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SUMMARY:Guido De Philippis (Courant Institute)
DTSTART;VALUE=DATE-TIME:20200408T190000Z
DTEND;VALUE=DATE-TIME:20200408T200000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234353Z
UID:PrincetonDG/1
DESCRIPTION:Title: Regularity of the free boundary for the two-phase Bernoulli problem\nby Guido De Philippis (Courant Institute) as part of Princeton differen
tial geometry and geometric analysis seminar\n\nLecture held in Fine 314.\
n\nAbstract\nI will illustrate a recent result obtained in collaboration
with L. Spolaor and B. Velichkov concerning the regularity of the free b
oundaries in the two phase Bernoulli problems. The new main point is the a
nalysis of the free boundary close to branch points\, where we show that i
t is given by the union of two $C ^ 1$ graphs. This complete the analysis
started by Alt Caffarelli Friedman in the 80’s.\n
LOCATION:https://researchseminars.org/talk/PrincetonDG/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guillermo Henry (Princeton University)
DTSTART;VALUE=DATE-TIME:20200422T190000Z
DTEND;VALUE=DATE-TIME:20200422T200000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234353Z
UID:PrincetonDG/2
DESCRIPTION:Title: Isoparametric functions and solutions of the Yamabe equation.\nby
Guillermo Henry (Princeton University) as part of Princeton differential g
eometry and geometric analysis seminar\n\nLecture held in Fine 314.\n\nAbs
tract\nIn this talk we will discuss the relationship between isoparametric
functions on closed Riemannian manifolds and solutions of the Yamabe equ
ation. I will show some results on the existence and multiplicity of posit
ive and nodal solutions of the Yamabe equation that have the property of b
eing constant along the level sets of an isoparametric function.\n
LOCATION:https://researchseminars.org/talk/PrincetonDG/2/
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SUMMARY:Nick Edelen (University of Notre Dame)
DTSTART;VALUE=DATE-TIME:20200429T190000Z
DTEND;VALUE=DATE-TIME:20200429T200000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234353Z
UID:PrincetonDG/3
DESCRIPTION:Title: Regularity of minimal surfaces near quadratic cones\nby Nick Edele
n (University of Notre Dame) as part of Princeton differential geometry an
d geometric analysis seminar\n\nLecture held in Fine 314.\n\nAbstract\nHar
dt-Simon proved that every area-minimizing hypercone $C$ having only an is
olated singularity fits into a foliation of $R^{n+1}$ by smooth\, area-min
imizing hypersurfaces asymptotic to $C$. We prove that if a minimal hypers
urface $M$ in the unit ball $B_1 \\subset R^{n+1}$ lies sufficiently close
to a minimizing quadratic cone (for example\, the Simons' cone)\, then $M
\\cap B_{1/2}$ is a $C^{1\,\\alpha}$ perturbation of either the cone itse
lf\, or some leaf of its associated foliation. In particular\, we show tha
t singularities modeled on these cones determine the local structure not o
nly of $M$\, but of any nearby minimal surface. Our result also implies th
e Bernstein-type result of Simon-Solomon\, which characterizes area-minimi
zing hypersurfaces in $R^{n+1}$ asymptotic to a quadratic cone as either t
he cone itself\, or some leaf of the foliation. \n\nThis is joint work wit
h Luca Spolaor\n
LOCATION:https://researchseminars.org/talk/PrincetonDG/3/
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SUMMARY:Xavier Cabre (ICREA and Universitat Politecnica de Catalunya)
DTSTART;VALUE=DATE-TIME:20200506T190000Z
DTEND;VALUE=DATE-TIME:20200506T200000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234353Z
UID:PrincetonDG/4
DESCRIPTION:Title: Stable solutions to semilinear elliptic equations are smooth up to dim
ension 9\nby Xavier Cabre (ICREA and Universitat Politecnica de Catalu
nya) as part of Princeton differential geometry and geometric analysis sem
inar\n\nLecture held in Fine 314.\n\nAbstract\nThe regularity of stable so
lutions to semilinear elliptic PDEs has been studied since the 1970's. In
dimensions 10 and higher\, there exist singular stable energy solutions. I
n this talk I will describe a recent work in collaboration with Figalli\,
Ros-Oton\, and Serra\, where we prove that stable solutions are smooth up
to the optimal dimension 9. This answers to an open problem posed by Brezi
s in the mid-nineties concerning the regularity of extremal solutions to G
elfand-type problems.\n
LOCATION:https://researchseminars.org/talk/PrincetonDG/4/
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BEGIN:VEVENT
SUMMARY:Ao Sun (MIT)
DTSTART;VALUE=DATE-TIME:20200513T210000Z
DTEND;VALUE=DATE-TIME:20200513T220000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234353Z
UID:PrincetonDG/5
DESCRIPTION:Title: Generalizations of mean curvature flow entropy\nby Ao Sun (MIT) as
part of Princeton differential geometry and geometric analysis seminar\n\
nLecture held in Fine 314.\n\nAbstract\nMean curvature flow entropy was in
troduced by Colding-Minicozzi\, and it is a very important quantity in the
study of mean curvature flow and related geometric problem. In this talk\
, I will discuss two generalizations of mean curvature flow entropy: one i
s a localized version of entropy\, another one is entropy in a closed mani
fold. I will discuss how to use these generalizations to study the regular
ity problem of mean curvature flow. In particular\, I will discuss how to
use these generalizations of entropy rule out some pathological asymptotic
behaviors.\n\nNote unusual time\n
LOCATION:https://researchseminars.org/talk/PrincetonDG/5/
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BEGIN:VEVENT
SUMMARY:Eden Prywes (Princeton University)
DTSTART;VALUE=DATE-TIME:20200527T190000Z
DTEND;VALUE=DATE-TIME:20200527T200000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234353Z
UID:PrincetonDG/6
DESCRIPTION:Title: Characterization of Branched Covers with Simplicial Branch Sets\nb
y Eden Prywes (Princeton University) as part of Princeton differential geo
metry and geometric analysis seminar\n\nLecture held in Fine 314.\n\nAbstr
act\nA branched covering $f \\colon \\mathbb R^n \\to \\mathbb R^n$ is an
open and discrete map. Branched coverings are topological generalizations
of quasiregular and holomorphic mappings. The branch set of $f$ is the se
t where $f$ fails to be locally injective. It is well known that the imag
e of the branch set of a PL branched covering between PL $n$-manifolds is
a simplicial $(n-2)$-complex. I will discuss a recent result that the reve
rse implication also holds. More precisely\, a branched covering with the
image of the branch set contained in a simplicial $(n-2)$-complex is equiv
alent up to homeomorphism to a PL mapping. This result is classical for $n
=2$ and was shown by Martio and Srebro for $n = 3$. This is joint work wi
th Rami Luisto.\n
LOCATION:https://researchseminars.org/talk/PrincetonDG/6/
END:VEVENT
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SUMMARY:Silvia Ghinassi (Institute for Advanced Study)
DTSTART;VALUE=DATE-TIME:20200603T190000Z
DTEND;VALUE=DATE-TIME:20200603T200000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234353Z
UID:PrincetonDG/7
DESCRIPTION:Title: Higher order Reifenberg theorems and the Analyst’s Traveling Salesma
n Theorem\nby Silvia Ghinassi (Institute for Advanced Study) as part o
f Princeton differential geometry and geometric analysis seminar\n\nLectur
e held in Fine 314.\n\nAbstract\nWe provide geometric sufficient condition
s for Reifenberg flat sets of any integer dimension in Euclidean space to
be parametrized by a Lipschitz map with Hölder derivatives. The condition
s use a Jones type square function and all statements are quantitative in
that the Hölder and Lipschitz constants of the parametrizations depend on
such a function. We use these results to prove sufficient conditions for
higher order rectifiability of sets and measures. Key tools for the proof
come from Guy David and Tatiana Toro’s parametrization of Reifenberg fla
t sets in the Hölder and Lipschitz categories. If time allows\, we will d
iscuss some related work in progress and an example that shows that the co
nditions are not necessary.\n
LOCATION:https://researchseminars.org/talk/PrincetonDG/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Zhu (Princeton)
DTSTART;VALUE=DATE-TIME:20200611T210000Z
DTEND;VALUE=DATE-TIME:20200611T220000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234353Z
UID:PrincetonDG/8
DESCRIPTION:Title: Mean convex mean curvature flow with free boundary\nby Jonathan Zh
u (Princeton) as part of Princeton differential geometry and geometric ana
lysis seminar\n\nLecture held in Fine 314.\n\nAbstract\nIn the class of me
an convex surfaces\, the mean curvature flow provides a useful geometric t
ool\, owing its power to the regularity and structure theory established b
y White and with subsequent developments by Haslhofer\, Kleiner and Hershk
ovits. In joint work with Edelen\, Haslhofer and Ivaki\, we generalise thi
s theory to the free boundary setting. We will discuss the analytic and ge
ometric issues that arise in the passage to free boundary as well as furth
er developments in progress.\n
LOCATION:https://researchseminars.org/talk/PrincetonDG/8/
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