## MATH 497B, Fall 2010

Course Title: Differential Equations from an Algebraic Perspective

Instructor: Nigel Higson

Assistant: Tyrone Crisp

Meeting Times: Mondays, Wednesdays and Fridays, 10:10-11:00, in 113 McAllister. In addition Tyrone will hold court on Thursdays, same time and place.

Office Hours: Mondays and Tuesdays, 3:30-4:30, in 228 McAllister

Overview: The main protagonist in the course will be the Weyl algebra of differential operators such as

d2/dx2 - x2 or d/dx + x or x d/dx + d/dx x

(these are linear operators with polynomial coefficients). Like the algebra of n by n matrices, the Weyl algebra is noncommutative.

Our aim is to understand some of the structure of the Weyl algebra and its modules, and to relate what we learn to differential equations. The course will obviously involve concepts from algebra and calculus; it will also make contact with algebraic geometry, which is concerned with commutative algebras and their modules.

We shall more or less follow the following book:

S.C. Coutinho. A primer of algebraic D-modules

London Math. Soc. Student Texts 33 (1995)

We shall cover roughly the first 11 chapters, more if we have time. But we shall begin with background material in commutative algebra and elementary affine algebraic geometry, so as to prepare ourselves for our adventure into the noncommutative world.

Prerequisites: The most important requirement will be some initial familiarity with linear and abstract algebra (in linear algebra: vector spaces, bases, subspaces and quotients, linear transformations; in abstract algebra: a bit about groups and rings). You don’t need to be an expert by any means, but if you’ve never seen any of these things before, then the course could prove to be very challenging for you. You also ought to know basic multivariable calculus and have some experience with inventing and writing proofs. Contact me if you have any questions or concerns.

MASS Program: This course will form part of the 2010 MASS program at Penn State University. See the MASS web pages for further information about the program. Students outside of the MASS program who wish to take this course ought to contact me.

Homework: There will be weekly homework assignments, due on Mondays.

Midterm: There will be a two-hour midterm exam on Monday October 11. The exact time of the exam is yet to be determined.

Final Exams: A final oral exam will be scheduled for each student on December 10, 13 or 15. The exam will be in the standard MASS format.

Project: Each student will be required to complete a project and present it during the final exam (this is part of the standard MASS exam format). A list of possible project titles will be distributed at about the time of the midterm, along with further details about the project requirement.

Academic Integrity: Students must adhere to the University's and the College's standards of academic integrity. The University defines academic integrity as "the pursuit of scholarly activity in an open, honest, and responsible manner." It further states that "Academic integrity includes a commitment not to engage in or tolerate acts of falsification, misrepresentation or deception. Such acts of dishonesty violate the fundamental ethical principles of the University community and compromise the worth of work completed by others." See this page for more information about the University and College academic integrity policies. For a compelling account of what honesty and integrity should especially mean for a scientist (or a mathematician), read this famous speech given by Richard Feynman.

Instructor: Nigel Higson

Assistant: Tyrone Crisp

Meeting Times: Mondays, Wednesdays and Fridays, 10:10-11:00, in 113 McAllister. In addition Tyrone will hold court on Thursdays, same time and place.

Office Hours: Mondays and Tuesdays, 3:30-4:30, in 228 McAllister

Overview: The main protagonist in the course will be the Weyl algebra of differential operators such as

d2/dx2 - x2 or d/dx + x or x d/dx + d/dx x

(these are linear operators with polynomial coefficients). Like the algebra of n by n matrices, the Weyl algebra is noncommutative.

Our aim is to understand some of the structure of the Weyl algebra and its modules, and to relate what we learn to differential equations. The course will obviously involve concepts from algebra and calculus; it will also make contact with algebraic geometry, which is concerned with commutative algebras and their modules.

We shall more or less follow the following book:

S.C. Coutinho. A primer of algebraic D-modules

London Math. Soc. Student Texts 33 (1995)

We shall cover roughly the first 11 chapters, more if we have time. But we shall begin with background material in commutative algebra and elementary affine algebraic geometry, so as to prepare ourselves for our adventure into the noncommutative world.

Prerequisites: The most important requirement will be some initial familiarity with linear and abstract algebra (in linear algebra: vector spaces, bases, subspaces and quotients, linear transformations; in abstract algebra: a bit about groups and rings). You don’t need to be an expert by any means, but if you’ve never seen any of these things before, then the course could prove to be very challenging for you. You also ought to know basic multivariable calculus and have some experience with inventing and writing proofs. Contact me if you have any questions or concerns.

MASS Program: This course will form part of the 2010 MASS program at Penn State University. See the MASS web pages for further information about the program. Students outside of the MASS program who wish to take this course ought to contact me.

Homework: There will be weekly homework assignments, due on Mondays.

Midterm: There will be a two-hour midterm exam on Monday October 11. The exact time of the exam is yet to be determined.

Final Exams: A final oral exam will be scheduled for each student on December 10, 13 or 15. The exam will be in the standard MASS format.

Project: Each student will be required to complete a project and present it during the final exam (this is part of the standard MASS exam format). A list of possible project titles will be distributed at about the time of the midterm, along with further details about the project requirement.

Academic Integrity: Students must adhere to the University's and the College's standards of academic integrity. The University defines academic integrity as "the pursuit of scholarly activity in an open, honest, and responsible manner." It further states that "Academic integrity includes a commitment not to engage in or tolerate acts of falsification, misrepresentation or deception. Such acts of dishonesty violate the fundamental ethical principles of the University community and compromise the worth of work completed by others." See this page for more information about the University and College academic integrity policies. For a compelling account of what honesty and integrity should especially mean for a scientist (or a mathematician), read this famous speech given by Richard Feynman.