Topics in Discrete Geometry

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The search for densest packings of congruent balls has a long and fascinating history in mathematics. Starting with investigations of Kepler and Gauss and continued by many others, the area of research was finally systematically established by the extensive research work of Coxeter, Delone, Fejes Toth, Rogers and Zassenhaus; and as a result the field of Discrete Geometry was born. Thus, one can briefly describe discrete geometry as the study of discrete arrangements of geometric objects in Euclidean, as well as in non-Euclidean spaces. In recent years renewed interest was generated by quite a number of breakthrough results. As a result discrete geometry has very strong connections to a number of research areas in pure mathematics such as convexity (see in particular, theory of convex bodies and polytopes), combinatorics (see for example, geometric graphs), rigidity (see in particular, flexibility of discrete geometric structures), geometric analysis (see for example, geometry of normed spaces), computational geometry (see for example, computation of volume), geometric groups (see for example, symmetries of polytopes), non-Euclidean geometry (see for example, spherical geometry) as well as to some research areas in communication and information technologies (see in particular, spherical codes) and in crystallography (see in particular, geometric lattices). In addition, there is demand from engineering, biology, and computer science for more emphasis on problems that are discrete in nature and do not fit into the usual continuous models. The research topics proposed represent the above concrete connections. On the other hand, the proposed research program is a combination of fundamental problems of discrete geometry that are connected to important problems of convex and non-Euclidean geometry, including the geometric theory of normed spaces with particular aspects in analysis and combinatorics. The timing seems to be perfect for achieving outstanding advances by intensive scientific collaboration, as well as by training a highly selected group of undergraduate and graduate students and postdoctoral fellows. In addition, the Center for Computational and Discrete Geometry within the Department of Mathematics and Statistics at U of C, which has been established and is supported by my Canada Research Chair (Tier 1) program, together with my proposed research program, has the potential to open a window into what research in modern discrete geometry is really like.