Summer 2004 Tetrahedrons

Hello. My name is Daniel Epstein. This summer, I worked with Professor Paul Gunnells, studying the nonsingular compactification of the space of simplexes determined by sets of 5 points in P^4, the 4-dimensional complex projective space.

We let P in P ^4 be a set of 5 labeled points in the space, and by taking all possible linear spans of subsets of P, we obtain a configuration of flats in P^5 arranged to form a simplex. We’ll call X_0 the set of all such configurations, and X the singular compactification of X_0. We are now concerned with constructing X_tilda, created by modifying X_0 so that it is non-singular.

A given point in X_tilda represents a 4-dimensional, 5-pointed simplex, the faces of which we encode in a diagram which we will call S. The points in the first polytope in S represent the 3-dimensional faces of our simplex, those in the second polytope its 2-dimensional faces (planes), those in its third polytope its 1-dimensional faces (lines), and those in its final polytope its points. By giving a value to the edge between a given pair of points in one of the polytopes in S, we describe the distance between these two faces of our simplex, so that if we set an edge to zero, our simplex has degenerated in such a way that the two points connected by that edge (representing 3d, 2d, 1d faces, or points) have become the same. For instance, if a line in the second polytope is set to zero, then the two planes that the points connected by that edge represent have merged into one. Of course, this degeneration may have ramifications in the other polytopes, representing faces of different dimensions, and, in general, a number of equations determine the effects of a given collapsed edge on the edges of neighboring polytopes.

Now, after first choosing those points that we suspect will non-singularly compactify X into X_tilda, we can demonstrate that by examining a finite number of the points in X_tilda, and showing them to be non-singular, we have proven that all of X_tilda is non-singular.

To demonstrate the non-singularity of this finite set of points, we must gather additional information about the points, which we can encode into a diagram that we will call S#. S# is determined from the original polytopes in S, and consists of a copy of each 4-dimensional polytope from S, as well as a disjoint copy of every face of every 4-dimensional polytope of dimension greater than 1; this includes 30 3-dimensional faces (tetrahedra and octahedra) and 80 2-dimensional faces (triangles). So, a given point in our space, with its unique coloring of edges in S (where “coloring” an edge means collapsing together the two faces that the edge’s end-points represent), will determine a coloring of its S# diagram, which in turn can be utilized to demonstrate that this point is non-singular.

Generating these S# diagrams has been my work, this summer. I have used Java programs to implement various algorithms that Professor Gunnells and I have created. These algorithms were composed of two primary tools, the first of which was “Triangle Comparisons.” Triangle Comparisons consist of comparing sets of “related triangles” in the various polytopes in S# to eachother, and coloring edges according to the rules of these comparisons (just as one can imagine how merging two lines together may imply consequences for our simplex’s planes or points).

We call our second tool, “Subspace Analysis.” To understand this method, first imagine the analog of our space, X_tilda, for P^3, the 3-dimensional complex projective space. This space, then, is the set of 3-dimensional, 4-pointed simplexes, or tetrahedra. An identical computation to the one that Professor Gunnells and I worked on this summer was performed over the last several years by Professor Gunnells and several of his colleagues for this space of tetrahedra. I, too, made this computation in preparation for this summer’s research. Since this computation had already been completed, then, I knew all of the allowable possibilities for the corresponding S and S# diagrams for the space of tetrahedra, which consist of tetrahedra, octahedra, and triangles. Now, as it turns out, there are copies of this tetrahedra space inside the space of our 5-pointed simplexes; specifically, in each S# in our X_tilda, there are 10 tetrahedron subspaces, each of which must be colored in one of these previously determined allowable patterns. So, by examining one of these subspaces individually, it may be possible to determine which coloration pattern this subspace must eventually agree with, or at least which set of patterns are still potential possibilities.

In my programs, I created progressively more sophisticated algorithms (they have become more abstracted, recursive rather than iterative, etc.), each using some combination of these two tools, to generate S#. None of these algorithms have so far proved sufficient to determine the set of S#’s for our points in X_tilda, but have come increasingly closer to doing so, and have provided new and relevant information to Professor Gunnells, as they have evolved.

Included in this project description are sample diagrams of colorations for the 4-dimensional faces of S#, screen shots of the relationships between the classes of my Java programs for both the space of tetrahedra and the space of 5-pointed simplexes, as well as the code itself for each of these classes.

I’ve enjoyed, and learned much from, my research experience this summer, and I recommend UMass’s REU program to all students who have the opportunity to participate in it.