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UID:bdb9731aa13c1a119d4a5f8593db42a6
CATEGORIES:Geometric Analysis Seminar
CREATED:20251029T080743
SUMMARY:Delayed structure for solutions to the Curve Shortening Flow
LOCATION:Hill Center 705
DESCRIPTION:Topping & the speaker proposed, in 2024, the principle that if two disj
 oint embedded (and sufficiently nice) Curve Shortening Flows (CSFs) bound a
 n evolving connected region of fixed area A, then the regularity of one sho
 uld be controlled after time A/π by the time, the area A and the regularity
  of the other. A handful of concrete results in the graphical setting which
  fit into the framework were provided.\nIn this talk I will discuss further
  work into this principle. Here we relax the grahicality constraint and ask
  the same question: Does the CSF regularise after a time depending on only 
 the area? I will present a positive result in the flavour of this direction
 . Under a relaxed notion of ‘swept area’, given a (sufficiently nice) non-g
 raphical curve, the CSF takes it to a graphical curve with controlled geome
 try after a time depending on only the swept area.\nTime permitting, I will
  explain some aspects of the proof; one of which is an ‘alternative’ Harnac
 k quantity/inequality for the CSF which does not require convexity.\nAll wo
 rk is joint with P. M. Topping.\n
X-ALT-DESC;FMTTYPE=text/html:<p><span style="font-family: Aptos; font-size: 16px;">Topping &amp; the spe
 aker proposed, in 2024, the principle that if two disjoint embedded (and su
 fficiently nice) Curve Shortening Flows (CSFs) bound an evolving connected 
 region of fixed area A, then the regularity of one should be controlled aft
 er time A/π by the time, the area A and the regularity of the other. A hand
 ful of concrete results in the <i>graphical</i>&nbsp;setting which fit into
  the framework were provided.</span><span style="font-family: Aptos, Aptos_
 MSFontService, -apple-system, Roboto, Arial, Helvetica, sans-serif; font-si
 ze: 12pt;"><br></span><span style="font-family: Aptos; font-size: 16px;">In
  this talk I will discuss further work into this principle. Here we relax t
 he grahicality constraint and ask the same question: Does the CSF regularis
 e after a time depending on only the area? I will present a positive result
  in the flavour of this direction. Under a relaxed notion of ‘swept area’, 
 given a (sufficiently nice) non-graphical curve, the CSF takes it to a grap
 hical curve with controlled geometry after a time depending on only the swe
 pt area.</span><span style="font-family: Aptos, Aptos_MSFontService, -apple
 -system, Roboto, Arial, Helvetica, sans-serif; font-size: 12pt;"><br></span
 ><span style="font-family: Aptos; font-size: 16px;">Time permitting, I will
  explain some aspects of the proof; one of which is an ‘alternative’ Harnac
 k quantity/inequality for the CSF which does not require convexity.</span><
 span style="font-family: Aptos, Aptos_MSFontService, -apple-system, Roboto,
  Arial, Helvetica, sans-serif; font-size: 12pt;"><br></span><span style="fo
 nt-family: Aptos; font-size: 16px;">All work is joint with P. M. Topping.</
 span></p>
CONTACT:Arjun Sobnack
DTSTAMP:20260827T204607
DTSTART;TZID=America/New_York:20251104T145000
DTEND;TZID=America/New_York:20251104T155000
SEQUENCE:0
TRANSP:OPAQUE
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