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Session S33 - Spectral Geometry

Tuesday, July 13, 12:00 ~ 12:20 UTC-3

Do the Hodge spectra detect orbifold singularities?

Carolyn Gordon

Dartmouth College, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak5feec1e3269f2991f29a5a34eb9972f4').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy5feec1e3269f2991f29a5a34eb9972f4 = 'csg&#111;rd&#111;n' + '&#64;'; addy5feec1e3269f2991f29a5a34eb9972f4 = addy5feec1e3269f2991f29a5a34eb9972f4 + 'd&#97;rtm&#111;&#117;th' + '&#46;' + '&#101;d&#117;'; var addy_text5feec1e3269f2991f29a5a34eb9972f4 = 'csg&#111;rd&#111;n' + '&#64;' + 'd&#97;rtm&#111;&#117;th' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak5feec1e3269f2991f29a5a34eb9972f4').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy5feec1e3269f2991f29a5a34eb9972f4 + '\'>'+addy_text5feec1e3269f2991f29a5a34eb9972f4+'<\/a>';

We will address the question: Does the spectrum of the Hodge Laplacian on $p$-forms distinguish closed Riemannian orbifolds with singularities from smooth Riemannian manifolds? This question remains open in the case of the Laplace-Beltrami operator (i.e., the case $p=0$), although many partial results are known. We show that the spectra of the Hodge Laplacians on functions and 1-forms together distinguish manifolds from orbifolds with sufficiently large singular set. We also obtain weaker affirmative results for the spectrum on 1-forms alone and show via counterexamples that some of these results are sharp.

Joint work with Katie Gittins (Durham University), Magda Khalile (Leibniz University), Ingrid Membrillo Solis (University of Southampton), Mary Sandoval (Trinity College) and Elizabeth Stanhope (Lewis & Clark College).