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Plateau's problem, raised from 18th century, has been developed by many mathematicians. Reifenberg introduced in 1960s the Plateau's problem with Čech homological conditions, and got a remarkable existence result. Plateau's problem has received much attention in recent years. In the talk, we will present a very general existence result, and show that all of the infimum, obtained by solving Plateau's Problem with homological conditions for various homologies, are equal.

A compact set E in the d-dimensional Euclidean space is said to be arithmetically thick if there exists a positive integer n so that the n-fold arithmetic sum of E has non-empty interior. We give a sufficient condition to guarantee the arithmetic thickness. Moreover, we verify this property for several classes of fractal sets, including all the self-similar sets and self-conformal sets that are not lying in a proper affine subspace. We also prove it for self-affine sets under mild assumptions. This is joint work with Yu-Feng Wu.

In this talk, we will introduce some results on the uniqueness of solutions to the heat equation on graphs. Moreover, we will discuss some results on the finiteness results on the dimension of ancient solutions of polynomial growth on graphs.

We consider a class of non-symmetric Cantor sets $C$ determined by $C=a_0C\cup (a_1C+1-a_1), a_0, a_1\in (0,1)$. Under certain conditions on $a_0$ and $a_1$ the upper and lower densities are obtained explictly for each $x\in C$.

The uniqueness of tangent behavior is an important regularity property, and has been widely investigated in many circumstances in geometric measure theory and calculus of variations. In this talk we discuss the unique tangent behavior for stationary 1-varifolds in arbitrary Riemannian manifolds. Stationary varifolds are weak solutions for Plateaus problem in the setting of measures, defined as critical points of measure while deforming along any vector fields. We will first introduce the background of the problem, following by basic definitions and examples. Then we will focus on tangent behavior for stationary varifolds, and discuss some recent results on this subject.

In this talk, we will talk about the weighted norm inequalities associated with the local Muckenhoupt weights for the local fractional integrals on Gaussian measure spaces. More precisely, we obtain the weighted boundedness of local fractional integrals of order $\beta$ from $L^p(\omega^p)$ to $L^q(\omega^q)$ for $p\in(1,\infty)$ and from $L^p(\omega^p)$ to $L^{q,\infty}(\omega^q)$ for $p=1$ under the condition of $\omega\in A_{p,q,a}$, where $1/q=1/p-\beta$, and also obtain the weighted boundedness of the local fractional integrals, local fractional maximal operators and local Hardy-Littlewood maximal operators on the Morrey-type spaces over Gaussian measure spaces.

Plateau’s problem in Euclidean space may be given many distinct formulations with solutions to most of them admitting an associated varifold. This includes Reifenberg’s approach based on sets and Čech homology as well as Federer and Fleming’s approach using integral currents and their homology. Thus, we employ the setting of varifolds to prove a priori bounds on the geodesic diameter in terms of boundary behaviour. This is ongoing joint work with C. Scharrer.

Lipschitz classification of fractals is an interesting and hard problem. It is initialed by a work of Falconer and Marsh in 1992 and a book by G. David and S. Semmes in 1989. In this talk, we review the recent progress on this problem, especially on the construction of various Lipschitz invariants, including groups, linear spaces, fields, relations, measure-preserving property and multifractal spectra of measures.

We study the mean curvature flow with given non-smooth forcing term $g$. In 1993, Ilmanen proved the existence of the Brakke flow with $g \equiv 0$, by using the phase field method. Generally, the most difficult part of the proof of the existence theorem is the estimate of the positive part of the discrepancy measure. To solve the problem, Ilmanen showed the non-positivity of the discrepancy measure via the maximum principle. However, in the case of $g \not\equiv 0$, the property does not hold for the usual phase field method of the problem. In this talk, we explain a new modified Allen-Cahn equation which satisfies the non-positivity of the discrepancy measure, and we prove the existence of the weak solution for the mean curvature flow with forcing term in suitable Sobolev spaces.

Concentration phenomena can be observed in many PDE problems, such as the bubbling phenomena in harmonic maps and many other geometric variational problems. In these problems, when the spatial dimension is above the critical one, the concentration set is usually a high dimensional set. The lower order information about this concentration set (e.g. rectifiability, energy identity) has been explored for more than two decades. In this talk I would like to discuss some problems related to higher order information, through two typical examples: Allen-Cahn equation and nonlinear heat equation.

We develop the Lawson-Osserman's works on minimal graphs. Firstly, we construct a constellation of uncountably many Lawson-Osserman spheres, which are minimal in Euclidean spheres and therefore generate Lawson-Osserman cones that correspond to Lipschitz but non-differentiable solutions to the minimal surface system. Then, by the theory of autonomous systems in plane, we find for each Lawson-Osserman cone an entire minimal graph having it as tangent cone at infinity. Further, in addition to the truncated Lawson-Osserman cones, we discover infinitely many analytic solutions to the Dirichlet problem of minimal surfaces system for boundary data induced by certain Lawson-Osserman spheres. As a corollary, those Lawson-Osserman cones are non-minimizing. These behaviors are observed for the first time. This is the joint work with Prof. Xiaowei Xu and Yongsheng Zhang.

In this talk, we will discuss measure estimates of nodal sets of solutions to some elliptic equations of higher order. By using elliptic estimates and frequency functions, we will show some monotonicity properties and doubling conditions, and give the measure upper bounds for nodal sets of these solutions. This is a joint work with Tian Long.

In this talk, we generalize an argument of L. Simon about the uniqueness of tangent cones to the setting of biharmonic maps. We also discuss a proof of the removable singularity theorem of biharmonic maps and show how the ideas in the two proofs are related.