Mar 21, 2022
Monday

07:55 AM  08:00 AM


Welcome

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08:00 AM  08:45 AM


Singularities and Diffeomorphisms
Tobias Colding (Massachusetts Institute of Technology)

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Comparing and recognizing metrics can be extraordinarily difficult because of the group of diffeomorphisms. Two metrics, that could even be the same, could look completely different in different coordinates. This is the gauge problem. The general gauge problem is extremely subtle for noncompact spaces. Often it can be avoided if one uses some additional structure of the particular situation. However, in many problems there is no additional structure. Instead we solve the gauge problem directly in great generality. We use them to solve a wellknown open problem in Ricci flow.
We solve the gauge problem by solving a nonlinear system of PDEs. The PDE produces a diffeomorphism that fixes an appropriate gauge in the spirit of the slice theorem for group actions. We then show optimal bounds for the displacement function of the diffeomorphism.
This is joint work with Bill Minicozzi.
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09:00 AM  09:45 AM


Degeneration of 7Dimensional Minimal Hypersurfaces with Bounded Index
Nick Edelen (University of Notre Dame)

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A 7D minimal and locallystable hypersurface will in general have a discrete singular set, provided it has no singularities modeled on a union of halfplanes. We show in this talk that the geometry/topology/singular set of these surfaces has uniform control, in the following sense: if $M_i$ is a sequence of 7D minimal hypersurfaces with uniformly bounded index and area, and discrete singular set, then up to a subsequence all the $M_i$ are biLipschitz equivalent, with uniform Lipschitz bounds on the maps. As a consequence, we prove the space of $C^2$ embedded minimal hypersurfaces in a fixed $8$manifold, having index $\leq I$, area $\leq \Lambda$, and discrete singular set, divides into finitelymany diffeomorphism types.
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09:45 AM  10:15 AM


Break

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10:15 AM  11:00 AM


MinMax Minimal Hypersurfaces with Higher Multiplicity
Xin Zhou (Cornell University)

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It is well known that minimal hypersurfaces produced by the AlmgrenPitts minmax theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the MarquesNeves Multiplicity One Conjecture. In this talk, we will exhibit a set of nonbumpy metrics on the standard (n+1)sphere, in which the minmax varifold associated with the second volume spectrum is a multiplicity two nsphere. Such nonbumpy metrics form the first set of examples where the minmax theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).
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11:00 AM  11:45 AM


Regularity of Free Boundary Minimal Surfaces in Locally Polyhedral Bomains
Chao Li (New York University, Courant Institute)

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We prove an Allardtype regularity theorem for free boundary minimal surfaces in Lipschitz domains locally modelled on convex polyhedra. We show that if such a minimal surface is sufficiently close to an appropriate freeboundary plane, then the surface is graphical over this plane. We apply our theorem to prove partial regularity results for freeboundary minimizing hypersurfaces, and isoperimetric regions. This is based on a joint work with Nick Edelen.
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Mar 22, 2022
Tuesday

08:00 AM  08:45 AM


Free Boundary Clusters with Two Phases
Bozhidar Velichkov (Università di Pisa)

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We will discuss a twophase free boundary problem in which the two state functions are not vanishing on the free interface between the two phases. The talk is based on joint works with Serena Guarino Lo Bianco and Domenico Angelo La Manna.
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09:00 AM  09:45 AM


Explicit Łojasiewicz Inequalities for Shrinking Solitons
Jonathan Zhu (Princeton University)

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Łojasiewicz inequalities have become a popular tool for studying the stability of geometric structures. For mean curvature flow, Schulze used Simon’s reduction to the classical Łojasiewicz inequality to study compact tangent flows. For round cylinders, Colding and Minicozzi instead used a direct method to prove Łojasiewicz inequalities. We’ll discuss similarly explicit Łojasiewicz inequalities and applications for other shrinking cylinders and products of spheres.
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09:45 AM  10:15 AM


Break

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10:15 AM  11:00 AM


Type II Smoothing in Mean Curvature Flow
Panagiota Daskalopoulos (Columbia University)

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In 1994 Vel\'azquez constructed a family of smooth \( O(4)\times O(4)\) invariant solutions to Mean Curvature Flow that form a typeII singularity at the origin. Stolarski has recently shown that the Vel\'azquez solutions have bounded Mean curvature at the singularity. Earlier, Vel\'azquez also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity.
Jointly with S. Angenent and N. Sesum, we establish the short time existence of Velázquez's formal continuation, and we verify that the Mean curvature is also uniformly bounded. Combined with the earlier results of Vel\'azquezStolarski we therefore show the existence of a Mean curvature flow solution \( \{ M_t^7\subset \R^8 \}_{t_0 < t < t_0 } \), that has an {\em isolated singularity} at the origin \( 0\in\R^8\) at time \(t=0\). Moreover, the {\em Mean curvature is uniformly bounded} on this solution, even though the second fundamental form is unbounded near the singularity.
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11:00 AM  11:45 AM


Boundary Regularity of AreaMinimizing Currents: a Linear Model with Analytic Interface
Zihui Zhao (University of Chicago)

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Given a curve Γ, what is the surface of least area spanning Γ? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or roughly speaking, orientable submanifolds. Since the 1960s a lot of work has been done by De Giorgi, Almgren, et al to study the regularity of these minimizers at the interior. Much less is known about regularity at the boundary (in the case of codimension greater than 1). Recently, De Lellis et al. have found surprising examples of boundary singularity even when the prescribed curve Γ is smooth. I will speak about some recent progress in this direction and my joint work with C. De Lellis.
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Mar 23, 2022
Wednesday

08:00 AM  08:45 AM


The Spherical Plateau Problem
Antoine Song (University of California, Berkeley)

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For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.
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09:00 AM  09:45 AM


Ancient Solutions and Translators in Lagrangian Mean Curvature Flow
Felix Schulze (University of Warwick)

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Tangent flows at singularities for locally almost calibrated solutions to Lagrangian mean curvature flow in C^2 are modelled on unions of static planes, and all singularities are of Type II. To understand the finer singularity structure at such singularities it is thus necessary to understand all possible limit flows. As an essential step in this direction we show that any ancient solution with a blowdown a union of two static multiplicity one planes, meeting along a line has to be a translator. Together with previous work together with Lambert and Lotay this gives a full picture of all ancient solutions with entropy less than three. This is joint work with J. Lotay and G. Szekelyhidi.
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09:45 AM  10:15 AM


Break

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10:15 AM  11:00 AM


Some Regularity Questions for the Special Lagrangian Equation
Connor Mooney (University of California, Irvine)

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The special Lagrangian equation is a fully nonlinear elliptic PDE whose solutions are potentials for volumeminimizing Lagrangian graphs. There exist continuous viscosity solutions to the Dirichlet problem for this equation, but many basic regularity questions (such as whether these solutions have bounded gradient) remain open. In this talk we will discuss some of these questions. We will use them to motivate the more general question of whether homogeneous functions with nowhere vanishing Hessian determinant can change sign when the degree of homogeneity is between zero and one, and we will answer this question in the negative.
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11:00 AM  11:45 AM


The Waist Inequality and Positive Scalar Curvature
Davi Maximo (University of Pennsylvania)

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The topology of threemanifolds with positive scalar curvature has been (mostly) known since the solution of the Poincare conjecture by Perelman. Indeed, they consist of connected sums of spherical space forms and S^2 x S^1's. In spite of this, their "shape" remains unknown and mysterious. Since a lower bound of scalar curvature can be preserved by a codimension two surgery, one may wonder about a description of the shape of such manifolds based on a codimension two data (in this case, 1dimensional manifolds). In this talk, I will show results from a recent collaboration with Y. Liokumovich elucidating this question for closed threemanifolds.
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Mar 24, 2022
Thursday

08:00 AM  08:45 AM


Entire Spacelike Hypersurfaces with Constant Curvature in Minkowski Space
Ling Xiao (University of Connecticut)

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We prove that, in the Minkowski space, if a spacelike, (n − 1)convex hypersurface M with constant $\sigma_{n−1}$ curvature has bounded principal curvatures, then M is convex. Moreover, if M is not strictly convex, after an R^{n,1} rigid motion, M splits as a product $M^{n−1}\times R.$ We also construct nontrivial examples of strictly convex, spacelike hypersurfaces M with constant $\sigma_k$ curvature and bounded principal curvatures. This is a joint work with Changyu Ren and Zhizhang Wang.
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09:00 AM  09:45 AM


The Space of Convex Ancient Solutions to Mean Curvature Flow
Natasa Sesum (Rutgers University)

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09:45 AM  10:15 AM


Break

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10:15 AM  11:00 AM


Translators for Mean Curvature Flow
Brian White (Stanford University)

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