and Goals: The purpose of this one credit Math
410 course is to introduce students to possible undergraduate research
projects in computational mathematics at William and Mary. The
format will consist mainly of weekly talks by faculty (approximately 30
minutes) followed by class discussions and/or exercises related to the
presented topics. The typical student in this course will be in
his or her sophomore or junior year and will have an interest in
pursuing a research project related to computational mathematics.
For many, this course can serve as a gateway to establishing a research
project through the CSUMS
program, with applications for this program due at the end of the
spring semester. If you have any questions about whether this
course could be appropriate for you, please contact me
or one of the faculty members listed on the CSUMS
website. (Students who have previously taken ``Math
410: Topics in Computational Mathematics'' are welcome to enroll again
this semester. Students may petition the Chair of the
Mathematics Department to have three 1 credit Math 410 courses count as
one 400 level course for the mathematics major.)
|Week 1 (1/19)
to be a winner: advices for students starting into research (Andre
Advice for undergraduates considering graduate schools (Phil Agre)
Does one have to be a genius to do mathematics? (Terry Tao)
The importance of stupidity in scientific research (Martin Schwartz)
|Week 2 (1/26)
||Ecology, Economics and Mathematics: for CSUMS||Junping Shi
||Complex systems: Ecology for bankers, Robert M. May, Simon A. Levin & George Sugihara, (2008) Nature, 451, 893-895.
Systemic risk in banking ecosystems, Andrew G. Haldane & Robert M. May, (2011), Nature, 469, 351-355.
Can ecological models explain global financial markets -- and make them more stable?, Katherine Harmon, (Jan 19, 2011), Scientific American.
|Week 3 (2/2)
||Synchronization of Biological Oscillators
||Coupled oscillators and biological synchronization, Steven Strogatz and Ian Stewart, Scientific American (1993).
Scale-free networks, Albert-Laszlo Barabasi and Eric Bonabeau, Scientific American (2003).
|Week 4 (2/9)
||Maximizing algebraic connectivity on graphs with a given degree distribution
||(See Discussion Board on Blackboard for reading material)
|Week 5 (2/16)
||Graph coloring||Gexin Yu|
|Week 6 (2/23)
||Population dynamics in multipatch models||Leah Shaw
||Chaotic desynchronization of multistrain diseases, Schwartz et al, Physical Review E (2005).
Asymmetry in the Presence of Migration Stabilizes Multistrain Disease Outbreaks, Simone Bianco and Leah B. Shaw, Bulletin of Mathematical Biology (2011).
|Week 7 (3/2)
||Dynamics via computational topology
||Three examples of applied and computational homology, Robert Ghrist, Nieuw Archief voor Wiskunde 5/9 no. 2 (2008).
|Week 8 (3/9)
||Spring Break (no class)|
|Week 9 (3/16)
||Computational Biology||Greg Smith
||Theoretical Neuroscience Rising, L.F. Abbott, Neuron 60 (2008).
Why Are Computational Neuroscience and Systems Biology So Separate?, Erik De Schutter, PLoS Computational Biology (2008), no. 4 (5).
COMPUTATIONAL STUDIES OF GENE REGULATORY NETWORKS: IN NUMERO MOLECULAR BIOLOGY, Jeff Hasty et al, Nature Reviews (2001).
Intracellular Signaling: Spatial and Temporal Control, Ion I. Moraru and Leslie M. Loew, Physiology 20: 169-179 (2005).
Bringing cartoons to life, John Tyson, Nature Vol 445 (2007).
|Week 10 (3/23)
||Patterns in Extinction in a Coupled Ricker Model||Ben Holman|
|Week 11 (3/30)
||Sophomore registration (no class)|
|Week 12 (4/6)
|Week 13 (4/13)
||Ecosystem Modeling in Marine Science||Mark Brush||Introduction to the special issue of Ecological Modelling: "Advances in Modeling Estuarine and Coastal Ecosystems: Approaches, Validation, and Applications", Mark Brush & Lora Harris, Ecological Modelling 221 (2010).|
|Week 14 (4/20)
||Computational Probability||Larry Leemis|
|Week 15 (4/27)
Colloquia/seminars suitable for undergraduate students:
February 2 (Wed), 4:00 pm, Applied Science Seminar (McGlothlin-Street Hall 020) -- pizza will follow talk.
Speaker: Timothy Blass (University of Texas at Austin)
Title: Plane-like minimizers and the Minimal Average Energy
Abstract: I will show how gradient descent techniques can be used to construct pane-like minimizers of energy functionals whose Euler-Lagrange equations are nonlinear elliptic PDEs. The method can be implemented numerically and used to compute the minimal average energy as a function of the "slope" of the minimizers. The minimal average energy has the physical interpretation as the surface tension of a crystal face, and its differential properties determine the shape of the crystal.
February 4 (Fri), 3:00 pm, Mathematics Colloquium (Jones Hall 131)
Speaker: Shitao Liu (University of Virginia)
Title: Global uniqueness and stability in inverse hyperbolic problems
Abstract: We provide a global uniqueness and stability result in determining both the damping and the potential coefficient of an inverse hyperbolic problem with non-homogeneous Neumann term, through the measurement of an additional Dirichlet boundary trace over an explicit portion of the boundary and over a close to optimal, computable time interval. The key ingredients of the proof include: (i) sharp and very general Carleman estimates for second order hyperbolic equations; (ii) a correspondingly implied Continuos Observability Inequality; (iii) Sharp/optimal interior and boundary regularity theory of second order hyperbolic equations with Neumann boundary datum. We will also unveil relations between inverse problems and control theory problems.
This is joint work with Roberto Triggiani.
February 9 (Wed), 4:00 pm, Applied Science Seminar (McGlothlin-Street Hall 020) -- pizza will follow talk.
Speaker: Jesse Berwald (Montana State University)
Title: A Novel Approach to Analyzing Multifractal Spectra of Dynamical Systems
Abstract: We introduce a novel, simplicial-based method for computing the multifractal spectrum of the attractor of a dynamical system. We motivate the work with a discussion of prior research involving the development and analysis of a computational model designed to simulate complex physiological systems.
February 16 (Wed), 4:00 pm, Applied Science Seminar (McGlothlin-Street Hall 020) -- pizza will follow talk.
Speaker: Gregory Scott Cochran (George Mason University)
Title: Homology Computations of Nodal Domains - Error Estimates, Validation, and Applications
Abstract: Pattern formation occurs throughout the natural sciences. Often, these patterns occur as certain nodal domains of certain functions. We can use algebraic topology and in particular homology to investigate coarse topological features of the nodal domains. However, usually we do not have direct access to the nodal domains either through discrete sampling or the patterns are experimentally determined. In either case, we cannot know if the computed homology is correct. I will present results that establish explicit probability bounds for making the correct homology computations for nodal domains of random fields in one and two spatial dimensions. I will also present a modified algorithm that guarantees we compute the correct homology. Lastly, I will present applications of these results.
February 18 (Fri), 1:00 pm, Mathematics Colloquium (Jones Hall 301)
Speaker: Qin Wang (Department of Statistical Sciences and Operations Research, Virginia Commonwealth University)
Title: Localized Sufficient Dimension Reduction
Abstract: The development of sufficient dimension reduction (SDR) in theory and methodology has provided us a powerful tool to tackle the challenging problem, high dimensional data analysis. It has been applied to many scientific fields in recent years. In this talk, I will introduce a novel approach to sufficient dimension reduction based on localization. A new method, k-nearest-neighbor sliced inverse regression (KNN-SIR), is developed to relax the restrictive linearity condition imposed on the predictor vector and the coverage condition normally needed for sliced inverse regression (SIR). KNN-SIR can exhaustively estimate the central subspace (CS), the target of inference for sufficient dimension reduction, without requiring particular distribution form of the predictor vector. The efficiency of proposed method is demonstrated via simulation studies and a real data analysis. This is a joint work with Prof. Xiangrong Yin (UGA), Prof. Bing Li (PSU) and Dr. Zhihui Tang (PSU).
February 25 (Fri), 3:00 pm, Mathematics Colloquium (Jones Hall 301)
Speaker: Ruhai Zhou (Old Dominion University)
Title: Attractors of liquid crystalline polymers
Abstract: Rigid, rod-like liquid crystalline polymers (or nematic polymers) are elements of many high-performance nano-composite materials. They are processed in liquid phase, and are engineered for a variety of properties in the final solid material, including mechanical, electrical, piezoelectric, thermal, and barrier properties. In this talk, I will review the basic concepts of flowing nematic polymers, along with the mathematical and computational issues. Then we survey a few computational results from kinetic model equations. We will study various stable sates of nematic polymers and their transition in monodomain case. A bifurcation diagram of bulk attractors is given versus the shear rate and molecular concentraton. We will also study the structure formation of nematic polymers in the plane Couette cell. An interesting phenomena, defect core, will be examined in one of the structure attractors.
March 18 (Fri), 3:00 pm, Mathematics Colloquium/CSUMS Lecture (Jones Hall 301)
Speaker: Rika Hagihara (St. Mary's College of Maryland)
April 1 (Fri), 3:00 pm, Mathematics Colloquium/CSUMS Lecture (Jones Hall 301)
Speaker: Joseph Bonin (George Washington University)
Title: An Introduction to Matroid Theory Through Lattice Paths
Abstract: The first (longer) part of this talk is based on joint work with Anna de Mier and Marc Noy of Universitat Polit`ecnica de Catalunya. In this part, we introduce matroid theory through concrete examples that arise from lattice paths. Fix lattice paths P and Q that go from (o,o) to (m, r), with P never going above Q, and consider the lattice paths from (o,o) to (m, r) that stay in the region that P and Q bound. We show that these paths can be identified with the bases of a special type of matroid. (Bases of matroids have properties that abstract those of bases of vector spaces.) For these matroids, called lattice path matroids, the fundamental operations of matroid theory have simple path interpretations. Many important invariants (such as the Tutte polynomial) are #P-hard to compute for arbitrary matroids, but their natural interpretations for lattice path matroids yield polynomial-time algorithms. Lattice path matroids form a small part of the deep and beautiful theory of matroids, a field that has important applications to, for instance, combinatorial optimization, arrangements of hyperplanes, coding theory, and physics. The second part of the talk provides glimpses of the broader field. Building on ideas in the first part, we give a bird's-eye view of some of the major research areas, such as representability over fields, well-quasi-ordering by minors, extremal matroid theory, matroid constructions, and more.
April 4 (Mon), 5:30 pm - 7:00 pm, Careers in Math Panel (Cohen Career Center, Presentation Rooms A&B)
Event announcement: http://events.wm.edu/mathematics/2011/04/04/526/
The Career Center asks that students register for this event at https://forms.wm.edu/form/view/1909/6e6c0e5343a81a15073cde4e7b3c6d29
April 8 (Fri), 2:00 pm, Mathematics Colloquium/CSUMS Lecture (Jones Hall 301)
Speaker: Rongsong Liu (University of Wyoming)
Title: Modeling the Dynamics of Woody Plant-Herbivore Interactions with Age-Dependent Toxicity
Abstract: We study the effects that woody plant chemical defenses may have on interactions between boreal hares that in winter feed almost entirely on twigs. We focus particularly on the fact that toxin concentration often varies with the age of twig segments. The model incorporates the fact that early in the growth of twigs, segments are often highly defended by toxins and are, therefore, highly unpalatable to hares. But that after a year or two, the toxin concentration of older twig segments is sufficiently reduced for their biomass to be more palatable. This age-dependent toxicity of twig segments is modeled using age-structured model equations which are reduced to a system of delay differential equations involving multiple delays in the woody plant - hare dynamics. A novel aspect of the modeling was that it had to account for mortality of non-consumed younger twig segment biomass when older twig biomass was bitten off and consumed. Basic mathematical properties of the model are established together with upper and lower bounds on the solutions. Necessary and sufficient conditions are found for the linear stability of the equilibrium in which the hare is extinct, and sufficient conditions are found for the global stability of this equilibrium. Numerical simulations confirmed the analytical results and demonstrated the existence of limit cycles over ranges of parameters reasonable for hares browsing on woody vegetation in boreal ecosystems.
April 8 (Fri), 3:00 pm, Mathematics Colloquium/CSUMS Lecture (Jones Hall 301)
Speaker: Jan Bouwe van den Berg (Vrije Universiteit, Amsterdam)
Title: Braids in dynamics
Abstract: Pieces of string or curves in three dimensional space may be knotted or braided. This physical idea can be used as a topological tool to study certain types of dynamical systems. In particular, such an approach leads to forcing theorems in the spirit of the famous "period three implies chaos" for interval maps. We discuss an application to a differential equation from the field of pattern formation. This involves several illuminating topological arguments, which are complemented by an illustrative computer-assisted approach.
April 8 (Fri), 5:00 pm, 2011 Pi Mu Epsilon Induction Ceremony (James Blair Hall 223)
Speaker: Charles Johnson
April 14 (Thu), 3:00 pm, Computer Science/CSUMS Lecture (McGlothlin-Street Hall 020)
Speaker: David Crandall (Indiana University)
Title: Reconstructing the World from Social Photo-sharing Websites
Abstract: The rapid rise of social photo-sharing websites has created immense collections of photographs online, with Flickr and Facebook alone now hosting over 50 billion images. The sheer size of these sites raises both challenges and opportunities. A major challenge is how to organize large photo collections effectively since modern photo-sharing sites rely on relatively primitive technology like keyword tags (causing untagged or poorly-tagged photographs to be essentially impossible to find). An opportunity is that these sites contain a vast amount of visual information about the world and its people, contributed by millions of photographers worldwide, that could be a new data source for scientists in a variety of disciplines. In this talk I'll describe some of our recent work both in organizing large-scale photo collections and in using these collections to mine for information about the world and human behavior. For example, I'll show how we use data mining techniques to automatically produce annotated maps of the world using nearly 100 million photographs downloaded from Flickr, and how we use computer vision techniques to efficiently reconstruct 3-d models of popular landmarks. I'll also show how combining computer vision, data mining, and machine learning lets us extract information from large photo collections for use in interdisciplinary studies with fields as diverse as sociology, economics, and ecology.
April 15 (Fri), 3:00 pm, Mathematics Colloquium/CSUMS Lecture (Jones Hall 301)
Speaker: Jan Bouwe van den Berg (Vrije Universiteit, Amsterdam)