The talks are held on Wednesdays from 4:00PM to 6:00PM, in Room 2-131 (unless otherwise noted).

- Title:
- Collapsing of Ricci-flat Calabi-Yau metrics
- Abstract:

- Title:
- TBA
- Abstract:
- TBA

- Title:
- Almost Rigidity Theorems with Nonnegative Scalar Curvature
- Abstract:

- Title:
- Uniqueness of closed self-similar solutions to the Gauss curvature flow
- Abstract:

- Title:
- Minimal surfaces in asymptotically flat 3-manifolds
- Abstract:

- Title:
- Geodesics via Allen-Cahn min-max on surfaces
- Abstract:

Monday, 4:00PM to 5:00PM, in 2-146

Monday, 4:00PM to 5:00PM, in 2-146

- Title:
- Embeddedness of the solutions to the H-Plateau Problem
- Abstract:

- Title:
- The local inverse problem for the geodesic X-ray transform on tensors and boundary rigidity
- Abstract:

CANCELLED

- Title:
- A PDE Approach to Prediction with Expert Advice
- Abstract:

Friday, 1:30pm in 2-147

- Title:
- Global existence of weak solution for volume preserving mean curvature flow via phase field method
- Abstract:

Friday, 2:00pm in 2-146

- Title:
- Non-Kaehler Ricci Flows that Converge to Kaehler-Ricci Solitons
- Abstract:

- Title:
- Free Boundary Regularity for Almost-Minimizers
- Abstract:

- Title:
- Min-max minimal hypersurfaces with free boundary
- Abstract:

- Title:
- The topology of min-max minimal surfaces
- Abstract:

- Title:
- The geometry of constant mean curvature surfaces in Euclidean space.
- Abstract:

- Title:
- Fill-ins, extensions, scalar curvature, and quasilocal mass
- Abstract:

- Title:
- CMC surfaces and CSC metrics with cylindrical ends
- Abstract:

- Title:
- Approximation of optimal constants and extremal functions for PoincarĂ©'s inequality
- Abstract:

- Title:
- The free-boundary Brakke flow
- Abstract:

- Title:
- Dynamical zeta functions and topology for negatively curved surfaces
- Abstract:

- Title:
- Existence of minimal hypersurfaces in complete manifolds of finite volume
- Abstract:

- Title:
- Regularity theory for 2-dimensional almost area minimizing currents
- Abstract:

- Title:
- Approaching Plateau's problem with minimizing sequences of sets
- Abstract:

- Title:
- The complexity of a minimal subvariety and the index conjecture
- Abstract:

- Title:
- An extension question for metrics and the Bartnik mass in General Relativity
- Abstract:

- Title:
- Four dimensional complete manifolds with positive isotropic curvature
- Abstract:

- Title:
- Boundary behaviour of solutions to singular elliptic equations
- Abstract:

- Title:
- Kaehler-Einstein metrics and higher alpha-invariants
- Abstract:
- I will describe a condition on the Bergman metrics of a Fano manifold M, which guarantees the existence of a Kaehler-Einstein metric on M. I will also discuss a conjectural relationship between this condition and M's higher alpha-invariants $\alpha_{m,k}(M)$, analogous to a 1991 theorem of Tian for $\alpha_{m,2}(M)$.

- Title:
- The geometry of constant mean curvature disks embedded in R3.
- Abstract:
- In this talk, I will survey several results on the geometry of constant mean curvature surfaces embedded in R3. Among other things, I will prove radius and curvature estimates for nonzero constant mean curvature embedded disks. It then follows from the radius estimate that the only complete, simply connected surface embedded in R3 with constant mean curvature is the round sphere. This is joint work with Bill Meeks.

- Title:
- Harmonic maps between hyperbolic spaces
- Abstract:
- We prove that any quasi-conformal map of the (n-1)-dimensional sphere can be extended to a harmonic quasi-isometry of the n-dimensional hyperbolic space which confirms the Schoen's Conjecture.

- Title:
- Weak harmonic map flows with singular targets
- Abstract:
- In this talk we consider two heat flow constructions from Euclidean domains into non-smooth spaces. The first, a nonlocal constrained heat flow into a metric tree, was originally motivated by a stationary eigenvalue partition problem. We prove spatial Lipschitz regularity of this flow, free interface regularity, and characterize the limit of the flow as time goes to infinity as a stationary solution of the partition problem. Next, we discuss the extension of this study to non-constrained heat flows into F-connected simplicial complexes, which are natural generalizations of trees to higher dimensions.

- Title:
- On the structure of the zero set of monochromatic random waves
- Abstract:
- There are several questions about the zero set of Laplace eigenfunctions that have proved to be extremely hard to deal with and remain unsolved. Among these are the study of the size of the zero set, the study of the number of connected components, and the study of the topology of such components. A natural approach is to randomize the problem and ask the same questions for the zero sets of random linear combinations of eigenfunctions. In this talk I will present some recent results in this direction.

- Title:
- Limits of convex real projective manifolds
- Abstract:
- We describe a compactification of the Hitchin component of $PSL(3,R)$ using certain geometric structures on a surface called mixed structures. Joint with Kelly Delp.

- Title:
- Min-max minimal hypersurfaces in noncompact manifolds
- Abstract:
- In this talk, I will discuss the existence of embedded closed minimal hypersurfaces in complete noncompact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For doing this, we develop a modified min-max theory for the area functional following Almgren-Pitts' setting, to produce minimal hypersurfaces with intersecting properties.

- Title:
- Singularities of Minimal Graphs
- Abstract:
- The speaker has recently obtained results describing singularities of two-valued (i.e. 'two-sheeted') minimal graphs. An early major breakthrough in the study of the regularity of weak solutions to non-linear PDE was De Giorgi's answer to the question: Can a codimension one minimal graph have singularities? (By "minimal" we mean "critical point of the area functional"). The answer is: No, such a graph must be smooth. The ideas used to obtain this answer were developed first by De Giorgi himself and later by Allard, Almgren and others into a robust philosophy capable of proving certain regularity results for very general classes of minimal submanifolds (the so-called "blow-up method"). One of the aims of this talk is to explain this philosophy. Typically however, singularities are unavoidable (e.g. in higher codimension minimal graphs) and understanding them becomes a central issue. The second aim of this talk is, with particular reference to the work of the speaker, to outline some of the ideas behind Simon's modifications of the blow-up method in the 90's that opened the door to more precise descriptions of singularities, including results about the regularity of the singular set.

- Title:
- Regularity scales and convergence of the Calabi flow
- Abstract:
- We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson's conjectural picture for the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra assumption that the scalar curvature is uniformly bounded. This is joint work with Bing Wang and Kai Zheng.

- Title:
- Flows of non-smooth vector fields and applications to PDEs
- Abstract:
- The classical Cauchy-Lipschitz theorem shows existence and uniqueness of the flow of any sufficiently smooth vector field in R^d. In 1989, Di Perna and Lions proved that Sobolev regularity for vector fields, with bounded divergence and a growth assumption, is sufficient to establish existence, uniqueness and stability of a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE. Their theory relies on a growth assumption on the vector field which prevents the trajectories from blowing up in finite time; in particular, it does not apply to fast-growing, smooth vector fields.

In this seminar we give an overview of the topic and we introduce a notion of maximal flow for non-smooth vector fields which allows for finite-time blow up of the trajectories. We show existence and uniqueness under only local assumptions on the vector field and we apply the result to the Vlasov-Poisson system, a classical model in physics used to describe the evolution of particles under their self-consistent electric or gravitational field. We show that, even for weak solutions, the Eulerian description provided by the equation physically corresponds to a Lagrangian evolution of the particles, by describing the solutions as transported by a suitable flow in the phase space. This allows, in turn, to prove existence of weak solutions for general initial data.

- Title:
- Quasi-local mass in general relativity
- Abstract:
- While the total energy of an isolated system in general relativity is well-studied, the concept of energy in general relativity remains a challenging problem because of the lack of a quasi-local description. In this talk, we survey several definition of quasi-local mass including the Hawking mass, the Brown-York mass, and their applications. We then describe a new proposal of quasi-local mass/energy and discuss its application and properties.

- Title:
- Concentrations in kinetic transport equations

- Title:
- Stationary Kirchhoff systems in closed manifolds
- Abstract:
- We discuss recent results we obtained for the critical stationary Kirchhoff systems in closed manifolds. We present results dealing with the question of the existence of nontrivial solutions to our system, with the dual question of getting nonexistence results in parallel to our existence results, and with the question of the stability of our systems. This is joint work with P.D. Thizy.

- Title:
- Windows, cores and skinning maps.
- Abstract:
- Thurston's skinning map played an important role in his original hyperbolization theorem. His Bounded Image Theorem, in the acylindrical case, gave control of the fixed-point problem on which the proof depends. We generalize this theorem to a relative version which holds for manifolds with cylinders, and along the way refine our understanding of how one constructs compact cores with uniform topological and geometric control in degenerating sequences of hyperbolic structures. This is joint work with J. Brock, K. Bromberg and R. Canary.

- Title:
- Minimal Surfaces with Arbitrary Topology in $H^2xR$
- Abstract:
- In this talk, we show that any open orientable surface can be embedded in H^2xR as a complete area minimizing surface. Furthermore, we will discuss the asymptotic Plateau problem in $H^2xR$, and give a fairly complete solution.

- Title:
- Shape of the min-max minimal hypersurface in manifold of positive Ricci curvature in all dimensions
- Abstract:
- We will discuss the shape of the min-max minimal hypersurface produced by Almgren and Pitts corresponding to the fundamental class of a Riemannian manifold of positive Ricci curvature. We will give a characterization of the Morse index, area and multiplicity. The min-max minimal hypersurface will have a singular set of codimension 7, and the talk will give new ideas to deal with the singularity issue.

- Title:
- A Weitzenbock formula for canonical metrics on four-manifolds and applications
- Abstract:
- In this talk, we will first provide an alternative proof of Derdzinski's Weitzenbock formula for Einstein four-manifolds, and extend it to a class of canonical metrics, which are called generalized quasi-Einstein metrics. As applications we show that a four-dimensional gradient shrinking Ricci soliton of half harmonic Weyl curvature is either Einstein, or a finite quotient of $S^3\times\mathbb{R}$, $S^2\times\mathbb{R}^2$, or $\mathbb{R}^4$ (joint with Jiayong Wu and William Wylie). We will also show that if a conformally Einstein metric has half nonnegative isotropic curvature, then the Einstein metric is either the standard metric on $S^4$ or a Hermitian-Einstein metric (joint with Jeffrey Case).

- Title:
- How mean curvature flow shapes cell structures
- Abstract:
- Geometric flows have played an important role in contemporary pure and applied mathematics. For a surface embedded in space, mean curvature flow (MCF) is the steepest descent gradient flow of the area functional. Many common materials -- including most metals and many ceramics -- have an underlying cellular structure, and have a free energy approximately proportional to the total area of all surfaces. Therefore, studying MCF on cellular microstructures can help us understand the evolution and long-time properties of these systems. In this talk I will describe some interesting numerical results about singularities of this flow, and about the "shapes" that result from this process. This talk is based on joint work with Robert MacPherson, Jeremy Mason, and David Srolovitz.

- Title:
- Shock Formation in Solutions to $3D$ Wave Equations
- Abstract:
- I will provide an overview of the formation of shock waves, developing from small, smooth initial conditions, in solutions to quasilinear wave equations in $3$ spatial dimensions. I will first describe prior contributions from many researchers including F. John, S. Alinhac, and especially D. Christodoulou. I will then describe some results from my recent book, in which I show that for two important classes of wave equations, a \emph{necessary and sufficient} condition for the phenomenon of small-data shock-formation is the failure of S. Klainerman's classic null condition. I will highlight some of the main ideas behind the analysis including the critical role played by geometric decompositions based on true characteristic hypersurfaces. Some aspects of this work are joint with G. Holzegel, S. Klainerman, and W. Wong.

- Title:
- Regularity scales of the Calabi flow
- Abstract:
- We introduce regularity scales to study the behavior of Calabi flow. Based on the estimates of regularity scales, we obtain convergence theorems of the Calabi flow on Kahler surfaces with the long time existence assumption.This is joint work with Bing Wang and Kai Zheng.

- Title:
- Volume distortion in homotopy groups: a safari
- Abstract:
- Given a nice compact metric space $X$, how can we use geometry to better understand elements $\alpha \in \pi_n(X)$? One way is by measuring distortion, that is, how geometric measurements of an optimal representative of $k\alpha$, such as Lipschitz constant or volume, grow as a function of $k$. The Lipschitz distortion of simply connected spaces is the subject of a conjecture of Gromov; volume distortion has not been previously studied. I will present examples of three ways that volume distortion can arise: from rational homotopy invariants, from the action of the fundamental group on higher homotopy groups, and from the geometry of the fundamental group. These three sources of distortion turn out to be enough to characterize those spaces which have no distorted elements.

- Title:
- Regularity for Almost Minimizers with Free Boundary
- Abstract:
- In recent work with Guy David we introduce the notion of almost minimizer for a series of functionals previously studied by Alt-Caffarelli and Alt-Caffarelli-Friedman. We prove regularity results for these almost minimizers and explore the structure of the corresponding free boundary. A key ingredient in the study of the 2-phase problem is the existence of almost monotone quantities. The goal of this talk is to present these results in a self-contained manner, emphasizing both the similarities and differences between minimizers and almost minimizers.

- Speaker:
- Bruno Martelli - 4PM
- Title:
- Combinatorial constructions of hyperbolic and Einstein four-manifolds.
- Abstract:
- We expose an algorithm which transforms a simple combinatorial object (a four-dimensional cubulation) into a cusped hyperbolic finite-volume 4-manifold tessellated by ideal hyperbolic regular 24-cells. The algorithm allows to construct easily plenty of cusped hyperbolic finite-volume 4-manifolds, including the first examples of hyperbolic 4-manifolds having only one cusp. By decorating the combinatorial object with numbers we may also encode various closed Einstein four-manifolds using the extended Dehn filling theorem recently proved by Anderson.

- Speaker:
- Matteo Novaga - 5PM
- Title:
- Mean curvature flow of multiple phase systems.
- Abstract:
- I will review some recent results on the evolution by mean curvature of multiple phase systems in two and three dimensions, and I will discuss some open problems.

- Title:
- Embedding Riemannian manifolds with heat kernels and eigenfunctions of the Laplacian
- Abstract:
- Several methods in nonlinear data analysis use eigenfunctions of the Laplace operator on manifolds on which data is supported to embed these manifolds in a Euclidean space. We show that under bounds on the Ricci curvature, injectivity radius and diameter, the complexity of the continuum versions of such methods can be controlled. That is, the number of eigenfunctions or heat kernels needed depends only on these geometric bounds, the dimension and a tolerance for the dilatation.

- Title:
- Geometric spaces of geometric structures
- Abstract:
- I'll give an expository introduction to spaces of geometric structures, and the power of considering such spaces as geometric objects in their own right. I'll start with classical examples like the space of inner products on a real vector space, the space of flat n-tori, and other (locally) symmetric spaces. I'll then discuss two sporadic but important spaces that arise only in low dimensions: "moduli space", the space of hyperbolic surfaces, and "Outer space", the space of metric graphs. A common thread will be the presence of negative / non-positive curvature, both coarse and fine. The talk is designed to be useful for people working in varied fields, including geometry, topology, and representation theory; no background is assumed.

- Title:
- Ricci flow through singularities
- Abstract:
- It has been a long-standing problem in geometric analysis to find a good definition of generalized solutions to the Ricci flow equation that would formalize the heuristic idea of flowing through singularities. I will discuss a notion in the 3-d case that has good analytical properties, enabling one to prove existence and compactness of solutions, as well as a number of structural results. It may also be used to partly address a question of Perelman concerning the convergence of Ricci flow with surgery to a canonical flow through singularities. This is joint work with John Lott.

- Title:
- Compactness theorems for weighted minimal surfaces
- Abstract:
- A hypersurface $\Sigma$ immersed in a Riemannian manifold $(M,\bar{g})$ is called an $f$-minimal hypersurface if its mean curvature $H$ satisfies that $$H=\langle \overline{\nabla} f,\nu\rangle,$$ where $f$ is a smooth function defined on $M$, $ \overline{\nabla} f$ denotes the gradient of $f$ on $M$, and $\nu$ is the outward normal vector of $\Sigma$. They appear naturally in the singularity analysis of mean curvature flow in Riemannian manifolds. One of nontrivial examples is self-shrinker in Euclidean space. In this talk, we will present some smooth compactness theorems for the space of complete embedded $f$-minimal surfaces with the uniform upper bounds of genus and the weighted volume in a complete $3$-manifold with Bakry-\'Emery Ricci curvature bounded below by a positive constant. The results generalize the compactness theorems for complete self-shrinkers in $\mathbb{R}^3$ by Colding-Minicozzi. This is a joint work with X. Cheng and T. Mejia.

- Title:
- Eigenvalues of the drifted Laplacian on complete metric measure spaces
- Abstract:
- The Lichnerowicz-Obata theorem states that if the Ricci curvature of a complete $n$-dimensional Riemannian manifold $M$ is bounded below by a positive constant $(n-1)a$, then the first nonzero eigenvalue of the Laplacian satisfies $\lambda_1 \geq na$. Moreover the equality holds if and only if the manifold is a round sphere. When a complete smooth metric measure space $(M^n,g, e^{-f}dv)$ has the Bakry-\'Emery Ricci curvature tensor $Ric+\nabla^2f\geq ag$, for some positive constant $a$, $M$ may be non-compact. It is known that the spectrum of the drifted Laplacian $\Delta_f=\Delta-\langle \nabla f, \nabla\cdot\rangle$ on $M$ is discrete and the first nonzero eigenvalue of $\Delta_f$ has lower bound $a$. In this talk, we will discuss the rigidity of this lower bound and prove that if it is achieved with multiplicity $k$, then $M$ is isometric to $\Sigma^{n-k}\times \mathbb{R}^k$ for some complete $(n-k)$-dimensional manifold $\Sigma$. One example is gradient shrinking solitons. We also discuss the case of self-shrinkers. This is a joint work with Detang Zhou.

- Title:
- Complete Ricci-flat Kahler manifolds with linear volume growth and applications
- Abstract:
- Yau's 1976 solution of the Calabi conjecture reduced the problem of understanding which smooth compact manifolds admit Ricci-flat Kahler metrics to a problem in complex algebraic geometry, ie to finding Kahler manifolds with vanishing first Chern class. In general no such reduction for the construction of Ricci-flat Kahler metrics on complete noncompact manifolds is known. We report on recent progress in this direction for complete Kahler Ricci-flat metrics with the minimal (linear) possible volume growth. In this case (modulo technicalities) we show that any Kahler Ricci-flat metric of linear volume growth can be complex analytically compactified to a projective orbifold by adding an (orbifold) divisor at infinity and that the compactifying divisor has very special geometry. In particular this answers a longstanding question of Yau's in the linear volume growth setting. Conversely we construct Kahler Ricci-flat metrics of linear volume growth in every admissible Kahler class on a orbifold pair satisfying the conclusions of our compactification result. Together these results reduce the problem of understanding all Kahler Ricci-flat metrics of linear volume growth to a problem in projective algebraic geometry. In the case of complex dimension three we explain how to find many solutions to the resulting problem in algebraic geometry and how this leads to the construction of hundreds of thousands of families of Ricci-flat Kahler 3-folds with linear volume growth. We outline some of the reasons for interest in such noncompact Kahler Ricci-flat 3-folds, especially for the construction of many new compact Ricci-flat 7-manifolds (so-called manifolds with G2 holonomy).

- Title:
- From tiling and packing to uniformization via combinatorial harmonic maps
- Abstract:
- The celebrated Riemann mapping theorem asserts that a non-empty simply connected open subset of the complex plane which is not the whole of it is conformally equivalent to the open unit disk in the complex plane. How should one visualize this conformal map? We will start with a remarkable conjecture by Thurston which was first proved by Rodin-Sullivan, continue with related work of Dehn, Schramm and Cannon-Floyd-Parry that involves tiling by squares, and then describe our current research directions aim at addressing more general questions such as: Given a surface endowed with some combinatorial structure, such as a triangulation, can one obtain effective versions of classical uniformization theorems by varying the triangulation?

- Title:
- Eigenvalue estimates and a compactness theorem for embedded minimal surfaces in the unit ball
- Abstract:
- Recently, A. Fraser and R. Schoen introduced an extremal eigenvalue problem for surfaces with boundary. They discovered an interesting relationship between extremal metrics and minimal surfaces in the unit ball satisfying a free boundary condition.In this talk, we will begin with some known results on the Steklov eigenvalue problem for the Dirichlet-to-Neumann map on compact manifolds with boundary. After giving some examples of minimal surfaces in the unit ball, we will prove a lower bound on the first Steklov eigenvalue for those which are embedded. We will then use this to show that the space of embedded minimal surfaces in the unit ball, with fixed topological type, is compact in the smooth topology. All these results apply if the ambient space is any compact 3-manifolds with nonnegative Ricci curvature and strictly convex boundary. In the end, we will mention some open problems in this direction. This is joint work with A. Fraser.

- Title:
- Regularity theory for area-minimizing currents
- Abstract:
- It was established by Almgren at the beginning of the eighties that area-minimizing $n$-dimensional currents in Riemannian manifolds are regular up to a singular set of dimension at most $n-2$. To reach this goal Almgren developed an entirely new regularity theory, which occupies a very large monograph, published posthumously. This talk is based on a series of joint works with Emanuele Spadaro, where we give alternative proofs to all Almgren's main steps, resulting into a much more manageable approach to his entire theory.

- Title:
- Dynamical stability and instability of Ricci-flat metrics
- Abstract:
- Let M be a compact manifold. A Ricci-flat metric on M is a Riemannian metric with vanishing Ricci curvature. Ricci-flat metrics are fairly hard to construct, and their properties are of great interest. They are the critical points of the Einstein-Hilbert functional, the fixed points of Hamilton's Ricci flow and the critical points of Perelman's lambda-functional.

In this talk, we are concerned with the stability properties of Ricci-flat metrics under Ricci flow. We will explain the following stability and instability results. If a Ricci-flat metric is a local maximizer of lambda, then every Ricci flow starting close to it exists for all times and converges (modulo diffeomorphisms) to a nearby Ricci-flat metric. If a Ricci-flat metric is not a local maximizer of lambda, then there exists a nontrivial ancient Ricci flow emerging from it. This is joint work with Robert Haslhofer.

- Title:
- A New Length Estimate for Curve Shortening Flow and Low Regularity Initial Data for Closed Curves
- Abstract:
- In this talk I'll introduce a crude geometric quantity that controls the length of a smooth curve as it evolves by curve shortening flow. The estimates obtained do not depend on the initial length of the curve, and we show that they can be used to control the evolution of any locally-connected compact set in the plane as it evolves by the related level set flow.

- Title:
- Closing Aubry Sets
- Abstract:
- Given a Hamiltonian H on a compact manifold, one can define the "projected Aubry set" associated to H: this is a subset of M which "captures" many important features of the Hamiltonian dynamics, and it is strongly related to the regularity of viscosity (sub)solutions to the Hamilton-Jacobi equation.

The Mane conjecture in $C^k$ topology states that, for a generic potential V(generic w.r.t. the $C^k$ topology), the Aubry set associated to H+V is either a fixed point or a periodic orbit.

In this talk I will describe how, given a Hamiltonian which possesses a sufficiently smooth viscosity (sub)solution to the Hamilton-Jacobi equation, for any $\epsilon >0$ there exists a potential $V_\epsilon$, whose $C^2-norm$ is bounded by $\epsilon$, such that the Aubry set associated to $H+V_\epsilon$ is either a fixed point of a periodic orbit. This represents a first step through the solution of the Mane Conjecture in $C^2$ topology. Moreover, we will see how these techniques allow to solve the Mane conjecture in C^1 topology. This is a joint work with Ludovic Rifford.

Wednesday December 5: Andre Neves (Imperial College London) and Igor Rodnianski (MIT) at 3PM in Building 4, Room 149

- Andre Neves' Title:
- Min-max Theory and the Willmore Conjecture
- Igor Rodnianski's Title:
- Propagation and Interaction of Impulsive Gravitational Waves

- Title:
- Properties of the Intrinsic Flat Convergence
- Abstract:
- The Intrinsic Flat distance between Riemannian manifolds has been applied to study the stability of the Positive Mass Theorem, the rectifiability of Gromov-Hausdorff limits of Riemannian manifolds, and smooth convergence away from singular sets. In this talk, we will present properties of Riemannian manifolds which are conserved under Intrinsic Flat convergence. The initial notion of the Intrinsic Flat distance and lower semicontinuity of mass and the continuity of filling volumes of balls is joint work with Stefan Wenger. The speaker will also present more recent work including an Arzela-Ascoli Theorem, a notion called the Sliced Filling Volume and, if time allows, the Tetrahedral Property.

- Title:
- Impulsive Gravitational Waves
- Abstract:
- We consider spacetimes satisfying the vacuum Einstein equations with impulsive Gravitational waves without symmetry assumptions. These are spacetimes such that the Riemann curvature tensor has a delta singularity across a null hypersurface. We prove local existence and uniqueness for the characteristic initial value problem with initial data that has a delta singularity in the curvature tensor. A precise description of the propagation of singularity is also given. The proof introduces a new type of energy estimates for the vacuum Einstein equations, allowing the L2 norm of some components of the curvature tensor to be infinite. The new estimate allows us to prove local existence and uniqueness for a general class of initial data which is non-regular along a null direction. We will also discuss extensions of this theorem, which can be applied to understand colliding impulsive gravitational waves and the formation of trapped surface. This is joint work with I. Rodnianski.

- Title:
- Rigidity results for conformal immersions in $R^n$
- Abstract:
- By a classical result of Codazzi every closed, totally umbilic surface in $R^n$ is a round sphere. De Lellis and Muller proved a rigidity statement corresponding to this result. More precisely, they showed that for every closed surface in $R^3$, whose traceless second fundamental form is "small" in $L^2$, there exists a conformal parametrization whose distance to a standard parametrization of a round sphere is small in $W^{2,2}$. In a recent joint work with H. Nguyen (Warwick) we were able to extend this result to arbitrary codimensions. Moreover, we obtained related rigidity results for inversions of the catenoid and Enneper`s minimal surface. In my talk I will review the analytic preliminaries (i.e. the results of Muller-Sverak and Kuwert-Li) and I will sketch the proof of the above mentioned results.

- Title:
- Boundary Rigidity and Minimal Surfaces: A Survey
- Abstract:
- A Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined by the boundary distance function, that is the restriction of the distance function to the boundary. Loosely speaking, this means that the Riemannian metric can be recovered from measuring distances between boundary points only. The goal is to show that certain classes of metrics are boundary rigid (and, ideally, to suggest a procedure for recovering the metric).

To visualize that, imagine that one wants to find out what the Earth is made of. More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point to point). The speed of sound depends on the material. One can "tap" at some points of the surface of the body and "listen when the sound gets to other points". The question is if this information is enough to determine what is inside.

This problem has been extensively studied from PDE viewpoint: the distance between boundary points can be interpreted as a "travel time" for a solution of the wave equation. Hence this becomes a classic Inverse Problem when we have some information about solutions of a certain PDE and want to recover its coefficients. For instance such problems naturally arise in geophysics (when we want to find out what is inside the Earth by sending sound waves), medical imaging etc.

In a on-going joint project with S. Ivanov we suggest an alternative geometric approach to this problem. In our earlier work, using this approach we were able to show boundary rigidity for metrics close to flat ones (in all dimensions), thus giving the first open class of boundary rigid metrics beyond two dimensions. We were now able to extend this result to include metrics close to a hyperbolic one.

The approach is grown up from another long-term project of studying surface area functionals in normed spaces, which we have been working on it for more than ten years. There are a number of related issues regarding area-minimizing surfaces in Riemannian manifold. The talk gives a non-technical survey of ideas involved. It assumes no background in inverse problems and is supposed to be accessible to a general math audience (in other words, we will sweep technical details under the carpet).

- Title:
- Rigidity of Self-shrinkers of Mean Curvature Flow
- Abstract:
- Recently, using the desingularization technique, Kapouleas-Kleene-Moller and independently Nguyen has successfully constructed a new family of smooth complete embedded self-shrinkers asymptotic to cones. These are the first non-rotationally symmetric examples after planes, spheres, cylinders and Angenent's torus.

In this talk, we report some new rigidity (at infinity) theorems of self-shrinkers. The results are two folds with emphasis on the asymptotically cylindrical case. First, we show the uniqueness of smooth properly embedded self-shrinkers asymptotic to any given regular cone in Euclidean space. Second, we discuss the optimal condition on the asymptotics of self-shrinkers, so that the uniqueness of self-shrinkers asymptotic to generalized shrinking cylinders holds true. This gives a partial affirmative answer to the cylinder rigidity conjecture. The point of our theorems is that we do not require completeness of self-shrinkers. One of the main ingredients of the proofs is the (anisotropic) Carleman estimates inspired by the work of Escauriaza-Seregin-Sverak.

Among applications, we obtain some non-existence results of self-shrinkers. Namely, except hyperplanes, there do not exist any other smooth complete properly embedded self-shrinkers with ends asymptotic to rotationally symmetric cones.

- Title:
- Three-manifolds of Constant Vector Curvature
- Abstract:
- A Riemannian manifold M has constant vector curvature k if every tangent vector to M belongs to a tangent plane of curvature k. When all sectional curvatures of M are additionally either bounded above or below by k, M is said to have extremal curvature k. I will discuss classification results for finite volume three-manifolds of constant vector curvature k and extremal curvature k. The talk will be based on joint work with Jon Wolfson.

- Title:
- Asymptotic Behavior of Degenerate Neckpinches in Ricci Flow
- Abstract:
- We discuss the detailed nature of the geometry of rotationally symmetric degenerate neckpinch singularities which develop in the course of Ricci flow.

- Title:
- Weyl Metrisability for Projective Surfaces
- Abstract:
- The existence problem for Riemannian metrics on a surface with prescribed unparametrised geodesics was first studied by R. Liouville. He observed that the problem can be formulated as a linear first order PDE system which in general will only admit trivial solutions. The necessary and sufficient conditions for local existence of nontrival solutions were found only recently by Bryant, Dunajski and Eastwood. Surprisingly the conditions are rather complicated. However if one looks for Weyl structures on surfaces with prescribed unparametrised geodesics the situation is different. In this talk I will use techniques from complex geometry to show that the corresponding PDE system always admits local solutions. I will also show that the Weyl structures on the 2-sphere whose geodesics are the great circles, are in one-to-one correspondence with the smooth quadrics without real points in the complex projective plane. If time permits, I will explain that the Weyl metrisability problem for projective surfaces has a natural analogue in all even dimensions.

- Speaker:
- Sean Timothy Paul (University of Wisconsin, Madison) 3:00PM
- Title:
- Hyperdiscriminants, Orbit Closures, and Lower Bounds on K-energy Maps
- Abstract:
- One of the main problems in complex geometry is to detect the existence of "canonical" Kahler metrics in a given Kahler class on a compact complex (Kahler) manifold. In particular one seeks necessary and sufficient conditions for the existence of a Kahler Einstein metric on a Fano manifold. In this case the presence of positive curvature makes this problem extremely difficult and has led to a striking series of conjectures--the "standard conjectures"-- which relate the existence of these special metrics (which are solutions to the complex Monge-Ampere equation, a fully non-linear elliptic p.d.e . ) to the algebraic geometry of the pluri-anticanonical images of the manifold. Yau speculated that the relevant algebraic geometry would be related (somehow) to Mumford's Geometric Invariant Theory. Eventually it was conjectured that K-energy bounds along Bergman potentials could be deduced from an appropriate notion of "semi-stability". Recently this conjecture has been completely justified by the speaker, building upon work of Gang Tian and Gelfand-Kapranov-Zelevinsky and Weyman-Zelevinsky. It is the aim of this talk to outline progress on the standard conjectures and to discuss the entire theory in the context of complex algebraic groups and dominance of rational representations of such groups.

- Speaker:
- Nikola Kamburov (MIT) 4:00PM
- Title:
- On a Free Boundary Variant of a Conjecture of De Giorgi
- Abstract:
- Drawing out a fascinating connection between Bernstein's problem, on the one hand, and the study of global, bounded and monotone solutions to the semilinear elliptic equation $\Delta u = u3 - u$ in $R^n$, on the other, a famous conjecture of De Giorgi states that the level sets of such solutions are hyperplanes, at least in dimension $n\leq 8$. The conjecture was verified for $n\leq 8$ by Savin. Recently, Del Pino, Kowalczyk and Wei constructed a counterexample in dimension $n=9$, using an intricate fixed point argument. In this talk, I would like to discuss the construction of such a counterexample for an appealing free boundary variant of De Giorgi's conjecture. Our approach uses the more transparent means of the method of barriers.

- Title:
- The Spacetime Positive Mass Theorem
- Abstract:
- I will discuss the proof of the spacetime positive mass theorem, recently proved in joint work with Eichmair, Huang, and Schoen. For dimensions less than $8$, we show that an asymptotically flat initial data set satisfying the dominant energy condition must have $E \ge |P|$, where $(E,P)$ is the ADM energy-momentum. Our proof follows the basic idea of the proof of the time-symmetric case, with marginally outer trapped hypersurfaces (MOTS) taking the place of minimal hypersurfaces, with some added complications.

- Title:
- Singularities in 4d Ricci Flow
- Abstract:
- In this talk, we discuss the formation of singularities in higher-dimensional Ricci flow without pointwise curvature assumptions. We show that the space of singularity models with bounded entropy and locally bounded energy is orbifold-compact in arbitrary dimensions. In dimension four, a delicate localized Gauss-Bonnet estimate even allows us to drop the assumption on energy in favor of (essentially) an upper bound for the Euler characteristic. We will also see how these results are part of a larger project exploring high curvature regions in 4d Ricci flow. This is all joint work with Reto Mueller.

- Title:
- Gluing of Self-shrinkers for Mean Curvature Flow: Complete, Embedded Examples w/ Genus
- Abstract:
- For mean curvature flow in any ambient Riemannian 3-manifold, the self-similar surfaces in R^3 form the basic "atoms" of the singularity theory. However, the only known complete, embedded examples are: 1) Flat planes, round cylinders, round spheres and 2) One (not round) torus found by Angenent in 1989.

We will discuss gluing constructions that yield new complete, embedded, self-shrinking surfaces of large genus g in R^3 (as expected from numerics by Tom Ilmanen in 1995) by fusing known examples (plane and sphere) while introducing genus. The analysis for non-compact ends is complicated by the unbounded geometry, and PDEs of Ornstein-Uhlenbeck type need to be understood well via Liouville-type results, which in turn enable compactifying parts of the problem, in order to construct resolvents and close the non-linear PDE system (w/ N. Kapouleas & S. Kleene).

- Title:
- Hypersurfaces with Nonnegative Scalar Curvature
- Abstract:
- Since the time of Gauss, geometers have been interested in the interplay between the intrinsic metric structure of hypersurfaces and their extrinsic geometry from the ambient space. For example, a result of Sacksteder tells us that if a complete hypersurface has non-negative sectional curvature, then its second fundamental form in Euclidean space must be positive semi-definite.

In a recent joint work with Damin Wu, we study hypersurfaces under a much weaker curvature condition. We prove that a hypersurface with nonnegative scalar curvature which is either closed or complete of finite many regular ends must be weakly mean convex. This result is optimal in the sense that the scalar curvature cannot be replaced by other k-th mean curvatures. The result and argument have applications to the mean curvature flow, positive mass theorem, and rigidity theorems.

- Title:
- Mean Curvature Flow of Two-Convex Hypersurfaces
- Abstract:
- We discuss mean curvature flow of two-convex hypersurfaces in Euclidean space: we explain the relationship between the Huisken-Sinestrari surgery program and the well-known weak solution, and we show how this theory can be used to prove regularity estimates for mean curvature flow.

- Title:
- On Inaudible Properties of Broken Drums - Isospectral Domains with Mixed Boundary Conditions
- Abstract:
- Since Kac raised the question "Can one hear the shape of a drum?", various families of non-smooth counterexamples have been constructed using the transplantation method which is based on a group-theoretic technique by Sunada. We apply the transplantation method to domains with mixed boundary conditions which can be interpreted as broken drums. The method is translated into graph theory which allowed for a computer-aided search for transplantable pairs, a classification in terms of induced representations, and the development of tools with which new pairs can be generated from given ones.

The talk finishes with a presentation of various new pairs among which there are 10 versions of the Gordon-Webb-Wolpert drums with mixed boundary conditions. In the end, we discuss inaudible properties and show the first example of a connected drum that sounds disconnected and of a broken drum that sounds unbroken, that is, a planar domain with mixed boundary conditions that is isospectral to a domain with Dirichlet boundary conditions. Above all, the latter example shows that an orbifold can be Dirichlet isospectral to a manifold.

- Title:
- The Toric Futaki Invariant on Quadrilaterals
- Abstract:
- I will first explain why, in the toric framework, Kahler and Sasaki metrics of constant scalar curvature corresponds to solutions of the same PDE problem on convex polytopes. Then, I will show that (the toric version of) the Futaki invariant is the only obstruction to the existence of such a solution whenever the polytope is a quadrilateral. Finally, I will explain how the Futaki invariant is related to a certain isoperimetric quotient in the Sasaki case and why this leads to (the first known) examples of non-isometric Sasaki metrics of constant scalar curvature, compatible with the same contact structure.

- Title:
- Valuations on Sobolov Spaces
- Abstract:
- Let $F$ be a space of real valued functions, for example, the Sobolev space ${W^{1,1}}({\mathbb R} ^n)$ and let ${\mathbb A} $ be an abelian semi-group. A function ${\operatorname{z}}: F\to {\mathbb A} $ is called a {\em valuation} if $${\operatorname{z}}(f\vee g) + {\operatorname{z}}(f\wedge g)= {\operatorname{z}}(f) + {\operatorname{z}}(g)$$ for all $f,g\in F$, where $f\vee g$ denotes the pointwise maximum and $f\wedge g$ the pointwise minimum of $f$ and $g$. If the abelian semi-group is given as the set of convex bodies (compact convex sets) in ${\mathbb R} ^n$ with Minkowski addition (defined by $K+L=\{x+y: x\in K, y\in L\}$), we talk about Minkowski valuations. After a brief excursion to the history of valuations in geometry, we give a complete classification of affinely contravariant Minkowski valuations on ${W^{1,1}}({\mathbb R} ^n)$ and show that every such valuation ${\operatorname{z}}$ is given as $${\operatorname{z}} (f) = c\, \Pi \langle f \rangle$$ for all $f\in{W^{1,1}}({\mathbb R} ^n)$ with a suitable constant $c\ge 0$. Here the convex body $\Pi \langle f \rangle$ is defined via its support function as $$h(\Pi \langle f \rangle, v)=\frac12\int_{{\mathbb R} ^n} |v\cdot \nabla f(x)|\,dx.$$ We discuss the connection of the convex bodies $\Pi\langle f\rangle$ and $\langle f \rangle$ (both introduced by Lutwak, Yang & Zhang) with the optimal norm in the sharp Sobolev inequality for a general norm, the affine Sobolev-Zhang inequality and the solution of the functional Minkowski problem.

- Title:
- Kahler-Einstein Metrics Singular along a Divisor
- Abstract:
- The simplest example of a Kahler-Einstein (KE) metric is a football. A European football corresponds to a smooth KE metric, while an American one corresponds to a KE metric with conical singularities. The existence of smooth KE metrics on compact Kahler manifolds was proven in the 70's by Aubin and Yau for nonpositive curvature, and in the early 90's by Tian for positive curvature, under some assumptions. In the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is bent at some angle along a complex hypersurface), motivated by applications to algebraic geometry. More recently, Donaldsonsuggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures. In this talk we will describe a proof of Tian's conjecture in the case the divisor is smooth, as well as a proof of the first of Donaldson's conjectures, obtained recently in joint work with T. Jeffres and R. Mazzeo.

- Title:
- The Newtonian Limit of Geometrostatics
- Abstract:
- Geometrostatics is an important subdomain of Einstein's General Relativity. It describes the mathematical and physical properties of static isolated relativistic systems such as stars, galaxies or black holes. For example, geometrostatic systems have a well-defined ADM-mass (Chrusciel, Bartnik) and (if this is nonzero) also a center of mass (Huisken-Yau, Metzger) induced by a CMC-foliation at infinity. We will present surface integral formulae for these physical properties in general geometrostatic systems. Together with an asymptotic analysis, these can be used to prove that ADM-mass and center of mass 'converge' to the Newtonian mass and center of mass in the Newtonian limit $c\to\infty$ (using Ehler's frame theory). We will discuss geometric similarities of geometrostatic and classical static Newtonian systems along the way.

- Title:
- Geometric Analysis Seminar: Sharp Lower Bounds for the First Eigenvalue of the P-laplacian with Ricci Curvature Assumptions
- Abstract:
- In this seminar, we will speak of sharp lower bounds for the first eigenvalue of the p-laplacian on a Riemannian manifold with a Ricci lower bound and some rigidity theorems. After a brief review of the results available for the usual laplacian, we will discuss how it is possible to generalize them for the generic p-case. In particular, we will focus on two techniques. The first one is based on Schoen-Yau type gradient estimates and in the usual case relies on the Bochner formula. As we will see it is possible to generalize this important theorem using the linearized p-laplacian. Another technique is based on Levy-Gromov isoperimetric ineaqualities and permits easily to obtain a generalization of the well-known Obata's theorem. Recent improvements have been made to isoperimetric inequalities by Milman, especially when the lower bound of the Ricci curvature is negative.

- Title:
- Collapsing of Abelian Fibred Calabi-Yaus and Hyperkahler Mirror Symmetry
- Abstract:
- We will address the problem of understanding the collapsing of Ricci-flat Kahler metric on abelian fibred projective Calabi-Yau manifolds. We will then explain an application of these results to the Strominger-Yau-Zaslow picture of mirror symmetry for some hyperkahler manifolds. Joint work with Mark Gross and Yuguang Zhang.

Location: 2-135 at 4:00 PM

- Title:
- A Variational Characterization of the Catenoid
- Abstract:
- We show that the catenoid is the unique surface of least area (suitably understood) within a geometrically natural class of minimal surfaces. The proof relies on a techniques involving the Weierstrass representation used by Osserman and Schiffer to show the sharp isoperimetric inequality for minimal annuli. An alternate approach that avoids the Weierstrass representation will also be discussed. This latter approach depends on a conjectural sharp eigenvalue estimate for a geometric operator and has interesting connections with spectral theory. This is joint work with C. Breiner.

- Title:
- The relaxed energy for harmonic maps from $B^3$ into $S^2$
- Abstract:
- I will discuss the relaxed Dirichlet energy for maps from $B^3$ into $S^2$ (definitions, meaning, etc... ), treating the regularity theory in particular. This field contains several long standing open questions. I will present a recent result which gives us a better understanding of the minimizers of this energy, and casts even more open questions.

- Title:
- Initial Time Singularities in Mean Curvature Flow
- Abstract:
- Let $M_0$ be a closed subset of $R^n+1$ that is a smooth hypersurface except for a finite number of isolated singular points. Suppose that M_0 is asymptotic to a regular cone near each singular point.

Can we flow M_0 by mean curvature?

Theorem (n<7): there exists a smooth mean curvature evolution starting at M_0 and defined for a short time 0 Such an initial M_0 might arise as the limit of a smooth mean curvature evolution defined earlier than t=0. Thus, the result allows us to flow through singularities in some cases. We use a monotonicity formula that complements the monotonicity formula of Huisken. The method applies to other geometric heat flows as well.

- Title:
- The Black Hole Stability Problem
- Abstract:
- The problem of nonlinear stability for the Kerr model of a rotating black hole is one of the central problems in general relativity. The analysis of linear fields of spins 0,1,2 on the Kerr spacetime is an important model problem for full nonlinear stability. In this talk, I will present recent work with Pieter Blue which makes use of the hidden symmetry related to the Carter constant to circumvent these difficulties, and give a "physical space" approach to estimates for the wave equation, including energy bounds, trapping, and dispersive estimates.

- Title:
- The Intrinsic Flat Distance between Oriented Riemannian Manifolds
- Abstract:
- We introduce the intrinsic flat distance between compact oriented Riemannian manifolds with finite volume. The limits of sequences with uniform upper bounds on their volume and diameter are integral current spaces: countably H^m rectifiable metric spaces with boundary. When the sequence has a uniform lower bound on Ricci curvature and volume, then by work of Cheeger-Colding, we see that the Gromov-Hausdorff and Intrinsic Flat limits agree. In general, Intrinsic Flat Limits are subsets of the GH limits. Intrinsic Flat limits may exist when there is no GH limit as it is a weaker notion of convergence. This is joint work with S. Wenger and is based on work of Ambrosio-Kirchheim. See http://comet.lehman.cuny.edu/intrinsicflat.html for relevant papers/preprints.

- Title:
- On Formation of Trapped Surfaces in General Relativity

- Title:
- Existence of an Approximate Second Fundamental Form for Singular Submanifolds with Generalized Mean Curvature
- Abstract:
- The objects considered are integral varifolds with locally bounded first variation. It is established that these are rectifiable of second order. The proof relies on a new differentiability criterion for functions in Lebesgue spaces phrased in terms of approximability by harmonic functions. Throughout the talk, concepts from geometric measure theory will be illustrated by many examples.

- Title:
- Some Geometric Aspects of Brownian Motion on a Minimal Surface
- Abstract:
- We demonstrate how stochastic methods can be applied to minimal surface theory via a concrete example requiring relatively few probabilistic technicalities. In particular, after explaining how the parabolicity and area growth of minimal ends have been previously studied using universal superharmonic functions, we describe an alternative approach, yielding stronger results, based on studying Brownian motion on the surface. We attempt to provide intuitive explanations of the stochastic techniques involved.

- Title:
- Asymptotic Geometry, Finite Generation of Fundamental Groups, and Harmonic Functions.

- Title:
- Complete Embedded MCF Self Shrinkers
- Abstract:
- (Joint work w/ Nicos Kapouleas and Niels Martin Moller): We present a new family of complete embedded self shrinking surfaces in $R^3$, extending the list beyond the plane, sphere, cylinder, and Angenent's Torus. They are the first and only such surfaces known outside the class of surfaces of revolution. The surfaces are constructed as highly symmetric singular perturbations of the sphere intersecting a plane through a great circle.