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UID:f144e15f-b336-4149-b445-d8646626e5c5
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X-WR-CALNAME:IAS Special Year
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DTSTAMP:20210622T230210Z
CREATED:20190516T150227Z
DESCRIPTION:During the 2021-22 academic year the School will have a special
\nprogram on the h-principle and its applications to problems in\nanalysis
and geometry\, organized by Camillo De Lellis and László\nSzékelyhidi Jr.
\, who will be the Distinguished Visiting Professor.\n\nConfirmed senior p
articipants are: Tristan Buckmaster\, Alexey\nCheskidov\, Kai Cieliebak\,
Diego Cordoba\, Mimi Dai\, Yakov Eliashberg\,\nDaniel Faraco\, Svitlana Ma
yboroda\, Gael Meigniez\, Emmy Murphy\, Dishant\nPancholi\, Alexander Shni
relman\, Vladimir Sverak.\n\nGromov’s h-principle provides a powerful fram
ework for dealing with\na large class of partial differential equations an
d inequalities\narising in differential geometry. Problems adhering to the
h-principle\nare typically “flexible”\, with uniqueness not expected even
under\nappropriate side conditions\, but instead solution spaces turn out
to\nbe extremely large. Whilst often the associated PDE are\nunderdetermi
ned\, Nash’s seminal work on isometric embeddings in the\n1950s provides a
prominent example where this need not be the case.\nThe Nash-Kuiper C1 is
ometric embedding theorem has lead to important\ndevelopments in several b
ranches of mathematics. Gromov developed the\ntechnique of convex integrat
ion for ample differential relations\, with\nimportant applications in sym
plectic geometry\, and more recently\nsimilar ideas have had surprising co
nsequences in classical fluid\ndynamics.\n\nIn the past 20 years there has
been considerable work on developing\nfurther the analytical side of Nash
’s iteration\, mostly on PDEs\nrelated to nonlinear elasticity and fluid d
ynamics\, with recent\nbreakthroughs such as the resolution of Onsager’s c
onjecture on the\nEuler equations and the construction\nof very weak solut
ions of the Navier-Stokes equations violating\nuniqueness and the energy i
nequality. More generally\, there is\nincreasing evidence that a form of h
-principle is closely related to\ncomplex phenomena associated with turbul
ence.\n\nAlso in the past 10 years there were a number of significant\ndis
coveries concerning applications of h-principles to\nhigh-dimensional symp
lectic and contact geometry. These have typically\ngiven existence theorem
s\, and also yielded geometric flexible/rigid\ndichotomies. Among the impo
rtant new results are general existence\ntheorem for contact structures\,
the theory of loose Legendrian knots\,\nwhich branched off into numerous o
ther results\, for instance the\ntheory of flexible Weinstein manifolds an
d new constructions of\nLagrangians and Liouville cobordisms.\n\nThe goals
of the special year are twofold: while expanding their\nknowledge on the
specific topics mentioned above\, the participants\nwill also aim to subst
antiate deeper connections between them\, with\nthe expectation that the t
ools developed in recent years will be\nuseful to push further the boundar
y\nof the class of problems adhering to the h-principle. A sample of\nsub-
themes of the special year is given by the following questions:\n\n * To w
hat extent are smooth flows of the Euler equations universal?\n * Can the
h-principle show anomalous dissipation in Navier-Stokes\nflows?\n * Do pse
udo-holomorphic curves and the Fukaya category detect\nsymplectic flexibil
ity?\n\n
DTSTART:20210901T040000Z
DTEND:20220430T040000Z
LAST-MODIFIED:20210423T151035Z
LOCATION:
SUMMARY:Special Year on h-Principle and Flexibility in Geometry and PDEs
URL:https://www.ias.edu/math/sp/hprinciple_flex_geometry_pdes
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