# Overview

Brittany Terese Fasy is a researcher in topological data analysis. She started a research group TDA at MSU in Fall 2015. The group meets weekly to discuss fundamental topics, recent research papers, and current research of group members. More information on specific research projects of Dr. Fasy and her collaborators can be found by navigating throughout this page, and in her formal research statement.

# Projects

## Quantifying Morphologic Phenotypes in Prostate Cancer - Developing Topological Descriptors for Machine Learning Algorithms (QuBBD)

The long-term goal of this project is to develop quantitative methodology for detecting geometric and topological features in point clouds extracted from (histology) images. Of particular relevance, this project considers the setting of prostate cancer classification, which is based on a pathologist grading of histology slides using the Gleason grading system. These pathology slides are a source of biomedical big data that are increasingly available as archived material. Developing these quantitative methods will be a significant advance towards a (semi-)automated quantification of prostate cancer aggressiveness. This award supports an interdisciplinary team of investigators in computational mathematics, computer science, biomedical engineering, and pathology to develop mathematical and computational tools based on topological descriptors and machine learning in order to distinguish between different morphological types of prostate cancer.

## Computer Science through Storytelling

Our team is developing and researching culturally responsive curriculum and teacher development that engage American Indian and rural Montana students in learning computer science and computing skills. Instead of creating a new stand-alone curriculum (and new standards for teachers to meet), the project infuses computer science across the grades 4-8 curriculum, which helps students understand that computing skills are relevant across disciplines and are important for a wide variety of professions in the work-force. Through a research practice partnership, this project is working directly with the Montana Office of Public Instruction, tribal entities, teachers, and other stakeholders to develop these culturally responsive resources, which will be aligned with the new Computer Science state content area standards and with Montana’s Indian Education for All curriculum.

## Map Construction and Comparison

The project aims to develop theoretically grounded, effective methods for analyzing data associated with road networks – using graphs that represent road networks as a framework for analyzing network data. Thanks to the spread of GPS-enabled devices, trajectory data has become ubiquitous. Many other sources, including census data and crime statistics, have addresses or geographic locations that link to an underlying road network. Algorithms with mathematical guarantees will be developed to align trajectories to the network under natural and realistic properties of true trajectories, to reconstruct road networks from trajectory and density data. It will also provide two frameworks for comparing data-endowed networks at different levels. While the problems of trajectory alignment, map reconstruction, and map comparison have attracted a lot of attention in the GIS community, most approaches are ad-hoc, provide no quality guarantees, and are limited to post-hoc analysis. This project will provide novel theoretical foundations combining approaches from computational topology and geometry, and will further advance the state-of-the-art of the field of topological / geometric data analysis.

## TopoStat

Statistical Approaches to Topological Data Analysis that Address Questions in Complex Data

## Other Projects

#### Modes of Gaussian Mixtures

To many researcher’s surprise, n+1 isotropic Gaussian kernels in R^n can have n+2 modes. I call these additional modes “ghost modes.” More generally, there exist finite configurations of isotropic Gaussian kernels with superlinearly many modes. Moreover, even the most simple configuration exhibits an exponential number of critical points.

#### Curves in Space

Researchers across many fields are interested in the topological and geometric properties of shapes in Euclidean space. In a 2011 Journal paper, I bound the difference of lengths of curves in Euclidean space by a function of the total curvature and the Fréchet distance between the curves.

#### Heat Equation Homotopy

Persistent homology is a tool used to classify topological features that are present in data sets and functions. I am interested in using persistence to explore the deep structure, or scale space, of images. The scale space of an image is a family of related images obtained through convolution with the Gaussian kernel. Viewing each image as a real-valued function, I stack the corresponding persistence diagrams. This creates a vineyard of curves that connect the points in the diagrams. I am interested in using the vineyard arising from the heat equation homotopy to define a distance between two images.

#### Computer Science Education

I created virtual worlds using a 3D interactive animation environment, Alice. Alice had previously been used as a program visualization tool for introductory computer science classes. The worlds I created enable use of this tool at the intermediate programming level by introducing the concepts of lists and arrays. This research has provoked interest in expanding the use of Alice to higher level CS courses.

#### Homotopy Classification Problems

A basic problem in mathematics is the classification problem. In group theory we have the Classification Problem for Groups: Given a collection G of groups, classify the groups G in G up to isomorphism. And in homotopy theory, we have the Homotopy Classification Problem for Spaces: Given a collection T of topological spaces, classify the spaces X in T up to homotopy equivalence. In some cases, these classification problems are actually equivalent. I am interested in understanding these cases. In particular, I look at examples classifying groups by the centralizers and classifying the path components of function spaces.