We will review some recent work on free boundary minimal
hypersurfaces. In particular, we will explain a geometric classification
of the critical catenoid (joint with Ivaldo Nunes) and discuss what
information about such hypersurfaces in a general ambient manifold one can
extract from the knowledge of their Morse index (joint with Alessandro
Carlotto and Ben Sharp).

We explore the applications of the pcf theory to cardinal estimates.

My group uses computational approaches in collaboration with experimental labs to develop better mechanistic understanding of cell decision processes. One of the major challenges in modeling these systems is that the molecular constituents of signaling networks interact in a multitude of ways to form densely connected networks involving hundreds to thousands (and beyond) of distinct biochemical species. Rule-based modeling is an approach to modeling complex biochemical networks in which signaling molecules are represented as structured objects whose interactions are governed by rules, which serve as generators of the species and reactions that comprise the network. This approach enables concise and precise encoding of known molecular biochemistry, freeing the modeler from having to explicitly enumerate the large number of possible species and reactions that can arise in such systems. BioNetGen, which is developed and maintained by my group, is one of several rule-based modeling platforms that enable scalable specification and simulation of large-scale models of signal transduction and other biochemical systems.  In recent years its capabilities for modeling, simulation, and analysis have been greatly expanded and it has been used to model and gain mechanistic understanding of number of important signaling processes. Here, in addition to providing a general introduction to rule-based modeling and describing some recent developments, I will present an application of combined computational and experimental approaches to further mechanistic understanding of signaling through the T cell antigen receptor to control differentiation of T helper cells in the immune system. I will also discuss some recent collaborative work to quantify information flow in a signaling through cytokine receptors. This work suggests that previous estimates of the limiting effects of noise on signal flow may have underestimated the capacity of these biochemical systems to transmit information within the cell.