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UID:cae9f8ef9f463581549eb838ac25c15d
CATEGORIES:Geometric Analysis Seminar
CREATED:20240930T084510
SUMMARY:General behavior of area-minimizing subvarieties
LOCATION:Hill-705 (Zoom)
DESCRIPTION:<p><span style="color: #242424; font-family: 'Segoe UI', 'Segoe UI Web (Wes
 t European)', -apple-system, 'system-ui', Roboto, 'Helvetica Neue', sans-se
 rif; font-size: 15px; font-style: normal; font-weight: 400; letter-spacing:
  normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: n
 one; widows: 2; word-spacing: 0px; white-space: normal; background-color: #
 ffffff; float: none;">We will review some recent progress on the general ge
 ometric behavior of homologically area-minimizing subvarieties, namely, obj
 ects that minimize area with respect to homologous competitors. They are pr
 evalent in geometry, for instance, as holomorphic subvarieties of a Kahler 
 manifold, or as special Lagrangians on a Calabi-Yau, etc. A fine understand
 ing of the geometric structure of homological area-minimizers can give far-
 reaching consequences for related problems.</span><br style="color: #242424
 ; font-family: 'Segoe UI', 'Segoe UI Web (West European)', -apple-system, '
 system-ui', Roboto, 'Helvetica Neue', sans-serif; font-size: 15px; font-sty
 le: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-alig
 n: start; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 
 0px; white-space: normal; background-color: #ffffff;"><br style="color: #24
 2424; font-family: 'Segoe UI', 'Segoe UI Web (West European)', -apple-syste
 m, 'system-ui', Roboto, 'Helvetica Neue', sans-serif; font-size: 15px; font
 -style: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-
 align: start; text-indent: 0px; text-transform: none; widows: 2; word-spaci
 ng: 0px; white-space: normal; background-color: #ffffff;"><span style="colo
 r: #242424; font-family: 'Segoe UI', 'Segoe UI Web (West European)', -apple
 -system, 'system-ui', Roboto, 'Helvetica Neue', sans-serif; font-size: 15px
 ; font-style: normal; font-weight: 400; letter-spacing: normal; orphans: 2;
  text-align: start; text-indent: 0px; text-transform: none; widows: 2; word
 -spacing: 0px; white-space: normal; background-color: #ffffff; float: none;
 ">Camillo De Lellis and his collaborators have proven that area-minimizing 
 integral currents have codimension two rectifiable singular sets. A pressin
 g next question is what one can say about the geometric behavior of area-mi
 nimizing currents beyond this. Almost all known examples and results point 
 towards that area-minimizing subvarieties are subanalytic, generically smoo
 th, and calibrated. It is natural to ask if these hold in general. In this 
 direction, we prove that all of these properties thought to be true general
 ly and proven to be true in special cases are totally false in general. We 
 prove that area-minimizing subvarieties can have fractal singular sets. Smo
 othable singularities are non-generic. Calibrated area minimizers are non-g
 eneric. Consequently, we answer several conjectures of Frederick J. Almgren
  Jr., Frank Morgan, and Brian White from the 1980s.</span></p>
CONTACT:Zhenhua Liu
DTSTAMP:20260829T215200
DTSTART;TZID=America/New_York:20241001T145000
DTEND;TZID=America/New_York:20241001T155000
SEQUENCE:0
TRANSP:OPAQUE
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