Topology in Plain English (26th Jan 1998).
Abstract: My aim is to introduce you to the basic ideas of topology without excessive formalism (but not without definitions). I will introduce the notions of topological space, homeomorphism, product space and identification space. All of this will provide us with enough background to move on to the definition of a manifold which is really what we want to understand.
Continuity in Plain English (2nd Feb 1998).
Abstract: There seems to be some need to review the notions of continuity and its relation to open sets. In this brief review I will begin with familiar notion of a limit and show how it relates to continuity. I will also introduce open sets in the real line and show how they relate to continuity. This is will all end by proving the main theorem on continuity. All of this will provide further motivation for studying topology and manifolds.
Further Topology in Plain English (9th Feb 1998).
Abstract: Now that we understand continuity and open sets, we will proceed with more topological considerations. We will discuss subspace topologies, product topologies, and identification spaces. The latter will generalize to the notion of a manifold.
There is an example of the the theorem on level sets (as identification spaces) worked out in the notes.
Manifolds in Plain English (16th Feb 1998).
Abstract: So far we looked at notion of continuity and generalized it so that we could investigate complicated spaces. Now, we will add to this generalization, the notion of differentiation. The most interesting complex spaces that we can investigate this way are called manifolds. Our aim is to understand the definition of manifolds and how it relates to the notion of an identification space.
We will look in detail at several manifolds that arise in classical computer vision problems.
Further Manifolds in Plain English (23th Feb 1998).
Abstract: In this talk, I will further explore the notion of a differential structure of a manifold. My aim is to show you that this notion is as important to the definition of a manifold as the notion of an open set is to a topological space. In fact it defines what we mean by a "derivative."
We will also define functions (in fact differentiable functions) between manifolds and investigate what their graphs look like. This will have rather surprising implications for medical imaging research (yes, the stuff we do).
Tangent Spaces in Plain English (2 March 1998).
Abstract: In this talk I will discuss the notion of tangent vectors and tangent spaces of manifolds. This is probably the single biggest obstacle to going from extrinsic to intrinsic differential geometry. So I will go over this slowly - we will spend the entire talk just discussing tangent vectors and spaces. Also, I expect to run over time - we have a lot to cover.
Affine Connections in Plain English (9th March 1998).
Abstract: After defining tangent spaces, the next major conceptual issue is that of generalizing the notion of parallel vectors and geodesics. All of this can be handled by defining a sort of derivative of tangent vectors called the affine connection. In this talk I will introduce the affine connection and discuss in detail.
Towards the end of the talk I will briefly survey Riemannian geometry using affine connections.
Differential Geometry of Curves (30th March 1998).
Abstract: While the most interesting and representive part of classical differential geometry is the study of surfaces, some local properties of curve appear more naturally. We will discuss several important concepts regarding curve characteristics, including parameterization, arc length, Frenet trihedron, the fundamental theorem of the local theory, and the local canonical form.
Regular Surfaces (6th April 1998).
Abstract:We will introduce the basic concept of regular surfaces, which are defined as sets rather maps. Some criteria are presented which are helpful in deciding whether a given sets is a regular surface.
Fundamental Forms of Surfaces (13th April 1998).
Abstract:We start out discussing the tangent planes of a regular surface. We then study the first and second fundamental forms, which will be continued in the following week(s).
Surface Curvatures (20th April 1998).
Abstract:We will look into the concepts of curvatures (normal, Gaussian, Mean, and the principal ones). Further, we will discuss how to calculate these curvatures within local coordinate representation, and from discrete surface representations. We will also touch upon surface classification using Koenderink shape index function.