 | Meditation on Homotopy of Embedding Pictures ........ Anatoly Fomenko Animation .... Lialia & Igor Nikitin Music ............ Hwost & Auction Text ............... Velimir Hlebnikov Video ............. Klaus Gunter Rautenberg Direction ....... Stanislav Klimenko & Martin Gobel |
 | Greedy Computation of a Homotopy Basis for a Genus 2 Surface Several tools from topology are useful for mesh processing and computer graphics. These tools often operate on the 1-skeleton of a surface, i.e., the graph of edges embedded in the surface. A common task is to find a collection of edges called a cut graph - cutting along these paths turns the surface into a shape which can be flattened into the plane. This kind of flattening is necessary for texture mapping, remeshing, etc. One way to find a cut graph is to find a set of loops, no two of which are homologous, which cut the surface into a disk when removed. Intuitively, two loops on a surface are homologous if one can be deformed into the other while always keeping it entirely on the surface. For a closed orientable surface with genus g (i.e., a torus with g handles), there are 2g classes of homologically independent loops. A homology basis consists of one loop from each class. Not every homology basis is a cut graph: some homology bases either disconnect the surface or cut it into a punctured sphere. However, a homotopy basis will cut the surface into a disk. This video shows the greedy homotopy basis for each vertex of the mesh (the magenta square is the current basepoint). The end of the video illustrates the total length of the homotopy basis at each point: the brightness of a vertex corresponds to the total length of the corresponding basis. For more information see http://www.cs.caltech.edu/~keenan/project_topology.html |
 | The fundamental Group This video illustrated the construction of the fundamental Group of a topological space. Inside a topological space (symbolized by the brown box) we consider paths. Paths that are homotopic are considered equal (symbolized by the wiggeling of the paths). Now two paths that share a beginning and an endpoint can be joined to form a new path. This will become the addition law of the fundamental group. The trivial path that goes from one point to the same point without ever moving can be added to a path without changing it. This will become the neutral element of the fundamental group. If we have a path we can consider its direction (red circles moving). If we add the same path with opposite orientation to we obtain a path that is homotopic to the trivial path. This reverse path will become the inverse element in the fundamental group. To obtain a true group we chose a fixed basepoint and consider only homotopy classes of paths that start and end there (not shown). This way we can add any two elements of the group. This Video was produces for a topology seminar at the Leibniz Universitaet Hannover. http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1 |
 | Natural transformations 1 The definition of natural transformations. An analogy with homotopy. |
 | Null-homotopic Paths A path is called null homotopic, if it can be contracted to a point. This Video was produces for a topology seminar at the Leibniz Universitaet Hannover. http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1 |
 | The fundamental Group of the Torus is abelian This video illustrates the proof of the Theorem in the title. The proof goes like this: Consider a rectangle. Then the path going up the left side of the rectangle and then along the top is homeomorphic to the path going first along the bottom and then up the right side. Gluing the rectancle to make a torus, this shows that going first around through the hole and then along the outside is homeomorphic to going first along the outside and then through the hole. Since these two path generate the fundamental group of the torus this proves that this group is abelan. q.e.d. Remark: This is a very special property. Many topological spaces have nonabelian fundamental groups. This video was produces for a topology seminar at the Leibniz Universitaet Hannover. http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1 |
 | Boy's surface Two constructions of Boy's immersion of the real projective plane are presented. One follows out from the triple point, and the other is given in movie form |
 | Not null-homotopic Paths Paths are null-homotopic if they can be contracted to a point. The path in the clip is not null-homotopic, since it can not wiggle past the extra point. This Video was produces for a topology seminar at the Leibniz Universitaet Hannover. http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1 |
 | Poodle-i-dicity The Math Muse of Doug Ravenel For those of you who have only seen the professional, reserved side of Doug, we thought it might be a treat in honour of his 60th birthday to show a silly film exposing -- and clearly exaggerating -- some of his domestic mannerisms. While making brilliant discoveries in the field of mathematics, our dad has a funny habit of pacing, rubbing his hands together with enthusiasm, and talking to his poodles in strange and repetitive statements (oftentimes even when the dogs aren't there). We have put together this little dramatization of our dad at work. Anna and Heidi hatched the plan and developed the story, 'G' plays our dad, and Heidi plays the math muse who manages to break through all the doggy talk to give him some ideas he can work with. Thank you all again for coming to the conference and celebrating the fruitful careers of Steve Wilson and Doug Ravenel! -Anna, Heidi, and "G' |
 | Homological Algebra Homological Algebra by Algirdas Javtokas |
 | Cutting a Torus into a Disk Several tools from topology are useful for mesh processing. These tools often operate on the 1-skeleton of a surface, i.e., the graph of edges embedded in the surface. A common task is to find a collection of edges called a cut graph - cutting along these paths turns the surface into a shape which can be flattened into the plane. This kind of flattening is necessary for texture mapping, remeshing, etc. This video shows a cut graph (in yellow) on a two-handled torus, which is flattened into a disk. For more information see http://www.cs.caltech.edu/~keenan/project_topology.html |
 | The fundamental Theorem of Algebra This video illustrates a proof of the Fundamental Theorem of Algebra: "Every polynomial degree at least 1 with complex coefficients has a least one complex zero". Proof: Let P(z)= z^k + a_k-1 z^k-1 + ... + a_1 z + a_0 be any polynomial (in the video P has degree 3). Substituting z=r exp(2πt) we obtain for fixed r a continuous map from the unit circle to C. For the special polynomial z^k the image of this map winds k times around the origin. Using the deformation z^k+λ(P(z)-z^k) we can continuously deform the image of zk into the image of P(z). Now consider the map of the unit disk D to C given by P (green areas in the video). If we can prove that 0 lies inside the image of D, we have proven that P has a zero. To do this we increase the radius r. If r is bigger than |a_k-1| + ... + |a_0| the difference between the image curves of z^k (in the video identified with the circle of radius r around the origin) and P(z) becomes smaller than the radius r (again shown in green). By the Theorem of Rouché both images must then have the same positive winding number k around 0. It follows that 0 must be inside the area the image of D (otherwise the image curve would have winding number 0). q.e.d. This video was produced by students I.Kenig, D.Tiessen, A.Timm and V.Wittman of Hans-Christian's topology seminar 2004 at Leibniz Universität Hannover. This Video was produces for a topology seminar at the Leibniz Universitaet Hannover. http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1 |
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