[iasmath-seminars] Reminder for tonight's Mathematical Conversations

[Kristina Phillips]

INSTITUTE FOR ADVANCED STUDY

School of Mathematics

Princeton, NJ 08540

 

Mathematical Conversations

Wednesday, March 13

 

About Mathematical Conversations: We meet in Harry's Bar* at 6pm, where free
drinks are provided. After 20 minutes, we move to the Dilworth room, where
the speaker gives a 20-minute talk, followed by 15 minutes of discussion
with the audience. After that we return to the bar for further discussions.
Website: https://www.math.ias.edu/math-conversations

 
*Please note this is a Buffet night so the bar will be set up in the dining
hall. Dining services has requested that guests from Math Conversations
gather in the Coffee Lounge after getting a drink from the bar.

 

To view mathematics in titles and abstracts, please click on the talk's
link.

 

 

Topic:                    Wiggling and wrinkling

Speaker:              Daniel Álvarez-Gavela, Member, School of Mathematics

Time/Room:       6:00pm - 7:30pm/Dilworth Room

Abstract Link:      <http://www.math.ias.edu/seminars/abstract?event=136648>
http://www.math.ias.edu/seminars/abstract?event=136648

 

The idea of corrugation goes back to Whitney, who proved that homotopy
classes of immersed curves in the plane are classified by their rotation
number. Generalizing this result, Smale and Hirsch proved that the space of
immersions of a manifold X into a manifold Y is (weakly) homotopy equivalent
to the space of injective bundle maps from TX to TY, whenever dim(X) <
dim(Y). One obtains surprising consequences, such as Smale's eversion of the
sphere. The key insight is that one can wiggle X in the extra dimensions
dim(Y)-dim(X), using the extra room for the corrugation. When dim(X)=dim(Y)
there is no extra room and we cannot hope for such a result. For example,
every real valued function on a circle has at least two critical points.
Nevertheless, one can always wrinkle X back and forth upon itself to create
the extra room. This inevitably produces some singularities, namely folds
along the wrinkle, however these singularities are very simple. This basic
principle underlies deep results of many mathematicians, including M.
Gromov, Y. Eliashberg, N. Mishachev, K. Igusa and E. Murphy among others. In
this talk we will learn how to wiggle and wrinkle manifolds, with a focus on
folded mappings from the sphere to the plane. This will lead us to consider
the surgery of folds and a beautiful picture discovered by J. Milnor.

 

 

http://www.math.ias.edu/seminars

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