Georgia Institute of Technology College of Sciences

In protein-protein interaction (PPI) networks, or more general protein-protein association networks, functional similarity is often inferred based on the some notion of proximity among proteins in a local neighborhood. In prior work, we have introduced diffusion state distance (DSD), a new metric based on a graph diffusion property, designed to capture more fine-grained notions of similarity from the neighborhood structure that we showed could improve the accuracy of network-based function-prediction algorithms. Boehnlein, Chin, Sinha and Liu have recently shown that a variant of the DSD metric has deep connections to Green's function, the normalized Laplacian, and the heat kernel of the graph. Because DSD is based on random walks, changing the probabilities of the underlying random walk gives a natural way to incorporate experimental error and noise (allowing us to place confidence weights on edges), incorporate biological knowledge in terms of known biological pathways, or weight subnetwork importance based on tissue-specific expression levels, or known disease processes. Our framework provides a mathematically natural way to integrate heterogeneous network data sources for classical function prediction and disease gene prioritization problems. This is joint work with Mengfei Cao, Hao Zhang, Jisoo Park, Noah Daniels, Mark Crovella and Ben Hescott.

In the early '90s, Gromov introduced a notion of hyperbolicity for geodesic metric spaces. The study of groups of isometries of such spaces has been an underlying theme in much of the work in geometric group theory since that time. Many geodesic metric spaces, while not hyperbolic in the sense of Gromov, nonetheless display some hyperbolic-like behavior. I will discuss a new invariant, the Morse boundary of a space, which captures this behavior. (Joint work with Harold Sultan and Matt Cordes.)

Given a plane field $dz-xdy$ in $\mathbb{R}^3$. A Legendrian knot is a knot which at every point is tangent to the plane at that point. One can similarly define a Legendrian knot in any contact 3-manifold (manifold with a plane field satisfying some conditions). In this talk, we will explore Legendrian knots in $\mathbb{R}^3$, $J^1(S^1)$, and $\#^k(S^1\times S^2)$ as well as a few Legendrian knot invariants. We will also look at the relationships between a few of these knot invariants. No knowledge of Legendrian knots will be assumed though some knowledge of basic knot theory would be useful.

Auckly gave two examples of irreducible integer homology spheres (one toroidal and one hyperbolic) which are not surgery on a knot in the three-sphere. Using an obstruction coming from Heegaard Floer homology, we will provide infinitely many hyperbolic examples, as well as infinitely many examples with arbitrary JSJ decomposition. This is joint work with Lidman.