The Programs

Summer 2008 Program in Matrix Analysis and Wavelet Theory

Mentored by Dave Larson

Wavelets are powerful tools used in signal analysis and is the subject of theoretical study as well as for its wide applications. The research area of the head mentor, Dr. Larson, involves the theory behind wavelets, although some participants in the past REU programs have worked on projects involving the use of wavelets in signal filtering and signal compression. To motivate wavelets, we begin with a few real-world examples:

Filtering. A sound signal is often corrupted by noise (i.e., frequencies different from those in the desirable parts of the signal). Signal analysis can be used to filter out this unwanted noise. A Dolby filter, which filters out tape-hiss on cassette tapes, is an example along these lines.

Data Compression. Digitized audio and video signals are usually quite large, and are difficult to transmit electronically. Efficient transmission of these signals often requires compression, a process that eliminates the less significant parts of a signal. Compression is used, for example, in transmitting fingerprints from a police squad car to FBI Headquarters (in Washington DC) to identify crime suspects.

Detection. Signals often have some feature that the user wants to detect. For example, the sound made by a mechanical device often changes when it does not operate correctly. A device that detects this change would be useful to the machine operator.

Fourier analysis and wavelets are two of the basic tools used, in signal analysis, to address the above issues.

Fourier Analysis. A Fourier series decomposes a signal f into its trigonometric components, which vibrate at various frequencies:



Here, is the time-domain, which can easily be adjusted to handle time intervals of different lengths. The Fast Fourier transform (FFT) is an efficient algorithm for calculating approximate values for the Fourier coefficients, a_n and b_n. The Fourier coefficients can then be manipulated according to the desired goal. If noise is to be filtered-out, then the Fourier coefficients corresponding to the unwanted frequencies can be eliminated. If the signal is to be compressed, then the Fourier coefficients that are smaller (in absolute value) than some specified tolerance can be discarded. Problems in detection can be addressed by matching a subset of the Fourier coefficients of f to a known profile of the type of signal to be detected.

Wavelets. One disadvantage of Fourier series is that the building blocks, sines and cosines, are periodic waves that continue forever. While this approach may be quite appropriate for filtering or compressing signals that have time-independent wave-like features, other signals may have more localized features that sines and cosines do not model very well. For example, suppose an isolated noisy ``pop'' to a sound signal is to be filtered-out. The graphs of sines and cosines do not resemble the pop's graph, an isolated bump. A different set of building blocks, called wavelets, are better suited to this type of signal. In a rough sense, a wavelet resembles a wave that travels for one or more periods and is nonzero only over a finite interval -- instead of propagating forever as do sines and cosines.

A wavelet can be translated forward or backwards in time. It also can be stretched or compressed, by scaling, to obtain low and high frequency wavelets. Once a wavelet function is constructed, it can be used to filter or compress signals in much the same manner as Fourier series. A given signal is first expressed as a sum of translations and scalings of the wavelet, and then the coefficients corresponding to the unwanted terms are removed or modified.

Care must be taken in the construction of a wavelet to ensure that its translates and rescalings satisfy orthogonality relationships -- analogous to those of sines and cosines -- so that efficient algorithms can be found for the computation of wavelet coefficients of a given signal.

Closely related to the orthonormal wavelets are the frame wavelets. Frame sequences have been used for many years by engineers for purposes of signal processing and data compression, in a manner much like the use of wavelet and other orthonormal bases. Frame wavelets are single vectors which generate frame sequences under the action of the wavelet unitary system.

Focus of this Program. There is a fascinating interplay between wavelets, frames, and operator theory (i.e. the theory of linear maps between vector spaces). This theoretical interplay will be the key topic of investigation during this summer program. We shall investigate the role of wandering vectors for unitary systems, both in finite and infinite dimensions. Related topics include introductory ideas in frame theory, sampling theory, and operator algebras, again both in finite and infinite dimensions. Additional topics include wavelet sets and minimally supported frequency wavelets. This special class of wavelets has an interesting internal structure, but also provides concrete examples of important ideas that will be discussed. Finally, we will investigate several interesting intrinsic problems dealing with wavelet sets.

Proposed Research Problems:

• Open problems regarding wandering vectors for unitary systems acting on Rⁿ or Cⁿ.
• Open problems concerning finite frames and their relationships with matrix analysis and operator theory.
• An open problem on the reflexivity of finite dimensional operator algebras which is purely algebraic in nature.
• Characterize interpolation pairs of minimally supported frequency wavelets.
• How can a wavelet set be perturbed to give rise to an interpolation pair?
• Does there exist a wavelet set in the support of the Fourier transform of any wavelet?
• An open problem in sampling theory.

Mentored by Paulo Lima-Filho, Jay Walton and Tom Wehrly

General Description. In this interdisciplinary program, we shall explore mathematical aspects of a wide variety of models arising in mathematical ecology and mathematical physiology. In the modeling of complex ecosystems, the main emphasis will lie on the impact of changes in the landscape topology on populations, including its distribution and dynamics. Two subareas will be emphasized - deterministic modeling using differential equations and stochastic modeling using techniques from statistics.

In the beginning of the program, various modeling problems and useful mathematical and statistical techniques will be presented, and participants will learn how to apply the mathematical techniques and theories to the biological problems. Participants will then choose research problems and form small research teams to work on the problems.

One topic we shall explore is the habitat fragmentation and re-connection question, which is concerned with the following issue. Decades of world wide wilderness land use policy decisions have resulted in many wilderness areas being broken into chains of isolated patches of national parks, forests, refuges or other protected zones. While the total area of these patches might seem considerable, their fragmented structure into relatively small patches might not provide viable habitats for many species, especially large predators. Many ecologists argue that the health of the predatory species at the top of the food chain provide a reliable barometer of the health of the entire ecological system. Consequently, there are large scale programs being proposed and in some cases already under way to connect fragmented habitats via corridors which will permit certain species, especially large predators, to travel between formerly isolated patches. The mathematical problem is to model this process in order to try to gauge the likelihood that it will be effective in making for more robust wildlife habitats, or whether it could cause a negative disruption to a habitat system and make matters worse.

Another important topic to be explored concerns the biodiversity/eco-stability debate which addresses the assertion that greater biodiversity leads to increased stability of an ecosystem, i.e. the more diverse the gene pool, the more robust the ecosystem is to perturbations. This has been a long running politically and ethically charged debate in need of rigorous quantitative scientific input.

Among the mathematical tools to be used are notions from finite dimensional dynamical systems, partial differential equations and differential geometry. Although sophisticated models could involve difficult concepts from these subjects, simplified models can be investigated by bright undergraduates with minimal background (but with a healthy enthusiasm to learn). The participants will be introduced to a variety of deterministic approaches to modeling ecosystems beginning with the classical Lotka/Volterra system of interacting species or predator/prey models. Very elementary topological ideas are used to quantify the fragmented structure of habitats from the point of view of individual species. More specifically, different species might see a given habitat system as having different connected components or patches. For example, certain bird species might see a habitat as connected whereas some plant or small, crawling animal species might see it as highly fragmented. Thus, the first task in studying an ecosystem is to determine its patches, or connected components, from the point of view of each of the species selected for study. Then one introduces interacting species models on the landscape.

The simplest level of modeling of species interaction dynamics on a fragmented habitat with corridors involves using ordinary differential equations for the densities of the species on the system of patches permitting migration of certain species between certain patches. These problems are ideal for undergraduate students to tackle, and preliminary results from the summer 2002 program suggest that selective migration can indeed qualitatively alter the dynamics over isolated patches without migration. Some groups of REU participants might want to choose projects extending these ODE models with migration to more than two patches and to more than two interacting species. Among the many possible questions to pursue is whether or not it matters which of the species has the ability to migrate? For example, in predator/prey type interactions, does one see similar dynamics if just the predator can migrate or if just the prey can migrate?

Other groups of participants might choose projects modeling the spatial distribution and movement of species within patches as well as between patches. These models will be of two types, continuous time partial differential equation models (the simplest being of diffusion type) and discrete time, convolution integral models (as proposed recently by Alan Hastings). Some groups will use simple diffusion operators to model the movement of species around a flat landscape, while other groups will study generalizations of these operators to non-flat surfaces using ideas from differential geometry.

As an alternative, some groups of the REU participants might wish to pursue statistical approaches to ecosystem modeling. Again, many avenues are open for these projects including stochastic population models. These are particularly useful for modelling the transient behavior of populations and also the behavior of small populations.

The basic ideas of stochastic population models will be introduced. The methodology is based on Markov models. Basic assumptions about the population lead to a system of Kolmogorov differential equations for the probability functions. Generating functions are useful in obtaining the moments or cumulants of the population size. The Kolmogorov differential equations lead naturally using the generating function approach to systems of differential equations for the cumulants. These can be solved analytically or numerically to describe the population. This methodology can be applied to models for a single population including both linear birth-immigration-death models and nonlinear birth-immigration-death models.

Our goal is to extend the basic ideas illustrated in single-population models to model multiple populations such as predators and prey. We also want to bring spatial aspects into the modelling by allowing for several locations. This leads us to developing models for multiple populations. Joint moments and cumulants for the sizes of multiple populations are introduced. In a manner analogous to that for a single population, bivariate Kolmogorov differential equations lead naturally using the generating function approach to systems of differential equations for the bivariate cumulants. This approach is illustrated by applying the methodology to multiple population models that can include births, deaths, immigration, and migration.

Much of our interest will be on the bivariate distribution of the number of predator and prey animals over the course of time. The above methodology enables us to obtain the cumulants of the bivariate distribution for any time. To approximate the corresponding bivariate density, such methods as saddlepoint approximations and series expansions will be used. Such research is helpful in the formulation of management strategies. Researchers can apply the stochastic models to determine the effects of various control procedures such as reducing birth rate, increasing a death rate, and restricting immigration on the bivariate distribution of the numbers of predators and prey.

The emphasis in mathematical physiology will be to the cardiovascular system. Among the topics discussed will be modeling the mechanical behavior of arteries and the heart muscle, and the role that mechanical behavior plays in the development and progression of cardiovascular disease. Another topic will be modeling the biology and chemistry involved with vascular diseases such as atherosclerosis and aneurisms. Recently, mathematical models have played a key role in understanding and predicting the onset, progression and treatment of these and many other diseases. The participants selecting to work in this area will be introduced to the relevant biological and chemical concepts as well as pertinent mathematical techniques and theories. As with the ecosystems modeling, they will then form small research groups to work on specific topics and problems.

Mentored by Luis Garcia and J. Maurice Rojas

Among the sciences, biology is unique in the subtlety of its mathematical foundations. This program focuses on recent applications of algebraic geometry to problems in protein folding and phylogenetic trees. Our main goal is to impart enough background to students so they have the freedom to be as algebraic and/or biological as they want to be.

Assuming NO background in algebraic geometry, we begin with a brief introduction to some of the computational tools from algebra, combinatorics, and geometry that we'll need. In parallel, we will also give an introduction to proteins and genomes. We then investigate some recent algebraic methods for understanding the shapes of proteins and the relation between genomes and the evolutionary hierarchy of organisms. Students will be expected to embark on computational experiments almost immediately.

A tentative list of the technical topics that we will cover is the following: Tools from Algebra, Combinatorics, and Computational Geometry (including Grobner Bases, Resultants, and Voronoi Diagrams), Basics on Kinematics of Molecules, Two-Way Contingency Tables, Toric Models and Markov Bases, Maximum Likelihood Estimation, Independence Models, Bayesian Networks, Phylogenetic Trees.

MAIN REFERENCES:

• [BT99] Introduction to Protein Structure, by Carl Branden and John Tooze, 2nd edition, Garland Publishing, Inc., NY, NY, 1999.
• [PS04] ``The Mathematics of Phylogenomics,'' by Lior Pachter and Bernd Sturmfels, Math ArXiV paper, downloadable from xxx.arxiv.org/math.ST/0409132.
• [Stu02] Solving Systems of Polynomial Equations, by Bernd Sturmfels, CBMS Lecture Series, AMS Press, 2002.

SUPPLEMENTAL REFERENCES:

• [CLO97] Ideals, Varieties, and Algorithms, by David A. Cox, John B. Little, and Donal O'Shea, Springer-Verlag, 1997.
• [GW91] The Cartoon Guide to Genetics, by Larry Gonick and Mark Wheelis, Harper Perennial, 1991.
• [PRW00] Algebraic Statistics: Computational Commutative Algebra in Statistics, by G. Pistone, E. Riccomagno, and H.P. Wynn, CRC Press (2000).