Amit Patel

Reeb spaces of piecewise linear mappings

Generalizing the concept of a Reeb graph, the Reeb space of a multivariate continuous mapping identifies points of the domain that belong to a common component of the preimage of a point in the range. We study the local and global structure of this space for generic, piecewise linear mappings on a combinatorial manifold.

Quantifying Transversality by Measuring the Robustness of Intersections

By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then we prove that the robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.

Categorified Reeb Graphs

The Reeb graph is a construction which originated in Morse theory to study a real-valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. Here, we exploit the fact that the category of Reeb graphs is equivalent to the category of a particular class of cosheaf. Using this equivalency, we can define an ‘interleaving’ distance between Reeb graphs which is stable under the perturbation of a function. Along the way, we obtain a natural construction for smoothing a Reeb graph to reduce its topological complexity. The smoothed Reeb graph can be constructed in polynomial time.

The Stability of the Apparent Contour of an Orientable 2-Manifold

The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f :𝕄→ℝ2, is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent. Assuming 𝕄 is compact and orientable and measuring difference with the erosion distance, we prove that the contour is stable.

The robustness of level sets

We define the robustness of a level set homology class of a function f : X → R as the magnitude of a perturbation necessary to kill the class. Casting this notion into a group theoretic framework, we compute the robustness for each class, using a connection to extended persistent homology. The special case X = R3 has ramifications in medical imaging and scientific visualization.

Generalized persistence diagrams

We generalize the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer to the setting of constructible persistence modules valued in a symmetric monoidal category. We call this the type

Efficacy and Safety of Ixabepilone and Capecitabine in Patients With Advanced Triple-negative Breast Cancer: a Pooled Analysis From Two Large Phase III, Randomized Clinical Trials

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Reeb spaces and the robustness of preimages

A persistence diagram of a persistence module. If the category is also abelian, then we define a second type