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** "A calculus for knot theory"**

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Wednesday, 04. Nov. 2015 |
Audio-only-Recording as MP3-File (smallest
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- Audio.mp3 (ca.28 Mb) ============================================ Video-Recording for any system with MP4-support:
- Video.mp4 (ca.203 Mb) |
15:20 – 16:20 |

Speaker :
Kathryn Hess (EPFL, Schweiz)

A
mathematical knot consists of an embedding of the circle in 3-space. Knot
theory - the mathematical study of knots –

has been
an active, important field of research for well over 100 years. When studying
knots, instead of considering

one knot
at a time, it can be very fruitful to analyze all at once the entire space of
knots, i.e., the set of all knots,

together
with a notion of when two knots are "close" to each other.

The space
of knots is an example of what is known as a space of embeddings. I will start
by reviewing the classical theory

of spaces
of embeddings, including famous results of Hirsch, Smale,
and Gromov, and explain the analogy between

these
results and the theory of linear functions. I will then describe how Goodwillie and Weiss extended this analogy,

establishing a "calculus" for spaces of embeddings. In particular their
theory enables us to construct

"polynomial approximations" to spaces of embeddings,
together with "converging

To conclude
I will sketch briefly a couple of recent applications of the Goodwillie-Weiss calculus, in work of Arone
and

Turchin and of Dwyer and myself.

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