Abstract: In the first part of my talk, I will explain my motivation to introduce the notions of δ-invariants on Riemannian manifolds and of ideal immersions; present some optimal inequalities involving the δ-invariants for isometric immersions; and provide some of their applications. In the second part of my talk, I will present some recent joint results with Franki Dillen, Alicia Prieto-Martin, Jeori Van der Veken and Luc Vrancken concerning improved optimal inequalities for Lagrangian submanifolds in complex space forms involving the δ-invariants

Abstract: In this talk, I will present some recent results in the study of p-harmonic maps on a complete noncompact manifold M with a weighted Poincare inequality such that the Ricci curvature of M has a lower bound depending on the weight function. We derive the Bochner's formula for solutions of a perturbed p-Lapace equation, obtain sharp Kato's inequality, and prove Liouville type theorem for weakly p-harmonic function with finite p-energy on M. This, in particular, yields information on topology, i.e. M has one p-hyperbolic end. We also prove the Liouville type theorems for p-harmonic functions with finite q-energy, and extend them to p-harmonic morphisms and conformal maps. This is joint work with Shu-Cheng Chang and Shihshu Walter Wei.

Abstract : We will consider the classical problem of trying to find warped product metrics with a fixed base that are also Einstein. There are by now many such examples and some of them are quite surprising. We'll indicate a new construction where the base is an arbitrary algebraic Ricci soliton. This is a joint work with Peter Petersen and William Wylie.

Abstract: Let X be a locally closed subset of Rⁿ so that all the tangent cones (in the sense of Federer), T_pX , are affine hyperplanes of Rⁿ, the dependence on p is continuous, and the measure theoretic multiplicity at each point is at most m < 3/2. Then X is an embedded C¹ hypersurface of Rⁿ. As applications it is shown (1) any convex real analytic hypersurface of Rⁿ is C¹ and (2) if X is real algebraic, strictly convex, and unbounded, then it is a graph of a C¹ function over a hyperplane. This is joint work with Mohammad Ghomi.

Abstract: More than 30 years ago, Schoen-Yau and later Witten made major breakthroughs in proving the positive mass theorem. It has become one of the most important theorems in general relativity and differential geometry. In the first part of the talk, I will introduce the positive mass theorem and present our recent work that extends the classical three-dimensional results to higher dimensions. In the second part, I will discuss how the observation from general relativity enables us to solve classical geometric problems related to the scalar curvature.

Abstract: For two comlpex vector bundles admitting a homomorphism with isolated singularities between them, we establish a Poincare-Hopf type formula for the difference of the Chern character number of these two vector bundles. As a consequence, we extend the original Poincare-Hopf index formula to the case of complex vector fields. This is a joint work with Huitao Feng and Weiping Zhang.

Abstract: In this talk, we present some recent results in the study of the L^p-Hodge decomposition on complete Riemannian manifolds. As applications, we prove some vanishing theorems of the L^p-cohomology on complete Riemannian manifolds, and establish some L^pestimates and existence theorems of the Cauchy-Riemann operator on complete Kahler manifolds. A probabilistic approach to the Riesz transforms and Riesz potential play an important role.

Abstract: I will report some recent progress on geometric inequalities on Riemannian manifolds and complex manifolds with applications. As a consequence, we introduce the notion of certain Banach spaces, and obtain generalized Cafferelli-Kohn-Nirenberg type and Hardy type inequalities on Riemannian manifolds for functions in these Banach spaces. We also give counter-examples of certain Hardy-type inequalites for smooth functions with compact support under a curvature assumption. Some applications to p-harmonic geometry will be discussed. This is based on joint work with Jui-Tang Chen and Shihshu Walter Wei, and joint work with Shihshu Walter Wei.

Abstract: We will discuss the behavior of Ricci-flat Kähler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We will prove a version of Candelas and de la Ossas conjecture: Ricci-flat Calabi-Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi-Yau manifolds and a compact metric space in the Gromov-Hausdorff topology. This is joint work with Yuguang Zhang.

Abstract: We study global aspects of the correspondence between isomondromic deformations and solutions of the Painleve VI ODE. Any solution of Painleve VI extends to a meromorphic function on a covering space of the projective line with three points removed. We are interested in algebraic solutions of Painleve VI, which live on finitely-sheeted covering spaces. We show that there exist isomondoromic deformations that are not globally isomorphic but that correspond to the same algebraic solution of Painleve VI. We discuss some parallels between isomonodromic deformations and moduli stacks of algebraic curves.

Abstract: The Schoen-Yau Positive Mass Theorem states that an asymptotically flat 3 manifold with nonnegative scalar curvature has positive ADM mass unless the manifold is Euclidean space. Here we examine sequences of such manifolds whose ADM mass is approaching 0. We assume the sequences have no interior minimal surfaces although we do allow them to have boundary if it is a minimal surface as is assumed in the Penrose inequality. It is known that such sequences need not converge in the smooth sense (as can be seen with a sequence of Schwarzschild spaces). Nor do they converge in the Gromov-Hausdorff sense (due to the possible existence of thin deep gravity wells). We conjecture that they do converge to Euclidean space in the pointed Intrinsic Flat sense for a well chosen sequence of points. The Intrinsic Flat Distance, introduced in joint work with Stefan Wenger, can be estimated using filling manifolds which allow one to control thin wells and small holes. Here we present joint work with Dan Lee constructing such filling manifolds explicitly and proving the conjecture in the rotationally symmetric case. We also discuss sequences of manifolds approaching equality in the Penrose Inequality, techniques that can be used to control the intrinsic flat distance, key examples in the nonrotationally symmetric case and open problems related to the conjecture stated above.

Abstract: In this talk I will discuss some recent works on the bubbling phenomena for approximate biharmonic maps in dimension 4. For general target manifolds, we obtain an energy identiy for weakly convergent sequences of approximate biharmonic maps whose bitension fields are bounded in L^p for p > 4/3, which can be improved to p > 1 when the target manifold is a round sphere. An application, the energy identity is obtained the heat flow of biharmonic maps at the time t = ∞.

Abstract: The laws of nature often provide information about curvature properties. Nature also endowed us with elegant symmetry, criticality, and minimal principle, and cherishes energy so much as to conserve them. As a unifying concept, p-harmonic geometry cuts across all mathematics. Examples of p-harmonic maps include: elementary functions such as exponential, logarithmic, trignometric, and polynomial functions, linear transformations, analytic functions, geodesics, isometries, isometric minimal immersions, Riemannian submersions with minimal fibers, harmonic maps, conformal maps , Hopf fibrations, holomorphic maps, etc. Most recently, Marques and Neves' proof of the Willmore Conjecture requires a clever use of a 3-harmonic map to transform Willmore functional on the space of surfaces in R³ to Area functional on the space of minimal surfaces in S³. Here minimal surfaces in S³ are p-harmonic maps for every p > 1. We'll discuss recent work on Curvature, Criticality, Convervation Laws, and Unity in p-Harmonic Geometry, their interconnectedness, applications, and related problems.