BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Robert Bryant (Duke University) DTSTART:20240605T160000Z DTEND:20240605T170000Z DTSTAMP:20260423T021226Z UID:VSGS/92 DESCRIPTION:Title: Af fine Bonnet surfaces\nby Robert Bryant (Duke University) as part of Vi rtual seminar on geometry with symmetries\n\n\nAbstract\nThe Bonnet proble m in Euclidean surface theory is well-known: Given a metric $g$ and a fun ction $H$ on an oriented surface $M^2$\, when (and in how many ways) can $ (M\,g)$ be isometrically immersed in $\\mathbb{R}^3$ with mean curvature $ H$? For generic data $(g\,H)$\, such an immersion does not exist and\, in the case that one does exist\, it is unique up to ambient isometry. Bonn et showed that\, aside from the famous case of surfaces of constant mean c urvature\, there is a finite-dimensional moduli space of $(g\,H)$ for whic h the space of such "Bonnet immersions" has positive dimension.\n\nThe cor responding problem in affine theory (a favorite topic of Eugenio Calabi) i s still not completely solved. After reviewing the results on the Euclide an problem by O. Bonnet\, J. Radon\, É. Cartan\, A. Bobenko and others\, I will give a report on affine analogs of those results. In particular\, I will consider the classification of the data $(g\,H)$ for which the spac e of "affine Bonnet immersions" has positive dimension\, showing a surpris ing connection with integrable systems in the case of data $(g\,H)$ for wh ich the space of affine Bonnet immersions has the highest possible dimensi on.\n LOCATION:https://researchseminars.org/talk/VSGS/92/ END:VEVENT END:VCALENDAR