The PI will continue studying the face numbers of various classes of cell complexes. Specifically, she proposes to work on centrally symmetric polytopes, simplicial complexes and posets whose geometric realization is a manifold, and cubical complexes. As previous works suggest, this will require applying existing and developing new combinatorial, algebraic, topological, and possibly analytical techniques, thus strengthening connections between different parts of Mathematics.
Cell complexes provide a convenient way to build various shapes from smaller and simpler blocks, such as for example pyramids or cubes. A natural question then is what restrictions on the number of building blocks are imposed by specific shapes. Variations of this question (e.g., what is the minimal number of blocks required) have practical importance when one wants to represent/store such a structure in a computer.
During the three year grant period, the PI (jointly with her graduate students and collaborators) worked on a few outstanding problems related to the face numbers of triangulations of various topological spaces. Triangulations divide a complex space into little (e.g. triangular) pieces, providing an easy way to represent a given shape in a computer. Their face numbers count the size of the representation. One of the main questions is then how various properties of the shape affect its possible face numbers. The PI jointly with Ed Swartz of Cornell University proved in the affirmative several conjectures on the face numbers of triangulations of manifolds (shapes that look like a disk if you look at only a small piece of them; for instance, the surface of a bagel). Among them a more than 20-years old conjecture due to Gil Kalai. Polytopes are high-dimensional analogs of polygons (such as pyramids, cubes, etc.). They are called "centrally symmetric" if they look the same when you look at them from the right and the left (or top and bottom, etc). Jointly with Alexander Barvinok and Seung Jin Lee (both from the University of Michigan at Ann Arbor) the PI investigated how large can the face numbers of centrally symmetric polytopes be. One of the outcomes of this (still ongoing project) is an explicit construction of centrally symmetric polytopes with a record number of faces. Besides being of intrinsic interest, centrally symmetric polytopes with many faces appear in problems of sparse signal reconstruction which in turn have applications in such subjects as optimization, computer vision, medical imaging, and digital communications. During the last three years the PI graduated four PhD students, all of whom went on to prestigious post-docs. The PI has continued to serve as an editor-in-chief of the Journal of Algebraic Combinatorics. In the past two years, she also co-organized two international conferences related to geometric and topological combinatorics.
