The Ricci iteration and its applications

In this Note we introduce and study dynamical systems related to the Ricci operator on the space of Kahler metrics as discretizations of certain geometric flows. We pose a conjecture on their convergence towards canonical Kahler metrics and study the case where the first Chern class is negative, zero or positive. This construction has several applications in Kahler geometry, among them an answer to a question of Nadel and a construction of multiplier ideal sheaves.Comment: v2: shortened introduction. v3: corrected some typos. v4: shortened to fit in C. R. Acad. Sci. Pari

The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan

We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge-Ampere equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of solutions with prescribed regularity. In the complex domain, we characterize the C^3 lifespan of the HCMA in terms of analytic continuation of Hamiltonian mechanics and intersection of complex time characteristics. We use a conservation law type argument to prove uniqueness of solutions of the Cauchy problem for the HCMA. We then prove that the Cauchy problem is ill-posed in C^3, in the sense that there exists a dense set of C^3 Cauchy data for which there exists no C^3 solution even for a short time. In the real domain we show that the HRMA is equivalent to a Hamilton--Jacobi equation, and use the equivalence to prove that any differentiable weak solution is smooth, so that the differentiable lifespan equals the convex lifespan determined in our previous articles. We further show that the only obstruction to C^1 solvability is the invertibility of the associated Moser maps. Thus, a smooth solution of the Cauchy problem for HRMA exists for a positive but generally finite time and cannot be continued even as a weak C^1 solution afterwards. Finally, we introduce the notion of a "leafwise subsolution" for the HCMA that generalizes that of a solution, and many of our aforementioned results are proved for this more general object

On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow

In this note we construct Nadel multiplier ideal sheaves using the Ricci flow on Fano manifolds. This extends a result of Phong, Sesum and Sturm. These sheaves, like their counterparts constructed by Nadel for the continuity method, can be used to obtain an existence criterion for Kahler-Einstein metrics.Comment: v2: 1. added details for the case n=1. 2. added some references. v3: minor changes. To appear in Transactions of the American Mathematical Societ

Reforming Public School Systems Through Sustained Union-Management Collaboration

Presents case studies of sustained collaboration between teachers' unions and management in school reform; common elements in initiating events, strategic priorities, supportive system infrastructure, and sustaining factors; and lessons learned