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BEGIN:VEVENT
SUMMARY:Leon Simon (Stanford University)
DTSTART:20201117T220000Z
DTEND:20201117T230000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/1
DESCRIPTION:Title: Stable minimal hypersurfaces in $\\R^{N+1+\\ell}$ with singular se
t an arbitrary closed $K\\subset\\{0\\}\\times\\R^{\\ell}$\nby Leon Si
mon (Stanford University) as part of NCTS international Geometric Measure
Theory seminar\n\n\nAbstract\nWith respect to a $C^{\\infty}$ metric which
is close 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 an
d strictly stable\, and which have singular set equal to an arbitrary prea
ssigned closed subset $K\\subset\\{0\\}\\times\\R^{\\ell}$.\n\nWe encourag
e everyone to employ the virtual venue to interact (through chat\, meeting
\, and boards) before and after the talk.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juncheng Wei (University of British Colombia)
DTSTART:20210120T143000Z
DTEND:20210120T153000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/2
DESCRIPTION:Title: Second order estimates for interfaces of Allen-Cahn\nby Junche
ng Wei (University of British Colombia) as part of NCTS international Geom
etric Measure Theory seminar\n\n\nAbstract\nIn this talk I will discuss a
uniform $C^{2\,\\theta}$ estimate for level sets of stable solutions to th
e singularly perturbed Allen-Cahn equation in dimensions $n \\leq 10$ (whi
ch is optimal). The proof combines two ingredients: one is a reverse appli
cation of the infinite dimensional Lyapunov-Schmidt reduction method which
enables us to reduce the $C^{2\,\\theta}$ estimate for these level sets t
o a corresponding one on solutions of Toda system\; the other one uses a s
mall regularity theorem on stable solutions of Toda system to establish va
rious decay estimates\, which gives a lower bound on distances between dif
ferent sheets of solutions to Toda system or level sets of solutions to Al
len-Cahn equation. (Joint work with Kelei Wang.)\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felix Schulze (University of Warwick)
DTSTART:20210317T121500Z
DTEND:20210317T131500Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/3
DESCRIPTION:Title: Mean curvature flow with generic initial data\nby Felix Schulz
e (University of Warwick) as part of NCTS international Geometric Measure
Theory seminar\n\n\nAbstract\nA well-known conjecture of Huisken states th
at a generic mean curvature flow has only spherical and cylindrical singul
arities. As a first step in this direction Colding-Minicozzi have shown in
fundamental work that spheres and cylinders are the only linearly stable
singularity models. As a second step toward Huisken's conjecture we show t
hat mean curvature flow of generic initial closed surfaces in $\\mathbb R^
3$ avoids asymptotically conical and non-spherical compact singularities.
We also show that mean curvature flow of generic closed low-entropy hypers
urfaces in $\\mathbb R^4$ is smooth until it disappears in a round point.
The main technical ingredient is a long-time existence and uniqueness resu
lt for ancient mean curvature flows that lie on one side of asymptotically
conical or compact self-similarly shrinking solutions. This is joint work
with Otis Chodosh\, Kyeongsu Choi and Christos Mantoulidis.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tatiana Toro (University of Washington)
DTSTART:20210519T133000Z
DTEND:20210519T143000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/4
DESCRIPTION:Title: Geometric measure theory: a powerful tool in potential theory\
nby Tatiana Toro (University of Washington) as part of NCTS international
Geometric Measure Theory seminar\n\n\nAbstract\nIn this talk I will descri
be a couple of instances in which ideas coming from geometric measure theo
ry have played a central role in proving results in potential theory. Unde
rstanding limits of measures associated to second order divergence form op
erators has allowed us to establish equivalences between boundary regulari
ty properties of solutions to these operators and the domains where they a
re defined.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Bate (University of Warwick)
DTSTART:20210721T121500Z
DTEND:20210721T131500Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/5
DESCRIPTION:Title: A non-linear Besicovitch–Federer projection theorem for metric s
paces\nby David Bate (University of Warwick) as part of NCTS internati
onal Geometric Measure Theory seminar\n\n\nAbstract\nThis talk will presen
t a characterisation of purely $n$-unrectifiable subsets $S$ of a complete
metric space with finite $n$-dimensional Hausdorff measure by studying no
n-linear projections (i.e. $1$-Lipschitz functions) into some fixed Euclid
ean space. We will show that a typical (in the sense of Baire category) no
n-linear projection maps $S$ to a set of zero $n$-dimensional Hausdorff me
asure. Conversely\, a typical non-linear projection maps an $n$-rectifiabl
e subset to a set of positive $n$-dimensional Hausdorff measure. These res
ults provide a replacement for the classical Besicovitch–Federer project
ion theorem\, which is known to be false outside of Euclidean spaces.\n\nT
ime permitting\, we will discuss some recent consequences of this characte
risation.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessio Figalli (ETH Zurich)
DTSTART:20210922T121500Z
DTEND:20210922T131500Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/6
DESCRIPTION:Title: Free boundary regularity in the Stefan problem\nby Alessio Fig
alli (ETH Zurich) as part of NCTS international Geometric Measure Theory s
eminar\n\n\nAbstract\nThe Stefan problem describes phase transitions\, suc
h as ice melting to water. In its simplest formulation\, this problem cons
ists of finding the evolution of the temperature off the water when a bloc
k of ice is submerged inside.\n\nIn this talk\, I will first discuss the c
lassical theory for this problem. Then I will present some recent results
concerning the fine regularity properties of the interface separating wate
r and ice (the so called "free boundary"). As we shall see\, these results
provide us with a very refined understanding of the Stefan problem's sing
ularities\, and they answer some long-standing open questions in the field
.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Otis Chodosh (Stanford University)
DTSTART:20211117T220000Z
DTEND:20211118T000000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/7
DESCRIPTION:Title: Stable minimal hypersurfaces in $\\mathbb R^4$\nby Otis Chodos
h (Stanford University) as part of NCTS international Geometric Measure Th
eory seminar\n\n\nAbstract\nI will explain why stable minimal hypersurface
s in $\\mathbb R^4$ are flat. This is joint work with Chao Li.\n\nGet-toge
ther (30 min) $\\cdot$ presentation Otis Chodosh (60 min) $\\cdot$ questio
ns and discussions (30 min).\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Minter (University of Cambridge)
DTSTART:20220119T123000Z
DTEND:20220119T143000Z
DTSTAMP:20260404T111135Z
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 M
easure Theory seminar\n\n\nAbstract\nThere are few known general regularit
y results for stationary integral varifolds aside from Allard’s celebrat
ed theory. The primary reason for this is the possibility of a degenerate
type of singularity\nknown as a branch point\, where at the tangent cone l
evel singularities vanish and are replaced with regions of higher multipli
city. In this talk I will discuss a recent regularity theory for branched
stable\nhypersurfaces which do not contain certain so-called classical sin
gularities\, including new tangent cone uniqueness results in the presence
of branch points. This theory can be readily applied to area\nminimising
hypercurrents mod p\, which resolves an old conjecture from the work of Br
ian White. Some results are joint with Neshan Wickramasekera.\n\nGet-toget
her (30 min) $\\cdot$ presentation Paul Minter (60 min) $\\cdot$ questions
and discussions (30 min).\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simone Steinbrüchel (Leipzig University)
DTSTART:20220316T120000Z
DTEND:20220316T140000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/9
DESCRIPTION:Title: A regularity theorem for area-minimizing currents at higher multip
licity boundary points\nby Simone Steinbrüchel (Leipzig University) a
s part of NCTS international Geometric Measure Theory seminar\n\n\nAbstrac
t\nThe boundary regularity theory for area-minimizing integral currents in
higher codimension has been completed in 2018 by a work of De Lellis\, De
Philippis\, Hirsch and Massaccesi proving the density of regular boundary
points. In this talk\, I will present our recent paper where we took a fi
rst step into analyzing area-minimizing currents with higher multiplicity
boundary. This question has first been raised by Allard and later again by
White. We focus on two-dimensional currents with a convex barrier and def
ine the regular boundary points to be those around which the current consi
sts of finitely many regular submanifolds meeting transversally at the bou
ndary. Adapting the techniques of Almgren\, we proved that every boundary
point is regular in the above sense. This is a joint work with C. De Lelli
s and S. Nardulli.\n\nGet-together (30 min) $\\cdot$ presentation Simone S
teinbrüchel (60 min) $\\cdot$ questions and discussions (30 min).\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro Pigati (New York University\, Courant Institute)
DTSTART:20220518T120000Z
DTEND:20220518T140000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/10
DESCRIPTION:Title: (Non-)quantization phenomena for higher-dimensional Ginzburg-Land
au vortices\nby Alessandro Pigati (New York University\, Courant Insti
tute) as part of NCTS international Geometric Measure Theory seminar\n\n\n
Abstract\nThe Ginzburg-Landau energies for complex-valued maps\, initially
introduced to model superconductivity\, were later found to approximate t
he area functional in codimension two.\n\nWhile the pioneering works of Li
n-Rivière and Bethuel-Brezis-Orlandi (2001) showed that\, for families of
critical maps\, energy does concentrate along a codimension-two minimal s
ubmanifold\, it has been an open question whether this happens with intege
r multiplicity. In this talk\, based on joint work with Daniel Stern\, we
show that\, in fact\, the set of all possible multiplicities is precisely
$\\{1\\} \\cup [2\,\\infty)$.\n\nGet-together (30 min) $\\cdot$ presentati
on Alessandro Pigati (60 min) $\\cdot$ questions and discussions (30 min).
\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Costante Bellettini (University College London)
DTSTART:20220720T120000Z
DTEND:20220720T140000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/11
DESCRIPTION:Title: Hypersurfaces with prescribed-mean-curvature: existence and prope
rties\nby Costante Bellettini (University College London) as part of N
CTS international Geometric Measure Theory seminar\n\n\nAbstract\nLet $N$
be a compact Riemannian manifold of dimension $3$ or higher\, and $g$ a Li
pschitz non-negative (or non-positive) function on $N$. In joint works wi
th Neshan Wickramasekera we prove that there exists a closed hypersurface
$M$ whose mean curvature attains the values prescribed by $g$. Except pos
sibly for a small singular set (of codimension $7$ or higher)\, the hypers
urface $M$ 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 ch
oice of unit normal. More precisely\, the immersion is a quasi-embedding\,
namely the only non-embedded points are caused by tangential self-interse
ctions: around any such non-embedded point\, the local structure is given
by two disks\, lying on one side of each other\, and intersecting tangenti
ally (as in the case of two spherical caps touching at a point). A special
case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when $g
$ is a constant\, in which the above result gives a CMC (constant-mean-cur
vature) hypersurface for any prescribed value of the mean curvature.\n\nGe
t-together (30 min) $\\cdot$ presentation Costante Bellettini (60 min) $\\
cdot$ questions and discussions (30 min).\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gábor Székelyhidi (Northwestern University)
DTSTART:20220921T120000Z
DTEND:20220921T140000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/12
DESCRIPTION:Title: Minimal hypersurfaces with cylindrical tangent cones\nby Gáb
or Székelyhidi (Northwestern University) as part of NCTS international Ge
ometric Measure Theory seminar\n\n\nAbstract\nI will discuss recent result
s on minimal hypersurfaces with cylindrical tangent cones of the form $C \
\times \\mathbb R$\, where $C$ is a minimal quadratic cone\, such as the S
imons cone over $\\mathbb S^3 \\times \\mathbb S^3$. I will talk about a u
niqueness result for such tangent cones in a certain non-integrable situat
ion\, as well as a precise description of such minimal hypersurfaces near
the singular set under a symmetry assumption.\n\nGet-together (30 min) $\\
cdot$ presentation Gábor Székelyhidi (60 min) $\\cdot$ questions and dis
cussions (30 min).\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antoine Song (California Institute of Technology)
DTSTART:20221123T130000Z
DTEND:20221123T150000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/13
DESCRIPTION:Title: The spherical Plateau problem: existence\, uniqueness\, stability
\nby Antoine Song (California Institute of Technology) as part of NCTS
international Geometric Measure Theory seminar\n\n\nAbstract\nConsider a
countable group $G$ acting on the unit sphere $S$ in the\nspace of $L^2$ f
unctions on $G$ by the regular representation. Given a\nhomology class $h$
in the quotient space $S/G$\, one defines the\nspherical Plateau solution
s for $h$ as the intrinsic flat limits of\nvolume minimizing sequences of
cycles representing $h$. Interestingly in\nsome special cases\, for exampl
e when $G$ is the fundamental group of a\nclosed hyperbolic manifold of di
mension at least $3$\, the spherical\nPlateau solutions are essentially un
ique and can be identified. However\nin general not much is known. I will
discuss the questions of existence\nand structure of non-trivial Plateau s
olutions. I will also explain how\nuniqueness of spherical Plateau solutio
ns for hyperbolic manifolds of\ndimension at least $3$ implies stability f
or the volume entropy\ninequality of Besson-Courtois-Gallot.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Filip Rindler (University of Warwick)
DTSTART:20230118T130000Z
DTEND:20230118T150000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/14
DESCRIPTION:Title: Singularities\, Rectifiability\, and PDE-constraints\nby Fili
p Rindler (University of Warwick) as part of NCTS international Geometric
Measure Theory seminar\n\n\nAbstract\nSurprisingly many different problems
of Analysis naturally lead to questions about singularities in (vector) m
easures. These problems come from both "pure" Analysis\, such as the quest
ion for which measures Rademacher's theorem on the differentiability of Li
pschitz functions holds\, and its non-Euclidean analogues\, as well as fro
m "applied" Analysis\, for example the problem to determine the fine struc
ture 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 (under-determined) PDE constraint\, e.g.\, divergence- or curl-freeness
. The crucial task is then to analyse the fine properties of these \nPDE-c
onstrained measures\, in particular to determine the possible singularitie
s that may occur. It turns out that the PDE constraint imposes strong rest
rictions on the shape of these singularities\, for instance that they can
only occur on a set of bounded Hausdorff-dimension\, or even that the meas
ure is k-rectifiable where its upper k-density is positive. The essential
difficulty in the analysis of PDE-constrained measures is that many standa
rd methods from harmonic analysis are much weaker in an L$^1$-context and
thus new strategies are needed. In this talk\, I will survey recent and on
going work on this area of research.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jie Zhou (Capital Normal University)
DTSTART:20230315T080000Z
DTEND:20230315T100000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/15
DESCRIPTION:Title: Bi-Lipschitz regularity of 2-varifolds with the critical Allard c
ondition\nby Jie Zhou (Capital Normal University) as part of NCTS inte
rnational Geometric Measure Theory seminar\n\n\nAbstract\nFor an integral
2-varifold in the unit ball of the Euclidean space passing through the ori
gin\, if it satisfies the critical Allard condition\, i.e.\, the mass of t
he varifold 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 locally the support of the varifold admits a bi-Lipschitz parameteri
zation from the unit disk. The presentation is based on a joint work with
Dr. Yuchen Bi.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christos Mantoulidis (Rice University)
DTSTART:20230517T120000Z
DTEND:20230517T140000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/16
DESCRIPTION:Title: Generic regularity of minimizing hypersurfaces in dimensions 9 an
d 10\nby Christos Mantoulidis (Rice University) as part of NCTS intern
ational Geometric Measure Theory seminar\n\n\nAbstract\nIn joint work with
Otis Chodosh and Felix Schulze we showed that the problem of finding a le
ast-area compact hypersurface with prescribed boundary or homology class h
as a smooth solution for generic data in dimensions 9 and 10. In this talk
I will explain the main steps of the proof.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Krummel (University of Melbourne)
DTSTART:20230712T080000Z
DTEND:20230712T100000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/17
DESCRIPTION:Title: Analysis of singularities of area minimizing currents\nby Bri
an Krummel (University of Melbourne) as part of NCTS international Geometr
ic Measure Theory seminar\n\n\nAbstract\nThe monumental work of Almgren in
the early 1980s showed that the singular set of a locally area minimizing
rectifiable current $T$ of dimension $n$ and codimension $\\geq 2$ has Ha
usdorff dimension at most $n-2$. In contrast to codimension 1 area minimi
zers (for which it had been established a decade earlier that the singular
set has Hausdorff dimension at most $n-7$)\, the problem in higher codime
nsion is substantially more complex because of the presence of branch poin
t singularities\, i.e. singular points where one tangent cone is a plane o
f multiplicity 2 or larger. Almgren's lengthy proof (made more accessible
and technically streamlined in the much more recent work of De Lellis--Spa
daro) showed first that the non-branch-point singularities form a set of H
ausdorff dimension at most $n-2$ using an elementary argument based on the
tangent cone type at such points\, and developed a powerful array of idea
s to obtain the same dimension bound for the branch set separately. In thi
s strategy\, the exceeding complexity of the argument to handle the branch
set stems 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 n
ew approach to this problem (joint work with Neshan Wickramasekera). In th
is approach\, the set of singularities (of a fixed integer density $q$) is
decomposed not as branch points and non-branch-points\, but as a set ${\\
mathcal B}$ of branch points where $T$ decays towards a (unique) plane fas
ter than a fixed exponential rate\, and the complementary set ${\\mathcal
S}$. The set ${\\mathcal S}$ contains all (density $q$) non-branch-point
singularities\, but a priori it could also contain a large set of branch p
oints. To analyse ${\\mathcal S}$\, the work introduces a new\, intrinsic
frequency function for $T$ relative to a plane\, called the planar frequen
cy function. The planar frequency function satisfies an approximate monoto
nicity property\, and takes correct values (i.e. $\\leq 1$) whenever $T$ i
s a cone (for which planar frequency is defined) and the base point is the
vertex of the cone. These properties of the planar frequency function to
gether with relatively elementary parts of Almgren’s theory (Dirichlet e
nergy minimizing multivalued functions and strong Lipschitz approximation)
imply that $T$ satisfies a key approximation property along $S$: near eac
h point of ${\\mathcal S}$ and at each sufficiently small scale\, $T$ is s
ignificantly closer to some non-planar cone than to any plane. This proper
ty together with a new estimate for the distance of $T$ to a union of non-
intersecting planes and the blow-up methods of Simon and Wickramasekera im
ply that $T$ has a unique non-planar tangent cone at $\\mathcal{H}^{n-2}$-
a.e. point of $\\mathcal{S}$ and that ${\\mathcal S}$ is $(n-2)$-rectifiab
le with locally finite measure. Analysis of ${\\mathcal B}$ using the plan
ar frequency function and the locally uniform decay estimate along ${\\mat
hcal B}$ recovers Almgren’s dimension bound for the singular set of $T$
in a simpler way\, and (again via Simon and Wickramasekera blow-up methods
) shows that ${\\mathcal B}$ (and hence the entire singular set of $T$) is
countably $(n-2)$-rectifiable with a unique\, non-zero multi-valued harmo
nic blow-up at $\\mathcal{H}^{n-2}$-a.e. point of ${\\mathcal B}$.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georg Weiss (University of Duisburg-Essen)
DTSTART:20230920T080000Z
DTEND:20230920T100000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/18
DESCRIPTION:Title: Rectifiability\, finite Hausdorff measure\, and compactness for n
on-minimizing Bernoulli free boundaries\nby Georg Weiss (University of
Duisburg-Essen) as part of NCTS international Geometric Measure Theory se
minar\n\n\nAbstract\nWhile there are numerous results on minimizers or sta
ble solutions of the Bernoulli problem proving regularity of the free boun
dary and analyzing singularities\, much less is known about $\\textit{crit
ical points}$ of the corresponding energy. Saddle points of the energy (or
of closely related energies) and solutions of the corresponding time-depe
ndent problem occur naturally in applied problems such as water waves and
combustion theory.\n\nFor such critical points $u\\text{---}$which can be
obtained as limits of classical solutions or limits of a singular perturba
tion problem$\\text{---}$it has been open since [Weiss03] whether the sing
ular set can be large and what equation the measure $\\Delta u$ satisfies\
, except for the case of two dimensions. In the present result we use rece
nt techniques such as a $\\textit{frequency formula}$ for the Bernoulli pr
oblem as well as the celebrated $\\textit{Naber-Valtorta procedure}$ to an
swer this more than 20 year old question in an affirmative way:\n\nFor a c
losed class we call $\\textit{variational solutions}$ of the Bernoulli pro
blem\, we show that the topological free boundary $\\partial \\{u > 0\\}$
(including $\\textit{degenerate}$ singular points $x$\, at which $u(x + r
\\cdot)/r \\rightarrow 0$ as $r\\to 0$) is countably $\\mathcal{H}^{n-1}$-
rectifiable and has locally finite $\\mathcal{H}^{n-1}$-measure\, and we i
dentify the measure $\\Delta u$ completely. This gives a more precise char
acterization of the free boundary of $u$ in arbitrary dimension than was p
reviously available even in dimension two.\n\nWe also show that limits of
(not necessarily minimizing) classical solutions as well as limits of crit
ical points of a singularly perturbed energy are variational solutions\, s
o that the result above applies directly to all of them.\n\nThis is a join
t work with Dennis Kriventsov (Rutgers).\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Haslhofer (University of Toronto)
DTSTART:20231115T120000Z
DTEND:20231115T140000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/19
DESCRIPTION:Title: Mean curvature flow with surgery\nby Robert Haslhofer (Univer
sity of Toronto) as part of NCTS international Geometric Measure Theory se
minar\n\n\nAbstract\nFlows with surgery are a powerful method to evolve ge
ometric shapes\, and have found many important applications in geometry an
d topology. In this talk\, I will describe a new method to establish exist
ence of flows with surgery. In contrast to all prior constructions of flow
s with surgery in the literature\, our new approach does not require any a
priori estimates in the smooth setting. Instead\, our approach uses geome
tric measure theory\, building in particular on the work of Brakke and Whi
te. We illustrate our method in the classical setting of mean-convex surfa
ces in R$^3$\, thus giving a new proof of the existence results due to Bre
ndle-Huisken and Kleiner and myself. Moreover\, our new method also enable
s the construction of flows with surgery in situations that have been inac
cessible with prior techniques\, including in particular the free-boundary
setting.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raanan Schul (Stony Brook University)
DTSTART:20240110T123000Z
DTEND:20240110T143000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/21
DESCRIPTION:Title: Uniformly rectifiable metric spaces\nby Raanan Schul (Stony B
rook University) as part of NCTS international Geometric Measure Theory se
minar\n\n\nAbstract\nIn their 1991 and 1993 foundational monographs\, Davi
d and Semmes characterized uniform rectifiability for subsets of Euclidean
space in a multitude of geometric and analytic ways. The fundamental geom
etric conditions can be naturally stated in any metric space and it has lo
ng been a question of how these concepts are related in this general setti
ng. In joint work with D. Bate and M. Hyde\, we prove their equivalence. N
amely\, we show the equivalence of Big Pieces of Lipschitz Images\, Bi-lat
eral Weak Geometric Lemma and Corona Decomposition in any Ahlfors regular
metric space. Loosely speaking\, this gives a quantitative equivalence bet
ween having Lipschitz charts and approximations by nice spaces. After givi
ng some background\, we will explain the main theorems and outline some ke
y steps in the proof (which will include a discussion of Reifenberg parame
terizations). We will also mention some open questions.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Richard Bamler (University of California\, Berkeley)
DTSTART:20240327T223000Z
DTEND:20240328T003000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/22
DESCRIPTION:Title: On the Multiplicity One Conjecture for Mean Curvature Flows of Su
rfaces\nby Richard Bamler (University of California\, Berkeley) as par
t of NCTS international Geometric Measure Theory seminar\n\n\nAbstract\nWe
prove the Multiplicity One Conjecture for mean curvature flows of surface
s in $\\mathbb R^3$. Specifically\, we show that any blow-up limit of such
mean curvature flows has multiplicity one. This has several applications.
First\, combining our work with results of Brendle and Choi-Haslhofer-Her
shkovits-White\, we show that any level set flow starting from an embedded
surface diffeomorphic to a 2-spheres does not fatten. In fact\, we obtain
that the problem of evolving embedded 2-spheres via the mean curvature fl
ow equation is well-posed within a natural class of singular solutions. Se
cond\, we use our result to remove an additional condition in recent work
of Chodosh-Choi-Mantoulidis-Schulze. This shows that mean curvature flows
starting from any generic embedded surface only incur cylindrical or spher
ical singularities. Third\, our approach offers a new regularity theory fo
r solutions of mean curvature flows that flow through singularities.\n\nTh
is talk is based on joint work with Bruce Kleiner.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniele Valtorta (University of Milano Bicocca)
DTSTART:20240515T120000Z
DTEND:20240515T140000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/23
DESCRIPTION:Title: Energy identity for stationary harmonic maps\nby Daniele Valt
orta (University of Milano Bicocca) as part of NCTS international Geometri
c Measure Theory seminar\n\n\nAbstract\nWe present the proof for Energy Id
entity for stationary harmonic maps. In particular\, given a sequence of s
tationary harmonic maps weakly converging to a limit with a defect measure
for the energy\, then $m-2$ almost everywhere on the support of this meas
ure the density is the sum of energy of bubbles. This is equivalent to say
ing that annular regions (or neck regions) do not contribute to the energy
of the limit.\n\nThis result is obtained via a quantitative analysis of t
he energy in annular regions for a fixed stationary harmonic map. The proo
f is technically involved\, but it will be presented in simplified cases t
o try and convey the main ideas behind it. (Preprint available on arXiv:24
01.02242)\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiangyu Liang (Beihang University)
DTSTART:20240717T080000Z
DTEND:20240717T100000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/24
DESCRIPTION:Title: Almgren minimals sets\, minimal cones\, unions and products\n
by Xiangyu Liang (Beihang University) as part of NCTS international Geomet
ric Measure Theory seminar\n\n\nAbstract\nThe notion of Almgren minimal se
ts is a way to try to solve\nPlateau’s problem in the setting of sets. T
o study local structures for\nthese sets\, one does blow-ups at each point
\, and the blow-up limits turn\nout to be minimal cones. People then would
like to know the list of all\nminimal cones.\n\nThe list of 1 or 2-dimens
ional minimal cones in $\\mathbb R^3$ are known\nfor over a century. For o
ther dimensions and codimensions\, much less is\nknown. Up to now there is
no general way to classify all possible\nminimal cones. One typical way i
s to test unions and products of known\nminimal cones.\n\nIn this talk\, w
e will first introduce basic notions and facts on Almgren\nminimal sets an
d minimal cones. Then we will discuss the minimality of\nunions and produc
ts of two minimal cones.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Piotr Hajłasz (University of Pittsburgh)
DTSTART:20240918T113000Z
DTEND:20240918T133000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/25
DESCRIPTION:Title: Constructing diffeomorphisms and homeomorphisms with prescribed d
erivative\nby Piotr Hajłasz (University of Pittsburgh) as part of NCT
S international Geometric Measure Theory seminar\n\n\nAbstract\nIn the tal
k I will prove that for any measurable mapping $T$ into the space of matri
ces with positive determinant\, there is a diffeomorphism whose derivative
equals $T$ outside a set of measure less than $\\varepsilon$. Using this
fact I will prove that for any measurable mapping $T$ into the space of ma
trices with non-zero determinant (with no sign restriction)\, there is an
almost everywhere approximately differentiable homeomorphism whose derivat
ive equals $T$ almost everywhere. The talk is based on my joint work with
P. Goldstein and Z. Grochulska.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhihan Wang (Cornell University)
DTSTART:20241120T123000Z
DTEND:20241120T143000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/26
DESCRIPTION:Title: Generic Regularity of Minimal Submanifolds\nby Zhihan Wang (C
ornell University) as part of NCTS international Geometric Measure Theory
seminar\n\n\nAbstract\nThe well-known Simons cone suggests that singularit
ies may exist in a stable minimal hypersurface in Riemannian manifolds of
dimension greater than 7\, locally modeled on minimal hypercones. It was c
onjectured that generically they can be perturbed away. In this talk\, we
shall present a way to resolve these singularities by perturbing metric in
an 8-manifold and hence obtain smoothness under a generic metric. We shal
l also talk about certain generalizations of this generic smoothness of mi
nimal submanifold in other dimensions and codimensions as well as their ap
plications.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicholas Edelen (University of Notre Dame)
DTSTART:20250115T110000Z
DTEND:20250115T130000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/27
DESCRIPTION:Title: Regularity of capillary minimal surfaces\nby Nicholas Edelen
(University of Notre Dame) as part of NCTS international Geometric Measure
Theory seminar\n\n\nAbstract\nA capillary surface is a hypersurface meeti
ng some container\nat a prescribed angle\, like the surface of water in a
cup. In this talk\nI describe some recent results concerning the boundary
regularity of\ncapillary surfaces which either minimize or are critical f
or their\nrelevant energy. The first result (joint with O. Chodosh and C.
Li) is\nan improved dimension bound for the boundary singular set of\nene
rgy-minimizers\, exploiting the connection between capillary minimal\nsurf
aces and the one-phase Bernoulli problem. The second (joint with L.\nde M
asi\, C. Gasparetto\, and C. Li) is an Allard-type regularity theorem\nfor
energy-critical capillary surfaces near capillary half-planes\, which\nim
plies regularity at generic boundary points of density $< 1$.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Fleschler (Princeton University)
DTSTART:20250319T110000Z
DTEND:20250319T130000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/28
DESCRIPTION:Title: A sharp extension of Allard’s boundary regularity theorem for a
rea minimizing currents with arbitrary boundary multiplicity\nby Ian F
leschler (Princeton University) as part of NCTS international Geometric Me
asure Theory seminar\n\n\nAbstract\nIn the context of area-minimizing curr
ents\, Allard boundary regularity\ntheorem asserts that an oriented curren
t with boundary that minimizes\narea cannot have boundary singularities of
minimum density. Indeed\, in a\nneighborhood of a point of minimum densit
y\, the surface must coincide\nwith a classical smooth minimal surface tha
t attaches smoothly to the\nboundary.\n\nIn this talk\, I will discuss a s
eries of papers\, one of them in\ncollaboration with Reinaldo Resende\, th
at extend Allard’s boundary\nregularity theory to a higher boundary mult
iplicity setting.\nSpecifically\, for an area-minimizing current with a mu
ltiplicity $Q$\nboundary\, we study density $Q/2$ boundary points. In this
context\, a\nregular point is one where smooth submanifolds with multipli
city attach\ntransversally to the boundary. We establish that the set of s
ingular\nboundary points of minimum density is of boundary codimension at
most 2\nand rectifiable\, extending the corresponding result in 2d by De L
ellis -\nSteinbrüchel - Nardulli to higher dimensional currents. The shar
pness of\nthis regularity theory is confirmed by my construction of a\n3-d
imensional area mininimizing current in $\\mathbb R^5$ with a singular\nbo
undary point of minimum density.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antonio De Rosa (Bocconi University)
DTSTART:20250514T120000Z
DTEND:20250514T140000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/29
DESCRIPTION:Title: Min-max construction of anisotropic minimal hypersurfaces\nby
Antonio De Rosa (Bocconi University) as part of NCTS international Geomet
ric Measure Theory seminar\n\n\nAbstract\nWe use the min-max construction
to find closed optimally\nregular hypersurfaces with constant anisotropic
mean curvature with\nrespect to elliptic integrands in closed $n$-dimensio
nal Riemannian\nmanifolds. The critical step is to obtain a uniform upper
bound for\ndensity ratios in the anisotropic min-max construction. This co
nfirms a\nconjecture posed by Allard [Invent. Math.\, 1983]. The talk is b
ased on\njoint work with G. De Philippis and Y. Li.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Costante Bellettini (University College London)
DTSTART:20250716T110000Z
DTEND:20250716T130000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/30
DESCRIPTION:Title: PDE analysis on stable minimal hypersurfaces: curvature estimates
and sheeting\nby Costante Bellettini (University College London) as p
art of NCTS international Geometric Measure Theory seminar\n\n\nAbstract\n
We consider properly immersed two-sided stable minimal hypersurfaces of di
mension $n$. We illustrate the validity of curvature estimates for $n \\le
q 6$ (and associated Bernstein-type properties with an extrinsic area grow
th assumption). For $n \\geq 7$ we illustrate sheeting results around "fla
t points". The proof relies on PDE analysis. The results extend respective
ly the Schoen-Simon-Yau estimates (obtained for $n \\leq 5$) and the Schoe
n-Simon sheeting theorem (valid for embeddings).\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gianmarco Caldini (University of Trento)
DTSTART:20250917T120000Z
DTEND:20250917T140000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/31
DESCRIPTION:Title: On smooth approximation of integral cycles\nby Gianmarco Cald
ini (University of Trento) as part of NCTS international Geometric Measure
Theory seminar\n\n\nAbstract\nThe natural question of how much smoother i
ntegral currents\nare with respect to their initial definition goes back t
o the late 1950s\nand to the origin of the theory with the seminal article
of Federer and\nFleming. In this seminar I will explain how closely one c
an approximate\nan integral current representing a given homology class wi
th a smooth\nsubmanifold. This is a joint study with William Browder and C
amillo De\nLellis\, based on some previous preliminary work of the former
author\ntogether with Frederick Almgren.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joaquim Serra (ETH Zurich)
DTSTART:20251119T100000Z
DTEND:20251119T120000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/32
DESCRIPTION:Title: Recent Progress on Stable Solutions of the Allen–Cahn Equation<
/a>\nby Joaquim Serra (ETH Zurich) as part of NCTS international Geometric
Measure Theory seminar\n\n\nAbstract\nI will present recent results and o
pen problems concerning stable\nsolutions of the Allen–Cahn equation and
its free boundary version. In\nparticular\, I will discuss the long-stand
ing problem of classifying\nstable solutions to the Allen–Cahn equation\
, both with and without area\nbounds\, in low dimensions\, and the consequ
ences of these\nclassifications. I will outline the classical results and
highlight more\nrecent developments\, emphasizing the main difficulties in
the problem\nand some of the key ideas underlying the proofs of our recen
t results.\n\nThe talk is based on two papers: one joint with Chan\, Figal
li\, and\nFernández-Real\, and another joint with Florit and Simon.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Minter (Stanford University)
DTSTART:20260121T093000Z
DTEND:20260121T113000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/33
DESCRIPTION:Title: Stationary Integral Varifolds near Multiplicity 2 Planes\nby
Paul Minter (Stanford University) as part of NCTS international Geometric
Measure Theory seminar\n\n\nAbstract\nA key open question in geometric mea
sure theory concerns the optimal regularity conclusion for stationary inte
gral varifolds. The primary difficulty for this lies in understanding bran
ch points\, namely non-immersed singular points where one tangent cone is
a plane with multiplicity at least 2. Both the uniqueness of such tangent
cones and the optimal dimension bound are not known (the latter is known f
or area minimising currents\, having been settled by the monumental work o
f Almgren).\n\nIn this talk\, I will discuss recent work with Spencer Beck
er-Kahn and Neshan Wickramasekera concerning these questions\, in which we
show that a simple topological structural condition on the varifold in
“flat density gaps” is sufficient to prove that the local structure ab
out density 2 branch points is given by a 2-valued function (with a regula
rity estimate). This is a consequence of a more general epsilon-regularity
theorem\, akin to Allard’s regularity theorem except in a multiplicity
2 setting.\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salvatore Stuvard (University of Milan)
DTSTART:20260318T093000Z
DTEND:20260318T113000Z
DTSTAMP:20260404T111135Z
UID:NCTS-GMT/34
DESCRIPTION:Title: The epsilon-regularity theorem for Brakke flows near triple junct
ions\nby Salvatore Stuvard (University of Milan) as part of NCTS inter
national Geometric Measure Theory seminar\n\n\nAbstract\nIn a pioneering p
aper published on JDG in 1993\, Leon Simon\nestablished a powerful method
to demonstrate\, among other things\, the validity of\nthe following resul
t: if a multiplicity one minimal $k$-dimensional surface (stationary\nvari
fold) is sufficiently close\, in the unit ball and in a weak measure-theor
etic\nsense\, to the stationary cone given by the union of three $k$-dimen
sional half-planes\nmeeting along a $(k-1)$-dimensional subspace and formi
ng angles of 120 degrees\nwith one another\, then\, in a smaller ball\, th
e surface must be a $C^{1\,\\alpha}$\ndeformation of the cone. In this tal
k\, I will present the proof of a parabolic\ncounterpart of this result\,
which applies to general classes of (possibly forced)\nweak mean curvature
flows (Brakke flows). I will particularly focus on the need of\nan assump
tion\, which is absent in the elliptic case\, and which\, on the other han
d\,\nis satisfied by both Brakke flows with multi-phase grain boundaries s
tructure and\nby Brakke flows that are flows of currents mod 3: these are
the main classes of\nBrakke flows for which a satisfactory existence theor
y is currently available and\ntriple junction singularities are expected.
In these cases\, the theorem holds true\nunconditionally\, and it implies
uniqueness of multiplicity-one\, backward-static triple\njunctions as tang
ent flows as well as a structure theorem on the singular set under\nsuitab
le Gaussian density restrictions.\nThis is a joint work with Yoshihiro Ton
egawa (Institute of Science Tokyo).\n
LOCATION:https://stable.researchseminars.org/talk/NCTS-GMT/34/
END:VEVENT
END:VCALENDAR