{"id":244,"date":"2025-08-27T09:44:36","date_gmt":"2025-08-27T13:44:36","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/thomas-z\/?page_id=244"},"modified":"2026-03-19T09:01:29","modified_gmt":"2026-03-19T13:01:29","slug":"my-work","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/thomas-z\/my-work\/","title":{"rendered":"My Work"},"content":{"rendered":"\r\n
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My Work<\/h1>\n\n\n\n

\"\"<\/a>My thesis work, “A Schauder Basis for Multiparameter Persistence”<\/a> (joint with\u00a0 Peter Bubenik<\/a>), develops a new method for mapping signed multiparameter persistence barcodes into Banach or Hilbert space. The main idea is to view these diagrams as elements of the dual space of compactly supported Lipschitz functionals<\/b> on a polyhedral pair \\((X,A)\\), where \\(X \\) is a polyhedron in Euclidean space, and \\(A\\) is a subspace representing ephemeral<\/em> features. This duality is illustrated in recent work by Peter Bubenik and Alex Elchesen, 2024<\/a>. An example of such a pairing is the space containing persistence diagrams from 1-parameter persistence, \\((\\mathbb{R}^2_\\leq, \\Delta)\\), where \\(\\mathbb{R}^2_\\leq = \\{(x,y) \\in \\mathbb{R}^2 \\ | \\ x \\leq y\\}\\) <\/span>and \\( \\Delta = \\{(x,x) \\ | \\ x \\in \\mathbb{R}\\). We construct a <\/span>Schauder basis<\/b> for the normed vector space of compactly-supported, Lipschitz functionals on \\(X\\) that are \\(0\\) on \\(A\\). We begin by g<\/span>enerating iteratively refined triangulations of the pair \\((X,A)\\). These triangulations may be chosen arbitrarily, or by taking into account the mass of a specified collection of signed barcodes.<\/span><\/p>\n\n\n\n\n\n

\"\"<\/a><\/strong><\/p>\n\n\n\n\n\n\n\n\n\n\n\n

Example<\/strong>:\\(X= \\mathbb{R}^4_\\leq = \\{(x_1,y_1, x_2 ,y_2) \\ | \\ x_1 \\leq y_1 \\ \\& \\ x_2 \\leq y_2)\\}\\) and \\(A = (\\mathbb{R}^2_\\leq \\times \\Delta) \\cup (\\Delta \\times \\mathbb{R}^2_\\leq)\\).This space contains an embedding of signed rectangle barcodes on 2-parameter persistence modules such as those byBotnan, Opperman, & Oudot, 2024<\/a>. <\/strong>In this case, the set \\(A\\) corresponds to features lived instantaneously in either of the two parameters (flat rectangles).<\/p>\n\n\n\n\n\n\n\n\n\n\n\n


\n\"\"<\/a>We may then define a list of piece-wise linear functionals on the pair \\((X,A)\\) by the nested sequence of triangulations. The collection we define of these functionals form a Schauder Basis of the vector space of compactly supported Lipschitz functionals on the pair \\((X,A)\\). That is, every compactly supported Lipschitz functional on \\((X,A)\\) can be approximated to arbitrary accuracy by a finite sum of scalar multiples of Schauder basis functionals.<\/p>\n\n\n\n\n\n

Evaluation of a persistence diagram against these basis elements produces a sequence of real numbers in \\(\\ell_1\\),\u00a0 which serves as our vector representation of the persistence diagram.<\/span><\/p>\n\n\n\n\n\n

We prove:<\/span><\/p>\n\n\n\n

  1. Injectivity:<\/b>\u00a0 the vectorization uniquely determines the persistence diagram<\/span><\/li>\n
  2. Lipschitz Stability:<\/b> the \\(\\ell_1\\)-distance between vectorizations is bounded above by a constant times the 1-Wasserstein distance between diagrams<\/span><\/li>\n
  3. Minimality:<\/b> in the broader setting of relative Radon measures on the pair (X,A), the constructed basis is minimal among all template systems that produce an embedding.<\/span><\/li><\/ol>\n\n\n\n

    \"\"<\/a>This work connects the algebraic nature of multiparameter persistence modules with analytic tools from functional analysis. Our vectorization method is general enough to handle:<\/span><\/p>\n\n\n\n