##### Topological Invariants for Enhanced Data Analysis

*This project is suppported by an NSF grant.*

The aim of the project is to create a collection of tools, together
with theoretical guarantees and algorithmic instruments, allowing one
to define and compute local topological structure of complex datasets.
In particular, we intend to develop mathematical tools for describing
local topological structures in data; develop methods for
understanding behavior of local topological invariants across scales;
study local topological invariants of random functions, which is an
important step towards quantifying local topological structure of pure
noise; develop robust, local topology based methods for capturing
transient behavior (e.g. phase transitions) in dynamical systems. An
additional goal is to employ the obtained results in several important
applied areas.

On the theoretical side, we have recently obtain a new result regarding
local homology recovery from fierly general stratified spaces. In
particular, we have proven that local homology groups of homology
stratified spaces that are neighborhood retracts can be correctly
computed at a quantifiable majority of points of a sufficiently dense
sample using the the same scales. The restrictions on the density of a
sample relax if the space is nicer, e.g. a Whitney stratified space
with positive weak feature size. This important result is being written
up as a research paper.

We have also shown that by employing discrete Morse cohomology
(developed by Forman) we can create a general framework for efficient
computation of persistent cohomology for any sequence of simplicial
complexes connected by simplicial maps. The basic idea is that one can
construct a simplicial complex efficiently representing the telescopic
union of the simplicial complexes in the sequence, and then define an
appropriate combinatorial vector field. This result, in a sense,
genralizes some previous works, and is also being written up as a
research paper.

On the application side, my students have been applying TDA techniques to
analyze the structure of various datasets, including the network of
autonomous systems on the Internet and the global climate data.

##### Analyzing the microbiome data from the Waimea ahupua'a

This project is a part of the Microbiome Initiative by the UH Manoa
Office of the Vice Chancellor for Academic Affairs. The goal is to study
the composition and dynamics of microbiome communities in the Waimea
Valley. I am a part of the group of mathematicians who try to help
biologists with this challenging task. We are still at the stage of
collecting data, the majority of which is expected to be measurements of
abundance of microorganisms at several sites in the valley. Our initial
approach to analyzing the data is to construct a similarity graph using one
class of objects (sites or spieces) and regard the other class as
functions on such a graph, employing a variety of techniques to study the
shape of the graph and the functions.

##### Predictive Agriculture through Data Integration

This is a collaborative effort with folks from the College of Tropical
Agriculter and Human Resources. The general goal of this project falls
within the well known problem of establishing a relashion between
organism's genotype, environment, and phenotype. In this particular case
the organisms of interest are corn plants, although the data analysis
component of the project should be extendible to other use cases. We are
currently trying to secure federal funding for this project.

##### Analysis of soil health and carbon sequestration data

This is a part of another collaborative effort with folks from the
College of Tropical Agriculter and Human Resources. The goal of this
project is to study how certain farm land practices affect soil health
and carbon sequestration over time. We are currently trying to secure
federal funding for this project.