BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Indranath Sengupta (IIT Gandhinagar) DTSTART:20210507T120000Z DTEND:20210507T130000Z DTSTAMP:20260513T222231Z UID:VCAS/81 DESCRIPTION:Title: So me Questions on bounds of Betti Numbers of Numerical Semigroup Rings\n by Indranath Sengupta (IIT Gandhinagar) as part of IIT Bombay Virtual Comm utative Algebra Seminar\n\n\nAbstract\nJ. Herzog proved in 1969 that the p ossible values of the first Betti number (minimal number of generators of the defining ideal) of numerical semigroup rings in embedding dimension 3 are 2 (complete intersection and Gorenstein) and 3 (the almost complete in tersection).\n\nIn a conversation about this work\, O. Zariski indicated a possible relation between Gorenstein rings and symmetric value semigroups . In response to that\, E.Kunz proved (in 1970) that a one-dimensional\, l ocal\, Noetherian\, reduced\, analytically irreducible ring is Gorenstein if and only if its value semigroup is symmetric. A question that remains o pen to date is whether the Betti numbers (or at least the first Betti numb er) of every numerical semigroup ring in embedding dimension e\, are bound ed above by a function of e.\n\nIn the years 1974 and 1975\, two interesti ng classes of examples were given by T. Moh and H. Bresinsky. Moh’s exam ple was that of a family of algebroid space curves and Bresinsky’s examp le was about a family of numerical semigroups in embedding dimension 4\, w ith the common feature that there is no upper bound on the Betti numbers. Therefore\, for embedding dimension 4 and above\, the Betti numbers (or at least the first Betti number) are not bounded above by some “good” fu nction of the embedding dimension e. A question that emerges is the follow ing: Is there a natural way to generate such numerical semigroups in arbit rary embedding dimension? In this talk\, we will discuss some recent obser vations in this direction\, which is a joint work of the author with his c ollaborators Joydip Saha and Ranjana Mehta.\n LOCATION:https://researchseminars.org/talk/VCAS/81/ END:VEVENT END:VCALENDAR