Abstract: In this lecture, we present a result on the d-bar Cauchy problem for weakly pseudo-convex domains in CPⁿ. Among other things, we derive new a-priori estimates for d-bar Cauchy problem for domains with C¹-smooth or Lipschitz boundaries. As an application, we show that there exists no Levi-flat real hypersurface in CP^n with n>=3, which is locally a graph of a Lipschitz function. This non-existence result on Levi-flat hypersurface greatly improved an earlier result of Professor Siu published in Annals of Mathematics recently. Our method is different from Siu's.
This is a joint work with Mei-Chi Shaw.

Abstract: We will discuss some recent work of Glickenstein on combinatorial analogue of the Yamabe flow on simplicial 3-manifolds.

Abstract: In 1960 Federer and Fleming developed a theory of rectifiable currents which allowed for a solution of least-area Plateau problems in arbitrary dimension and codimension in Euclidean space. To account for nonorientable chains like a minimal Mobius band and other examples, Fleming introduced a theory of flat and rectifiable chains with coefficients each of the finite groups Z/jZ. In 1999, B.White generalized this to essentially the largest possible class of groups allowing compactness of rectifiable cycles in Euclidean space. Also in 2000, L. Ambrosio and B.Kirchheim generalized much of the Federer-Fleming work using their notion of currents in a metric space. Here we describe some new elementary definitions and arguments that allows one to treat both metric spaces and other coefficient groups simultaneously. We use some of the best ideas from the previous works.

Abstract: Let G(n,k) be the Grassmann manifold of all k dimensional linear subspaces of Rⁿ. Then each convex body K in Rⁿ defines a function, V_k(K| ), on G(n,k) where V_k(K|L) is the k dimensional volume of the orthogonal projection of K onto L. This is the k-th projection function, or k-brightness function, of K. Results about the extent that a convex body is determined by one or more of its projection functions will be surveyed. While there has been some recent progress in settling some long standing problems, such as Nakajima's problem from 1926 of showing that a convex body in three dimensional space with constant width and constant brightness is a Euclidean ball, there are still a large number of natural questions remaining open.

Abstract: Following Jacobi's reformulation of Lagrange's least action principle of classical mechanics, global geometry on the trajectories of a given mechanical system, such as an n-body system, can be reduced to the study of global geometry of geodesics with respect to a specific metric structure on its configuration space, nowadays called the Morse theory on geodesics of such a Riemannian manifold. In this talk, we shall focus on the outstanding special topic of 3-body motions in celestial mechanics.
I shall begin with a systematic discussion on the kene-matic geometry of mass-triangles and then proceed to prove some basic theorems on the global geometry of 3-body motions with vanishing angular momentum, which enable us to reduce the study of such motions to that a specific type of curves in S²(1). I shall also discuss some open problem which naturally present themselves for further in depth study of 3-body problem.

Abstract: We will present a Margulis lemma for compact Lie groups (small elements in any finite subgroup generate an abelian group) and some applications. A main tool used in this work is the curvature comparison in Riemannian geometry. This is a joint work with Marcin Maur and Yusheng Wang.

Abstract: While there have been many advances in the understanding of the topology of complete noncompact manifolds of nonnegative Ricci curvature, this area is wide open for further study. In particular, Milnor's famous 1969 conjecture that such a manifold has a finitely generated fundamental group is still open.
The speaker will survey a number of theorems and examples, including her work with Zhongmin Shen classifying the codimension one integer homology of these manifolds and her proof of the Milnor Conjecture when the manifolds are assumed to have small linear diameter growth. Unlike the algebraic approach in Burkhard Wilking's reduction of the Milnor conjecture to manifolds with abelian fundamental groups and the analytic proof by Shing-Tung Yau that a manifold with positive Ricci curvature has trivial codimension one real homology, the proofs of these results are purely geometric and can be described with a few key diagrams and lemmas.

Abstract: In this talk, I will discuss the heat flow of (extrinsic) biharmonic maps in 4-dimensions and show the existence of global weak solutions which are smooth except finitely many singular times. As a corollary, we prove that if the 4th fundamental group of the target manifold is zero, then any free homotopy class contains a minimizing biharmonic map. I will also indicate the existence of smooth biharmonic map flows in dimension at most 8 for small initial data.

Abstract: The quasi-local mass is a quantity associated with a compact space-like region in the space-time. It is expected that this information can be de- rived from the boundary, which is a two-dimensional space-like surface. By Throne's hoop conjecture, the quasi-local mass is supposed to be closely related to the formation of black holes in the enclosed region. We shall discuss some recent developments in this direction which include Shi-Tam's proof of the pos- itivity of the Brown-York mass, Liu-Yau's quasi-local mass and its positivity, and a generalization by Wang-Yau to a quasi-local energy-momentum space- like vector. The construction relies on the solutions of some canonically defined elliptic and parabolic equations associated to the geometry of the surface, and the application of the positive mass theorem.

Abstract: Stability issue comes up naturally in variational problems. One of the most important geometric variational problems is that of the total scalar curvature functional whose critical points corresponding to Einstein metrics. We call an Einstein metric stable if the second variation of the total scalar curvature functional is non-positive in the direction of changes in conformal structures. Using spin^c structure we prove that Kähler-Einstein metrics with nonpositive scalar curvature are stable. Moreover if all infinitesimal complex deformation of the complex structure are integrable, then the Kähler-Einstein metric is a local maximal of the Yamabe invariant, and its volume is a local minimum among all metrics with scalar curvature bigger or equal to the scalar curvature of the Kähler-Einstein metric. This is joint work with X. Dai and X. Wang.

Abstract: We will briefly discuss several aspects of p-harmonic geometry - its past, present, and future. They arise from the needs of daily life and exist in abundance. Today they lie at the foundation of the most intensive research in a surprisingly broad spectrum of fields.
We will make various sharp global integral estimates by a unified method, and find a dichotomy between constancy and `infinity' of weak sub- and supersolutions of a large class of degenerate and singular nonlinear partial differential equations on complete noncompact Riemannian manifolds, by introducing the concepts of their corresponding p-finite, p-mild, p-obtuse, p-moderate, and p-small growth, and their counter-parts p-infinite, p-severe, p-acute, p-immoderate, and p-large growth. These lead naturally to p-harmonic generalizations of the Uniformization Theorem and Bochner's Method, and an iterative method, by which we approach various geometric and variational problems in complete noncompact manifolds of general dimensions. For instance, we make p-harmonic applications to minimal surfaces and to biharmonic maps between complete noncompact manifolds. In particular, we give some partial answers to the Generalized Bernstein Conjecture, the Chen Conjecture, and problems of Kobayashi, and Lin and Wang. Certain interactions among p-harmonic geometry, complex and Riemannian geometry, conformal geometry, nonlinear degenerate potential theory, the isometric immersion of warped products in manifolds with curvatures bounded above by nonpositive constants, the geometry and the topology of 4-dimensional noncompact manifolds via Seiberg-Witten theory, and gauge field theory in physics will be made. Some open problems and conjectures will be discussed.

Abstract: We discuss the construction of exotic containers in a gravitational field. These are containers which, for certain volumes of fluid, possess a continua of geometricly distinct equilibria. Prior work on this topic has been done by R. Gulliver-S. Hildebrandt and subsequently by P. Concus-R. Finn. Besides describing our construction we are also interested in questions of stability.

Abstract: I will discuss the equations involving the Schouten tensor in conformal geometry. There is a nearly complete theory of existence and regularity for such equations in low dimensions. I will discuss the question of compactness for the space of such metrics. In particular talk about the consequence of the bubbling analysis for the space of solutions of such equation on a restricted class of 4-manifolds.

Abstract: We develop a new method for the calculation of the heat trace asymptotics of the Laplacian on symmetric spaces that is based on a representation of the heat semigroup in form of an average over the Lie group of isometries and obtain a generating function for the whole sequence of all heat invariants.

Abstract: The induced Lorentzian metric on a immersed 2-manifold M into a semi-Riemannian 3-manifold determines a consformal structure on M. Using this conformal structure, the notion of split-Kahler structure can be introduced analogously to Kahler structure on Riemann surfaces. The split-Kahler structure enables us to define Lorentz holomorphic maps on M. In this talk, we show that timelike surfaces of constant mean curvature 1 in anti-de Sitter (AdS) 3-space can be constructed from a pair of Lorentz holomorphic and Lorentz anti-holomorphic null curves in PSL(2,R) via Weierstrass type representation formula. The existence of such a representation is a consequence of Lawson correspondence between timelike constant mean curvature surfaces in AdS 3-space and Minkowski 3-space. The hyperbolic Gauss map plays an important role in the study of timelike surfaces of constant mean curvature 1 in AdS 3-space. The relationship between the Lorentz holomorphicity of the hyperbolic Gauss map and timelike surfaces of constant mean curvature 1 surfaces is also discussed.

Abstract: In this talk, we will introduce a p-harmonic approach to the Generalized Bernstein Conjecture. We will discuss a p-harmonic generalization of the Uniformization theorem with applications.

Abstract: The existence problem is solved for quasilinear and Hessian equations of Lane--Emden type, including the following two model problems: $$-\\Delta_p u = u^q + \\mu, \\qquad F_k[-u] = u^q + \\mu,\\qquad u \\ge 0, $$ on $\\mathbb{R}^n$, or on a bounded domain $\\Omega \\subset \\mathbb{R}^n$. Here $\\Delta_p$ is the $p$-Laplacian defined by $\\Delta_p u = {\\rm div} \\, (\\nabla u |\\nabla u|^{p-2})$, and $F_k[u]$ is the $k$-Hessian defined as the sum of $k\\times k$ principal minors of the Hessian matrix $D^2 u$ ($k=1,2, \\ldots, n$); $\\mu$ is a nonnegative measurable function (or measure) on $\\Omega$.
The solvability of these classes of equations has been an open problem even for good data $\\mu \\in L^s (\\Omega)$, $s>1$. Such results are deduced from our existence criteria with the sharp exponents $s = \\frac {n(q-p+1)} {pq}$ for the first equation, and $s = \\frac{n(q-k)}{2kq}$ for the second one. Furthermore, a complete characterization of removable singularities is also given for the corresponding homogeneous equations.
This talk is based on joint work with Igor E. Verbitsky.

Abstract: A method is given for calculation of a distribution of small particles, embedded in a medium, so that the resulting medium would have a desired radiation pattern for the plane wave scattering by this medium.

Abstract: We shall use arithmetic subgroups of generalized modular groups that act totally discontinuously on upper half space in Rⁿ to construct examples of some conformally flat manifolds. These manifolds are natural generalizations of Riemann surfaces. Dirac type operators will be introduced in these settings along with explicit formulas for their fundamental solutions. Some of their properties will be described.

Abstract: This talk is based on joint work with my student James Ryan Brown. Details appears in his PhD Dissertation, University of Missouri, May 2006.
The existence of a complex structure on the six-dimensional sphere is a famous open probelm in differential geometry, which has been addressed by Hirzebruch, Yau, and Chern among others. A complex structure on the six-sphere would would imply the existence of an exotic CP^3, meaning a complex manifold which is diffeomorphic but not biholomorphic to the standard CP^3. If an exotic CP^3 were to exist, it would have some unusual properties. We survey some known and some new results about the properties of a hypothetical CP^3.

Abstract: In this talk, we will discuss representing homotopy classes by p-harmonic maps with applications in minimal submanifolds in ellipsoids of general type.

Abstract: The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in Rⁿ with smaller volume of all k-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to this question is negative if k>3. The problem is still open for k=2,3. We study this problem in the hyperbolic space Hⁿ and show that in this case the answer is negative for all 2 <= k <= n-1.

Abstract: We construct symmetric convex bodies that are not intersection bodies, but all of their central hyperplane sections are intersection bodies. This result extends the studies by Weil in the case of zonoids and by Neyman in the case of subspaces of L_p.