R. Buzano

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

A mean curvature flow starting from a closed embedded surface must necessarily form a singularity in finite time. The formation of singularities is therefore one of the central themes in the study of mean curvature flow. By work of Huisken, Ilmanen, and White, such singularities are modelled on surfaces which shrink self-similarly along the flow and are hence called self-shrinkers. These are minimal surfaces in Gaussian space. Noncompact self-shrinkers are particularly interesting as they model local singularities. In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends. This is joint work with Huy Nguyen and Mario Schulz.

T. Colding

Geometry of PDEs

Optimal geometric structures and the evolution of shapes are governed by partial differential equations. These same types of equations come up over and over again across many diverse areas in science, engineering and mathematics. The geometric invariance makes the equations canonical, and means that they also describe phenomena seemingly unrelated to geometry. Often the geometry unlocks the structure of the equation and leads to fundamental tools in PDE. Conversely, analysis has played a central role in the development of geometry. Understanding the equations and their fundamental properties requires simultaneous insight into both analysis and geometry and the interplay between the two. In this talk we will discuss this principle for several fundamental equations. We start by seeing how a long-standing problem in geometry leads to optimal regularity for viscosity solutions of a degenerate elliptic PDE, then turn to using PDE to understand optimal shapes and geometric evolution.

J. Fine

Knots, minimal surfaces and J-holomorphic curves

Let K be a knot or link in the 3-sphere, thought of as the ideal boundary of hyperbolic 4-space. I will describe a programme to define a link invariant of K by counting 2D minimal surfaces in H^4 which have K as their ideal boundary. In other words, this count should depend only on the isotopy class of K. I will explain how to prove this when counting minimal discs filling knots and discuss the obstacles to overcome in extending the results to minimal surfaces with more complicated topology. An important role is played by the Eells-Salamon correspondence: minimal surfaces in H^4 are in 1-1 correspondence with J-holomorphic curves in the twistor space. From this perspective, the minimal surface counts are Gromov-Witten invariants. An important difference with the normal theory however is that the equations (for either J-holomorphic curves or minimal surfaces) become degenerate at the boundary. This means that both the Fredholm and compactness parts of the story have to be rebuilt from scratch. One potential pay-off is that it should be possible to define Gromov-Witten invariants for a new class of infinite symplectic manifolds which are “asymptotically twistorial”. This could lead in turn to invariants of links in other 3-manifolds, as well as having applications to 4-dimensional Riemannian geometry.

L. Foscolo

Complete non-compact manifolds with holonomy G2 and ALC asymptotics

G2 manifolds are the Ricci-flat 7-manifolds with holonomy G2. Until recently there was only a handful of known examples of complete non-compact G2 manifolds, all highly symmetric and arising from explicit solutions to ODE systems. In joint work with Haskins and Nordström, we produced infinitely many G2 manifolds on total spaces of principal circle bundles over asymptotically conical Calabi-Yau manifolds. The asymptotic geometry of the G2 metrics we produced is analogous to the geometry of 4-dimensional ALF (asymptotically locally flat) spaces and has been labelled ALC (asymptotically locally conical) in the physics literature. In this talk, I will discuss some further joint work on this class of manifolds, in particular consequences of the good deformation theory of ALC G2 manifolds and the construction of new examples with a slightly more complicated ALC asymptotic geometry analogous to the well-known Atiyah-Hitchin metric in 4-dimensional hyperkähler geometry.

A. Fraser

Eigenvalue optimization among metrics with symmetries

We will discuss an eigenvalue problem on manifolds with boundary and consider the question of the degenerations of Riemannian manifolds under which the eigenvalues are continuous. This question is important when one attempts to construct metrics which optimize an eigenvalue. We show that imposing symmetry prevents certain degenerations from occurring when maximizing higher eigenvalues for surfaces. This is joint work with R. Schoen and P. Sargent.

M. Ghomi

Geometric inequalities in spaces of nonpositive curvature

We will discuss total mean curvatures, i.e., integrals of symmetric functions of the principle curvatures, of hypersurfaces in Riemannian manifolds. These quantities are fundamental in geometric variational problems as they appear in Steiner’s formula, Brunn-Minkowski theory, and Alexandrov-Fenchel inequalities. We will describe a number of new inequalities for these integrals in non positively curved spaces, which are obtained via Reilly’s identities, Chern’s formulas, and harmonic mean curvature flow. As applications we obtain several new isoperimetric inequalities, and Riemannian rigidity theorems. This is joint work with Joel Spruck.

N. Gigli

Lipschitz continuity of harmonic maps from RCD to CAT(0) spaces

In `classical' geometric analysis a celebrated result by Eells-Sampson grants Lipschitz continuity of harmonic maps from manifolds with Ricci curvature bounded from below to simply connected manifolds with non-negative sectional curvature. All these concepts, namely lower Ricci bounds, upper sectional bounds and harmonicity, make sense in the setting of metric-measure geometry and is therefore natural to ask whether the same sort of regularity holds in this more general setting. In this talk I will survey a series of recent papers that ultimately answer affirmatively to this question.

Y. Liokumovich

Parametric inequalities and the Weyl law for the volume spectrum

The isoperimetric inequality is one of the most important results in geometry. However, the parametric version of the isoperimetric inequality is an open problem. Namely, given a family of k-dimensional boundaries (in some n-dimensional space for n>k+1) does there exist a family of fillings, whose size is controlled in terms of the size of the boundaries (in a certain optimal way)? It turns out that the answer to this question is key to proving the Weyl for the volume spectrum, an important result about asymptotic distribution of volumes of certain singular minimal submanifolds. I will describe proofs of the parametric isoperimetric inequality and parametric coarea inequality in low dimensions and how they lead to the proof of the Weyl law for 1-cycles in 3-manifolds. My talk will be based on a joint work with Fernando Marques and Andre Neves and a joint work with Larry Guth.

J. Lott

Complex Alexandrov spaces.

I will discuss the geometry of Kaehler manifolds with a lower bound on the holomorphic bisectional curvature, along with their pointed Gromov-Hausdorff limits. Some of the proofs use Ricci flow.

A. Malchiodi

Prescribing Morse scalar curvatures in high dimensions

We consider the classical question of prescribing the scalar curvature of a manifold via conformal deformations of the metric, dating back to works by Kazdan and Warner. This problem is mainly understood in low dimensions, where blow-ups of solutions are proven to be "isolated simple". We find natural conditions to guarantee this also in arbitrary dimensions, when the prescribed curvatures are Morse functions. As a consequence, we improve some pinching conditions in the literature and derive existence and non-existence results of new type. This is joint work with M. Mayer.

P. Petersen

Curvature and Betti Numbers via the Bochner Technique

The talk will explain the basic set-up of the Bochner technique on forms. This will be followed by a survey of recent results by the speaker, Matthias Wink, and for the most recent results Jan Nienhaus. The over all idea is to find the weakest possible conditions on curvature operators that guarantee control over the Weitzenböck term in the Bochner formula. In the case of the curvature operator this intertwines nicely with the holonomy group. In the case of the curvature of the second kind we obtain conditions that do not imply nonnegative Ricci curvature.

F. Schulze

Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

We investigate the question of stability for a class of Ricci flows which start at possibly non-smooth metric spaces. We show that if the initial metric space is Reifenberg and locally bi-Lipschitz to Euclidean space, then two solutions to the Ricci flow whose Ricci curvature is uniformly bounded from below and whose curvature is bounded by ct^{-1} converge to one another at an exponential rate once they have been appropriately gauged. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott. This is joint work with A. Deruelle and M. Simon.

C. Sormani

Curvature and Integral Current Spaces

We will discuss the properties of an integral current space, a class of rectifiable metric spaces Sormani first defined with Wenger when defining intrinsic flat convergence applying Ambrosio-Kirchheim theory. Wenger proved that if one has a sequence of oriented Riemannian manifolds with boundary with a uniform upper bound on diameter, volume, and boundary area, then a subsequence converges in the intrinsic flat sense to an integral current space, possibly the 0 space. Matveev-Portegies proved that if the sequence has no boundary and is non-collapsing with a uniform lower bound on sectional or Ricci curvature then the intrinsic flat limit agrees with the measured Gromov-Hausdorff limit. Jaramillo-Perales-Rajan-Searle-Siffert proved that any oriented Alexandrov space without boundary is an integral current space. Various mathematicians are studying which CD and RCD spaces are integral current spaces. Gromov has suggested that non-collapsing sequences of Riemannian manifolds with lower bounds on their scalar curvature might have limits which satisfy his prism inequality. Nobody has yet studied the setting with boundary in depth yet although Perales and Allen-Perales have some intriguing work in this direction. For a bibliography of papers on this topic see https://sites.google.com/site/intrinsicflatconvergence/

L. Wang

Relative expander entropy

In this talk, I will discuss a notion of relative entropy motivated by self-expanding solutions to mean curvature flow. I will also discuss some basic properties of the relative entropy and applications to the study of mean curvature flow coming out of a cone. This is based on joint work with Jacob Bernstein.

G. Wei

Universal Covers of Ricci Limit Spaces are Simply Connected

By Gromov's precompactness theorem, any sequence of n-dim manifolds with uniform Ricci curvature lower bound has a convergent subsequence. The limit spaces are referred to as Ricci limit spaces. Cheeger-Colding-Naber developed great regularity and geometric properties for Ricci limit spaces. However, unlike Alexandrov spaces, these spaces could locally have infinite topological types. About twenty years ago, joint with C. Sormani, we gave the first topological result by showing the universal cover of Ricci limit spaces exists. Here the universal cover is in the sense of a universal covering map (need not be simply connected). I will present a series of recent work of J. Pan-G.Wei, J.Pan-J.Wang and J.Wang showing that the Ricci limit spaces are semilocally simply connected, therefore the universal covers are simply connected.

B. Wilking

Positive sectional curvature, torus actions and matroids

This is a report on joint work with Lee Kennard and Michael Wiemeler. Among other things we study positively curved manifold M with an effective isometric action of $T^9$ without finite isotropy groups of even order. We show that these manifolds are rationally equivalent to a rank one symmetric space. I will also explain how a seemingly unrelated new result enters the proof: A finite graph $G$ with an inner metric and first Betti number $ b_1(G)=9$ satisfies $sys(G)\le vol(G)/4$, where sys(G) is the systole of G and vol(G) its one dimensional volume.

W. Ziller

Exposition of Karsten Grove's work