math490topics

Math 490 seminar course topics

Two of the three proposed topics will be offered in the Spring semester of 2007.

Semester	Instructor	Title	Brief Description
Spring 2007 tentative	Rex Kincaid rrkinc@math.wm.edu	Discrete Optimization	Discrete optimization problems are those problems with decisions that are logical (yes/no) or countable. Both exact and heuristic methods for discrete optimization models will be presented in the course. Topics include relaxation techniques, constructive heuristics, improving search techniques (simplex method, simulated annealing, tabu search and genetic algorithms), branch and bound schemes, and valid inequalites for branch and cut methods.
Spring 2007 tentative	Chi-Kwong Li ckli@math.wm.edu	Posing questions, solving problems, and doing research in mathematics	Solving problems is important in learning, teaching, and doing research in mathematics. We will select different topics in mathematics and discuss how one can formulate questions, develop problem solving techniques, and explore open problems.
Spring 2007 tentative	Leiba Rodman lxrodm@math.wm.edu		Students will study a narrow topic of mathematics, make oral presentations for discussion in class, and write a report. The topic need not be the same for all students. The choice of topic(s) is flexible, subject to instructor's approval, and may reflect particular interests of each student.
Fall 2006	Ilya Spitkovsky ilya@math.wm.edu	Topics in analysis.	We will choose among several topics, based in particular on the number and interests of the students enrolled. The selection includes (but is not limited to) numerical ranges of matrices and operators, topology, and cryptography.
Fall 2006	Charles Johnson crjohnso@math.wm.edu	Seminar for prospective teachers	Contact the instructor for details.
Spring 2006	Leiba Rodman lxrodm@math.wm.edu		Students will study a narrow topic of mathematics, make oral presentations for discussion in class, and write a report. The topic need not be the same for all students. The choice of topic(s) is flexible, subject to instructor's approval, and may reflect particular interests of each student.
Spring 2006	Junping Shi shij@math.wm.edu	Mathematical Biology and Partial Differential Equations	Reaction-diffusion (R-D) systems are some of the most widely used models in situations where spatial dispersal plays significant role. We will learn some theory of reaction-diffusion equations in the context of models in mathematical biology. Applications to be discussed in class include spatial spread of genes and of diseases, random dispersal of population, random and chemotactic motion of microorganisms, cellular maturation, pattern formations in developmental biology and morphogenesis, and animal coat patterns (why zebra has the stripes......). While introducing many biology models, we will also develop related mathematical theory and methods like, diffusion mechanism, waves, bifurcation theory, Turing's instability mechanism. Computation and simulation of solutions will be used throughout the class. (We are going to use software Matlab or/and Maple, but no prior knowledge is required.) The prerequisites of the course are Math 111, 112 (Calculus I and II), and Math 302 (Differential equations) or Math 410 (Mathematical Models in Biology). Math 211 (multivariable calculus) and 212 (linear algebra) are not required, but suggested.
Fall 2005	Chi-Kwong Li ckli@math.wm.edu and Jinchuan Hou, 2005-06 Freeman Visiting Fellow	Teacher seminar on posing questions, solving problems, and doing research in mathematics	Solving problems is important in learning and teaching mathematics. However, teachers and students may not be excited by the problems imposed on them. We discuss how K-12 teachers and students can pose their own questions in elementary mathematics. Studying these problems would require a deeper understanding of their known mathematical knowledge and new techniques. Some of the questions may have open ends leading to research projects.
Fall 2005	Junping Shi shij@math.wm.edu	Problem solving techniques	Techniques of solving mathematical problems will be reviewed, and the seminar will also prepare the students for college level mathematical competitions, like Putnam Exam, and Virginia Tech competion. Parallel to a systematic review of knowledge on algebra, combinatorics, number theory, calculus, geometry and other topics, students will also solve problems from selected problem sets, present and discuss their solutions in seminar. Some related material can be found from 2004 problem solving group webpage. Pre-requisites: Math 214. (Math 412 Number theory and Math 432 combinatorics are recommended, but not necessary.)
Spring 2005	Ilya Spitkovsky ilya@math.wm.edu	Topics in analysis.	We will choose among several topics, based in particular on the number and interests of the students enrolled. The selection includes (but is not limited to) numerical ranges of matrices and operators, topology, and cryptography.
Spring 2005	Rex Kincaid rrkinc@math.wm.edu	Discrete Optimization	pre-requisites: Math 323, CSCI 241 In the late 1990s a number of researchers noticed that networks in biology, sociology, and telecommunications exhibited similar characteristics unlike standard random networks. In particular, researchers found that the cummulative degree distributions of these graphs followed a power law rather than a binomial distribution and that their clustering coefficients tended to a nonzero constant as the number of nodes, n, became large rather than O(1/n). Moreover, these networks shared an important property with traditional random graphs---as n becomes large the average shortest path length scaled with log n. This latter property has been coined the small-world property. When taken together these three properties---small-world, power law, and constant clustering coefficient---describe what are now most commonly referred to as scale-free networks. My plan for the course is to cover roughly one chapter of the book LINKED each week. You will be given discussion questions for each chapter and will be expected to actively participate in classroom discussion. In addition I plan to provide supportive material for many of the chapters. For example, chapter 2 is about random graphs but does not give any of the mathematical details.
Fall 2004	Charles Johnson crjohnso@math.wm.edu	Seminar for prospective teachers	Contact the instructor for details.
Fall 2004	Michael Lewis buckaroo@math.wm.edu	Nonlinear optimization	pre-requisites: Math 311 See this link for details.
Spring 2004	Junping Shi shij@math.wm.edu	Mathematical Biology and Partial Differential Equations	Reaction-diffusion (R-D) systems are some of the most widely used models in situations where spatial dispersal plays significant role. We will learn some theory of reaction-diffusion equations in the context of models in mathematical biology. Applications to be discussed in class include spatial spread of genes and of diseases, random dispersal of population, random and chemotactic motion of microorganisms, cellular maturation, pattern formations in developmental biology and morphogenesis, and animal coat patterns (why zebra has the stripes......). While introducing many biology models, we will also develop related mathematical theory and methods like, diffusion mechanism, waves, bifurcation theory, Turing's instability mechanism. Computation and simulation of solutions will be used throughout the class. (We are going to use software Matlab or/and Maple, but no prior knowledge is required.) The prerequisites of the course are Math 111, 112 (Calculus I and II), and Math 302 (Differential equations) or Math 410 (Mathematical Models in Biology). Math 211 (multivariable calculus) and 212 (linear algebra) are not required, but suggested.
Spring 2004	Ilya Spitkovsky ilya@math.wm.edu	Selected topics in linear algebra	We will consider some topics in linear algebra leading to (and associated with) the notion of the so called numerical range of matrices. The properties of the latter will be considered, both classical (going back to the beginning of the twentieth century) and new (established in the last few years). It is hoped that the course will lead to further new results in this direction.
Fall 2003 490-01	George Rublein gtrubl@math.wm.edu	Seminar for prospective teachers	This seminar is designed for prospective secondary mathematics teachers. There are three things to be accomplished. 1. Some parts of the History of Mathematics will be discussed. a) Students will do a literature search so as to prepare themselves to present NCTM-dictated topics on ethno-mathematics. b) Students will be required to read a general history of Mathematics and a biography of some mathematician. Students will prepare a book report for each of these, which report must pass musteras good writing. In so doing, they will satisfy 2. The Concentration writing requirement in Mathematics. 3. The discipline-specific content of the seminar will focus on mathematical problems external to mathematics: ``Why are we doing this ?'', high school students will ask. Mathematics intrudes into many other subjects in school: physics, chemistry, earth science, music, economics and business among them. If we include statistical applications, then sociology and political science should also be in our list. All high-school mathematics teachers should understand the mathematical content of any of its uses found in school. As a start toward this goal, seminar participants will prepare a knowledgeable presentation of one such application. Students will make a written and oral account of the structure of the model employed and its intuitive underpinning, sufficient for the understanding of all other seminar participants. Mathematical tools for solution of the problem and the contextual interpretation of that solution will be presented.
Fall 2003 490-02	Nahum Zobin zobin@math.wm.edu	Seminar on Techniques of problem solving	In this class we shall study a wide range of techniques used in problem solving. Our main inspiration comes from the problems of the recent Putnam math contests. We shall try to classify the problems, find common ideas to approach them. We shall also cover quite a lot of additional material which will help us to cope with such problems. Though no significant prerequisites are necessary, the class assumes some level of mathematical maturity and very good knowledge of elementary calculus, algebra and geometry. There is no textbook required for this class.

Possible Future Topics

Instructor	Title	Brief Description
Michael Trosset trosset@math.wm.edu	Theory and Applications of Distance Geometry	Distance geometry is the study of the pairwise distances that arise in a configuration of points. Distance geometry encompasses elegant mathematical theory, has important applications in a variety of disciplines (statistics, psychology, chemistry, etc.), and poses interesting computational challenges. We will read several journal articles and book chapters that address various topics in distance geometry. Students will give classroom lectures, write papers, and give presentations at an end-of-semester "conference". Remark: Although 490 courses are usually intended for seniors, this course will introduce various research problems that might lead to a senior honors thesis. Interested juniors should consider enrolling.