In this talk I will show the global (in time) well-posedness for the 3D viscous primitive equations of atmospheric and oceanic dynamics for all initial data. Motivated by strong anisotropic turbulence mixing I will also show the global well-posedness of this model with only horizontal viscosity and diffusion. Similar results also hold for the case of full viscosity, but only vertical diffusion. On the other hand, I will show that in the inviscid case there is a class of initial data for which the corresponding smooth solutions of the inviscid primitive equations develop singulary (blow-up) in finite-time.

 This talk is based on different joint results with C. Cao, S. Ibrahim, J. Li and K. Nakanishi.

In this talk,  I will present some results on axially symmetric solutions to the Allen-Cahn equation in entire spaces.  In particular,  a  complete branch of axially symmetric entire solutions to the Allen-Cahn equation in $\mathbb{R}^{3}$ will be constructed.  The nodal sets of these solutions behave asymptotically like catenoids at the two ends of the axis. These solutions are monotone in the radial direction and even in the direction of the axis after possible translation, and have finite Morse indices. The compactness of solutions and the linearized equations are carefully investigated and play important roles in the analysis.

Discrete exponential families are now widely used to model social and other networks with heterogeneity and/or dependence among edge variables.  Models for graphs written in this way are called exponential family random graph models, or ERGMs.  Although the ERGM framework is extremely flexible, few techniques other than Markov chain Monte Carlo have historically been available for studying ERGM behavior.  By contrast, random graphs with independent edge variables (i.e., the Bernoulli graphs) are the subject of a large literature, and much is known regarding their properties.  In this talk, I describe a method for exploiting this knowledge by constructing families of Bernoulli graphs that bound the behavior of an arbitrary ERGM in a well-defined sense.  By determining the properties of these Bernoulli graphs (either analytically or via simulation), one can thus constrain the properties of the associated ERGM.  I show how this technique can be used to recapitulate the well-known ``density explosion'' of the edge-triangle model, and also demonstrate the application of this technique to the problem of checking the robustness of a spatial network model to an omitted source of edgewise dependence.

Network Science is a rapidly growing interdisciplinary area
at the intersection of mathematics, computer science, and a
multitude of disciplines from the natural and life sciences
to the social sciences and even the humanities. Network
analysis methods are now widely used in proteomics, in the
study of social networks (both human and animal), in finance,
in ecology, in bibliometric studies, in archeology, and in a
host of other fields.

In this talk I will introduce the audience to some of the
mathematical and computational problems and methods of complex
networks, with an emphasis on the basic notions of centrality
and communicability. More specifically, I will describe some of
the problems in large-scale numerical linear algebra arising in
this area, and how they differ from the corresponding problems
encountered in more traditional applications of numerical analysis.

The talk will be accessible to students, requiring only a modest
background in linear algebra and graph theory.