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This talk focuses on recent progress regarding the collapsing theory of Einstein spaces. Specifically, the talk consists of the following primary parts: (1) We will introduce some regularity and structure theorems for higher dimensional collapsing manifolds with Ricci curvature bounds which works in very general settings. (2) This talk is also concerned with the geometry of collapsed Ricci-flat Kähler manifolds. Besides exhibiting new collapsing Einstein families, we will establish the correspondence between complex structures (Type II) degeneration and the metric collapsing.

In 1950’s, Nash-Kuiper built up the C^1 isometric embedding for any surface into $/mathbb{R}^3$, this can be viewed as analysis side of isometric embedding. On the other hand, there is obstruction for the existence of $C^2$ isometric embedding of surface into $/mathbb{R}^3$ known since Hilbert, which reflects the geometry flavor of isometric embedding. What’s happening from $C^1$ to $C^2$ (from analysis to geometry)? We will present our partial progress along this direction. The talk will be accessible to audience with basic knowledge of analysis and differential geometry.

The subspace of symmetric tensors and the subspace of anti-symmetric tensors are two natural reducing subspaces of tensor product A⊗I+I⊗A and A⊗A for any bounded linear operator A on a complex separable Hilbert space H. We show the set of operators A such that these two subspaces are the only (nontrivial) reducing subspaces of A⊗I+I⊗A is a dense G_δ set in B(H). This generalizes Halmos’s theorem that the set of irreducible operators is a dense G_δ set in B(H). The same question for A⊗A is still open.

In this presentation, we will give some recent progress on the geometry of compact complex manifolds with RC-positive tangent bundles including confirming a long-standing open problem of Yau on rational connectedness and various rigidity theorems of harmonic maps and holomorphic maps.

For a complex Hilbert space H, the d-copy tensor product of H is denoted by H^⊗d. For a class of tensor products of operators on H^⊗d which are invariant under a subgroup of the permutation group of d element, we identify their reducing subspaces. These reducing subspaces are formally (or implicitly) known through Schur-Weyl duality in the group representation theory where finite dimensional vectors spaces and the invertible similarity are general used. We state these results in the operator theoretic framework which deals with infinite dimensional complex Hilbert spaces and uses the unitary similarity. We explicitly display some of these reducing subspaces. Most importantly we initiate the investigation of the question for which operator these reducing subspaces are minimal.

We study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any integer k>0, as the genus g goes to infinity, the smallest k-th eigenvalue of a Riemann surface in any thick part of moduli space of Riemann surfaces of genus g is uniformly comparable to 1/g^2 in g. This is a joint work with Yuhao Xue.

We first discuss some relations between positivity, geometric PDEs and Hodge-index type theorems. It is noted that using complex Hessian equations and the concavity inequalities for elementary symmetric polynomials inspires a generalized form of Hodge index inequality. Inspired by this result, we obtain a mixed Hodge-Riemann bilinear relation. The new feature is that this Hodge-Riemann bilinear relation holds with respect to classes in which some satisfy particular positivity condition, but could be degenerate along some directions. We also discuss its some connnections with the Hodge theory of algebraic maps.

The dual Brunn-Minkowski theory, initiated by Lutwak, provides powerful tools to solve the long-standing Busemann-Petty problem in the 1990's. Among those deep results, the dual Brunn-Minkowski and dual Minkowski inequalities are the most important. In this talk, I will discuss the newly introduced dual Orlicz-Brunn-Minkowski theory, a nontrivial extension of the dual Brunn-Minkowski theory. In particular, I will talk about the dual Orlicz-Minkowski and dual Orlicz-Brunn-Minkowski inequalities. These inequalities are based on the Orlicz-addition of star bodies, and are thought to