BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Steve Oudot (INRIA Saclay - France.) DTSTART:20230210T160000Z DTEND:20230210T170000Z DTSTAMP:20260423T022915Z UID:GEOTOP-A/46 DESCRIPTION:Title: Signed rank decompositions for multi-parameter persistence: from Moebius inversion to relative homological algebra\nby Steve Oudot (INRIA Sacl ay - France.) as part of GEOTOP-A seminar\n\n\nAbstract\nA question that c omes up repeatedly in recent developments on\nmulti-parameter persistence is to define mathematically sound and\ncomputationally tractable notions o f approximation for multi-parameter\npersistence modules. As $\\mathbb{R}^ n$ is of wild representation type\, one\nseeks to approximate arbitrary (s ay\, finitely presentable) modules by\nmodules coming from some subcategor y that is easier to work with in\npractice. An obvious candidate subcatego ry is the one of\ninterval-decomposable modules\, whose summands are indic ator modules of\nintervals (i.e. convex\, connected subsets of $\\mathbb{R }^n$\, equipped\nwith the product order). Indeed\, interval-decomposable m odules are\nconvenient to work with\, since they are easy to encode and ma nipulate on\na computer\, and to interpret visually. Several notions of mo dule\napproximation using this subcategory have been proposed\, among whic h the\nmost common one seeks to preserve the rank invariant when switching from\nthe original module to its interval-decomposable approximation. The \nmotivation is that\, the rank invariant being one of the weakest\ninvari ants available to us\, preserving it is considered to be a minimum.\nAs it turns out\, this is not always possible\, however one can always\ndecompo se the rank invariant of the module as a $\\mathbb{Z}$-linear\ncombination of rank invariants of interval modules. Thus\, a weaker form\nof preserva tion of the rank invariant is possible\, in which the interval\nsummands a re signed (hence the name "signed rank decomposition"). This\nfact can be viewed as a consequence of the Moebius inversion formula\,\nbut more funda mentally\, it can be obtained by working in the\nGrothendieck group relat ive to an appropriate exact structure\, where the\nrank invariant of the m odule becomes equal to the alternating sum of the\nrank invariants of the various terms in the module's minimal relative\nprojective resolution. Thi s alternative proof strategy offers some\nsignificant benefits: (1) it lin ks the coefficients in the decomposition\nto the structure of the module\, as in the 1-parameter setting\; (2) it\nprovides a roadmap to study their bottleneck stability\; (3) it connects\nmulti-parameter persistence to re lative homological algebra\, thereby\npaving the way towards the definitio n of more refined invariants for\nmulti-parameter persistence modules usin g larger classes of projectives.\nThe purpose of my talk will be to tell t his story.\n LOCATION:https://researchseminars.org/talk/GEOTOP-A/46/ END:VEVENT END:VCALENDAR