The Algebra/Number Theory/Combinatorics Seminar meets at Pomona College, Tuesdays at 12:15pm in Millikan 211. Brown bag lunch.

The Analysis Seminar meets at Pomona College, Mondays at 3:00pm in Millikan 213.

The Topology-Geometry Seminar continues at Pomona College Tuesdays at 3:00pm, Millikan 211.

The Spring 2009 Math Colloquium will be held at Harvey Mudd College, Beckman B126, Wednesdays at 4:00pm. Ellis Cumberbatch and Mario Martelli are Colloquia co-chairs for 2008-09.


January 21, Asuman Aksoy, Claremont McKenna College
Title: "Best Approximation in Metric Trees"
Abstract: The study of injective envelopes of metric spaces, also known as metric trees (R- trees or T-theory), has its motivation in many subdisciplines of mathematics as well as biology/medicine and computer science. Its relationship with biology and medicine stems from the construction of phylogenetic trees. Concepts from \string matching" in computer science are closely related with the structure of metric trees. A metric tree is a metric space (M; d) such that for every x; y in M there is a unique arc between x and y and this arc is isometric to an interval in R. In this talk, we examine convexity and compact structures in metric trees and apply these results to show the existence and uniqueness of best approximations. Several applications of best approximation in metric trees will also be discussed.
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January 28, Peter Blomgren, San Diego State University
Title: "Structure Enhancement Diffusion and Contour Extraction for Electron Tomography of Mitochondria"
Abstract: The interpretation and measurement of the structural architecture of mitochondria depend heavily upon the availability of good software tools for filtering, segmenting, extracting, measuring and classifying the features of interest. Images of mitochondria contain many flow-like patterns and they are usually corrupted by large amounts of noise. Thus, it becomes necessary to enhance them by denoising and closing interrupted structures. We introduce a new approach based on anisotropic nonlinear diffusion and bilateral filtering for electron tomography of mitochondria. It allows noise removal and structure closure at certain scales, while preserving both the orientation and magnitude of discontinuities. This technique facilitates image enhancement for subsequent segmentation, contour extraction, and improved visualization of the complex and intricate mitochondrial morphology. We perform the extraction of the structure- defining contours by employing a variational level set formulation. The propagating front for this approach is an approximate signed distance function which does not require expensive re-initialization. The behavior of the combined approach is tested for visualizing the structure of a HeLa cell mitochondrion and the results we obtain are very promising. Joint work with: Carlos Bazan (San Diego State University), and Michelle Miller (Illumina Corp
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February 4, Stefan Llewellyn Smith, University of California, San Diego
Title: "Vortex Rings with Swirl"
Abstract: Vortex rings have been a source of fascination in fluid mechanics since the time of Helmholtz. Asymptotic results for the velocity of axisymmetric vortex rings exist in the limit of narrow cores, including the effect of swirl (flow ”around” the ring). Exact solutions are however more difficult to find, in particular for the unsteady case. Axisymmetric vortex rings with azimuthal vorticity proportional to distance from the axis of symmetry have a contour dynamical formulation as shown by Pozrikidis and Shariff, and steadily- propagating solutions had been found earlier by Norbury. We consider the effect of adding swirl. Taking the swirl inversely proportional to distance from the axis so that the swirl is irrotational leads again to a contour dynamics formulation, but it becomes necessary to add a vortex sheet at the boundary of the rings. The steady case requires an extra constraint. Steady and unsteady results are discussed.
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February 11, Ricardo Carretero, San Diego State University
Title: "Faraday Waves in Bose-Einstein Condensates"
Abstract: Traditional Faraday waves appear in a layer of liquid that is shaken vertically. These patterns can take the form of horizontal stripes, close-packed hexagons, or even squares or quasiperiodic patterns. Faraday waves are commonly observed as fine stripes on the surface of wine in a wineglass that is ringing like a bell when periodically forced. Motivated by recent experiments on Faraday waves in Bose-Einstein condensates we investigate both analytically and numerically the dynamics of cigar-shaped Bose-condensed gases subject to periodic modulation of the strength of the transverse confinement’s trap. We offer a fully analytical explanation of the observed parametric resonance yielding the pattern periodicity versus the driving frequency. These results, corroborated by numerical simulations, match extremely well with the experimental observations.
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February 18, Elissa Schwartz, Washington State University
Title: "A Stochastic Model of HIV-1 Escape from the Cytotoxic T Lymphocyte Response"
Abstract: Knowledge of the correlates of viral escape from the cytotoxic T lymphocyte (CTL) response is needed for HIV-1 vaccine development. Mathematical models of this process have been constructed using deterministic equations, but the appearance of CTL escape mutants reported in the published data is not well approximated by deterministic models of viral escape. This finding motivated us to model viral escape as a stochastic process, with the aim of predicting parameter sets likely to prevent escape. Our model takes into account viral infection, mutation, CTL killing, and viral production and includes parameters for viral burst size, mutation rate, and probabilities of recognition and elimination by CTLs. We used the model to simulate viral production by both wild type and mutant strains in 50 to 500 individuals over 25 years. We found that our model reproduced the CTL escape phenomena seen in clinical data, with varying waiting times before the emergence of escape mutants. The model can be used to determine under what conditions we see escape frequencies like those observed in the data. We can also estimate the number of mutations needed for an escape mutant that arises after x years. Model results were consistent with a scenario in which mutant virus has a competitive advantage when CTL pressure against the wild type strain is strong, and when the number of infected cells is greater. In this way, a vaccine that stimulates a CTL response that incompletely eliminates wild type infected cells can promote escape by mutant virus. Such results may aid in the development of vaccines.
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February 25, Robert L. Devaney, Boston University
Title: "The Fractal Geometry of the Mandelbrot Set"
Abstract: In this lecture we describe several folk theorems concerning the Mandelbrot set. While this set is extremely complicated from a geometric point of view, we will show that, as long as you know how to add and how to count, you can understand this geometry completely. We will encounter many famous mathematical objects in the Mandelbrot set, like the Farey tree and the Fibonacci sequence. And we will find many soon-to-be-famous objects as well, like the "Devaney" sequence. There might even be a joke or two in the talk.
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March 4, Brian Hopkins, St. Peter's College
Title: "The Bridges of Königsberg: Euler, Hierholzer, Fleury, and Beyond"
Abstract: Euler's 1736 article on the bridges of Königsberg is commonly cited as the first result in graph theory, but Euler did not mention edges and vertices or draw the common "dots and lines" picture of a graph. We will look at correspondence leading to the article and see how Euler actually solved the problem. His solution is only partial, in that it does little to address how to construct what we now call an Euler path when one is possible. That part of the problem was not solved until the late 1800s by Hierholzer and later by Fleury, who developed distinct algorithms. We will also look at their original work.
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March 11, Wai Kiu Chan, Wesleyan University
Title: "Integral Quadratic Forms with Finitely Many Exceptions"
Abstract: An integral quadratic form is called regular if it represents all integers which cannot be ruled out by congruence conditions. Examples are the sum of four squares and other universal quadratic forms. The goal of this talk is to describe some of the recent advances in the study of regular quadratic forms and generalizations, with emphasis put on various finiteness results.
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March 25, Alfonso Castro Harvey Mudd College
Title: "Radial Steady States of Stars"
Abstract: All the radial solutions to steady states of partial differential equations of models that include diffusion, heat generation caused by nuclear fusion, and radiation are classified. Stability, bifurcation, and free boundary problems will be discussed.
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April 1, Ellis Cumberbatch, Claremont Graduate University
Title: "Transistor Modeling for the Semi-conductor Industry"
Abstract: Software packages, under the generic name SPICE, simulate C-MOS integrated circuits. With >10⁶ transistors on a chip, each transistor must have a simple representation in SPICE. The current/voltage characteristics of a single transistor are derived from the flow of electrons and holes, governed by non-linear partial differential equations, the so-called drift-diffusion equations (in the classical limit). Simple approximations were found suitable in early models, and as device sizes reduced these have been adapted empirically to fit data so much so that now there can be as many as 400 fitting parameters needed. This is expensive it requires lots of measurements and parameter extraction. We re-visit the physics, and obtain better approximations with much fewer empirical constants. After a brief introduction to the physics of semi-conductors, I shall describe approaches to the basic equations that yield accurate formulae. Also I shall describe enhanced equations that have been introduced to model quantum effects that are present in ultra-small devices. Our results are compared with exact numerical solutions and with data.
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April 8, Eric Rains, California Institute of Technology
Title: "Elliptic Hypergeometric Integrals"
Abstract: Euler’s beta (and gamma) integral and the associated orthogonal polynomials lie at the core of much of the theory of special functions, and many generalizations have been studied, including multivariate analogues (the Selberg integral; also work of Dixon and Varchenko), q-analogues (Askey-Wilson, Nasrallah-Rahman), and both (work of Milne-Lilly and Gustafson; Macdonald and Koornwinder for orthogonal polynomials). In 2000, van Diejen and Spiridonov conjectured a further generalization of the Selberg integral, going beyond q to the elliptic level (replacing q by a point on an elliptic curve). I’ll discuss two proofs of their conjecture, and the corresponding elliptic analogue of the Macdonald and Koornwinder orthogonal polynomials. In addition, I’ll discuss a further generalization of the elliptic Selberg integral with a (partial) symmetry under the exceptional Weyl group E₈, and its relation to Sakai’s elliptic Painlevé equation.
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April 15, Michael Orrison, Harvey Mudd College
Title: "Algebraic Voting Theory"
Abstract: If the results of your election procedure can be realized as a matrix- vector product, then the representation theory of the symmetric group can probably say something interesting about the way you are voting. In this talk, I'll describe some of the recent work that my students and I have been doing to better understand voting and voting paradoxes from an algebraic perspective. Along the way, I'll describe how our work fits inside the more general framework of harmonic analysis on finite groups.

The originally scheduled speaker, Robert Rovetti, of Loyola Marymount University was compelled to cancel at the last minute. Special thanks to Mike Orrison for so ably stepping in.


April 22, Jim Hoste, Pitzer College
Title: "Lissajous, Fourier, and Chebyshev Knots"
Abstract: Ask a knot theorist to "tell" you a specific knot and he will probably draw you a picture. Since the beginning of the subject, knot diagrams have been the standard way of depicting knots. But a knot is a simple closed curve in space, so why not describe it with parametric equations?
In this talk I will describe three different, but similar, ways to do just that. We'll see that every knot can be described parametrically using Chebyshev polynomials and that this allows every knot to be described by a 4-tuple (a,b,c, φ) where a, b and c are positive integers and φ is a real number.
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April 29, Chris Rasmussen, San Diego State University
Title: "The Inquiry Oriented Differential Equations Project:
Addressing Challenges Facing Undergraduate Mathematics Education "
Abstract: Undergraduate mathematics education today faces a number of new challenges and difficulties. One way to address these challenges is to build on promising theoretical advances and instructional approaches, even those not originally developed with undergraduate mathematics in mind. The Inquiry Oriented Differential Equations Project (IODE) is one such effort, which can serve as model for other undergraduate course innovations. In this presentation I describe central characteristics of the IODE approach, report on results of a comparison study, and detail the emergence of a bifurcation diagram, a surprising and illustrative example of student reinvention. I use the bifurcation diagram reinvention example to develop the notion of brokering, which speaks to the unique role of the instructor in student reinvention of significant mathematical ideas. The notion of brokering, which generalizes beyond differential equations, highlights how teaching and learning mathematics is a cultural practice, one that is mediated by and coordinated with the broader mathematics community, the local classroom community, and the small groups that comprise the classroom community.
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The Fall 2008 Math Colloquium archived abstracts are below

September 10, Shahriar Shahriari, Pomona College
Title: "Matchings and Marriage, Chains and Dominance"
Abstract: The subsets of a finite set, the divisors of a positive integer, the subspaces of a finite dimensional vector space over a finite field, and the subgroups of a finite group are examples of finite partially ordered sets (poset). Now assume that you have a poset with 22 elements. What conditions on the poset would assure that you can partition the elements into four chains (totally ordered subsets) of sizes 6, 6, 6, and 4, as well as into five chains of sizes 5, 5, 5, 5, and 2. Partitions of posets into chains have been the object of a number of long standing conjectures (two of them by Pomona alumnus Jerry Griggs). In this talk, I will survey recent results on these conjectures, and propose a new and more general conjecture. My focus will be on a certain class of posets called normalized matching posets, and I will present a number of results obtained in collaboration with undergraduate students Elinor Escamilla, Andreea Nicolae, Andrew Pearsall, Paul Salerno, and Jordan Tirrell. This talk has no prerequisite and should be accessible to all. Click here for the flyer.


September 17, Student Research Poster Session
Summer undergraduate and graduate research poster session at the Marian Miner Cook Athenaeum, Claremont McKenna College, sponsored by REBMI and CCMS.

September 24, Robert J. Sacker, University of Southern California
Title: "Semigroups of Maps and Periodic Difference Equations"
Abstract: A collection M of monotonic maps from the positive reals to the positive reals is defined. Each map is linearly bounded, has non-negative Schwarzian and is either concave increasing or convex decreasing. It is shown that M is a semigroup under composition that contains the sub-semigroup of fractional linear maps and each function in M that is uni-linearly bounded has a globally attracting exponentially asymptotically stable fixed point. Thus we obtain a condition under which a periodic difference equation (mapping system) will have a periodic solution having the same properties. Certain restricted algebraic operations are valid in M and the structure of M is explored together with conjectures regarding the interlacing of roots of a rational function in M.
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October 1, Adolfo Rumbos, Pomona College
Title: "Periodic Solutions to a Piece-wise Linear Second Order Ordinary Differential Equation "
Abstract: Consider the problem of determining values of $\mu$ and $\nu$ for which the second order ordinary differential equation (ODE) \begin{equation}\label{AbsEqn1} -u''=\mu u^+ - \nu u^- \end{equation} has non--trivial solutions of period $2\pi$, where $u^+$ and $u^-$ denote the positive and negative parts, respectively, of the function $u\colon\mathbb{R}\to\mathbb{R}$. The set of pairs $(\mu,\nu)$ for which the ODE in (\ref{AbsEqn1}) has non--trivial, $2\pi$--periodic solutions is called the {\em Fu\v cik--Dancer spectrum} for boundary value problem (BVP) \begin{equation}\label{AbsBvp1} \left \{ \begin{array}{l} -u'' =\mu u^+ - \nu u^- \qquad\mbox{for }0 Click here for the flyer.


October 8, Anna Bargagliotti, Univeristy of Memphis
Title: "Rating System Design: Transforming Individual Preferences to Rating Scores"
Abstract: We use rating systems in our daily lives to represent people's opinions. At a university, we are asked to use a scale to rate our professors, departments, and students. On-line, markets such as Ebay or Amazon, depend on their rating systems to create a trustworthy environment. Can these rating systems provide accurate descriptions of our opinions? Why don't all markets use the same rating system? Is one system better than the other? We show that different rating systems may translate the same opinion to different conclusions. Using simple mathematics, we characterize the differences in the systems and provide simple tools to determine whether rating systems are consistent.
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October 15, Raanan Schul, University of California, Los Angeles
Title: "Harmonic Analysis and Analyzing Data"
Abstract: The topic of this talk lies on the interface between Harmonic Analysis or Geometric Measure Theory and Applied Mathematics. In particular, in many cases, one is given a large data set, represented as a subset of a metric space, such as R^d for a large dimension d. One seeks to faithfully represent a large portion of this data set as a subset of Rk for dimension k much smaller than d. We will discuss mathematical aspects of three different methods of doing so. Click here for the flyer.


October 22, Ali Nadim, Claremont Graduate University
Title: "Electrowetting and Digital Microfluidics"
Abstract: Electrowetting actuation of individual liquid droplets on a solid surface, known as digital microfluidics, has a variety of interesting applications. These include liquid lenses without mechanical moving parts (www.varioptic.com), novel displays for consumer electronics (www.liquavista.com), and liquid handling without the need for channels, pumps or valves (www.liquid-logic.com). In this talk, we review our group's progress in this area and describe some of the mathematical models we have developed that help us estimate the magnitudes of forces and speeds that can be achieved by electrowetting. Our models focus on the problem of electrowetting actuation of individual sessile drops on a patterned array of electrodes with a thin dielectric coating. For both the case when the drop is electrically grounded from below and when it is floating, we compute the electric field in the vicinity of the drop over a range of frequencies and use the traction derived from the Maxwell stress tensor to calculate the effective electrowetting force on the drop. At low frequencies when the drop behaves like a perfect conductor, the results are compared with previously derived lumped parameter models for the electrowetting force.
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October 29, David Costa, Univeristy of Nevada, Las Vegas
Title: "Maxima, Minima, and other Extrema"
Abstract: Starting with a historical account of some classical results in minimization or maximization, we motivate the use of variational techniques for solving problems on differential equations. In particular, the so-called Mountain-Pass Theorem and Saddle-Point Theorem will be introduced.
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November 5, Glen Van Brummelen, Quest Univeristy Canada
Title: "Computing without Computers: Serving the Needs of Mathematical Astronomy in Medieval Islam"
Abstract: Prior to the birth of calculus, and hence of most numerical approximation techniques used today, medieval astronomers were often faced with fierce computational predicaments. Their solutions, often brilliant and usually practical, presaged various modern methods. We shall glimpse a few of these, including constructing a sine table with one's bare hands, solving equations using iterative methods, using interpolation schemes to approximate single- and double-argument functions, and various approximate solutions to mathematical problems related to Islamic religious ritual.
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November 12, David Pengelley, New Mexico State Univeristy
Title: "Dances Between Continuous and Discrete: Euler's Summation Formula"
Abstract: Euler developed and used his summation formula to estimate the sum of reciprocal squares to 14 digits --- a value mathematicians had been competing for since Leibniz's astonishing discovery that the alternating sum of the reciprocal odd numbers is exactly π/4. This competition became known as the Basel Problem, and Euler's approximation probably spurred his spectacular solution, π^2/6. Subsequently he connected his summation formula to Bernoulli numbers and many other topics, masterfully circumventing that it almost always diverges. He applied it to estimate harmonic series partial sums, the gamma constant, and sums of logarithms, thereby calculating large factorials (Stirling's series) with ease. He even commented that his approximation of π was surprisingly accurate for so little work.
I will illustrate Euler's achievements in his own (translated) words, and discuss an undergraduate teaching unit of original sources about the search for formulas for sums of numerical powers in relation to integration, seen through writings of Archimedes, Fermat, Pascal, Jakob Bernoulli, and Euler. I will show Euler's idea for deriving his summation formula, how he applied the formula, and discuss exercises for students, perhaps including use of computing software.
This talk requires two solid semesters of calculus.
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November 19, Jo Hardin, Pomona College
Title: "Outliers when Clustering Microarray Data"
Abstract: Microarray data are well known to be noisy and rife with outliers. The outliers are sometimes interesting in their own right, but often they are simply poor quality measurements that should be removed from the analysis. Unlike many other statistical techniques, clustering methods will always give you cluster outputs regardless of the structure of the data. Though clustering results can be enormously informative, the results can also be misleading if the data have outlying values. In particular, when clustering genes with only tens of samples, a few outlying values can easily change the direction of the relationship between a pair of genes. We provide mechanisms for robust clustering that minimize unwanted noise.
No background in microarrays or clustering needed for this talk.
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December 3, Hongkai Zhao, Univeristy of California, Irvine
Title: "A Fast Forward Solver for Radiative Transfer Equation with Applications in Optical Imaging"
Abstract: Radiative transport equation (RTE) is one of the most fundamental equations for modeling particle, such as neutron and photon, transport in complex media. Although the RTE is linear, the main challenge for numerical computing comes from high dimensionality of space plus phase variables and the scattering among different directions.
In this talk I will present an efficient forward solver for steady-state RTE on both structured and unstructured grid. First we introduce a novel angular discretization in phase space that can capture scattering among different directions more physically and efficiently. Second, for a steady-state RTE, a large coupled linear system has to be solved efficiently. We develop a Gauss-Seidel iterative scheme that incorporates dominant scattering and angular dependent ordering effectively. The iterative method is then used naturally as an efficient relaxation scheme for multigrid solver in both spatial and angular space. I will demonstrate both efficiency and accuracy of our forward solver by extensive tests and applications in various optical imaging regimes. The fundamentals of particle transport will be explained at the beginning of the talk.
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December 10, Aparna Higgins, University of Dayton
Title: "Demonic Graphs and Undergraduate Research"
Abstract: Working with undergraduates on mathematical research has been one of the most satisfying aspects of my professional life. This talk will highlight some of the beautiful and interesting research done by my former undergraduate students on line graphs and pebbling on graphs. We will consider line graphs, some pioneering results in pebbling graphs, and pebbling numbers of line graphs. This work has inspired other students to investigate questions in these areas, and it has contributed to my research as well.
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