09:00 ‐ 09:30 
Steve Oudot 
On Inverse Problems in TDA (slides)
Abstract: While the Topological Data Analysis pipeline is known to yield provably stable descriptors for data, its discrimination power has remained mostly unexplored to date from the theoretical point of view. Simple examples show that, in general, significantly different data sets can have the same persistence diagram. However, what happens when only small perturbations of the data are considered? Or when a single persistence diagram is replaced by a collection thereof? Gameiro et al. on the one hand, Turner et al. on the other hand, have opened the way for the study of these two questions. The common thread among their contributions is that they consider data equipped with an ambient metric. In this talk I will focus on intrinsic metric spaces, and as a starter, I will restrict the focus to a dense subset thereof, for which fairly precise injectivity statements can be made.
This is joint work with Elchanan Solomon (Brown University).

09:40 ‐ 10:10 
Ulrich Bauer 
Persistence Diagrams as Diagrams (slides)
Abstract: We explore the perspective of viewing persistence diagrams, or persistence barcodes, as diagrams in the categorical sense. Specifically, we consider functors indexed over the reals and taking values in the category of matchings, which has sets as objects and partial bijections as morphisms.
This yields a categorical structure on barcodes, turning the bottleneck distance into an interleaving distance, and allowing for a simple reformulation of the induced matching theorem, which has been used to prove the algebraic stability of persistence barcodes.
We will also discuss an explicit construction of a barcode for a pointwise finitedimensional persistence module that doesn’t require a decomposition into indecomposable interval summands, and that is actually functorial on monomorphisms of persistence modules (along with a dual construction, which is functorial on epimorphisms).
This is joint work with Michael Lesnick (Princeton).

10:20 ‐ 10:50 
Vin de Silva 
Operations on Reeb Graphs (slides)
Abstract: Interpreting Reeb graphs as cosheaves over the site of compact intervals over the real line, [dS, Munch, Patel] introduced a metric on the class of Reeb graphs together with a topological smoothing operation that, in the limit, converts a given Reeb graph to a twosided tree that combines the merge tree and the split tree of the graph. In the present paper, I will discuss a combinatorial interpretation of this smoothing operation, developed in collaboration with two Pomona College undergraduates, Dmitriy Smirnov and Song Yu.

10:50 ‐ 11:30 
Coffee 
11:30 ‐ 12:00 
Peter Bubenik 
Multiparameter Persistence and Generalized Morse Theory (slides, 3D image)
Abstract: We consider multiparameter persistence from the viewpoint of geometric and differential topology. Specifically, we show how multiparameter persistence modules arise from parametrized families of realvalued smooth functions on a compact manifold. Careful analysis of cobordisms arising in this construction allows us to reduce this persistence module to the representation of a quiver. We give examples in which this representation can be explicitly calculated.
This is joint work with Michael Catanzaro.

12:10 ‐ 12:40 
Wojtek Chacholski 
What is Persistence? (slides)
Abstract: Persistance has been about assigning meaning to indecomposable summands of certain modules. This ad hoc interpretation is in my opinion the key reason behind the lack of progress towards extending the notion of persistence to multidimensional situations. My aim is to present a new approach to persistence that does not require any decomposability. This novel approach provides new ways of interpreting persistence even in the one parameter case. My aim is describe this approach and illustrate how it can be used for shape recognition.

12:40 ‐ 15:00 
Lunch 
15:00 ‐ 15:30 
Facundo Memoli 
Stable Signatures for Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence (slides)
Abstract: When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. In a similar vein, studying the dynamics of social networks leads to the problem of characterizing groups/communities as they form and disperse throughout time.
Motivated by this, we study the problem of obtaining persistent homology based summaries of timedependent data. Given a finite dynamic graph (DG), we first construct a zigzag persistence module arising from linearizing the dynamic transitive graph naturally induced from the input DG. Based on standard results, we then obtain a persistence diagram or barcode from this zigzag persistence module. We prove that these barcodes are stable under perturbations in the input DG under a suitable distance between DGs that we identify.
More precisely, our stability theorem can be interpreted as providing a lower bound for the distance between DGs. Since it relies on barcodes, and their bottleneck distance, this lower bound can be computed in polynomial time from the DG inputs.
Along the way, we propose a summarization of dynamic graphs that captures their timedependent clustering features which we call formigrams. These setvalued functions generalize the notion of dendrogram, a prevalent tool for hierarchical clustering. In order to elucidate the relationship between our distance between two DGs and the bottleneck distance between their associated barcodes, we exploit recent advances in the stability of zigzag persistence due to Botnan and Lesnick, and to Bjerkevik.
This is joint work with Woojin Kim and Zane Smith.

15:40 ‐ 16:10 
Brittany Fasy 
LocalitySensitive Searching in the Space of Persistence Diagrams
Abstract: Persistence diagrams are important tools in the field of topological data analysis that describe the presence and magnitude of features in a filtered topological space. However, the current approach for comparing a persistence diagram to a set of other persistence diagrams is linear in the number of diagrams. In this paper, we apply concepts from localitysensitive hashing to support approximate nearest neighbor search in the space of persistence diagrams. Given a set \(\Gamma\) of \(n\) persistence diagrams, each bounded and with at most \(m\) points, we snapround the points of each diagram to points on a cubical lattice and produce a key for each possible snaprounding. Specifically, we fix a grid over each diagram at varying resolutions and consider the snaproundings of each diagram to the four nearest lattice points. Then, we propose a data structure that stores all snaproundings of each persistence diagram in \(\Gamma\) at each resolution. This data structure has size \(O(5^m tn)\) to account for the \(t\) lattice resolutions as well as snaproundings and the deletion of points with low persistence. To search for a persistence diagram, we compute a key for a query diagram by snapping each point to a lattice and deleting points of low persistence. Furthermore, as the lattice parameter decreases, searching our data structure yields a fourapproximation of the nearest diagram. This talk presents joint work with Xiaozhou He, Zhihui Liu, Samuel Micka, David L. Millman, and Binhai Zhu.

16:10 ‐ 16:30 
Coffee 
16:30 ‐ 17:00 
Javier Arsuaga 
Identification of Genetic Interactions Using Computational Homology (slides)
Abstract: Genomic technologies have revolutionized the field of genetics and provided new methodologies to identify thousands of genetic/molecular signals that are associated to specific phenotypes. In particular, the identification of genetic elements has greatly benefited from the introduction of Genome Wide Association Studies (GWAS), which allow for the testing of thousands of loci simultaneously. The mathematical approaches used in GWAS studies however fall short when interactions between genetic elements are of interest.
In this talk we present a methodology that associates a point cloud to each genotype and performs an association between the homology of the filtration of the point cloud and a predetermined phenotype. This topological encoding of the genotype maps some of the interactions between genetic elements to generators of homology groups hence greatly facilitating their identification.
We tested our method on copy number data from breast cancer patients. When stratifying patients according to the molecular subtypes we identified regions of coamplification in chromosome 17q in ERBB2 amplified patients, 9q in Luminal B patients, and 1q, 4q, 5p, 6p, 9p, 9q, 10q, 18q in basals. Some of these coamplifications have been previously reported and are associated to different breast cancer diseases.

17:10 ‐ 17:40 
Herbert Edelsbrunner 
Stochastic Geometry with Topological Flavor (slides)
Abstract: We study classical questions in stochastic geometry, such as the expected density of \(p\)simplices in the Delaunay mosaic of a Poisson point process in \(d\)dimensional Euclidean space. Using a discrete Morse theory approach, we distinguish between critical and noncritical of the radius function and determine their expected densities dependent on a radius threshold. We generalize the analytic results to weighted Delaunay mosaics and to order\(k\) Delaunay mosaics, and we present experimental result for wrap complexes and for weighted Voronoi tessellations.

19:00 
Dinner 