Omer Bobrowski: Topology of Random Geometric Complexes
A simplicial complex is a generalization of a graph, consisting of vertices, edges, triangle tetrahedral, and higher dimensional simplexes. A random geometric complex is a simplicial complex whose vertices are generated by a random point process in a metric space, and higher-order simplexes are added according to a set of rules that depend on the geometric configuration of the vertices. In this talk we will review recent advances in the study of the homology of random geometric complexes. Loosely speaking, homology is a topological-algebraic structure that contains information about cycles of various dimensions in the complex. We will discuss phase transitions related to the appearance and vanishing of homology, as well the limiting distributions for the Betti numbers of these complexes (the number of cycles in different dimensions).
Magnus Bakke Botnan: Generalized Interleavings
The theory of interleavings lies at the very core of the theoretical foundations of persistent homology. With rising interest in more generalized indexing categories (zigzag, commutative ladders, circle valued maps, etc. ) comes the desire to extend this theory. In the first part of the talk I will survey interleavings in ordinary persistent homology, and, in particular, the celebrated algebraic stability theorem. Then I will report on recent work with Michael Lesnick (Princeton U.) where we extend this to zigzag persistence. Lastly I will show how the notion of interleavings can be generalized to general posets by means of cosheaves. The last part is joint work with Justin Curry (Duke) and Elizabeth Munch (U Albany).
Bei Wang: Topological Thinking in Visualization
Large and complex data arise in many application domains, such as oceanography, combustion simulation, material science, nuclear engineering, astrophysics and brain imaging. However, their explosive growth in size and complexity is more than enough to exhaust our ability to apprehend them directly. Topological techniques which capture the “shape of data” have the potential to extract salient features and to provide robust descriptions of large and complex data. Such a versatile approach connects naturally with and provides infrastructures for data visualization. In this talk, I will discuss some of our recent efforts in understanding the shape of data with topological data analysis and visualization. I will give some examples of how complex forms of data, such as vectors, tensors, brain networks and astronomical data cubes, could be reimagined via topological thinking.
Primoz Skraba: An Approximate Nerve Theorem
The Nerve Theorem is an implicit tool in most application of topological data analysis relating the topological type of a suitably nice space with a combinatorial description of the space, namely, the nerve of a cover of that space. It is required that it is a good cover, that each element and intersection is contractible or at least acyclic. In this talk, I will describe a weaker condition we call an epsilon-acyclic cover. It encodes the idea that if a cover is almost a good cover, the persistent homology of a filtration computed on the nerve is a good approximation of the persistent homology of a filtration on the underlying space. The main application of this result is to reduce the computational burden for computing persistence by allowing the use of coarser representations of the space (e.g. smaller simplicial complexes). I will also describe how to obtain explicit error bounds from local computations.
Anthea Monod: Quantifying Cancer Images via the Smooth Euler Characteristic Transform Predicting Clinical Outcome in Glioblastoma Multiforme
In cancer research, images of tumors are an important source of information with limited computational utility as they are difficult to quantify in a comprehensive manner. In this paper, we propose a statistic, the smooth Euler characteristic transform (SECT), which quantifies the shapes of tumors based on topological summaries. The SECT allows us to represent shapes as a collection of vectors with little to no loss in information. As a result, we are now able to integrate shape information with other biomarkers to predict clinical outcomes with greater accuracy. We demonstrate the utility of the SECT by showing that the inclusion of topological features of brain tumor MRI images in standard statistical models greatly improves the predictive accuracy of disease-free survival for patients with glioblastoma multiforme (GBM). We provide empirical evidence that the utilization of topological summaries to control for the physical heterogeneity of tumors can provide an increased statistical power over molecular features, such as gene expression. This is joint work with Lorin Crawford, Andrew X. Chen, Sayan Mukherjee, and Raul Rabadan.