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BEGIN:VEVENT
SUMMARY:Leon Simon (Stanford University)
DTSTART;VALUE=DATE-TIME:20201117T220000Z
DTEND;VALUE=DATE-TIME:20201117T230000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/1
DESCRIPTION:Title: Stable minimal hypersurfaces in $\\R^{N+1+\\ell}$ with singular set an ar
bitrary closed $K\\subset\\{0\\}\\times\\R^{\\ell}$\nby Leon Simon (St
anford University) as part of NCTS international Geometric Measure Theory
seminar\n\n\nAbstract\nWith respect to a $C^{\\infty}$ metric which is clo
se to the standard Euclidean metric on $\\R^{N+1+\\ell}$\, where $N\\ge 7$
and $\\ell\\ge 1$ are given\, we construct a class of embedded $(N+\\ell)
$-dimensional hypersurfaces (without boundary) which are minimal and stric
tly stable\, and which have singular set equal to an arbitrary preassigned
closed subset $K\\subset\\{0\\}\\times\\R^{\\ell}$.\n\nWe encourage every
one to employ the virtual venue to interact (through chat\, meeting\, and
boards) before and after the talk.\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juncheng Wei (University of British Colombia)
DTSTART;VALUE=DATE-TIME:20210120T143000Z
DTEND;VALUE=DATE-TIME:20210120T153000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/2
DESCRIPTION:Title: Second order estimates for interfaces of Allen-Cahn\nby Juncheng Wei
(University of British Colombia) as part of NCTS international Geometric M
easure Theory seminar\n\n\nAbstract\nIn this talk I will discuss a uniform
$C^{2\,\\theta}$ estimate for level sets of stable solutions to the singu
larly perturbed Allen-Cahn equation in dimensions $n \\leq 10$ (which is o
ptimal). The proof combines two ingredients: one is a reverse application
of the infinite dimensional Lyapunov-Schmidt reduction method which enable
s us to reduce the $C^{2\,\\theta}$ estimate for these level sets to a cor
responding one on solutions of Toda system\; the other one uses a small re
gularity theorem on stable solutions of Toda system to establish various d
ecay estimates\, which gives a lower bound on distances between different
sheets of solutions to Toda system or level sets of solutions to Allen-Cah
n equation. (Joint work with Kelei Wang.)\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felix Schulze (University of Warwick)
DTSTART;VALUE=DATE-TIME:20210317T121500Z
DTEND;VALUE=DATE-TIME:20210317T131500Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/3
DESCRIPTION:Title: Mean curvature flow with generic initial data\nby Felix Schulze (Univ
ersity of Warwick) as part of NCTS international Geometric Measure Theory
seminar\n\n\nAbstract\nA well-known conjecture of Huisken states that a ge
neric mean curvature flow has only spherical and cylindrical singularities
. As a first step in this direction Colding-Minicozzi have shown in fundam
ental work that spheres and cylinders are the only linearly stable singula
rity models. As a second step toward Huisken's conjecture we show that mea
n curvature flow of generic initial closed surfaces in $\\mathbb R^3$ avoi
ds asymptotically conical and non-spherical compact singularities. We also
show that mean curvature flow of generic closed low-entropy hypersurfaces
in $\\mathbb R^4$ is smooth until it disappears in a round point. The mai
n technical ingredient is a long-time existence and uniqueness result for
ancient mean curvature flows that lie on one side of asymptotically conica
l or compact self-similarly shrinking solutions. This is joint work with O
tis Chodosh\, Kyeongsu Choi and Christos Mantoulidis.\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tatiana Toro (University of Washington)
DTSTART;VALUE=DATE-TIME:20210519T133000Z
DTEND;VALUE=DATE-TIME:20210519T143000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/4
DESCRIPTION:Title: Geometric measure theory: a powerful tool in potential theory\nby Tat
iana Toro (University of Washington) as part of NCTS international Geometr
ic Measure Theory seminar\n\n\nAbstract\nIn this talk I will describe a co
uple of instances in which ideas coming from geometric measure theory have
played a central role in proving results in potential theory. Understandi
ng limits of measures associated to second order divergence form operators
has allowed us to establish equivalences between boundary regularity prop
erties of solutions to these operators and the domains where they are defi
ned.\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Bate (University of Warwick)
DTSTART;VALUE=DATE-TIME:20210721T121500Z
DTEND;VALUE=DATE-TIME:20210721T131500Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/5
DESCRIPTION:Title: A non-linear Besicovitch–Federer projection theorem for metric spaces\nby David Bate (University of Warwick) as part of NCTS international Ge
ometric Measure Theory seminar\n\n\nAbstract\nThis talk will present a cha
racterisation of purely $n$-unrectifiable subsets $S$ of a complete metric
space with finite $n$-dimensional Hausdorff measure by studying non-linea
r projections (i.e. $1$-Lipschitz functions) into some fixed Euclidean spa
ce. We will show that a typical (in the sense of Baire category) non-linea
r projection maps $S$ to a set of zero $n$-dimensional Hausdorff measure.
Conversely\, a typical non-linear projection maps an $n$-rectifiable subse
t to a set of positive $n$-dimensional Hausdorff measure. These results pr
ovide a replacement for the classical Besicovitch–Federer projection the
orem\, which is known to be false outside of Euclidean spaces.\n\nTime per
mitting\, we will discuss some recent consequences of this characterisatio
n.\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessio Figalli (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20210922T121500Z
DTEND;VALUE=DATE-TIME:20210922T131500Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/6
DESCRIPTION:Title: Free boundary regularity in the Stefan problem\nby Alessio Figalli (E
TH Zurich) as part of NCTS international Geometric Measure Theory seminar\
n\n\nAbstract\nThe Stefan problem describes phase transitions\, such as ic
e melting to water. In its simplest formulation\, this problem consists of
finding the evolution of the temperature off the water when a block of ic
e is submerged inside.\n\nIn this talk\, I will first discuss the classica
l theory for this problem. Then I will present some recent results concern
ing the fine regularity properties of the interface separating water and i
ce (the so called "free boundary"). As we shall see\, these results provid
e us with a very refined understanding of the Stefan problem's singulariti
es\, and they answer some long-standing open questions in the field.\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Otis Chodosh (Stanford University)
DTSTART;VALUE=DATE-TIME:20211117T220000Z
DTEND;VALUE=DATE-TIME:20211118T000000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/7
DESCRIPTION:Title: Stable minimal hypersurfaces in $\\mathbb R^4$\nby Otis Chodosh (Stan
ford University) as part of NCTS international Geometric Measure Theory se
minar\n\n\nAbstract\nI will explain why stable minimal hypersurfaces in $\
\mathbb R^4$ are flat. This is joint work with Chao Li.\n\nGet-together (3
0 min) $\\cdot$ presentation Otis Chodosh (60 min) $\\cdot$ questions and
discussions (30 min).\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Minter (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20220119T123000Z
DTEND;VALUE=DATE-TIME:20220119T143000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/8
DESCRIPTION:Title: A structure theory for branched stable hypersurfaces\nby Paul Minter
(University of Cambridge) as part of NCTS international Geometric Measure
Theory seminar\n\n\nAbstract\nThere are few known general regularity resul
ts for stationary integral varifolds aside from Allard’s celebrated theo
ry. The primary reason for this is the possibility of a degenerate type of
singularity\nknown as a branch point\, where at the tangent cone level si
ngularities vanish and are replaced with regions of higher multiplicity. I
n this talk I will discuss a recent regularity theory for branched stable\
nhypersurfaces which do not contain certain so-called classical singularit
ies\, including new tangent cone uniqueness results in the presence of bra
nch points. This theory can be readily applied to area\nminimising hypercu
rrents mod p\, which resolves an old conjecture from the work of Brian Whi
te. Some results are joint with Neshan Wickramasekera.\n\nGet-together (30
min) $\\cdot$ presentation Paul Minter (60 min) $\\cdot$ questions and di
scussions (30 min).\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simone Steinbrüchel (Leipzig University)
DTSTART;VALUE=DATE-TIME:20220316T120000Z
DTEND;VALUE=DATE-TIME:20220316T140000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/9
DESCRIPTION:Title: A regularity theorem for area-minimizing currents at higher multiplicity
boundary points\nby Simone Steinbrüchel (Leipzig University) as part
of NCTS international Geometric Measure Theory seminar\n\n\nAbstract\nThe
boundary regularity theory for area-minimizing integral currents in higher
codimension has been completed in 2018 by a work of De Lellis\, De Philip
pis\, Hirsch and Massaccesi proving the density of regular boundary points
. In this talk\, I will present our recent paper where we took a first ste
p into analyzing area-minimizing currents with higher multiplicity boundar
y. This question has first been raised by Allard and later again by White.
We focus on two-dimensional currents with a convex barrier and define the
regular boundary points to be those around which the current consists of
finitely many regular submanifolds meeting transversally at the boundary.
Adapting the techniques of Almgren\, we proved that every boundary point i
s regular in the above sense. This is a joint work with C. De Lellis and S
. Nardulli.\n\nGet-together (30 min) $\\cdot$ presentation Simone Steinbr
üchel (60 min) $\\cdot$ questions and discussions (30 min).\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro Pigati (New York University\, Courant Institute)
DTSTART;VALUE=DATE-TIME:20220518T120000Z
DTEND;VALUE=DATE-TIME:20220518T140000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/10
DESCRIPTION:Title: (Non-)quantization phenomena for higher-dimensional Ginzburg-Landau vort
ices\nby Alessandro Pigati (New York University\, Courant Institute) a
s part of NCTS international Geometric Measure Theory seminar\n\n\nAbstrac
t\nThe Ginzburg-Landau energies for complex-valued maps\, initially introd
uced to model superconductivity\, were later found to approximate the area
functional in codimension two.\n\nWhile the pioneering works of Lin-Rivi
ère and Bethuel-Brezis-Orlandi (2001) showed that\, for families of criti
cal maps\, energy does concentrate along a codimension-two minimal submani
fold\, it has been an open question whether this happens with integer mult
iplicity. In this talk\, based on joint work with Daniel Stern\, we show t
hat\, in fact\, the set of all possible multiplicities is precisely $\\{1\
\} \\cup [2\,\\infty)$.\n\nGet-together (30 min) $\\cdot$ presentation Ale
ssandro Pigati (60 min) $\\cdot$ questions and discussions (30 min).\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Costante Bellettini (University College London)
DTSTART;VALUE=DATE-TIME:20220720T120000Z
DTEND;VALUE=DATE-TIME:20220720T140000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/11
DESCRIPTION:Title: Hypersurfaces with prescribed-mean-curvature: existence and properties\nby Costante Bellettini (University College London) as part of NCTS int
ernational Geometric Measure Theory seminar\n\n\nAbstract\nLet $N$ be a co
mpact Riemannian manifold of dimension $3$ or higher\, and $g$ a Lipschitz
non-negative (or non-positive) function on $N$. In joint works with Nesh
an Wickramasekera we prove that there exists a closed hypersurface $M$ who
se mean curvature attains the values prescribed by $g$. Except possibly f
or a small singular set (of codimension $7$ or higher)\, the hypersurface
$M$ is $C^2$ immersed and two-sided (it admits a global unit normal)\; the
scalar mean curvature at $x$ is $g(x)$ with respect to a global choice of
unit normal. More precisely\, the immersion is a quasi-embedding\, namely
the only non-embedded points are caused by tangential self-intersections:
around any such non-embedded point\, the local structure is given by two
disks\, lying on one side of each other\, and intersecting tangentially (a
s in the case of two spherical caps touching at a point). A special case o
f PMC (prescribed-mean-curvature) hypersurfaces is obtained when $g$ is a
constant\, in which the above result gives a CMC (constant-mean-curvature)
hypersurface for any prescribed value of the mean curvature.\n\nGet-toget
her (30 min) $\\cdot$ presentation Costante Bellettini (60 min) $\\cdot$ q
uestions and discussions (30 min).\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gábor Székelyhidi (Northwestern University)
DTSTART;VALUE=DATE-TIME:20220921T120000Z
DTEND;VALUE=DATE-TIME:20220921T140000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/12
DESCRIPTION:Title: Minimal hypersurfaces with cylindrical tangent cones\nby Gábor Szé
kelyhidi (Northwestern University) as part of NCTS international Geometric
Measure Theory seminar\n\n\nAbstract\nI will discuss recent results on mi
nimal hypersurfaces with cylindrical tangent cones of the form $C \\times
\\mathbb R$\, where $C$ is a minimal quadratic cone\, such as the Simons c
one over $\\mathbb S^3 \\times \\mathbb S^3$. I will talk about a uniquene
ss result for such tangent cones in a certain non-integrable situation\, a
s well as a precise description of such minimal hypersurfaces near the sin
gular set under a symmetry assumption.\n\nGet-together (30 min) $\\cdot$ p
resentation Gábor Székelyhidi (60 min) $\\cdot$ questions and discussion
s (30 min).\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antoine Song (California Institute of Technology)
DTSTART;VALUE=DATE-TIME:20221123T130000Z
DTEND;VALUE=DATE-TIME:20221123T150000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/13
DESCRIPTION:Title: The spherical Plateau problem: existence\, uniqueness\, stability\nb
y Antoine Song (California Institute of Technology) as part of NCTS intern
ational Geometric Measure Theory seminar\n\n\nAbstract\nConsider a countab
le group $G$ acting on the unit sphere $S$ in the\nspace of $L^2$ function
s on $G$ by the regular representation. Given a\nhomology class $h$ in the
quotient space $S/G$\, one defines the\nspherical Plateau solutions for $
h$ as the intrinsic flat limits of\nvolume minimizing sequences of cycles
representing $h$. Interestingly in\nsome special cases\, for example when
$G$ is the fundamental group of a\nclosed hyperbolic manifold of dimension
at least $3$\, the spherical\nPlateau solutions are essentially unique an
d can be identified. However\nin general not much is known. I will discuss
the questions of existence\nand structure of non-trivial Plateau solution
s. I will also explain how\nuniqueness of spherical Plateau solutions for
hyperbolic manifolds of\ndimension at least $3$ implies stability for the
volume entropy\ninequality of Besson-Courtois-Gallot.\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Filip Rindler (University of Warwick)
DTSTART;VALUE=DATE-TIME:20230118T130000Z
DTEND;VALUE=DATE-TIME:20230118T150000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/14
DESCRIPTION:Title: Singularities\, Rectifiability\, and PDE-constraints\nby Filip Rindl
er (University of Warwick) as part of NCTS international Geometric Measure
Theory seminar\n\n\nAbstract\nSurprisingly many different problems of Ana
lysis naturally lead to questions about singularities in (vector) measures
. These problems come from both "pure" Analysis\, such as the question for
which measures Rademacher's theorem on the differentiability of Lipschitz
functions holds\, and its non-Euclidean analogues\, as well as from "appl
ied" Analysis\, for example the problem to determine the fine structure of
slip lines in elasto-plasticity. It is a remarkable fact that many of the
(vector) measures that naturally occur in these questions satisfy an (und
er-determined) PDE constraint\, e.g.\, divergence- or curl-freeness. The c
rucial task is then to analyse the fine properties of these \nPDE-constrai
ned measures\, in particular to determine the possible singularities that
may occur. It turns out that the PDE constraint imposes strong restriction
s on the shape of these singularities\, for instance that they can only oc
cur on a set of bounded Hausdorff-dimension\, or even that the measure is
k-rectifiable where its upper k-density is positive. The essential difficu
lty in the analysis of PDE-constrained measures is that many standard meth
ods from harmonic analysis are much weaker in an L$^1$-context and thus ne
w strategies are needed. In this talk\, I will survey recent and ongoing w
ork on this area of research.\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jie Zhou (Capital Normal University)
DTSTART;VALUE=DATE-TIME:20230315T080000Z
DTEND;VALUE=DATE-TIME:20230315T100000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/15
DESCRIPTION:Title: Bi-Lipschitz regularity of 2-varifolds with the critical Allard conditio
n\nby Jie Zhou (Capital Normal University) as part of NCTS internation
al Geometric Measure Theory seminar\n\n\nAbstract\nFor an integral 2-varif
old in the unit ball of the Euclidean space passing through the origin\, i
f it satisfies the critical Allard condition\, i.e.\, the mass of the vari
fold in the unit ball is close to the area of a flat unit disk and the L$^
2$ norm of the generalized mean curvature is small enough\, we show that l
ocally the support of the varifold admits a bi-Lipschitz parameterization
from the unit disk. The presentation is based on a joint work with Dr. Yuc
hen Bi.\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christos Mantoulidis (Rice University)
DTSTART;VALUE=DATE-TIME:20230517T120000Z
DTEND;VALUE=DATE-TIME:20230517T140000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/16
DESCRIPTION:Title: Generic regularity of minimizing hypersurfaces in dimensions 9 and 10\nby Christos Mantoulidis (Rice University) as part of NCTS international
Geometric Measure Theory seminar\n\n\nAbstract\nIn joint work with Otis C
hodosh and Felix Schulze we showed that the problem of finding a least-are
a compact hypersurface with prescribed boundary or homology class has a sm
ooth solution for generic data in dimensions 9 and 10. In this talk I will
explain the main steps of the proof.\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Krummel (University of Melbourne)
DTSTART;VALUE=DATE-TIME:20230712T080000Z
DTEND;VALUE=DATE-TIME:20230712T100000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/17
DESCRIPTION:Title: Analysis of singularities of area minimizing currents\nby Brian Krum
mel (University of Melbourne) as part of NCTS international Geometric Meas
ure Theory seminar\n\n\nAbstract\nThe monumental work of Almgren in the ea
rly 1980s showed that the singular set of a locally area minimizing rectif
iable current $T$ of dimension $n$ and codimension $\\geq 2$ has Hausdorff
dimension at most $n-2$. In contrast to codimension 1 area minimizers (f
or which it had been established a decade earlier that the singular set ha
s Hausdorff dimension at most $n-7$)\, the problem in higher codimension i
s substantially more complex because of the presence of branch point singu
larities\, i.e. singular points where one tangent cone is a plane of multi
plicity 2 or larger. Almgren's lengthy proof (made more accessible and tec
hnically streamlined in the much more recent work of De Lellis--Spadaro) s
howed first that the non-branch-point singularities form a set of Hausdorf
f dimension at most $n-2$ using an elementary argument based on the tangen
t cone type at such points\, and developed a powerful array of ideas to ob
tain the same dimension bound for the branch set separately. In this strat
egy\, the exceeding complexity of the argument to handle the branch set st
ems in large part from the lack of an estimate giving decay of $T$ towards
a unique tangent plane at a branch point. \n\nWe will discuss a new appr
oach to this problem (joint work with Neshan Wickramasekera). In this appr
oach\, the set of singularities (of a fixed integer density $q$) is decomp
osed not as branch points and non-branch-points\, but as a set ${\\mathcal
B}$ of branch points where $T$ decays towards a (unique) plane faster tha
n a fixed exponential rate\, and the complementary set ${\\mathcal S}$. T
he set ${\\mathcal S}$ contains all (density $q$) non-branch-point singula
rities\, but a priori it could also contain a large set of branch points.
To analyse ${\\mathcal S}$\, the work introduces a new\, intrinsic frequen
cy function for $T$ relative to a plane\, called the planar frequency func
tion. The planar frequency function satisfies an approximate monotonicity
property\, and takes correct values (i.e. $\\leq 1$) whenever $T$ is a con
e (for which planar frequency is defined) and the base point is the vertex
of the cone. These properties of the planar frequency function together
with relatively elementary parts of Almgren’s theory (Dirichlet energy m
inimizing multivalued functions and strong Lipschitz approximation) imply
that $T$ satisfies a key approximation property along $S$: near each point
of ${\\mathcal S}$ and at each sufficiently small scale\, $T$ is signific
antly closer to some non-planar cone than to any plane. This property toge
ther with a new estimate for the distance of $T$ to a union of non-interse
cting planes and the blow-up methods of Simon and Wickramasekera imply tha
t $T$ has a unique non-planar tangent cone at $\\mathcal{H}^{n-2}$-a.e. po
int of $\\mathcal{S}$ and that ${\\mathcal S}$ is $(n-2)$-rectifiable with
locally finite measure. Analysis of ${\\mathcal B}$ using the planar freq
uency function and the locally uniform decay estimate along ${\\mathcal B}
$ recovers Almgren’s dimension bound for the singular set of $T$ in a si
mpler way\, and (again via Simon and Wickramasekera blow-up methods) shows
that ${\\mathcal B}$ (and hence the entire singular set of $T$) is counta
bly $(n-2)$-rectifiable with a unique\, non-zero multi-valued harmonic blo
w-up at $\\mathcal{H}^{n-2}$-a.e. point of ${\\mathcal B}$.\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georg Weiss (University of Duisburg-Essen)
DTSTART;VALUE=DATE-TIME:20230920T080000Z
DTEND;VALUE=DATE-TIME:20230920T100000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/18
DESCRIPTION:Title: Rectifiability\, finite Hausdorff measure\, and compactness for non-mini
mizing Bernoulli free boundaries\nby Georg Weiss (University of Duisbu
rg-Essen) as part of NCTS international Geometric Measure Theory seminar\n
\n\nAbstract\nWhile there are numerous results on minimizers or stable sol
utions of the Bernoulli problem proving regularity of the free boundary an
d analyzing singularities\, much less is known about $\\textit{critical po
ints}$ of the corresponding energy. Saddle points of the energy (or of clo
sely related energies) and solutions of the corresponding time-dependent p
roblem occur naturally in applied problems such as water waves and combust
ion theory.\n\nFor such critical points $u\\text{---}$which can be obtaine
d as limits of classical solutions or limits of a singular perturbation pr
oblem$\\text{---}$it has been open since [Weiss03] whether the singular se
t can be large and what equation the measure $\\Delta u$ satisfies\, excep
t for the case of two dimensions. In the present result we use recent tech
niques such as a $\\textit{frequency formula}$ for the Bernoulli problem a
s well as the celebrated $\\textit{Naber-Valtorta procedure}$ to answer th
is more than 20 year old question in an affirmative way:\n\nFor a closed c
lass we call $\\textit{variational solutions}$ of the Bernoulli problem\,
we show that the topological free boundary $\\partial \\{u > 0\\}$ (includ
ing $\\textit{degenerate}$ singular points $x$\, at which $u(x + r \\cdot)
/r \\rightarrow 0$ as $r\\to 0$) is countably $\\mathcal{H}^{n-1}$-rectifi
able and has locally finite $\\mathcal{H}^{n-1}$-measure\, and we identify
the measure $\\Delta u$ completely. This gives a more precise characteriz
ation of the free boundary of $u$ in arbitrary dimension than was previous
ly available even in dimension two.\n\nWe also show that limits of (not ne
cessarily minimizing) classical solutions as well as limits of critical po
ints of a singularly perturbed energy are variational solutions\, so that
the result above applies directly to all of them.\n\nThis is a joint work
with Dennis Kriventsov (Rutgers).\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Haslhofer (University of Toronto)
DTSTART;VALUE=DATE-TIME:20231115T120000Z
DTEND;VALUE=DATE-TIME:20231115T140000Z
DTSTAMP;VALUE=DATE-TIME:20230925T234812Z
UID:NCTS-GMT/19
DESCRIPTION:Title: Mean curvature flow with surgery\nby Robert Haslhofer (University of
Toronto) as part of NCTS international Geometric Measure Theory seminar\n
\n\nAbstract\nFlows with surgery are a powerful method to evolve geometric
shapes\, and have found many important applications in geometry and topol
ogy. In this talk\, I will describe a new method to establish existence of
flows with surgery. In contrast to all prior constructions of flows with
surgery in the literature\, our new approach does not require any a priori
estimates in the smooth setting. Instead\, our approach uses geometric me
asure theory\, building in particular on the work of Brakke and White. We
illustrate our method in the classical setting of mean-convex surfaces in
R$^3$\, thus giving a new proof of the existence results due to Brendle-Hu
isken and Kleiner and myself. Moreover\, our new method also enables the c
onstruction of flows with surgery in situations that have been inaccessibl
e with prior techniques\, including in particular the free-boundary settin
g.\n
LOCATION:https://researchseminars.org/talk/NCTS-GMT/19/
END:VEVENT
END:VCALENDAR