Professor:	Alex Kasman kasmana@cofc.edu
Office:	336 Robert Scott Small Building, 953-8018
Office Hours:	M 10:00-11:00, W 2:00-3:00, Th 1:30-2:30 Please visit me during these times if possible. I am often in my office at other times and do not mind at all if you drop by to talk, though I cannot guarantee that I will always be there or have time to meet. If you are unable to see me during my office hours and cannot find me at other times, just contact me by phone or e-mail and I will find an alternative time to meet with you. These would be also good times to ask me to let you into the computer lab if you need access to Mathematica. Come to my office and I can let you into 200 Maybank.
Class Meeting Times:	We meet in room 223 Maybank two times per week: TR 10:50-12:05PM.
Handouts:	I will prepare a handout for each class reviewing the key ideas and assigning homework problems. They are available below as PDF files: 10 January (Course Information) 15 January (Linear Differential Equations) 17 January (Nonlinear Differential Equations) 22 January (KdV: The Story of Solitons) What is Soliton Theory? 24 January (Differential Algebra I) 31 January (A bit more about solitons and a bit more about differential algebra) 5 February (Lax Pairs) 7 February (Lax Pairs - take two) 12 February (Elliptic Curves) 14 February (Elliptic Curves and KdV) 19 February (Review and Reading Assignment) 11 March (The KP Equation) 13 March (Bilinear KP and Tau-Functions) 18 March (Wedge Products and the Grassmann Cone) 20 March (Relating Wedge Products and Bilinear KP) 10 April (Higher Time Flows and "Dark" Soliton Profiles for KdV)
Graded Work:	Your grade in this class will be based on: effort, exams, homework, independent reading, an independent project, and oral presentation. Effort: We will be working on hard questions and difficult subjects in this class. There is no guarantee that we will be able to achieve all of the goals I set. Consequently, unlike many undergraduate math courses in which outcome is all that matters, effort will make a difference towards your grade. As long as a student is making a serious effort to comply with the requirements of the course, that student will receive a final course grade of C or higher. Only those students who seem to "not be trying" will face the possibility of a D or F. (Note: Class participation will be considered part of the evidence of effort. So, students should be sure to answer some questions in class when I ask them.) Exams: There will only be two exams. A midterm (date to be determined) and a final exam (8AM on Tuesday, April 29th). It is difficult to be precise about what will be on these exams as the class is experimental. They may test any topic covered in the class, from historical facts to mathematical techniques. Homework: Homework will be assigned at most class meetings. It is due at the next Tuesday class meeting (regardless of whether it was assigned on Tuesday or Thursday). You are allowed to collaborate on homework. You can talk to each other, ask me questions, and even ask others for help. However, your final answer should be your own, in your own words and your own handwriting (or typesetting). The homework questions will vary in difficulty from very easy to superhard. Some of the homework questions will require you to work with a computer algebra package such as Mathematica. (If your work was done in Mathematica, you may prefer to e-mail me a file in ``.nb'' format rather than turning in a printout.) Independent Reading: Often in undergraduate math classes, it is possible to learn all of the important material from lecture without doing much reading. I will make sure that this is not the case here. The ability to read mathematical texts is an important skill in a math major. Consequently, I will assign some readings on topics that I have not previously discussed. Some of these will be common for the whole class (such as chapters from textbooks) and others will be different for each student (associated with the project). A grade will be determined for this based on your ability to answer questions and discuss what you have read during class meetings. Independent Project: In the second half of the course, rather than working together on one subject, each student will pick a subject to investigate on their own. (I will offer many suggestions in case you cannot come up with one independently.) Working on it will involve independent reading, some computation to demonstrate what you've learned, possibly proving some theorems, writing a short paper to summarize all of the work, and finally making a presentation to the class about your topic. Oral Presentation: It will be my goal to get students up at the board as much as possible. When a problem comes up in class that you should be able to answer, when a homework assignment has been completed, and as you are working on your independent projects, I may call you in front of the class to present some mathematics. A record of all of these presentations will be kept so that a grade can be given for each individual.
Final Grade:	I will come up with a letter grade for you on each of the criteria listed above. Then, they will be combined in the following proportions to give a course grade: Midterm Exam 20% Final Exam 20% Homework 20% Independent Project 20% Independent Reading 10% Oral Presentation 10% Total 100%
Class Website:	http://math.cofc.edu/kasman/MATH495/
Readings:	There is no textbook that you must buy for this class. Most of the readings will be distributed by the professor. Quite a few of the ``readings'' will actually be lecture notes that I will write specifically for this class. I will also copy a few sections out of the books ``Soliton Equations and Hamiltonian Systems'' by Leonid Dickey and ``Solitons: An Introduction'' by Drazin and Johnson. We will read some papers together, including ``Solving Differential Equations by Symmetry Groups'' by John Starrett. Then, there may be reading assignments (papers and/or books) associated to the individual student's choice of a project.
Material to be Covered:	Soliton Theory is an active area of research in mathematics which combines mathematical physics, calculus, algebra, and geometry in a beautiful and surprising way. We cannot hope to cover more than just a tiny bit of it in this one-semester, undergraduate course. In particular, I hope that the students will have educational encounters with (a) the idea that some nonlinear partial differential equations have extra structure (symmetries and conservation laws) which allows them to be solved (b) the remarkable fact that these equations have solutions that seem to blur the distinction between ``waves'' and ``particles'' (c) algebraic methods for solving these equations such as Lax operators, Darboux transformation, tau-functions and nonlinear superposition and (d) geometric methos for solving these equations including spectral curves and grassmannian manifolds.