public 01:00:48

Yanir Rubinstein : Einstein metrics on Kahler manifolds

  -   Geometry and Topology ( 232 Views )

The Uniformization Theorem implies that any compact Riemann surface has a constant curvature metric. Kahler-Einstein (KE) metrics are a natural generalization of such metrics, and the search for them has a long and rich history, going back to Schouten, Kahler (30's), Calabi (50's), Aubin, Yau (70's) and Tian (90's), among others. Yet, despite much progress, a complete picture is available only in complex dimension 2. In contrast to such smooth KE metrics, in the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is `bent' at some angle along a complex hypersurface), motivated by applications to algebraic geometry and Calabi-Yau manifolds. More recently, Donaldson suggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures. In this talk I will try to give an introduction to Kahler-Einstein geometry and briefly describe some recent work mostly joint with R. Mazzeo that resolves some of these conjectures. One key ingredient is a new C^{2,\alpha} a priori estimate and continuity method for the complex Monge-Ampere equation. It follows that many algebraic varieties that may not admit smooth KE metrics (e.g., Fano or minimal varieties) nevertheless admit KE metrics bent along a simple normal crossing divisor.

public 01:34:50

Amit Einav : Entropic Inequality on the Sphere

  -   Geometry and Topology ( 222 Views )

It is an interesting well known fact that the relative entropy with respect to the Gaussian measure on $\mathbb{R}^N$ satisfies a simple subadditivity property. Namely, if $\Pi_1^{(i)}(F_N)$ is the first marginal of the density function F_N in the i-th variable then \begin{equation} \sum_{i=1}^N H(\Pi_1^{(i)}(F_N) | \gamma_1) \leq H(F_N | \gamma_N), \end{equation} where $\gamma_k$ is the standard Gaussian on $\mathbb{R}^k$. Surprisingly enough, when one tries to achieve a similar result on $\mathbb{S}^{N-1}(\sqrt{N})$ a factor of 2 appears in the right hand side of the inequality (a result due to Carlen, Lieb and Loss), and the constant is sharp. Besides a deviation from the simple equivalence of ensembles principle in equilibrium Statistical Mechanics, this entropic inequality on the sphere has interesting ramifications in other fields, such as Kinetic Theory. In this talk we will present conditions on the density function F_N, on the sphere, under which we can get an ‘almost’ subaditivity property; i.e. the factor 2 can be replaced with a factor of $1+\epsilon_N$, with $\epsilon_N$ given explicitly and going to zero. The main tools to be used in order to proved this result are an entropy conservation extension of F_N to $\mathbb{R}^N$ together with comparison of appropriate transportation distances such as the entropy, Fisher information and Wasserstein distance between the marginal of the original density and that of the extension. Time permitting, we will give an example, one that arises naturally in the investigation of the so-called Kac Model, to many families of functions that satisfy these conditions.