Tag: Morse function

2010-04-11




 

Some things that I really would like to make good drawings of

Very bad grammar in the title, sorry.

The ``things that I really would like to make good drawings of'' are unfoldings of two codimension-$ 3$ singularities in the stratified space of smooth functions from $ \mathbb{R}^n$ to $ \mathbb{R}$ . The two are the the butterfly singularity, or hyperbolic umbilic, and the elliptic umbilic, or monkey saddle. In fact, the butterfly singularity lies in the space of functions from $ \mathbb{R}$ to $ \mathbb{R}$ and the monkey saddle in the space of functions from $ \mathbb{R}^2$ to $ \mathbb{R}$ , but by adding sums of squares in the other coordinates, these give codimension-$ 3$ singularities for maps from $ \mathbb{R}^n$ to $ \mathbb{R}$ . Incidentally, the best pictures I know of in print are in Hatcher and Wagoner's ``Pseudo-isotopies of compact manifolds'', but we should be able to do better now with our increased computational power.

The butterfly singularity is, well, pretty simple: $ f(x) = x^5$ . Seems strange to give it such a special name, but this becomes more meaningful when you think about its ``universal unfolding''. This function can of course be perturbed in many different ways to make it Morse. The ``universal unfolding'' is $ f_{a,b,c}(x) = x^5 + a x + b x^2 + c x^3$ . For generic values of $ a$ , $ b$ and $ c$ this will be a Morse function, and as we move around in $ (a,b,c)$ parameter space we see the singularities coming together and cancelling in various ways. What I want is a nice way to draw, for a given plane $ P$ in $ (a,b,c)$ -space, the immersed surface $ S$ in $ \mathbb{R}\times P$ defined by $ (z,(a,b,c)) \in S$ if $ z=f_{a,b,c}(x_0)$ for some $ x_0$ with $ f'_{a,b,c}(x_0) = 0$ . This shouldn't be too hard, right? Then I want to animate this as we move this plane $ P$ through $ (a,b,c)$ -space.

The monkey saddle is ... arggh, lost some text here when editing this post later... moral: don't edit your posts after you've written and posted them... will try to fix it later...






 

Draft of paper with Rob Kirby ready, and some cool drawings

This is a short post to get back in the flow of posting blog entries. First off, Rob Kirby and I finally have a publicly accessible draft of our paper on indefinite Morse 2-functions, which you can access here as a PDF file. We welcome any comments you may have, and will probably post it to the arXiv in a week or so, after Rob visits and we hash out some remaining technical issues. Incidentally, we are not entirely settled on terminology - finding the right names for things in math is worse than finding a name for a rock band, it seems. So you're welcome to weigh in on terminology, if you care.

The second thing I want to share is a sequence of drawings I made and scanned, here as a PDF file for now. This is a drawing of a 1-parameter family of Morse 2-functions from a solid torus to a disk. In my next post I plan to break up the PDF file into lots of little images and explain - for now, just enjoy and see if it makes any sense.







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