Georgia Institute of Technology College of Sciences

Graphs, which in their simplest forms are vertices connected by edges,
are widely used in high performance computing, machine learning and
network science. This talk will use recent progresses on two
well-studied algorithmic problems in static and dynamic graph,
max-flows and dynamic matchings, to discuss a methodology for
designing faster algorithm for large graphs. This approach is
motivated by a fundamental phenomenon in data structures: the
advantages of offline data structures over online ones.

I will start by describing how work on max-flows led to a focus on
finding short paths in residual graphs, and how investigating more
global notions of progress in residual graphs led to a more
sophisticated and general understanding of iterative methods and
preconditioning. I will then discuss a similar phenomenon in dynamic
graphs, where maintaining a large matching seems to require the online
detection of short augmenting paths, but can once again be
circumvented through the offline construction of smaller equivalent
graphs.

Cutting a polyhedron along some spanning tree of its edges will yield an isometric immersion of the polyhedron into the plane. If this immersion is also injective, we call it an unfolding. In this talk I will give some general results about unfoldings of polyhedra. There is also a notion of pseudo-edge unfolding, which involves cutting on a pseudo edge graph, as opposed to an edge graph. A pseudo edge graph is a 3-connected graph on the surface of the polyhedron, whose vertices coincide with the vertices of the polyhedron, and whose edges are geodesics. I will explain part of the paper "Pseudo-Edge Unfoldings of Convex Polyhedra," a joint work of mine with Professor Ghomi, which proves the existence of a convex polyhedron with a pseudo edge graph along which it is not unfoldable. Finally, I will discuss some connections between pseudo edge graphs and edge graphs. 

Phylogenetic trees  are  the fundamental  mathematical  representation  of evolutionary processes in biology. As data objects, they are characterized by the challenges associated with "big data," as well as the  complication that  their  discrete  geometric  structure  results  in  a  non-Euclidean phylogenetic  tree  space,  which  poses  computational  and   statistical limitations.

In this  talk, I  will compare  the geometric  and statistical  properties between a  well-studied framework  -  the BHV  space, and  an  alternative framework that  we  propose, which  is  based on  tropical  geometry.  Our framework exhibits analytic,  geometric, and  topological properties  that are desirable for  theoretical studies in  probability and statistics,  as well  as  increased  computational  efficiency.  I  also  demonstrate  our approach on an example of seasonal influenza data.