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UID:4daa1ffd2aae47792fe9f58bea74a6fe
CATEGORIES:Math and Data Seminar
CREATED:20250424T073754
SUMMARY:Cohomological Invariants in Topological Data Analysis: Persistent Cup-Length and Cup Modules
LOCATION:Hill 705
DESCRIPTION:<p style="margin: 0px; color: #222222; font-family: Arial, Helvetica, sans-
 serif; font-size: small; font-weight: 400; letter-spacing: normal; orphans:
  2; text-align: start; text-indent: 0px; text-transform: none; widows: 2; w
 ord-spacing: 0px; white-space: normal; background-color: #ffffff;">In topol
 ogical data analysis, cohomological ideas have recently been incorporated t
 o enhance computations of persistence diagrams in software like Ripser. The
  cup product endows cohomology with a graded ring structure that captures r
 icher topological information than the vector space structure alone. The cu
 p-length, defined as the maximum number of cocycles with nontrivial cup pro
 duct, serves as a useful invariant of the cohomology ring.</p><p style="mar
 gin: 0px; color: #222222; font-family: Arial, Helvetica, sans-serif; font-s
 ize: small; font-weight: 400; letter-spacing: normal; orphans: 2; text-alig
 n: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 
 0px; white-space: normal; background-color: #ffffff;">In this talk, we lift
  cup-length to a persistent invariant that tracks the evolution of the coho
 mology ring structure across a filtration. We provide a polynomial-time alg
 orithm for its computation from representative cocycles and prove its stabi
 lity under interleaving distances. To further organize the information enco
 ded by cup products, we introduce the persistent cup module, a two-paramete
 r persistence module indexed by filtration and cup-length, and show its Gro
 mov-Hausdorff stability. In addition, we compare the distinguishing power o
 f these invariants with persistent homology and with each other, highlighti
 ng cases where they better distinguish between metric spaces.</p>
CONTACT:Ling Zhou (Duke)
DTSTAMP:20260926T091846
DTSTART;TZID=America/New_York:20250508T103000
DTEND;TZID=America/New_York:20250508T234500
SEQUENCE:0
TRANSP:OPAQUE
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