January 7 Chuu-Lian Terng

The Korteweg de Vries (KdV) equation models shallow water wave propagation in a narrow channel. It started the theory of solitons (solitary waves). Later, many important partial differential equations in applied mathematics, mathematical physics, and geometry were found to share many of the remarkable properties of KdV and are now referred to as soliton equations. In this talk I will explain some of the history and properties of KdV. Then discuss geometries associated to various soliton equations, and also some recent results concerning minimal spheres with special geometric structures.

January 21 Xiaowei Wang

The interplay between algebraic geometry and differential geometry has long history start from hodge, Kodaira, Yau, Tian and more recently Donaldson. The existence of special metric on a vector bundle or a manifold helps us to get many insights into their algebro-geometric property and vice versa. In this talk, I will discuss a metric characterization of whether a vector bundle over a projective manifold being stable or not in GIT sense, and it's relation with the classical Donaldson-Uhlenbeck-Yau picture.

January 28 Pengzi Miao

We will first discuss a generalized Positive Mass Theorem on a class of non-smooth asymptotically flat manifolds with broken mean curvature across a hypersurface. Then we will use it to modify Bartnik's quasi-local mass definition so that one can apply Corvino's scalar curvature deformation result to show that a minimal mass extension, if exists, must be a static metric. Finally, we will prove that, for any metric that is close enough to the Euclidean metric on a ball and has reflection invariant boundary data, there always exists a static metric extension with a geometric boundary condition.

Feburary 11th Jian Song

The global holomorphic invariant alpha invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metric. We apply the result of Tian, Lu and Zelditch on polarized Kahler metrics to approximate plurisubharmonic functions and compute the alpha invariant of CP^2 blowup at two points.

Feburary 14th Huaidong Cao

In the study of the Ricci flow on a compact Kaehler manifold $M^n$ with positive bisectional curvature, the most important problem, which is open for more than 15 years, is to give a proof of the Frankel conjecture via the Ricci flow method. The crucial step in doing so is to obain a uniform estimate on the curvature of the solution metric. In this talk we'll present a new uniform estimate in our joint work with B. Chen and X. Zhu. In contrast to the recent work of X. Chen and G. Tian, our proof does not rely on the exsitence of Kaehler-Einstein metrics on such a manifold $M^n$, but instead on the Harnack estimate for the scalar curvature of the solution metric which is a consequence of Li-Yau-Hamilton estimate for the K\"ahler-Ricci flow I obtain previously, and a very recent local injectivity radius estimate of Perelman for the Ricci flow.

Feburary 18th Bing Cheng

We will discuss different differential Harnack inequalities with focus on the Ricci flow, and in particular a new differential Harnack for the Ricci flow.

Feburary 27th Mark Gross

Mirror symmetry was a phenomenon discovered around 1990 by string theorists, suggesting a deep but mysterious relationship between apparently unrelated pairs of certain 3-dimensional complex manifolds called Calabi-Yau manifolds. I will discuss this phenomenon, and show how it can be at least partially explained using the Strominger-Yau-Zaslow conjecture, by dualizing torus fibrations. One obtains a beautiful picture in some explicit examples of how mirror symmetry arises at the topological level.

March 4th Jim Isenberg

A globally hyperbolic solution of Einstein's gravitational field equations is determined by specifying initial data on a 3-manifold S. The data must satisfy a set of constraint equations. Is it true that, for any manifold S, there is a set of data on that manifold which solves the constraints and therefore generates a spacetime solution?

We answer this question for each of the three main types of data which are of interest: spatially compact, asymptotically hyperbolic, and asymptotically flat. Gluing techniques play a key role in our analysis.

March 7th Scot Adams

We develop the M-S-Y Bochner formula and use it to prove Gromov's result that, for any compact quotient M of the globally symmetric space (SL_3(R))/(SO(3)), there is, up to diffeomorphism and rescaling, only one metric of nonpositive curvature on M, namely the locally symmetric one. If time permits, I'll explain how, with additional ergodic theoretic arguments, one may prove a foliated form of that result. The foliated form is joint work with Luis Hernandez.

March 11th Jean Steiner

We describe two mass-like quantities arising from the Green's function for the Laplacian operator on surfaces. The Robin's mass is obtained by regularizing the logarithmic singularity of the Green's function. We show that the Robin's mass is connected to a spectral invariant. On spheres, we introduce a "geometrical mass", which is, a priori, a smooth function on the sphere. The goemetrical mass is shown to be independent of the point on the sphere, and it is also a spectral invariant. Moreover, a connection to a Sobolev-type inequality reveals that it is minimized at the standard round metric. The definition of the geometrical mass is inspired by the role played by the Green's function for the conformal Laplacian and the Positive Mass Theorem in the solution to the Yamabe Problem.