We propose a general method that parameterizes general surfaces with complex (possible branching) topology using Riemann surface structure. Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilinear coordinate systems on the surfaces. We can then automatically partition the surface using a critical graph that connects zero points in the global conformal structure on the surface. The trajectories of iso-parametric curves canonically partition a surface into patches. Each of these patches is either a topological disk or a cylinder and can be conformally mapped to a parallelogram by integrating a holomorphic I-form defined on the surface. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. For surfaces with similar topology

We introduce a new concept of strong vanishing property and investigate this property for minimally elliptic singularities. Acomparison theorem about strong vanishing property is also obtained. We use Zariski decomposition to give a formula of plurigenera for Gorenstein singularities and use it to test whether a divisor is f-nef.

No abstract uploaded!

We study the nonnegative part \overline {G_ {> 0}} of the De Concini-Procesi compactification of a semisimple algebraic group \overline {G_ {> 0}} , as defined by Lusztig. Using positivity properties of the canonical basis and parametrization of flag varieties, we will give an explicit description of \overline {G_ {> 0}} . This answers the question of Lusztig in Total positivity and canonical bases, Algebraic groups and Lie groups (ed. GI Lehrer), Cambridge Univ. Press, 1997, pp. 281-295. We will also prove that \overline {G_ {> 0}} has a cell decomposition which was conjectured by Lusztig.

When people in a society want to make inference about some parameter, each person may want to use data collected by other people. Information (data) exchange in social networks is usually costly, so to make reliable statistical decisions, people need to weigh the benefits and costs of information acquisition. Conflicts of interests and coordination problems will arise in the process. Classical statistics does not consider peoples incentives and interactions in the data-collection process. To address this imperfection, this work explores multi-agent Bayesian inference problems with a game theoretic social network model. Motivated by our interest in aggregate inference at the societal level, we propose a new concept, <i>finite population learning</i>, to address whether with high probability, a large fraction of people in a given finite population network can make good inference. Serving as a foundation, this concept enables