: (Special Colloquium)

A classical problem in representation theory, motivated by abstract harmonic analysis and number theory, is the study of unitary representations of reductive algebraic groups (for example the general linear, symplectic, or orthogonal groups) defined over real and p -adic fields. A unitary representation of a group G is a continuous homomorphism π from G to the group of unitary operators on a complex Hilbert space. One defines the irreducible unitary representations to be those without proper closed invariant subspaces. Of particular interest is the identification of the spherical irreducible unitary representations of G, that is, those which have nontrivial fixed vectors under the action of a maximal compact subgroup K. The main motivation for their study comes from the theory of automorphic forms. In this talk, I will attempt to present the background for this problem, and report on joint work with D. Barbasch on the identification of spherical unitary representations.

I'll talk about comparisons that can be made in geometric group theory between classical arithmetic groups such as SL(n,Z) and arithmetic groups over function fields such as SL(n,F[t]) where F is a finite field. I'll also try to explain how sometimes these classes of groups exhibit their similar tendencies through contrasting behavior. Topics to be covered include quasi-isometric rigidity, finiteness properties, and if time permits, Dehn functions of groups.

Microbes live in fluctuating environments that are often limiting for growth. They have evolved several sophisticated mechanisms to sense changes in important environmental parameters such as light and nutrients. Most bacteria also have complex appendages that allow them to move, so they can swim or crawl into optimal conditions. This combination of sensing changes in the immediate environment and transducing these changes to the motion organisms, allows for movement in a particular direction: a phenomenon known as ''chemotaxis'' or ''phototaxis''. Using time-lapse video microscopy we have monitored the movement of Cyanobacteria (which are phototaxis, i.e., bacteria that move towards light). These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important. The various patterns of movement that we observe appear to be a complex function of cell density, surface properties and genotype. Very little is known about the interactions between these parameters. In this talk, we will present a hierarchy of new models for describing the motion of phototaxis, that were constructed based on the experimental observations. The first model is a stochastic model that describes the locations of bacteria, the group dynamics, and the interaction between the bacteria and the medium in which it resides. The second model is a new multi-particle system that is obtained from a discretization of the first model. Our third model is obtained as the continuum limit of the second model, and as such it is a system of nonlinear PDEs. Our main theorems clarify the sense in which the system of PDEs can be considered as the limit dynamics of the multi-particle system. We conclude with several numerical simulations that demonstrate the properties of our models. This is a joint work with Devaki Bhaya (Department of Plant Biology, Carnegie Institute) and Tiago Requeijo (Math, Stanford).

I will consider algorithms and methodologies for the numerical solution of problems of scattering by complex bodies in three-dimensional space, and I will focus on a newly developed class of high-order algorithms for these problems. These methods, which are based on integral equations, high-order integration, fast Fourier transforms and highly accurate high-frequency methods, can be used in the solution of problems of electromagnetic and acoustic scattering by surfaces and penetrable scatterers - even in cases in which the scatterers contain geometric singularities such as corners and edges. In all cases the solvers exhibit high-order convergence, they run on low memories and reduced operation counts, and they result in solutions with a high degree of accuracy. In this talk I will describe the basic methodologies inherent in these approaches. In particular, I will touch upon a new class of high-order surface representation methods introduced recently which, starting from point clouds or CAD data, can produce high-order-accurate surface parametrizations and associated surface/volume finite-element representations - suitable for high-order numerical simulations - of complex engineering surfaces in three-dimensional space.

Auditory pathway from sound to perception is a multi-level information processing system. It is modeled as a transform with uniform and fine frequency resolution at low frequencies, yet nonuniform and coarse frequency resolution towards higher frequencies. Such transforms are built with discrete Fourier transform and ear characteristics and called auditory transforms. They are invertible either mathematically or perceptually. Inversion from perception to sounds is non-unique and involves optimization. Similar problem in blind source separation will be illustrated as well.

I will give an introductory talk about a theory of cobordism for algebraic varieties defined by M. Levine and F. Morel from Quillen's axiomatic perspective. My point of view will be rather concrete with the goal of explaining a new geometric presentation of algebraic cobordism via the simplest normal crossings degenerations. Applications to computations in algebraic geometry will also be discussed. The talk is based on joint work with Levine.

We will see that the 1959 Kadison-Singer Problem in C *-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and engineering. This gives all these research areas common ground on which to interact as well as explaining why each of them has volumes of literature on their respective problems without a satisfactory resolution. We will look at some of the equivalences of KS in operator theory, Banach space theory, harmonic analysis, and applied math/engineering.

This talk will present some snapshots from the history of numerals, mostly photographs of various systems, dating from Sumer (3400 B.C.) to the early Renaissance.