Organizers: Henry Adams and Ziqin Feng
The "ham sandwich" theorem states that any $n$ finite Borel measures on $\mathbb{R}^{n}$ can be simultaneously bisected by a single hyperplane, provided each measure is absolutely continuous with respect to Lebesgue measure. In 1984, Cox & McKelvey showed that even for discontinuous measures, there exists a single hyperplane such that at most half of each measure lies on each side. In this talk, we consider the problem of minimizing the differences of the measures of the two open half-spaces determined by a chosen hyperplane, where the measures may be discontinuous. We show that if the dimension of $\mathbb{R}^{n}$ is much larger than the number of measures, then there exists a hyperplane that divides the measures more fairly than in Cox & McKelvey's result.
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Bestvina-Brady discrete Morse theory is a topological tool that has historically been most useful in geometric group theory. In this talk I will discuss a version of Bestvina-Brady Morse theory that is particularly conducive to understanding topological properties of Vietoris-Rips complexes of metric spaces, and has applications not only to geometric group theory, but also to applied topology and topological data analysis. In particular I will discuss a recent short proof of a result of Virk, that says the metric space $\mathbb{Z}^n$ with the usual $L^1$ metric has contractible Vietoris-Rips complexes.
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We introduce the notion of a discrete approximate circle bundle, as well as theory and algorithms to estimate characteristic classes. We apply these tools to study a benchmark optical flow dataset, where we confirm the toroidal model proposed by Adams et al. and discover larger spaces in other density regimes.
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One of the most common distances used to compare two persistence diagrams is the bottleneck distance. When the persistence diagrams of a shape in $\mathbb{R}^d$ are computed from every direction in $\mathbb{S}^{d-1}$, we obtain the persistent homology transform (PHT). An efficient way of comparing two PHTs remains unexplored. In this work, we develop a new kinetic data structure to compute the bottleneck distance between two PHTs obtained from shapes in $\mathbb{R}^2$ from every direction. We provide the events and necessary updates to maintain the distance between the diagrams using this structure. Our resulting algorithm runs in $O(n^2\log^2n)$. This is compared to the naive algorithm where $d_B$ is computed at a finite number of smartly chosen directions, which is $O(n^{7/2}\log n)$ complex. It is important to note that our algorithm provides an exact distance in every direction, while the latter is an approximation. Furthermore, we show that this data structure is not limited to the directional transform setting since the techniques apply to more general vineyard structures.
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Deciding whether or not two curves are congruent under rotations and translations is a classical, but surprisingly subtle problem. In addition to its theoretical interest, this problem has numerous applications in computer vision and image processing, automated assembly, signal processing, and more. To address this, as well as more general congruence problems, the signature curve parameterized by differential invariants was introduced by Calabi, Olver, Shakiban, Tannenbaum, and Haker (1998). While congruent curves have identical signatures, the converse is not true, as shown in Muso and Nicolodi (2009). In a joint work with Eric Geiger (2021), we presented a mechanism for constructing non-congruent, non-degenerate curves with identical signatures. We also introduced a notion of the signature quiver and used it to formulate a congruence criterion for non-degenerate curves with non-simple signatures.
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Suppose one wants to monitor a domain with sensors, each sensing a small ball-shaped region, but the domain is hazardous enough that one cannot control the placement of the sensors. A prohibitively large number of randomly placed sensors could be required to obtain static coverage. Instead, one can use fewer sensors by providing mobile coverage, a generalization of the static setup wherein every possible evader is detected by the moving sensors in a bounded amount of time. Here, we use topology in order to implement algorithms certifying mobile coverage that use only local data to solve the global problem. Our algorithms do not require knowledge of the sensors' locations, only their connectivity information. We experimentally study the statistics of mobile coverage in two dynamical scenarios. We allow the sensors to move independently (billiard dynamics and Brownian motion), or to locally coordinate their dynamics (collective animal motion models). Our detailed simulations show, for example, that collective motion can enhance performance: The expected time until the mobile sensor network achieves mobile coverage is lower for the D'Orsogna collective motion model than for the billiard motion model. Further, we show that even when the probability of static coverage is low, all possible evaders can nevertheless be detected relatively quickly by the mobile sensor network.
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The Gromov-Hausdorff distance, a dissimilarity measure between metric spaces, is used in computational topology and geometry to compare datasets that can be represented as metric spaces. Despite the computational obstructions to its practical use, it still provides a theoretical framework to quantify invariants' stability and information loss. In this talk, we focus on a particular problem regarding the Gromov-Hausdorff distance: Given an object and a sample of it, under what conditions do their Hausdorff and Gromov-Hausdorff distances coincide? As the Gromov-Hausdorff distance describes how far they are from being isometric, and the Hausdorff distance measures the density of the sample, we can less formally restate the question as follows: When is a sample dense enough to describe the original object’s geometry faithfully? In particular, we discuss the case of metric graphs providing both negative and positive results.
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Given a metric space $X$, the Vietoris-Rips complex VR$(X)$ is a classical simplicial complex obtained from $X$, and a group $G$ acting properly by isometries yields another metric space $X/G$ of the orbits of $X$ under $G$. There is a canonical way in which $G$ can act on VR$(X)$, so instead using the Vietoris-Rips metric thickening VR$^m(X)$ allows a meaningful comparison between VR$^m(X)/G$ and VR$^m(X/G)$ as metric spaces. This talk will survey a variety of properties which a group action on a metric space can have with some examples, and culminate with a discussion of the strong $r$-diameter action, which guarantees that under certain scale parameters VR$^m(X)/G\simeq$ VR$^m(X/G)$. I also discuss a strictly weaker condition and present some open questions concerning the connection between the two. Finally, I will briefly mention some analogous results for the Cech metric thickening.
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We will discuss the topological properties of the independence complex of Kneser graphs, Ind(KG$(n, k))$, with $n\geq 3$ and $k\geq 1$. By identifying one kind of maximal simplices through projective planes, we obtain homology generators for the $6$-dimensional homology of the complex Ind(KG$(3, k))$. Using cross-polytopal generators, we provide lower bounds for the rank of $p$-dimensional homology of the complex Ind(KG$(n, k))$ where $p=1/2\cdot {2n+k\choose 2n}$.
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When studying mechanisms of DNA repair, short mutations often arise at the repair site, frequently manifesting as the insertion of short nucleotide sequences from the alphabet {A, C, G, T}. Each of these insertions occurs across millions of DNA molecules, generating a set of short words with varying frequencies. Our goal is to identify a suitable mathematical object to analyze these word sets and distinguish patterns across different experimental conditions. In this talk, we introduce the Insertion Chain Complex, a higher-dimensional generalization of insertion graphs, where homology serves as a measure of the complexity of a set of words. We present its construction, fundamental properties, and applications to biological data. In our case study, we analyze data from human cells in which DNA breaks were induced and the repaired sequences were sequenced. Our findings demonstrate that counting the highest-dimensional cells in these insertion complexes effectively distinguishes between different break locations.
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There is a lot to be gained by using topological data analysis (TDA) in conjunction with domain knowledge. As an example, consider the task where one wants to cluster brain states based on the underlying neural activity. Electroencephalograms (EEGs) are a common tool used to investigate neural activity by detecting electrical signals through sensors affixed to the scalp. EEG is relatively easy to use and provides high temporal resolution. However, it has low spatial resolution and prone to contamination with artifacts of movement or signals from external sources. Thus, for tasks like clustering brain states, it is difficult to capture the underlying structure and connectivity of individual states from EEG data. TDA, specifically the Mapper algorithm, has been used successfully in these types of problem spaces to pull important and relevant information from datasets. But, when applied directly to EEG data, Mapper does not reveal any structure or information. Luckily, there is a plethora of research and tools that have been developed to process and examine EEG data. Specifically, researchers have found that looking at the signal in the frequency domain can often provide insight into the neural activity. As such, we use the power spectral density paired with Mapper to create MapperEEG (MEEG). MEEG is neuroscience-infused topological tool that can cluster brain states without any pre-labeling or prior knowledge. In this talk, we will illustrate the importance of using prior domain knowledge within the EEG context, introduce the MEEG algorithm as an example of combining domain knowledge and TDA, and demonstrate its use on clustering brain state during a teaming task.
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Vietoris-Rips complexes play an important role in geometric topology, geometric group theory, and topological data analysis. For a given scale parameter $r>0$ and a metric space $X$, a Vietoris-Rips complex, $\mathrm{VR(}X,r)$, is defined as a simplicial complex with the vertex set $X$, and so that the simplices are finite collections of points from $X$ of diameter $< r$. One of the main difficulties in working with Vietoris-Rips complexes is that $\mathrm{VR}(X,r)$ is not metrizable unless $X$ is discrete. Moreover, the space $X$, in general, cannot be viewed as naturally embedded in $\mathrm{VR} (X,r)$. To remedy these problems, one can consider so-called metric thickening $\mathrm{VR}^m(X,r)$ of $\mathrm{VR}(X,r).$ To this end, we can view $\mathrm{VR}(X,r)$ as a set of all finitely supported measures with diameter of support $ < r$, and endow it with the Wasserstein metric. The main focus of this talk will be on the relation between the homotopy types of $\mathrm{VR} (X,r)$ and $\mathrm{VR}^m(X,r).$ A recent result by Gillespie implies that $\mathrm{VR}(x,r)$ and $\mathrm{VR}^m(X,r)$ are weekly homotopy equivalent. Therefore, to conclude that they are homotopy equivalent it is sufficient to show that $\mathrm{VR}^m(X,r)$ is an ANR. It has been previously demonstrated by Adams, Frick, and Virk that $\mathrm{VR}^m(X,r)$ is locally contractible. Using different method, we prove that $\mathrm{VR}^m(X,r)$ is strongly locally contractible for a compact metric space $X.$ We also show that if such X is finite-dimensional then $\mathrm{VR}^m(X,r)$ is an ANR. (Note: this is a joint work with Henry Adams and Ziga Virk).
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Recent work by Wayland, Coupette, and Rieck (2024) proposes a method to characterize and compare the latent embedding spaces arising from machine learning models. Their method is based on persistent homology and allows variability and sensitivity analysis of various hyperparameter choices for these models. Inspired by this idea but focusing on the case of classification problems, we would like to develop tools for a similar analysis. In this talk, we define a variant of multiparameter persistence landscapes, which can be seen as a generalization of the definition in the recent work by Vipond (2020). For practical applications, we are interested in the landscapes that are defined over a poset that is a product of $\mathbb R$ and a subposet of an inclusion poset. We discuss the properties of this definition, the theoretical challenges, and future directions.
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We establish the foundations of the theory of persistent cohomology operations, derive decomposition formulas for wedge sums and products, and prove their Gromov–Hausdorff stability. We use these results to construct pairs of Riemannian pseudomanifolds for which the Gromov-Hausdorff estimates derived from persistent cohomology operations are strictly sharper than those obtained using persistent homology. This work is joint with Anibal M. Medina-Mardones.
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In this talk I will discuss a new model for random simplicial complexes. Unlike other models, which are generically simply-connected when sufficiently dense, the complexes in this model generically have fundamental group $\mathbb{Z}/2\mathbb{Z}$. I'll describe results on the asymptotic behavior of the homology and homotopy groups in these complexes as well as how these results imply a "random Borsuk--Ulam theorem". This talk is based on joint work with Florian Frick.
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In geometric and topological reconstruction of compact length spaces embedded in some metric space, one needs an appropriate notion of distortion of the embedding. We consider variants of the classical notion of distortion, by controlling the coarseness of the distance scale of the ambient space and the discreteness of the coarse paths used to generate the length structure. In addition to discussing the stability and convergence of these notions of distortion, we compare them with existing notions of sampling parameters used in shape reconstruction and show some applications of our approach. This is based on joint work with Rafal Komendarczyk and Sushovan Majhi.
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Although Vietoris--Rips complexes are frequently used in topological data analysis to approximate the “shape” of a dataset, their theoretical properties are not fully understood. In the case of the circle, these complexes exhibit a surprising progression of homotopy types (from $S^1$ to $S^3$ to $S^5$, etc.) as the scale increases. However, much less is known about the Vietoris--Rips complexes of higher-dimensional spheres. I will present work that explores Vietoris--Rips complexes of the $n$-sphere $S^n$ and shows how the appearance of nontrivial homotopy groups of $\mathrm{VR}(S^n; t)$ can be controlled by covering properties of $S^n$ and real projective space $\mathbb{R}P^n$. Specifically, if the first nontrivial homotopy group of $\mathrm{VR}(S^n; \pi-t)$ occurs in dimension $k$, then $S^n$ can be covered by $2k+2$ balls of radius $t$, but there is no covering of $\mathbb{R}P^n$ by $k$ balls of radius $t/2$. This is joint work with Henry Adams and Žiga Virk.
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Topological data analysis is naturally suited to “data with shape”. In this talk, I will use a recent joint project with Jakini Auset Kauba as a demonstration of how TDA can uncover shape in geospatial data. In our project, we looked at persistence diagrams given by the demographics of 100 U.S. cities, and used them to perform various investigations and comparisons. Towards the end of the talk, I will highlight some of the pitfalls of using persistent homology on this kind of data, and pitch some challenges for those interested in TDA and geospatial data.
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We address the problem of homotopy-type reconstruction of compact shapes $X\subset\mathbb{R}^N$ that are $\operatorname{CAT}(\kappa)$ in the intrinsic length metric. The reconstructed spaces take the form of Vietoris–Rips complexes, computed from a compact sample $S$ that is Hausdorff-close to the unknown shape $X$. Instead of employing the Euclidean metric on the sample, our reconstruction technique utilizes a path-based metric to compute these complexes. Naturally emerging in the reconstruction framework, we also explore the Gromov–Hausdorff topological stability and the finiteness problem for general compact $\operatorname{CAT}(\kappa)$ spaces. Our techniques offer novel sampling conditions as alternatives to the existing and commonly used methods based on the weak feature size and $\mu$-reach.
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We discuss the problem of homotopy-type reconstruction of compact shapes $X\subset\mathbb{R}^N$ that are $\mathrm{CAT}(\kappa)$ in the intrinsic length metric. The reconstructed spaces are Vietoris–Rips complexes computed from a compact sample $S$, Hausdorff–close to the unknown shape $X$. Instead of the Euclidean metric on the sample, our reconstruction technique leverages a path-based metric to compute these complexes. As naturally emerging in the reconstruction framework, we also study the Gromov–Hausdorff topological stability and finiteness problem for general compact $\mathrm{CAT}(\kappa)$ spaces. Our techniques provide novel sampling conditions as an alternative to the existing and commonly used techniques using weak feature size and $\mu$–reach. In particular, we introduce a new parameter, called the restricted distortion, which is a generalization of the well-known global distortion of embedding. We show examples of Euclidean subspaces, for which the known parameters such as the reach, $\mu$–reach and weak features size vanish, whereas the restricted distortion is finite, making our reconstruction results applicable for such spaces.
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A plane set X admits an inscribed polygon P, if every vertex of a polygon similar to P lies in X. It is still not known whether every Jordan curve admits an inscribed square. In 1977 H.Vaughan proved that every homeomorphic copy of $S^1$ in $ \mathbb{R}^2$ admits at least one inscribed rectangle. In this talk, we present an algorithm implemented in Python that helps us visualize Vaughan's function, and we classify locally connected plane continua that inscribe rectangles.
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Topological Data Analysis (TDA) is an emerging field that aims to extract the shape and structure of the data. The key idea here is to build a higher-dimensional graph by connecting more than two nearby data points- resulting in simplicial complexes. There are different ways to build a simplicial complex on a metric space including Vietoris-Rips Complexes, Čech complexes. In this talk, we specifically examine Čech complexes constructed from the finite union of finite metric spaces at scales 2 and 3, using the symmetric difference metric. Our primary focus is on determining the homotopy types of these Čech complexes. Using these homotopy types, we also derived a precise formula for the homotpy types of Čech complex of a hypercube graph. This is a joint work by me and my PhD advisor Dr. Ziqin Feng.
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