| Date | Speaker | Title and Abstract |
| Tuesday June 12 | Ioana Mihaila California State Polytechnic University, Pomona |
Title: The Peculiar Cantor Set Abstract: What is the Cantor Set, and what is peculiar about it? What do the elements of this strange set look like? This talk will present a characterization of the rational numbers in the triadic Cantor set, and ask some questions about the irrational ones. Afterwards we will look at possible generalizations involving other Cantor-like sets. |
| Thursday June 21 | Emmanuel Candes California Institute of Technology |
Title: Compressive Sampling: Sense-Less but Smart! Abstract: One of the central tenets of signal processing and data acquisition is the Shannon/Nyquist sampling theory: the number of samples needed to capture a signal is dictated by its bandwidth. This talk introduces a novel sampling or sensing theory which goes against this conventional wisdom. This theory now known as "Compressed Sensing" or "Compressive Sampling" allows the faithful recovery of signals and images from what appear to be highly incomplete sets of data, i.e. from far fewer measurements than traditional methods use. We will present the key mathematical ideas underlying this new sampling or sensing theory, and will survey some of the most important results. We will emphasize the practicality and the broad applicability of this technique, and discuss what we believe are far reaching implications; e.g. procedures for sensing and compressing data simultaneously and much faster. |
| Tuesday June 26 | Adolfo Rumbos Pomona College |
Title: Nonlinear Resonance Abstract: Linear resonance is a phenomenon familiar to all who have taken a College Physics course. The classical textbook example used to illustrate this phenomenon is that of the collapse of the Tacoma Narrows suspension bridge in 1940. Research by Lazer and McKenna in the 1980s has revealed that the linear resonance model is not adequate to explain the failure of the bridge. The equations modeling bridge oscillations are intrinsically non-linear, and therefore the methods of nonlinear analysis are more appropriate for the study of this particular phenomenon. In this talk, weÕll study what ÒresonanceÓ means in the context of nonlinear problems and illustrate some of what is known about those problems. WeÕll contrast results for nonlinear problems to those of the linear counterpart. |
| Tuesday July 3 | Mike O'Neill Claremont McKenna College |
Title: An inverse theorem in additive number theory. Abstract: The Cauchy-Davenport inequality gives a lower bound for the size of sum sets in cyclic groups of prime order. The inverse theorem (due to Vosper) characterizes subsets A and B for which equality is attained. Pollard gave a generalization of the Cauchy-Davenport inequality which takes into account the number of times an element of the sum set is represented as a sum of something in A and something in B. In joint work with Claremont McKenna College undergraduates, Mike OÕBrien, Eva Nazarewicz and Carolyn Staples we prove the inverse theorem for PollardÕs inequality. I will also discuss some other research directions in this area which are appropriate for undergraduates and some connections between additive number theory and analysis. |
| Tuesday July 10 | Marc Chamberland Grinnell College |
Title: Mathematical Discovery via Computer Experimentation Abstract: The use of computer packages has brought us to a point where the computer can be used for many tasks: discover new mathematical patterns and relationships, create impressive graphics to expose mathematical structure, falsify conjectures, confirm analytically derived results, and perhaps most impressively for the purist, suggest approaches for formal proofs. This is the thrust of experimental mathematics. This talk will give some examples to discover or prove results concerning goemetry, integrals, binomial sums, and infinite series. |
| Tuesday July 17 | Charles Lee California State University, Fullerton |
Title: Cancer Detection Using Component Analysis Methods on DNA Microarray Data Abstract: The Principal Component Analysis (PCA) has been used widely as an effective tool for pattern recognition and feature extraction in many areas such signal enhancing for large array antennas, model reduction for simulating and controlling fluid flows, characteristics identification in criminology, etc. In this talk, the mathematics for the PCA along with its application on DNA microarray data in cancer detection will be discussed. Studies based on two sets of liver and bladder cancers will be presented. Following this spirit, another feature extraction technique, called the Independent Component Analysis (ICA) will also be discussed. Its primary advantage, in contrast to the correlation-based PCA, is that not only can the ICA de-correlate the 2nd-order statistics of the signals, but it can also produce higher-order statistical dependencies, attempting to make the signals as independent as possible. The ICA is known for its capability to identify multiple blind signals in speech recognition systems and medical signal processing. The latest results on using the ICA for cancer diagnosis will also be reported. This is joint work with David J. Peterson, Michael Vodhanel and Nasser Abbasi of California State University Fullerton. |