Networks of nodes connected by links are a useful abstraction in many areas -- from sociology and biology to engineering and transportation. Networks are used to model causal structures, hierarchies, and scheduling. Although networks are typically drawn as node-link diagrams, several areas use geometric representations of networks (molecules in chemistry or floor-plans in engineering). This project explores intersection and contact representation of networks, where nodes are geometric objects (e.g., rectangles, segments) and links are realized by intersections (e.g., segments crossing) or contacts between the objects (e.g., rectangular duals). A geometric representation is more than just a way to display the network; it reveals underlying combinatorial structures which can often be described only using geometry. The goal of this project is to leverage these connections and build new bridges between geometry and combinatorics, in order to develop algorithms for intersection and contact representations of networks, with broader impact in information visualization, network analysis, and computational cartography. Educational activities, such as new coursework development, graduate and undergraduate student advising and outreach are integral parts of this project. Finally, continuing the investigator's tradition of implementing new algorithms and deploying software, dissemination of results will not only be accomplished by publishing in scientific journals, organizing workshops and seminars, but also by making software and data available. The specific research agenda is to identify connections between geometry and combinatorics in the context of intersection and contact representations of graphs. As a result of studying the classes of graphs that can be represented as contacts between simple polygons (e.g., triangles and quadrilaterals), the combinatorial properties of these graph classes will be used to design algorithms for constructing such representations. Efficient algorithms will be designed and implemented for constructing proportional (value-by-area) contact representations and for creating static and dynamic cartograms. These problems are addressed on two fronts: (a) by finding necessary and sufficient conditions for specific types of intersection and contact representation, developing new representation algorithms, and establishing tight bounds for the problems; and (b) by directly applying the theoretical results to practical problems in information visualization and cartography, as well as implementing and experimentally evaluating the new methods, which are made available as fully functional online systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.