Overview

Homology is the notion from the mathematical field of algebraic topology that characterizes how a space is connected. This notion, while not as strict as some other similar concepts of connectivity, is the most amenable to computation, and hence to applications in real life. While efficient algorithms have existed for some time to characterize the homology of a data set (or a space in general) by identifying a member in each homology class, one often wants to identify an optimal member in the class (and not just any member). The need to find optimal homology chains is motivated both from applications in science and engineering, and from more fundamental questions that arise in other related fields of mathematics such as geometric analysis. The PI and his team made many mathematical and computational advances in the understanding of how to efficiently solve such optimal homology problems.

On the more directly applied side of this Project, the PI collaborated with orthopedic surgeons and biochemists to identify structure in and gain insights from biomedical data sets using techniques from topology, optimization, machine learning, and statistics.

Overall, work supported by this Project resulted in eleven research publications.


Intellectual Merit

The obstruction to efficient solvability of many optimal homology problems is the presence of "Mobius strips" in the simplicial complex. These structures make the boundary matrix of the complex not totally unimodular (TU), due to which polynomial time solvability of the problem is not guaranteed. In this context, the PI and his PhD student have characterized a highly interesting class of spaces termed non-TU neutralized complexes, on which efficient solvability of the homology problem is guaranteed even in the presence of Mobius strips! The figure illustrates one such instance.

The PI and team have provided mathematical characterizations of the effects of edge contractions on homology of a simplicial complex. Edge contractions are standard operations used to simplify computations on meshes - in both theory and practice. Hence it is natural to consider the effects of such operations on homology computations. The PI and team have proposed sufficient conditions that are local for the edge in question, and guarantee the polynomial time solution of several classes of optimal homology problems subsequent to the edge contraction.

The flat norm is a widely studied measure of distance between currents, or generalized surfaces, in geometric measure theory (GMT), with applications to many theoretical questions (e.g., area minimizing surfaces) as well as practical applications (e.g., image denoising). A fundamental question in this context is to determine when a current with integer multiplicities is guaranteed to have an integral flat norm decomposition. The PI and coworkers posed this question in the finite simplicial setting, for which the answer is yes in codimension 1 (e.g., 1-current in 2D space). They developed an analysis framework to transfer this result to the continuous infinite setting. This result answered a question that was open in GMT for many years, using techniques that are quite different from ones that are typically employed in GMT. The PI is also using the framework of currents to study average of shapes. A new median shape of input shapes defined as currents has been defined, which can be efficiently computed by solving a linear program.

The PI used topological techniques to identify new groups of 6-12 genes that could potentially have implications in many cancers including brain, breast, and leukemia. Analysis of surgical datasets have identified factors that are critical to predicting the strength of bone allograft pieces. In particular, age, gender, and bone mineral density were not critical factors. The PI and collaborators have identified disparities based on race and gender in how surgical treatment of humerus (arm) fractures are administered.



Broader Impact

The PI's results on effects of edge contractions on simplicial homology could imply the polynomial time solution of several classes of optimal homology problems. The work on integral decomposition of currents and on median shapes has had a direct impact on the otherwise unrelated field of geometric measure theory.

Identification of groups of genes implicated in various cancers promises to have a huge impact in the effort to understand and treat the cancer. Discovery that donor age and bone mineral density are not critical to predict the strength of allograft bone could result in a marked increase of the supply of allograft bone.Identification of disparities in surgical treatment administration based on race and gender potentially impact procedures and policies followed, resulting in better outlook for patients, and even saving more lives in the long run.

Six PhD students and four undergraduate students were supported on research activities by this Project. Two of them are Hispanic, two are female, and one is physically disabled. Their work has motivated other students from underrepresented communities to take up research-oriented careers.

Last Modified: 10/31/2016
Modified by: Bala Krishnamoorthy