Distance Geometry (DG) has a rich mathematical history, rooted in Heron's theorem for computing the area of a triangle from the lengths of its sides. DG was further developed in the 1800s and 1900s by Cayley, Maxwell, Menger, and Isaac Schoenberg, who gave, among other things, an algebraic proof of the equivalence between distance matrices and Gram matrices. The essence of Schoenberg's proof is now used to show the validity of the well-known Multidimensional Scaling technique.
DG today is a research area bridging mathematics and computer science with applicability to practical problems in a wide range of disciplines. In the majority of DG applications, we are given an incomplete list of distances between pairs of objects, and we seek positions in Rn realizing those distances. Classical applications of DG include such topics as protein conformation determination and sensor network localization, while emerging applications range from the study of molecular nanostructure to the adaptation of human movements in simulated environments. DG is also used in important data science applications such as compressed sensing, low rank matrix completion, and visualization of high-dimensional data.
Although the natural statement of a DG problem is as a constraint satisfaction problem, most solution methods are based on formulating a DG problem as an optimization problem. Depending on the instance at hand, either a continuous formulation as a semidefinite program or a combinatorial formulation might be preferred. Thus, DG applications reap benefits from progress in both the continuous and discrete domains.
This workshop will: 1) highlight important optimization challenges in distance geometry; 2) draw connections to closely related problems in graph rigidity, semidefinite programming, and matrix completion, among others; 3) investigate complementary continuous and discrete approaches to distance geometry, with the aim of developing new efficient hybrid methods; and 4) involve researchers who are applying DG to in a wide range of fields. While solution methods for DG problems are mainly developed by researchers in mathematics, computer science, and operations research, novel applications emerge from a myriad of fields, such as biology, chemistry, materials science, engineering, robotics, and data and information sciences.
The workshop will include two tutorial presentations to foster interdisciplinary engagement. One will be a general overview of DG that is mostly aimed at students and other newcomers, and the second will highlight on emerging applications of the distance geometry, such as, for example, nanostructure problems in materials science. This workshop builds on the 2016 DIMACS Workshop on Distance Geometry: Theory and Applications.