BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Francesco Fidaleo (Università di Roma "Tor Vergata") DTSTART:20220314T150000Z DTEND:20220314T160000Z DTSTAMP:20260420T052909Z UID:NYC-NCG/89 DESCRIPTION:Title: Modular Spectral Triples and deformed Fredholm modules (Part I)\nby F rancesco Fidaleo (Università di Roma "Tor Vergata") as part of Noncommuta tive geometry in NYC\n\n\nAbstract\nDue to possible applications to the at tempt to provide a set of equations which unify the four elementary intera ctions in nature (the grand-unification) and in another\, perhaps connecte d\, direction in proving the long-standing\, still unsolved\, Riemann conj ecture about the zeroes of the $\\zeta$-function\, Connes’ non- commutat ive geometry grew up rapidly in the last decades.\n\nAmong the main object s introduced (by A. Connes) for handling noncommutative geometry there are the so called spectral triples\, encoding most of the properties enjoyed by the (quantum) ”manifold” into consideration\, and the associated Fr edholm modules.\n\nOn the other hand\, the so-called Tomita modular theory is nowadays assuming an increasingly relevant role for several applicatio ns in mathematics and in physics. Such a scenario suggests the necessary n eed to take the modular data into account in the investigation of quantum manifolds. In such a situation\, the involved Dirac operators should be su itably deformed (by the use of the modular operator)\, and should come fro m non-type $II_1$ representations.\n\nTaking into account such comments\, we discuss the preliminary necessary step consisting in the explicit const ruction of examples of non type $II_1$ representations and relative spectr al triples\, called modular. This is done for the noncommutative 2-torus $ A_{\\alpha}$\, provided α is a (special kind of) Liouville number\, where the nontrivial modular structure plays a crucial role.\n\nFor such repres entations\, we briefly discuss the appropriate Fourier analysis\, by provi ng the analogous of many of the classical known theorems in harmonic analy sis such as the Riemann-Lebesgue lemma\, the Hausdorff-Young theorem\, and the $L_p$-convergence results associated to the Cesaro means (i.e. the Fe jer theorem) and the Abel means reproducing the Poisson kernel. We show ho w those Fourier transforms ”diagonalise” appropriately some examples o f the Dirac operators associated to the previous mentioned spectral triple s.\n\nFinally\, we provide a definition of a deformed generalisation of ”Fredholm module”\, i.e. a suitably deformed commutator of the ”phas e” of the involved Dirac operator with elements of a subset (the so-call ed Lipschitz $\\star$-algebra or Lipschitz operator system) which\, depend ing on the cases under consideration\, is either a dense $\\star$-algebra or an essential operator system. We also show that all models of modular spectral triples for the noncommutative 2-torus mentioned above enjoy the property to being also a deformed Fredholm module. This definition of defo rmed Fredholm module is new even in the usual cases associated to a trace\ , and could provide other\, hopefully interesting\, applications.\n\nThe p resent talk is based on the following papers:\n\n[1] F. Fidaleo and L. Sur iano: Type $III$ representations and modular spectral triples for the nonc ommutative torus\, J. Funct. Anal. 275 (2018)\, 1484-1531.\n\n[2] F. Fidal eo: Fourier analysis for type III representations of the noncommutative to rus\, J. Fourier Anal. Appl. 25 (201)\, 2801-2835.\n\n[3] F. Ciolli and F. Fidaleo: Type $III$ modular spectral triples and deformed Fredholm module s\, preprint.\n LOCATION:https://researchseminars.org/talk/NYC-NCG/89/ END:VEVENT END:VCALENDAR