Sergei Yakovenko's blog: on Math and Teaching

Monday, December 11, 2017

Lectures 4-7, Nov. 21, 28, Dec. 5, 12

Continuity and limits

In these four lectures (sorry for the delay with posting the notes) we have introduced and discussed the notions of continuous functions. Contrary to the usual practice, we switch immediately to the case of functions of several variables, where pictures are much more illustrative.

We discuss the first topological notions: open/closed sets, accumulation/interior points, limits of functions as a way to extend functions continuously beyond the “natural” way of their definition by formulas.

Then we pass to more involved notions as compactness, connectivity (in two flavors) and finally end up by our first really nontrivial topological result, the fixed point theorem for 2-dimensional disk.

The (preliminary) lecture notes are available here: please note that there are over 30 problems approximately of the same sort that will appear on the exam. A more extended version will appear later, besides, you are always invited to recycle the lecture notes from the past years, available from this blog.


Sunday, March 23, 2008

2-Sphere eversion in 3D-space

Filed under: links — Sergei Yakovenko @ 12:23
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If a smooth curve  embedded in the plane \mathbb R^2  is deformed allowing self-intersections but remaining smooth, then there is a natural integral invariant, the rotation number, which prevents eversion of a circle (deformation of the oriented circle into another circle with an opposite orientation). For two-dimensional surfaces smoothly embedded in \mathbb R^3 a similar invariant of deformations exists, yet this invariant does not preclude eversion of the sphere inside out.

 The possibility of such deformation was discovered bt S. Smale in 1958. Relatively recently W. Thurston invented a general algorithm of smoothening, which yields an explicit sphere eversion. All these spectacular things are discussed on the level accessible to high school students in the most fascinating animation (21 min.) discovered on the web by Dmitry Novikov (thanks!). A much shorter animation (mere 22 sec.) does not easily reveal the mistery, so the longer one is really worth its time!

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