Title
On smooth approximation of integral cycles

Speaker
Gianmarco Caldini (University of Trento)

The natural question of how much smoother integral currents are with respect to their initial definition goes back to the late 1950s and to the origin of the theory with the seminal article of Federer and Fleming. In this seminar I will explain how closely one can approximate an integral current representing a given homology class with a smooth submanifold. This is a joint study with William Browder and Camillo De Lellis, based on some previous preliminary work of the former author together with Frederick Almgren.

Title
PDE analysis on stable minimal hypersurfaces: curvature estimates and sheeting

Speaker
Costante Bellettini (University College London)

We consider properly immersed two-sided stable minimal hypersurfaces of dimension n. We illustrate the validity of curvature estimates for n ≤ 6 (and associated Bernstein-type properties with an extrinsic area growth assumption). For n ≥ 7 we illustrate sheeting results around "flat points". The proof relies on PDE analysis. The results extend respectively the Schoen-Simon-Yau estimates (obtained for n ≤ 5) and the Schoen-Simon sheeting theorem (valid for embeddings).

Title
Min-max construction of anisotropic minimal hypersurfaces

Speaker
Antonio De Rosa (Bocconi University)

We use the min-max construction to find closed optimally regular hypersurfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed $n$-dimensional Riemannian manifolds. The critical step is to obtain a uniform upper bound for density ratios in the anisotropic min-max construction. This confirms a conjecture posed by Allard [Invent. Math., 1983]. The talk is based on joint work with G. De Philippis and Y. Li.

Title
A sharp extension of Allard’s boundary regularity theorem for area minimizing currents with arbitrary boundary multiplicity

Speaker
Ian Fleschler (Princeton University)

In the context of area-minimizing currents, Allard boundary regularity theorem asserts that an oriented current with boundary that minimizes area cannot have boundary singularities of minimum density. Indeed, in a neighborhood of a point of minimum density, the surface must coincide with a classical smooth minimal surface that attaches smoothly to the boundary.

In this talk, I will discuss a series of papers, one of them in collaboration with Reinaldo Resende, that extend Allard’s boundary regularity theory to a higher boundary multiplicity setting. Specifically, for an area-minimizing current with a multiplicity Q boundary, we study density Q / 2 boundary points. In this context, a regular point is one where smooth submanifolds with multiplicity attach transversally to the boundary. We establish that the set of singular boundary points of minimum density is of boundary codimension at most 2 and rectifiable, extending the corresponding result in 2d by De Lellis - Steinbrüchel - Nardulli to higher dimensional currents. The sharpness of this regularity theory is confirmed by my construction of a 3-dimensional area mininimizing current in R⁵ with a singular boundary point of minimum density.

Title
Regularity of capillary minimal surfaces

Speaker
Nicholas Edelen (University of Notre Dame)

A capillary surface is a hypersurface meeting some container at a prescribed angle, like the surface of water in a cup.  In this talk I describe some recent results concerning the boundary regularity of capillary surfaces which either minimize or are critical for their relevant energy.  The first result (joint with O. Chodosh and C. Li) is an improved dimension bound for the boundary singular set of energy-minimizers, exploiting the connection between capillary minimal surfaces and the one-phase Bernoulli problem.  The second (joint with L. de Masi, C. Gasparetto, and C. Li) is an Allard-type regularity theorem for energy-critical capillary surfaces near capillary half-planes, which implies regularity at generic boundary points of density ﹤１.

Title
Generic Regularity of Minimal Submanifolds

Speaker
Zhihan Wang (Cornell University)

The well-known Simons cone suggests that singularities may exist in a stable minimal hypersurface in Riemannian manifolds of dimension greater than 7, locally modeled on minimal hypercones. It was conjectured 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 shall also talk about certain generalizations of this generic smoothness of minimal submanifold in other dimensions and codimensions as well as their applications.

Title
Constructing diffeomorphisms and homeomorphisms with prescribed derivative

Speaker
Piotr Hajłasz (University of Pittsburgh)

In the talk I will prove that for any measurable mapping T into the space of matrices with positive determinant, there is a diffeomorphism whose derivative equals T outside a set of measure less than ε. Using this fact I will prove that for any measurable mapping T into the space of matrices with non-zero determinant (with no sign restriction), there is an almost everywhere approximately differentiable homeomorphism whose derivative equals T almost everywhere. The talk is based on my joint work with P. Goldstein and Z. Grochulska.

Title
Almgren minimals sets, minimal cones, unions and products

Speaker
Xiangyu Liang (Beihang University)

The notion of Almgren minimal sets is a way to try to solve Plateau’s problem in the setting of sets. To study local structures for these sets, one does blow-up at each point, and the blow-up limits turn out to be minimal cones. People then would like to know the list of all minimal cones.

The list of 1 or 2-dimensional minimal cones in ℝ3 are known for over a century. For other dimensions and codimensions, much less is known. Up to now there is no general way to classify all possible minimal cones. One typical way is to test unions and products of known minimal cones.

In this talk, we will first introduce basic notions and facts on Almgren minimal sets and minimal cones. Then we will discuss the minimality of unions and products of two minimal cones.

Title
Energy identity for stationary harmonic maps

Speaker
Daniele Valtorta (University of Milano Bicocca)

We present the proof for Energy Identity for stationary harmonic maps. In particular, given a sequence of stationary harmonic maps weakly converging to a limit with a defect measure for the energy, then m ⎯２ almost everywhere on the support of this measure the density is the sum of energy of bubbles. This is equivalent to saying that annular regions (or neck regions) do not contribute to the energy of the limit.

This result is obtained via a quantitative analysis of the energy in annular regions for a fixed stationary harmonic map. The proof is technically involved, but it will be presented in simplified cases to try and convey the main ideas behind it. (Preprint available on arXiv:2401.02242)

Title
On the Multiplicity One Conjecture for Mean Curvature Flows of Surfaces

Speaker
Richard Bamler ( University of California, Berkeley)

We prove the Multiplicity One Conjecture for mean curvature flows of surfaces 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-Hershkovits-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 flow equation is well-posed within a natural class of singular solutions. Second, 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 spherical singularities. Third, our approach offers a new regularity theory for solutions of mean curvature flows that flow through singularities.

This talk is based on joint work with Bruce Kleiner.

Title
Uniformly rectifiable metric spaces

Speaker
Raanan Schul (Stony Brook University)

In their 1991 and 1993 foundational monographs, David and Semmes characterized uniform rectifiability for subsets of Euclidean space in a multitude of geometric and analytic ways. The fundamental geometric conditions can be naturally stated in any metric space and it has long been a question of how these concepts are related in this general setting. In joint work with D. Bate and M. Hyde, we prove their equivalence. Namely, we show the equivalence of Big Pieces of Lipschitz Images, Bi-lateral Weak Geometric Lemma and Corona Decomposition in any Ahlfors regular metric space. Loosely speaking, this gives a quantitative equivalence between having Lipschitz charts and approximations by nice spaces. After giving some background, we will explain the main theorems and outline some key steps in the proof (which will include a discussion of Reifenberg parameterizations). We will also mention some open questions. 

Title
Mean curvature flow with surgery

Speaker
Robert Haslhofer (University of Toronto)

Flows with surgery are a powerful method to evolve geometric shapes, and have found many important applications in geometry and topology. 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 measure theory, building in particular on the work of Brakke and White. We illustrate our method in the classical setting of mean-convex surfaces in R3 , thus giving a new proof of the existence results due to Brendle-Huisken and Kleiner and myself. Moreover, our new method also enables the construction of flows with surgery in situations that have been inaccessible with prior techniques, including in particular the free-boundary setting.

Title
Rectifiability, finite Hausdorff measure, and compactness for non-minimizing Bernoulli free boundaries

Speaker
Georg Weiss (University of Duisburg-Essen)

While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less is known about critical points of the corresponding energy. Saddle points of the energy (or of closely related energies) and solutions of the corresponding time-dependent problem occur naturally in applied problems such as water waves and combustion theory.

For such critical points u–which can be obtained as limits of classical solutions or limits of a singular perturbation problem–it has been open since [Weiss03] whether the singular set can be large and what equation the measure ∆u satisfies, except for the case of two dimensions. In the present result we use recent techniques such as a frequency formula for the Bernoulli problem as well as the celebrated Naber-Valtorta procedure to answer this more than 20 year old question in an affirmative way: 

For a closed class we call variational solutions of the Bernoulli problem, we show that the topological free boundary ∂{u > 0} (including degenerate singular points x, at which u(x + r·)/r → 0 as r → 0) is countably H n-1-rectifiable and has locally finite H n-1-measure, and we identify the measure ∆u completely. This gives a more precise characterization of the free boundary of u in arbitrary dimension than was previously available even in dimension two. 

We also show that limits of (not necessarily minimizing) classical solutions as well as limits of critical points of a singularly perturbed energy are variational solutions, so that the result above applies directly to all of them. 

This is a joint work with Dennis Kriventsov (Rutgers).

Title
Analysis of singularities of area minimizing currents

Speaker
Brian Krummel (University of Melbourne)

The 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 ≥ 2 has Hausdorff dimension at most n − 2. In contrast to codimension 1 area minimizers (for which it had been established a decade earlier that the singular set has Hausdorff dimension at most n − 7), the problem in higher codimension is substantially more complex because of the presence of branch point singularities, i.e. singular points where one tangent cone is a plane of multiplicity 2 or larger. Almgren’s lengthy proof (made more accessible and technically streamlined in the much more recent work of De Lellis-Spadaro) showed first that the non-branch-point singularities form a set of Hausdorff dimension at most n − 2 using an elementary argument based on the tangent cone type at such points, and developed a powerful array of ideas to obtain the same dimension bound for the branch set separately. In this 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.

We will discuss a new approach to this problem (joint work with Neshan Wickramasekera). In this 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 B  of branch points where T decays towards a (unique) plane faster than a fixed exponential rate, and the complementary set S. The set S contains all (density q) non-branch-point singularities, but a priori it could also contain a large set of branch points. To analyze S, the work introduces a new, intrinsic frequency function for T relative to a plane, called the planar frequency function. The planar frequency function satisfies an approximate monotonicity property, and takes correct values (i.e. ≤ 1) whenever T is 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 together with relatively elementary parts of Almgren’s theory (Dirichlet energy minimizing multivalued functions and strong Lipschitz approximation) imply that T satisfies a key approximation property along S: near each point of S and at each sufficiently small scale, T is significantly closer to some non-planar cone than to any plane. This property 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 imply that T has a unique non-planar tangent cone at H n-2-a.e. point of S and that S is (n − 2)-rectifiable with locally finite measure. Analysis of B using the planar frequency function and the locally uniform decay estimate along 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 B (and hence the entire singular set of T ) is countably (n − 2)-rectifiable with a unique, non-zero multi-valued harmonic blow-up at    H n-2-a.e. point of B.

Title
Generic regularity of minimizing hypersurfaces in dimensions 9 and 10

Speaker
Christos Mantoulidis (Rice University)

In joint work with Otis Chodosh and Felix Schulze we showed that the problem of finding a least-area compact hypersurface with prescribed boundary or homology class has a smooth solution for generic data in dimensions 9 and 10. In this talk I will explain the main steps of the proof.

Wednesday, 15  March 2023, 4:00-6:00p.m. (Taipei time)

Title
Bi-Lipschitz regularity of 2-varifolds with the critical Allard condition

Speaker
Jie Zhou (Capital Normal University)

Abstract
For an integral 2-varifold in the unit ball of the Euclidean space passing through the origin, if it satisfies the critical Allard condition, i.e., the mass of the varifold in the unit ball is close to the area of a flat unit disk and the L2 norm of the generalized mean curvature is small enough, we show that locally the support of the varifold admits a bi-Lipschitz parameterization from the unit disk. The presentation is based on a joint work with Dr. Yuchen Bi.

Title
Singularities, Rectifiability, and PDE-constraints

Speaker
Filip Rindler (University of  Warwick)

Surprisingly many different problems of Analysis 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 "applied" 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 (under-determined) PDE constraint, e.g., divergence- or curl-freeness. The crucial task is then to analyse the fine properties of these PDE-constrained measures, in particular to determine the possible singularities that may occur. It turns out that the PDE constraint imposes strong restrictions on the shape of these singularities, for instance that they can only occur on a set of bounded Hausdorff-dimension, or even that the measure is k-rectifiable where its upper k-density is positive. The essential difficulty in the analysis of PDE-constrained measures is that many standard methods from harmonic analysis are much weaker in an L1-context and thus new strategies are needed. In this talk, I will survey recent and ongoing work on this area of research.

Title
The spherical Plateau problem: existence, uniqueness, stability

Speaker
Antoine Song (California Institute of Technology)

Consider a countable group G acting on the unit sphere S in the space of L2  functions on G by the regular representation. Given a homology class h in the quotient space S/G, one defines the spherical Plateau solutions for h as the intrinsic flat limits of volume minimizing sequences of cycles representing h. Interestingly in some special cases, for example when G is the fundamental group of a closed hyperbolic manifold of dimension at least 3, the spherical Plateau solutions are essentially unique and can be identified. However in general not much is known. I will discuss the questions of existence and structure of non-trivial Plateau solutions. I will also explain how uniqueness of spherical Plateau solutions for hyperbolic manifolds of dimension at least 3 implies stability for the volume entropy inequality of Besson-Courtois-Gallot.

Wednesday, 21  September 2022, 8:00-10:00 p.m. (Taipei time)

Title
Minimal hypersurfaces with cylindrical tangent cones

Speaker
Gábor Székelyhidi (Northwestern University)

I will discuss recent results on minimal hypersurfaces with cylindrical tangent cones of the form C × R, where C is a minimal quadratic cone, such as the Simons cone over S3 × S3.  I will talk about a uniqueness result for such tangent cones in a certain non-integrable situation, as well as a precise description of such minimal hypersurfaces near the singular set under a symmetry assumption.

Wednesday, 20 July  2022,  8:00-10:00 p.m. (Taipei time)

Title
Hypersurfaces with prescribed-mean-curvature: existence and properties

Speaker
Costante Bellettini (University College London).

Let N be a compact Riemannian manifold of dimension 3 or higher, and g a Lipschitz non-negative (or non-positive) function on N. In joint works with Neshan Wickramasekera we prove that there exists a closed hypersurface M whose mean curvature attains the values prescribed by g. Except possibly for a small singular set (of codimension 7 or higher), the hypersurface M is  C2   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 (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-curvature) hypersurface for any prescribed value of the mean curvature.

Title
(Non-)quantization phenomena for higher-dimensional Ginzburg-Landau vortices

Speaker
Alessandro Pigati (New York University, Courant Institute)

The Ginzburg-Landau energies for complex-valued maps, initially introduced to model superconductivity, were later found to approximate the area functional in codimension two.

While the pioneering works of Lin-Rivière and Bethuel-Brezis-Orlandi (2001) showed that, for families of critical maps, energy does concentrate along a codimension-two minimal submanifold, it has been an open question whether this happens with integer 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} U [2,∞).

Title
A Regularity Theorem for Area-minimizing Currents at Higher Multiplicity Boundary Points

Speaker
Simone Steinbrüchel (Leipzig University)

Abstract
The 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 first 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 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 is regular in the above sense. This is a joint work with C. De Lellis and S. Nardulli.

Title
A Structure Theory for Branched Stable Hyper-surfaces

Speaker
Paul Minter (University of Cambridge)

Abstract
There are few known general regularity results for stationary integral varifolds aside from Allard’s celebrated theory. The primary reason for this is the possibility of a degenerate type of singularity known as a branch point, where at the tangent cone level singularities vanish and are replaced with regions of higher multiplicity. In this talk I will discuss a recent regularity theory for branched stable hypersurfaces which do not contain certain so-called classical singularities, including new tangent cone uniqueness results in the presence of branch points. This theory can be readily applied to area minimising hypercurrents mod p, which resolves an old conjecture from the work of Brian White. Some results are joint with Neshan Wickramasekera.

Title
Stable minimal hypersurfaces in R^4

Speaker
Otis Chodosh (Stanford University)

Abstract 
I will explain why stable minimal hypersurfaces in R^4 are flat. This is joint work with Chao Li.

Title
Free boundary regularity in the Stefan problem

Speaker
Alessio Figa lli (ETH Zurich)

Abstract
The Stefan problem describes phase transitions, such as ice melting to water. In its simplest formulation, this problem consists of finding the evolution of the temperature off the water when a block of ice is submerged inside.

In this talk, I will first discuss the classical theory for this problem. Then I will present some recent results concerning the fine regularity properties of the interface separating water 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 singularities, and they answer some long-standing open questions in the field.

Title
A non-linear Besicovitch–Federer projection theorem for metric spaces

Speaker
David Bate (University of Warwick)

Abstract 
This talk will present a characterisation of purely n-unrectifiable subsets S of a complete metric space with finite n-dimensional Hausdorff measure by studying non-linear projections (i.e. 1-Lipschitz functions) into some fixed Euclidean space. We will show that a typical (in the sense of Baire category) non-linear projection maps S to a set of zero n-dimensional Hausdorff measure. Conversely, a typical non-linear projection maps an n-rectifiable subset to a set of positive n-dimensional Hausdorff measure. These results provide a replacement for the classical Besicovitch–Federer projection theorem, which is known to be false outside of Euclidean spaces.

Time permitting, we will discuss some recent consequences of this characterisation.

Title
Geometric Measure Theory: a powerful tool in Potential Theory

Speaker
Tatiana Toro (University of Washington)

Abstract 
In this talk, I will describe a couple of instances in which ideas coming from geometric measure theory have played a central role in proving results in potential theory. Understanding limits of measures associated to second order divergence form operators has allowed us to establish equivalences between boundary regularity properties of solutions to these operators and the domains where they are defined.

Title
Mean Curvature Flow with Generic Initial Data

Speaker
Felix Schulze (University of Warwick)

Abstract 
A well-known conjecture of Huisken states that a generic mean curvature flow has only spherical and cylindrical singularities. 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 that mean curvature flow of generic initial closed surfaces in R^3 avoids asymptotically conical and non-spherical compact singularities. We also show that mean curvature flow of generic closed low-entropy hypersurfaces in R^4 is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result 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.

Title
Second Order Estimates for Interfaces of Allen-Cahn 

Speaker
Juncheng Wei (University of British Colombia)

Abstract 
In this talk,  I will discuss a uniform $C^{2, \theta}$ estimate for level sets of stable solutions to the singularly perturbed Allen-Cahn equation in dimensions $n \leq 10$ (which is optimal). The proof combines two ingredients: one is a reverse application of the infinite dimensional Lyapunov-Schmidt reduction method which enables us to reduce the $C^{2, \theta}$ estimate for these level sets to a corresponding one on solutions of Toda system; the other one uses a small regularity theorem on stable solutions of Toda system to establish various decay estimates, which gives a lower bound on distances between different sheets of solutions to Toda system or level sets of solutions to Allen-Cahn equation. (Joint work with Kelei Wang.)

Title
Stable minimal hypersurfaces in $\R^{N+1+\ell}$ with singular set an arbitrary closed $K\subset\{0\}\times\R^{\ell}$. 

Speaker
Leon Simon (Stanford University)

Abstract 
With 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 and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset $K\subset\{0\}\times\R^{\ell}$.