Teichmüller Theory and the Statistics of Shapes

Size: px

Start display at page:

Download "Teichmüller Theory and the Statistics of Shapes"

Jessica Cameron
5 years ago
Views:

1 Brown University Teichmüller Theory and the Statistics of Shapes What I talk about when I talk about Geometric Topology Yitzchak Elchanan (Isaac) Solomon MathSLAM 2016 Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

2 Inspiration Out of nothing I have created a strange new universe. (A reference to the creation of a non-euclidean geometry.) János Bolyai (1802 to 1860) Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

3 Inspiration Out of nothing I have created a strange new universe. (A reference to the creation of a non-euclidean geometry.) János Bolyai (1802 to 1860) For God s sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life. (A letter to his son János urging him to give up work on non-euclidean geometry.) Farkas Bolyai (1775 to 1856) Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

4 Describing Shapes The world is full of shapes that are not triangles, boxes, or any classical Euclidean construction. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

5 Describing Shapes The world is full of shapes that are not triangles, boxes, or any classical Euclidean construction. Examples range from physics and computer science to biology and medicine. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

6 Describing Shapes The world is full of shapes that are not triangles, boxes, or any classical Euclidean construction. Examples range from physics and computer science to biology and medicine. We need ways to describe, compare, and compute with these objects. Where would we be without trigonometry, which tells us which measurements determine a triangle (e.g. SSS), and how to compute unknown lengths and angles from known ones? Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

7 Narrowing our focus Of course, we can t set about describing any shape. It s easy to draw something arbitrarily wild and messy. Also, we will focus our attention here to two-dimensional shapes: surfaces. Two surfaces are called conformal if there is an angle-preserving transformation from one to the other. We can study conformal classes of surfaces, so that we are really extracting and studying the angle information. Definition Two metrics g and h on a surface S are called conformal if g = λ 2 h, for some smooth function λ : S R. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

8 Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

9 The Teichmüller Space At this point, we ve focused in on trying to understand the various conformal structures one can place on a surface. Actually, since we re trying to compare surfaces, we also need to record the possible functions from one surface to another. Definition Fix a reference surface S. A point in the Teichmüller space T (S) will be a conformal structure X on the surface, together with a map f : S X. Two points (X, f ) and (Y, g) are considered equivalent if g f 1 : X Y can be deformed to be conformal. T (S) has a number of interesting metrics that I won t define here. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

10 A quick definition The genus g of a surface counts the number of holes it has. Below is a genus 3 surface. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

11 How to think about conformal classes? This seems like a more manageable task than arbitrary surfaces, but it is quite abstract, since we are now thinking about equivalence classes of shapes, and not a concrete geometric object. Theorem (Riemann Uniformization) Every conformal class of metrics on a surface S has a unique representative of constant curvature +1, 0 or 1. For g = 0 (the sphere), the curvature is +1. For g = 1 (the torus), the curvature is 0. For g 2, the curvature is 1. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

Constant curvature surfaces The remarkable Riemann Uniformization theorem tells us that every surface can be stretched in an angle-preserving way to have constant curvature.

12 Constant curvature surfaces The remarkable Riemann Uniformization theorem tells us that every surface can be stretched in an angle-preserving way to have constant curvature. Most of the time, it ends up being hyperbolic, which tells us that hyperbolic geometry is somehow quite flexible. To contrast that, hyperbolic geometry is also rather rigid. We will exploit this to coordinatize the Teichmüller space. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

13 Hyperbolic Pants A pair of pants is, topologically, a sphere with three holes cut out. Any hyperbolic metric on a pair of pants is entirely determined by the length of the three boundary curves. In fact, any triple of positive lengths is possible. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

14 Pants Decompositions Given any hyperbolic surface S of genus g 2, one can find a collection of 3g 3 curves cutting it into pairs of pants. If we know the lengths of these (3g 3) curves, we know the geometry of each pair of pants. The remaining question is to measure how they are glued together. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

15 Twisting To understand the way that the pants are glued together, we need to measure the relative twisting. Each twisting parameter gives a number in R. It might seem that there are actually only a circle s worth of twists, but integer twists (and the choice of pants decomposition) are related to the map g : S X. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

16 Fenchel-Nielsen Coordinates We see that we can associate to a point (X, f ) in Teichmüller space a value in R 3g 3 + R 3g 3. Actually, this is a homeomorphism, and these are called the Fenchel-Nielsen coordinates. A total of 6g 6 numbers can uniquely specify a conformal/hyperbolic structure on a surface! This gives us a precise, computational way of navigating the universe of these non-euclidean geometries. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

17 A curious meta-coincidence? The one-holed torus has a pants decomposition with a single curve, so its Fenchel-Nielsen coordinates live in R R +, the upper half-plane. This is also the Teichmüller space of (flat structures on) the torus. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

18 The Teichmüller Space of the Torus Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

19 The Teichmüller Space of the Torus Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

20 The Teichmüller Space of the Torus Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

So the geometry on the space of possible hyperbolic geometries on the once-holed torus is.

21 Meta-geometry The Teichmüller metric on the Teichmüller space of the torus (or the once-holed torus) is exactly the hyperbolic metric! So the geometry on the space of possible hyperbolic geometries on the once-holed torus is...hyperbolic. A meta-coincidence? Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

22 Concluding Thoughts There is much more to say about Teichmüller theory and geometric structures in two dimensions. My own research focuses on the three-dimensional case, which is closely related but with additional challenges. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

23 Concluding Thoughts There is much more to say about Teichmüller theory and geometric structures in two dimensions. My own research focuses on the three-dimensional case, which is closely related but with additional challenges. Still, what we have developed so far can be used to apply coordinates to surfaces for the purposes of analysis and statistics. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

24 Computing Teichmuller Shape Space, 2009, Jin, Zeng, Luo, Gu Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

25 Computing Teichmuller Shape Space, 2009, Jin, Zeng, Luo, Gu Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

26 Thanks! Thanks to the organizers, and to you, for listening! All pictures shamelessly stolen from the internet. Including the one above. Yitzchak Elchanan (Isaac) Solomon Teichmüller Theory and the Statistics of Shapes MathSLAM / 22

Systems of Orthogonal Circles and Poincarè Geometry, on the TI-92

Proceedings of the Third DERIVE/TI-92 Conference Systems of Orthogonal Circles and Poincarè Geometry, on the TI-92 Paul Beem Indiana University South Bend, IN pbeem@iusb.edu When we encounter hyperbolic