We discuss about uniqueness of ancient mean curvature flow whose rescaled flow stays in a one-side of a shrinker. The rescaled flow locally and smoothly converges to the shrinker as time goes back. Hence, the flow is the graph of a positive function defined on the shrinker. Therefor, a parabolic Liouville’s theorem for positive entire ancient solutions gives us the uniqueness.

The fundamental gap is the difference of the first two eigenvalues of the Laplacian, which is important both in mathematics and physics. We will review many recent fantastic results for convex domains in $\mathbb R^n, \mathbb S^n, \mathbb H^n$ with Dirichlet boundary conditions. Then we will present a very recent estimate for the convex domain in surfaces with positive curvature. The last result is joint with G. Khan, H. Nguyen, M. Tuerkoen.

The question of whether stable minimal surfaces are holomorphic under suitable geometric conditions has been much studied, beginning with work of Lawson-Simons in complex projective space, and the proof of the Frankel conjecture by Siu-Yau. A theorem of Micallef shows that a stable minimal immersion of a complete oriented parabolic surface into Euclidean 4-space is holomorphic with respect to an orthogonal complex structure, and the same result in all dimensions if additionally the minimal surface has finite total curvature and genus zero. However, Arezzo, Micallef and Pirola gave a counterexample in general. It is therefore necessary to strengthen the stability condition in the general question. We will discuss some recent progress on this question. This is joint work with R. Schoen.

In this talk we establish two surprising types of Weyl’s laws for some compact $\mathrm{RCD}(K, N)$/Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for $\mathrm{RCD}(K, N)$ spaces. Our results depends crucially on analyzing and developing important properties of the examples constructed by Pan-Wei. Of independent interest, this also allows us to provide a counterexample to a conjecture by Cheeger-Colding. This is a joint work with Xianzhe Dai (University of California, Santa Barbara), Jiayin Pan (University of California, Santa Cruz) and Guofang Wei (University of California, Santa Barbara).

If $\Sigma_{i}$ is a stable submanifold of $M_{i}$, for $i=1,2$, then $\Sigma_{1}\times \Sigma_{2} $ is a stable submanifold in $M_{1}\times M_{2}$. Are all the stable submanifolds in $M_{1}\times M_{2}$ like that? We will show that is the case for specific dimensions and codimensions in the product of a complex or quaternionic projective space with any other Riemannian manifold. We will also talk about the behaviour of stable submanifolds under a complex structure of the product of two complex projective spaces. We will describe how our proofs were motivated by work that has be done by Simons, Lawson, Ohnita, Torralbo and Urbano. Part of this work is joint with Shuli Chen (Stanford).

$Q$-valued maps minimizing a suitably defined Dirichlet energy were introduce by Almgren in his proof of the optimal regularity of area minimizing currents in any dimension and codimension. In this talk I will discuss the extension of Almgren’s result to stationary $Q$-valued maps in dimension $2$.

This is joint work with Jonas Hirsch (Leipzig).

In this talk, I will explain how to use mean curvature flow to obtain explicit lower bound for the density of topologically nontrivial minimal cones. This is joint with Jacob Bernstein.

The Allen-Cahn equation is a semi-linear elliptic equation arising in the van der Waals-Cahn-Hilliard theory of phase transitions. Earlier fundamental work by De Giorgi, Modica, Sternberg etc. revealed intriguing relationship between the Allen-Cahn equation and the theory minimal surfaces. Based on the deep regularity theory by Hutchinson, Tonegawa and Wickramasekera, Guaraco recently introduced a new PDE approach to the existence of minimal surfaces via the Allen-Cahn equation and there have been substantial progress along this direction in the past few years. In this talk, we will consider the Allen-Cahn equation on bounded domains and describe some geometric and analytic aspects of the boundary behaviour of the limit-interfaces. This work is substantially supported by research grants from Hong Kong Research Grants Council and National Science Foundation China.

In this talk we give survey what is currently known for Chen’s flow, and discuss some very recent results. Chen’s flow is the biharmonic heat flow for immersions, where the velocity is given by the rough Laplacian of the mean curvature vector. This operator is known as Chen’s biharmonic operator and the solutions to the elliptic problem are called biharmonic submanifolds. The flow itself is very similar to the mean curvature flow (this is essentially the content of Chen’s conjecture), however proving this requires quite different strategies compared to the mean curvature flow. We focus on results available in low dimensions – curves, surfaces, and 4-manifolds. We provide characterisations of finite-time singularities and global analysis. The case of curves is particularly challenging. Here we identify a new shrinker (the Lemniscate of Bernoulli) and use some new observations to push through the analysis. Some numerics is also presented. The work reported on in the talk is in collaboration with Yann Bernard, Matthew Cooper, Philip Schrader and Glen Wheeler.

For a balanced bistable reaction-diffusion equation, an axisymmetric traveling front has been well studied. We prove that an axially asymmetric traveling front with any positive speed does exist in a balanced bistable reaction-diffusion equation. Our method is as follows. We use a pyramidal traveling front for an imbalanced reaction-diffusion equation whose cross section has a major axis and a minor axis. Preserving the major axis and the minor axis to be given constants and taking the balanced limit, we obtain an axially asymmetric traveling front in a balanced bistable reaction-diffusion equation. This traveling front is monotone decreasing with respect to the traveling axis, and its cross section is a compact set with a major axis and a minor axis.