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DTSTART;TZID=America/Halifax:20201127T160000
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DTSTAMP:20240423T055553
CREATED:20200904T115630Z
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UID:5472-1606492800-1606496400@aarms.math.ca
SUMMARY:Dalhousie-AARMS AAMP Seminar: Graham Cox (Memorial University)
DESCRIPTION:Title: Nodal deficiency via equipartition energy functionals and the Dirichlet-to-Neumann map\nAbstract: A classic result in differential equations is that the nth eigenfunction\nof a Sturm-Liouville boundary value problem has precisely n-1 zeros.\nCourant’s nodal domain theorem provides a natural generalization of this\nresult to higher dimensions\, but it is generally not sharp. The lack of\nsharpness is measured by the “nodal deficiency” of an eigenfunction.\nDespite over a century of intensive study\, this quantity is still not\nvery well understood. \nThe first explicit formula for the nodal deficiency was obtained in 2012\nby Berkolaiko\, Kuchment and Smilansky\, using an energy functional\ndefined on the space of equipartitions. More recently\, with Jones and\nMarzuola\, I obtained another formula for the nodal deficiency\, in terms\nof Dirichlet-to-Neumann operators defined on the eigenfunction’s nodal\ndomains. While originally derived using symplectic methods\, this result\ncan also be understood using the spectral flow generated by a family of\nboundary conditions imposed on the nodal set. In this talk I will\ndescribe this flow\, and explain how it provides a concrete mechanism by\nwhich low energy eigenfunctions do or do not contribute to the nodal\ndeficiency. I will also describe recent progress relating these two\nformulas for the nodal deficiency\, and hint at some applications to the\ntheory of spectral minimal partitions. \nThis talk represents joint work with Thomas Beck\, Gregory Berkolaiko\,\nIsabel Bors\, Yaiza Canzani\, Grace Conte\, Christopher Jones and Jeremy\nMarzuola. \nThe Dalhousie-AARMS Analysis-Applied Math-Physics Seminar takes place on Fridays from 4 – 5 pm Atlantic Time over Zoom. If you would like to attend\, please email the organizers for connection details.
URL:https://aarms.math.ca/event/dalhousie-aarms-aamp-seminar-2020-10-16-2-2-2/
LOCATION:Zoom seminar
CATEGORIES:AAMP Seminar
ORGANIZER;CN="Suresh%20Eswarathasan":MAILTO:sr766936@dal.ca
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