Shih-Kai Chiu 邱詩凱 (University of Notre Dame): Calabi-Yau metrics with maximal volume growth

We survey recent existence results, due to Conlon-Rochon, Y. Li and Szekelyhidi, of Calabi-Yau metrics on C^n with maximal volume growth. Such metrics are asymptotic to singular Calabi-Yau cones, thus disproving a conjecture of Tian, which says that every Calabi-Yau metric on C^n with maximal volume growth must be the Euclidean metric. We closely follow the construction of Szekelyhidi. If time permits, we will give a generalization of these results to constructing new families of such Calabi-Yau metrics, and discuss how to distinguish them.

Po-Ning Chen 陳泊寧 (University of California, Riverside): Supertranslation invariance of angular momentum

LIGO's successful detection of gravitational waves has revitalized the theoretical understanding of the angular momentum carried away by gravitational radiation. An infinite-dimensional supertranslation ambiguity has presented an essential difficulty for decades of study. Recent advances were made to quantify the supertranslation ambiguity in the context of binary coalescence. In this talk, we will present the first definition of angular momentum in general relativity that is completely free from supertranslation ambiguity. The new definition of angular momentum is derived from the limit of the quasilocal angular momentum we defined previously.

This talk is based on joint work with Jordan Keller, Mu-Tao wang, Ye-Kai Wang, and Shing-Tung Yau.

Tang-Kai Lee 李堂愷 (Massachusetts Institute of Technology): Self-shrinkers and Entropy Minimizers for the Mean Curvature Flow

The analysis of self-shrinkers plays an important role in the study of the mean curvature flow. Among many invariants of self-shrinkers, entropy is a natural geometric quantity measuring the complexity of a self-shrinker. In this talk, we will survey some rigidity results of hypersurface self-shrinkers in terms of entropy, and see an attempt toward the higher-codimension case.

Kyeongsu Choi 최경수 (Korea Institute for Advanced Study): Ancient mean curvature flow and spectral analysis

In this talk, we first discuss how to classify ancient mean curvature flows which converge to a shrinker after rescaling as time goes back.

The linearized operator of the flow over a shrinker has finitely many unstable eigenfunctions, and they provide corresponding ancient flows.

Next, if time permits, we will consider the ancient flows corresponding to the first eigenfunction and study their nice properties.

Then, we will talk about how to use those properties to establish the avoidance theory of unstable singularities.

Man Chun Lee 李文俊 (Chinese University of Hong Kong): d_p convergence and epsilon-regularity theorems for entropy and scalar curvature lower bound

In this talk, we consider Riemannian manifolds with almost non-negative scalar curvature and Perelman entropy. We establish an epsilon-regularity theorem showing that such a space must be close to Euclidean space in a suitable sense. We will illustrate examples showing that the result is false with respect to the Gromov-Hausdorff and Intrinsic Flat distances, and more generally the metric space structure is not controlled under entropy and scalar lower bounds. We will introduce the notion of the d_p distance between (in particular) Riemannian manifolds, which measures the distance between W^{1,p} Sobolev spaces, and it is with respect to this distance that the epsilon regularity theorem holds. This is joint work with A. Naber and R. Neumayer.

Sławomir Kolasiński (University of Warsaw): Geometric ellipticity

By a geometric variational problem I mean a problem of minimising a functional defined on k-dimensional geometric objects, like currents or varifolds, lying in n-dimensional ambient space. I focus on functionals defined as integrals, where the integrand depends on the point and the tangent k-plane at that point. One example is the k-dimensional Hausdorff measure generated by some non-Euclidean norm on 𝐑ⁿ.

Ellipticity (AE), introduced by Almgren in the 1960s, is a condition on the functional ensuring existence and partial regularity of minimisers. The atomic condition (AC) was defined a few years ago by G. De Philippis, A. De Rosa, and F. Ghiraldin so to ensure rectifiability of critical points. Together with A. De Rosa we proved that (AC) implies (AE).

The problem with this theory is that there are virtually no specific non-trivial examples of functionals satisfying any of (AE) or (AC). A. De Rosa and R. Tione proposed recently the scalar atomic condition (SAC) which might be easier to verify.

In my talk I shall review definitions, properties, and relations between conditions (AE), (AC), and (SAC). I shall talk about my joint work with A. De Rosa (CPAM 2020) and also about ongoing work with my student Mariusz Janosz.

Zhizhang Wang 王志张 (Fudan University): Entire space like hyper-surfaces with constant curvature in Minkowski space

In this talk, we discuss the existence of smooth, entire, strictly convex, spacelike, constant σk curvature hypersurfaces with prescribed lightlike directions and boundary data in Minkowski space. We further consider the prescribed curvature problem and downward translated soliton equations in Minkowski space.

Hikaru Yamamoto 山本光 (University of Tsukuba): An Example of the Noncompact Yamabe Flow having the Infinite-time Incompleteness

I explain a recent result on the noncompact Yamabe flow which is joint work with Jin Takahashi at Tokyo Institute of Technology. The noncompact Yamabe flow is complicated compared to the compact case. There are many unexpected phenomena from the viewpoint of the compact Yamabe flow. One of the remaining questions is the following. If each Riemannian metric is complete under the Yamabe flow on a noncompact manifold for all time and the long time limit exists, then is the limit also complete? I give the negative answer to this question by giving a counterexample.

Sean McCurdy (NTNU): Cusps and the degenerate Alt-Caffarelli functional

The focus of this talk is new results on the partial regularity of the free-boundary for local minimizes of a class of degenerate Alt-Caffarelli functionals. These degenerate functionals arise naturally in variational formulations of the Stokes wave problem. But, because of their degeneracy, the standard tools for proving weak geometric regularity (interior ball conditions) leave open the possibility that cusps may form. Recently, this question was answered in joint work with Lisa Naples.

Da Rong Daren Cheng 程大容 (University of Waterloo): Existence of constant mean curvature 2-spheres

This talk is based on joint work with Xin Zhou (Cornell). I'll describe our work on the existence of constant mean curvature 2-spheres. In particular, we show that in a 3-sphere equipped with a Riemannian metric of positive Ricci curvature, for all H there exists a non-trivial, branched immersed 2-sphere with constant mean curvature H. Time permitting, I'll also talk about our more recent work applying some of the ideas involved in the proof of the above result to study the existence of curves with constant geodesic curvature in Riemannian 2-spheres.

Kotaro Kawai 河井 公大朗 (Gakushuin University): Deformed Donaldson-Thomas connections

The deformed Donaldson-Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a $G_2$-manifold satisfying certain nonlinear PDEs. This is considered to be the mirror of a calibrated (associative) submanifold via mirror symmetry. As the name indicates, the dDT connection can also be considered as an analogue of the Donaldson-Thomas connection ($G_2$-instanton).

In this talk, after reviewing these backgrounds, I will show that dDT connections indeed have properties similar to associative submanifolds and $G_2$-instantons. I would also like to present some related problems. This is joint work with Hikaru Yamamoto.

Yuchin Sun 孫有慶 (University of California, Santa Cruz): Morse index bound of minimal two torus

Min-max construction of minimal spheres using harmonic replacement is introduced by Colding and Minicozzi and generalized by Zhou to conformal harmonic torus. The difference between spheres and tori is that tori has varying conformal structures. We construct deformation with respect to the varying conformal structures and prove that the Morse index of the min-max conformal harmonic torus is bounded by one.

Yu-Shen Lin 林昱伸 (Boston University): Torelli theorem of ALH^* gravitational instantons

Gravitational instantons are non-compact complete hyper-Kahler 4-manifolds with L^2 curvatures. They are introduced by Hawkings as the building blocks of his quantum gravity theory. The gravitational instantons are classified by their volume growths into types ALE, ALF, ALG, ALH, and ALG^*, ALH^*. In this talk, I will give a short proof of the Torelli theorem of gravitational instantons of type ALH^*, namely their corresponding hyperKahler triples are completely determined by the cohomology classes of their hyper-Kahler triple. The proof is inspired by the SYZ mirror symmetry of log Calabi-Yau surfaces. This is a joint work with T. Collins and A. Jacob.