Claremont Center for the Mathematical Sciences

One of the central tenets of signal processing is the Shannon-Nyquist sampling theory: the numbers of samples needed to reconstruct a signal without error is dictated by its bandwith, namely the shortest interval which contains the support of the spectrum of the signal under study. Veryrecently, an alternative sampling or sensing theory has emerged which goes against the conventional wisdom. This theory allows the faithfulrecovery of signals and images from what appear to be highly incomplete sets of data, i.e. from far fewer data that tradinal methods use. Underlying this methodology is a concrete protocol for sensing and compressing data simultaneously.
This talk will present the key mathematical ideas underlying this new sampling or sensing theory, and will survey some of the most important results. We will argue that this is a robust mathematical theory; not only it is possible to recover signals accurately from just an incomplete sets of measurements, but it is also possible to do so when the measurements are unrealiable and corrupted by noise.

The sphere packing problem in dimension N asks for an arrangement of non-overlapping spheres of equal radius which occupies the largest porportion of the corresponding Euclidean space.

The Dirichlet-type spaces are Hilbert spaces of analytic functions on the unit disk which encompass many of the standard spaces (e.g., Hardy, Bergman, Dirichlet). We discuss a few aspects of the backward shift operator (i.e., shift the sequence of Taylor coefficients backward one step) on these spaces. This talk will be mostly expository (i.e., no difficult proofs) and is aimed at introducing these fascinating spaces to general analysts.

In 1806, Joseph-Louis Lagrange (1736-1813) read a memoir proving Euclid's parallel postulate to the Institut de France in Paris, but stopped, as the story goes, saying "I have to think about this some more." We'll look at Lagrange's still unpublished Paris manuscript on this subject, and place this activity in the context of his mathematical career. We will look also at how the ideas in this manuscript are related to Lagrange's philosophy of mathematics, Newton's physics, and Leibniz's Principle of Sufficient Reason. Finally, we will reflect on what this episode tells us about eighteenth-century attitudes toward geometry, space, and the universe.

Can mathematical models be truly helpful in medical applications? In this talk I will present a new mathematical model developed with immunologist Sarah Hook that uses delay-differential equations to model the immune kinetics in response to tumor antigen. Differential equations with delays occur in many applications, and, mathematically,

they pose considerable challenges in their analysis. In particular, the introduction of delays makes it difficult to analytically apply control techniques to this problem.

The aim of this work is to model the T cell immune response to a vaccine with the goal that the model can be used to aid in the optimisation of vaccine-induced cellular immune responses. Since these responses are much more difficult to measure than antibody levels, often the only option available for researchers and clinicians is the "Goldilocks'' approach to cancer vaccination: give not too much of the vaccine, or too little and give it not too often or too few times.

In this talk I will discuss some of the mathematical difficulties presented by delay-differential models, and will describe several mathematical methods that can be used to suggest immunization protocols that would optimize the immune response.

No previous knowledge of immunology or control theory will be assumed.

Principles for the existences of critical points will be discussed and applied to the solvability of boundary value problems. Terms such as deformation lemma, mountain pass lemma, Palais-Smale condition, and saddle point principle will be explained.

The theory of the structure of weights, which reached a peak in the 1980 with the eponymous "factorization theorem'' of Peter Jones, is by now no longer young. Yet despite its middle age, it is still capable of some aesthetically satisfying surprises. We will present a brief introduction to the theory and describe how some recent insights have simplified our understanding of the later refinement (Cruz-Uribe-Neugebauer, 1995) of Jones's result to simultaneously incorporate "reverse Hölder'' class information.

What is the trajectory of a projectile launched into the air? Archimedes, Tartaglia, Galileo, Torricelli, Descartes, Newton and Bernoulli are just a few among the many mathematicians who had investigated this question by the time it caught the attention of Leonhard Euler. Not to mention that his boss at the time was more than a little interested in specific applications of the solution, such as determining the horizontal range of a musket ball with 3/4 inch diameter and an initial velocity of 1700 feet per second when fired at an angle of 45 degrees.

The 18^th century saw significant advances in the analysis of the trajectory of a projectile in air. In the first half of this century, Benjamin Robins, a British mathematician and military engineer, published New Principles of Gunnery, the first book to deal extensively with external ballistics. Robins' work motivated a deeper mathematical analysis of projectile motion and invited ``commentary'' from Euler. In this talk we consider the influence of Robins' work and look at how Euler used it to attack the problem of projectile motion.

“Now is a time of great interest in K-12 mathematics education. Student performance, curriculum, and teacher education are the subjects of much scrutiny and debate. Studies of the mathematical knowledge of prospective and practicing U.S. teachers suggest ways to improve their mathematical educations” (Conference Board of the Mathematical Sciences, 2001)

Elementary school teachers are the public face of mathematics. (Phillip Wagreich, Mathematician)

In this talk, I will share findings from studies indicating that we should be concerned about U.S. teachers’, including prospective teachers’, mathematical content knowledge for teaching. Using recent research findings and vignettes from classroom lessons, I will elaborate on the notion of mathematical content knowledge for teaching and illustrate how teaching mathematics, even to young children, may entail more mathematical thinking on the teacher’s behalf than one might expect. I will then consider recent recommendations for the mathematical preparation of teachers and how mathematics departments have enacted these recommendations. In so doing, I hope to convey to the audience the distinct and important contribution that mathematicians are uniquely qualified to make toward the improvement of the mathematics education of teachers and consequently, school age children.

The theory of harmonic measure describes the way in which a random particle in a domain first hits the boundary. It also, and equivalently, gives a description of the boundary behavior of solutions to Laplace's equation in a given, perhaps very complicated, domain.

A brief survey of harmonic measures in the plane and in space will be followed by a discussion of some analogs of classical theorems about exit behavior of Brownian motion in simply connected planar domains to the case of multiply connected planar domains and to some classes of "nice'' domains in three dimensions. There are many interesting open questions in the area and we will try to talk about a few of them. The talk will be accessible to undergraduates.