Tree properties are a family of combinatorial principles that characterize large cardinal properties for inaccessibles, but can consistently hold for "small" (successor) cardinals such as $\aleph_2$.  It is a classic theorem of Magidor and Shelah that if $\kappa$ is the singular limit of supercompact cardinals, then $\kappa^+$ has the tree property.  Neeman showed how to force $\kappa^+$ to become $\aleph_{\omega+1}$ while maintaining the tree property.  Fontanella generalized these results to the strong tree property.

We show (in ZFC) that if $\kappa$ is a singular limit of supercompact cardinals, then $\kappa^+$ has the super tree property (this jump from "strong" to "super" is analogous to the jump in strength from strongly to supercompact cardinals).  We remark on how to get the super tree property at $\aleph_{\omega+1}$, and on some interesting consequences for the existence of guessing models at successors of singulars.  This is joint work with Dima Sinapova.

The ligand-receptor binding/unbinding is a complex biophysical process in which water plays a critical role. To understand the fundamental mechanisms of such a process, we have developed a new and efficient approach that combines our level-set variational implicit-solvent model with the string method for transition paths, and have studied the pathways of dry-wet transition in a model ligand-receptor system. We carry out Brownian dynamics simulations as well as Fokker-Planck equation modeling with our efficiently calculated potentials of mean force to capture the effect of solvent fluctuations to the binding and unbinding processes. Without the description of individual water molecules, we have been able to predict the binding and unbinding kinetics quantitatively in comparison with the explicit-water molecular dynamics simulations. Our work indicates that the binding/unbinding can be controlled by a few key parameters, and provides a tool of efficiently predicting molecular recognition with application in drug design.

In this continuation of my talk from last week, I will introduce the notion of a spectral gap subalgebra of a tracial von Neumann algebra and show how it connects to the definability of relative commutants.  I will also mention some applications of these results.  I will introduce all notions needed from the theory of von Neumann algebras.

In this talk, we shall give a mathematical setting of the Random Backpropogation  (RBP) method in unsupervised machine learning. When there is no hidden layer in the neural network, the method degenerates to the usual least square method. When there are multiple hidden layers, we can formulate the learning procedure as a system of nonlinear ODEs. We proved the short time, long time existences as well as the convergence of the system of nonlinear ODEs when there is only one hidden layer. This is joint work with Pierre Baldi in Neural Networks 33 (2012) 136-147, and with Pierre Baldi, Peter Sadowski in Neural Networks 95 (2017) 110-133 and in Artificial Intelligence 260 (2018), 1-35.