City University of New York

A common theme in the definitions of larger large cardinals is the existence of elementary embeddings from the universe into an inner model. In contrast, smaller large cardinals, such as weakly compact and Ramsey cardinals, are usually characterized by their combinatorial properties such as existence of large homogeneous sets for colorings. It turns out that many familiar smaller large cardinals have elegant elementary embedding characterizations. The embeddings here are correspondingly ‘small’; they are between transitive set models of set theory, usually the size of the large cardinal in question. The study of these elementary embeddings has led us to isolate certain important properties via which we have defined robust hierarchies of large cardinals below a measurable cardinal. In this talk, I will introduce these types of elementary embeddings and discuss the large cardinal hierarchies that have come out of the analysis of their properties. The more recent results in this area are a joint work with Philipp Schlicht.

Phylogenies, or evolutionary histories, play a central role in modern biology, illustrating the interrelationships between species, and also aiding the prediction of structural, physiological, and biochemical properties. The reconstruction of the underlying evolutionary history from a set of morphological characters or biomolecular sequences is difficult since the optimality criteria favored by biologists are NP-hard, and the space of possible answers is huge. Phylogenies are often modeled by trees with n leaves, and the number of possible phylogenetic trees is $(2n-5)!!$. Due to the hardness and the large number of possible answers, clever searching techniques and heuristics are used to estimate the underlying tree.

We explore the combinatorial structure of the underlying space of trees, under different metrics, in particular the nearest-neighbor-interchange (NNI), subtree- prune-and-regraft (SPR), tree-bisection-and-reconnection (TBR), and Robinson-Foulds (RF) distances.  Further, we examine the interplay between the metric chosen and the difficulty of the search for the optimal tree.

The talk will address two recent results concerning the Doi-Smoluchowski equation and the Onsager model for nematic liquid crystals. The first result concerns the existence of inertial manifolds for the Smloluchowski equation both in the presence and in the absence of external flows. While the Doi-Smoluchowski equation as a PDE is an infinite-dimensional dynamical system, it reduces to a system of ODEs on a set coined inertial manifold, to which all other solutions converge exponentially fast.  The proof uses a non-standard method, which consists in circumventing the restrictive spectral-gap condition, which the original equation fails to satisfy by transforming the equation into a form that does. 

The second result concerns the isotropic-nematic phase transition for the Onsager model on the circle using more complicated potentials than the Maier-Saupe potential. Exact multiplicity of steady-states on the circle is proven for the two-mode truncation of the Onsager potential.