This course is a tour through fundamental mathematical techniques used to model continuous objects and events, both deterministic and random, in the physical world. It was originally conceived as a preparation for courses in artificial intelligence, robotics, and computer vision, but has turned out to be useful also to students in several disciplines of engineering and the biological sciences, and in general to anyone who wants to develop a firm intuitive understanding of the most applicable aspects of continuous mathematics.
This course emphasizes the geometric, physical, and intuitive meaning of the various concepts over the details of formal proofs. Classical numerical algorithms are viewed as black boxes, rather than by attempting to describe the details of how they work. For instance, you may already have seen the definition of eigenvalue, and know how to compute the eigenvalues of a matrix A. (Hint: type eig(A) in Matlab. This is all we will study about the algorithm). However, what matters to users is the concept itself: what an eigenvalue is, geometrically and physically, why it is important, and what you can do with it. Once you know this, you will also know why the basic algorithms for computing eigenvalues work, when they succeed and fail, and how you can tell.
We will use this same approach for a rather wide array of topics: After an introduction to the geometric meaning of singular values, the pseudo-inverse, and eigenvalues, the course covers geometric transformations (rotations, similarities, affine and projective transformations), elements of numerical unconstrained optimization, stochastic processes and random fields, deterministic and probabilistic dynamic systems, stochastic filtering and state estimation. Time permitting, the course will also cover elements of tensor fields, variational principles, and flow problems in two dimensions.
The class requires some prior exposure to linear algebra, introductory calculus, and elementary probability, but is otherwise self-contained.
Tuesdays and Thursdays 11:40÷12:55 AM in the Levine Science Research Center (LSRC), room D243. Look for the big, red LSRC in a small map, or in a larger one.
