Abstract: We present a new topological data analysis tool for nonlinear dimensionality reduction. The method (TALLEM) utilizes decomposition and coordinates in a topologically nonlinear fashion. Treating the data analogously to a fiber bundle, local coordinates are aligned along common domains -- which defines a Čech cocycle -- and then assembled along the base space accordingly. The results are the TALLEM coordinates: a single coherent mapping which is globally consistent with the underlying topology of the data. In the presentation, we'll describe the TALLEM algorithm through its connection to cohomology and fiber bundles. This perspective allows us to better convey the larger context of TALLEM, discuss other kinds of topological assembly, and analyze the impact of imperfect alignment. We also include implementation for topologically nontrivial examples, showing that the method -- which utilizes approximate cocycles -- often works very precisely in practice.

Abstract: Persistent homology is an algebraic and computational tool from topological data analysis. This talk is meant to be a self-sufficient introduction to persistent homology. We will discuss the general algebraic framework, specific notions of persistence, and a couple of real-world applications.

Abstract: Multi-dimensional persistence theory is a natural generalization of one-dimensional persistence theory (for example, we can interpret a one-dimensional persistence module as a 'slice' of some two-dimensional persistence module) and it has wide application in data science. Unlike the one-dimensional case, finite-dimensional indecomposable multipersistence modules are not classifiable, so multidimensional persistence is still in an early stage of development and there are multiple perspectives. In this talk, we will discuss some basic notions of the multidimensional persistence theory, along with some examples. This is a self-contained talk and everyone is welcome to attend.

Abstract: We use Čech's closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses the discrete homology of metric spaces, the singular homology of topological spaces, and the homology of (directed) clique complexes, along with their respective homotopy theories. We obtain six homology and six homotopy theories of closure spaces, that satisfy analogues of classical theorems from algebraic topology. We show how metric spaces and more general structures such as weighted directed graphs produce filtered closure spaces. For filtered closure spaces, our homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance to filtered closure spaces and use it to prove that our persistence modules and their persistence diagrams are stable. We also extend the definitions Vietoris-Rips and Čech complexes to closure spaces and prove that their persistent homology is stable.

Abstract: 

Bifurcations in a dynamical system are drastic behavioral changes, thus being able to detect the parameter values for which these bifurcations occur is essential to understanding the system overall. We develop a one-step method to study and detect bifurcations using zigzag persistent homology. While standard persistent homology has been used in this setting, it usually requires analyzing a collection of persistence diagrams, which in turn drives up the computational cost. Using zigzag persistence, we can capture topological changes in the state space of the dynamical system in only one persistence diagram.

Abstract: Actin filaments are polymers that interact with motor proteins inside cells and play important roles in cell motility, shape, and development. Depending on its function, this dynamic network of interacting proteins reshapes and organizes in a variety of structures, including bundles, clusters, and contractile rings. Motivated by observations from the reproductive system of the roundworm C. elegans, we use an agent-based modeling framework to simulate interactions between actin filaments and motor proteins inside cells. We also develop tools based on topological data analysis to understand time-series data extracted from these filamentous network interactions. We use these tools to compare the filament organization resulting from motors with different properties. We are currently interested in gaining insights into myosin motor regulation and the resulting actin architectures during cell cycle progression. This work also raises questions about how to assess the significance of topological features in topological summaries such as persistence diagrams.

Abstract: Many natural phenomena are characterized by their periodic nature, including animal locomotion, biological processes, pendulums, etc. Part of understanding periodic process is being able to differentiate them from quasiperiodic occurrences. Many advances have been made to use topological data analysis to classy and understand quasiperiodic signals. Some of these methodologies make use of persistent 2-dimensional homology to obtain quasiperiodic scores that indicate the degree of periodicity or quiasiperiodicity of a signal. There is a significant computational disadvantage in this approach since it requires, the often expensive, computation of 2-dimensional persistent homology.

Our contribution in this area uses the algebraic structure of the cohomology ring to obtain classes in the 2-dimensional persistent diagram by only using classes in dimension 1, saving valuable computational time in this manner and obtaining more reliable quasiperiodicity scores. We develop an algorithm that allow us to effectively compute the cohomological death and birth of a persistent cup product expression. This allows us to define a quasiperiodic score that reliably separates periodic from quasiperiodic time series.