NASA SCaN The Mathematics of the Near Space Network (SIP)
NASA’s Near Space Network provides mission-critical communications and navigation services, enabling the transmission of science and exploration data to and from space. As a single point of service for missions in the near-space region — out to two million kilometers away — the network connects users with either government or commercial service providers.
The fundamental goal of this internship is to create a venue for students to apply mathematics that are traditionally pure (e.g. algebraic topology, algebraic geometry, homotopy type theory, etc.) to problems in networked communications. We will work in a team of students to use the tools and products of the Near Space Network to drive various research projects and studies as suggested below. The summer contributions are taken very seriously by the project, and typically we try to get some publications out based on the work.
The network modeling effort is where we try to understand the global structure of a network. Traditionally such a structure is represented as a graph, however a space network is a vast generalization - hence we need deeper tools to accurately model the time-varying, heterogeneous structure of these networks. As commercial services become more and more common, the overlay-nature of space networks will become more pronounced, and to achieve a returns to scale with communicating assets deeper machinery needs to be involved. Moreover, we must consider the modeling the traffic over these networks, as network loading cannot be truly decoupled from the underlying topology.
• The networks can be thought of as temporal (hyper)graphs. Routing over these graphs can be modeled naturally using sheaves – for example, Dijkstra’s algorithm can be written in the language of sheaves, and it has been shown that the pullback of various routing sheaves gives rise to a routing sheaf. One project is to take simulations of space networks and to model the combination of dynamic and static routing using this pullback. This algorithmic approach could eventually lead to routing algorithms implemented for testing in NS3.
• Graphs can be realized as varieties. Recent work suggests this can be extended to hypergraphs, which can among other things more accurately model multicast and broadcast communications. One project here is to express hypergraphs as schemes and to model these in such software as Macaulay2. From here, coherent sheaf cohomology, deformation theory, and other tools can be considered to probe the geometry of communications.
• Tools from computational (zigzag) homology can be used to simplify networks, and in particular, to suggest structures analogous to autonomous systems in traditional networking. These tools can be used to explore ways to simplify routing across space networks, and perhaps even to suggest means to addressing. Resolving scalability enables providing networking and routing as a service.
• We wish to explore applications of higher sheaf theory to graphs and hypergraphs. Example angles of attack include graphs as varieties/schemes, moduli spaces/stacks, and simplicial sets.
• Networks and traffic flows can be modeled in the setting of tropical geometry. Following parametric graph optimization work from last summer, we wish to explore ways to propose optimal updates to existing networks.
• Along the tropical geometry lines, we can also ask about correspondence between tropical varieties/schemes and temperate varieties/schemes, and potential correspondences between their moduli spaces/deformations.
• We can also model link-saturation using parametric temporal graphs, which could be used to study the nature of prioritized traffic.
• It has been shown that game theory, particularly mixed cooperative/non-cooperative games, can be used to model temporal networks with a view towards congestion and hence load balancing.
• U.S. Citizen
• GPA 3.0+ (on a 4.0 scale)
• Currently enrolled full-time student
The Space Communications and Navigation Internship Project (SIP) is an intensive add-on program which complements the core NASA intern experience. SIP offers additional professional development growth opportunities in addition to technical work experience.The Space Communications and Navigation Internship Project (SIP) is an intensive add-on program which complements the core NASA intern experience. SIP offers additional professional development growth opportunities in addition to technical work experience.