Research Areas
|
|
Computational Group Theory, Cryptography and Applications to System Security Computational group theory is the study of groups by means of computers. It is concerned with designing and analysing algorithms and data structures to compute information about groups. Specific groups that I am interested in include: free groups, the automorphism group of free groups, and one relator groups.
Equations over free groups. We are developing software which, given a system of equations S over a free group, computes a complete description of the set of all solutions of S, and provides a formal mathematical proof that this description is complete and correct. Such computational software will significantly impact the development of the general theory of algebraic geometry over free groups, leading to the discovery of new hypotheses, the confirmation and exploration of existing conjectures, and the construction of databases of equations with interesting properties. The software will be instrumental in bringing us closer to the resolution of several open problems in combinatorial group theory that can be formulated in terms of such equations. Examples include questions regarding the faithfulness of the Gassner representation for Braid groups, the cardinality of the Burnside group of exponent 5, and the Hanna Neumann conjecture bounding the rank of the intersections of subgroups of free groups.
Current Students. Shana Hyvat (Ph.D candidate).
Grants/Collaborations. NSA Young Investigator Award (Fall 2007-Summer 2008)
Transparent security for memory. We are developing software technology that permits us to transparently add memory encryption to existing software, without requiring complicated software redesign or additional costly hardware.
Current Students. Jamie Levy (M.Sc candidate).
Graph Theory and Graph AlgorithmsWe explore fundamental problems in graph theory, with an emphasis on metric, and algebraic structures on the set of all graphs.
We consider graph embedding problems which arise naturally from the context of multi-agent systems, distributed computing, wireless networks, and optical networks.
Past Students: Birendro Roy (B.Sc).
Grants/Collaborations. NRL Center for Computational Sciences.
Optical Network Protocols for Signalling, Routing, and Management Physical topology design. We are developing effective methods for design of optimal physical topologies for large-scale Dense Wavelength Division Multiplexing (DWDM) networks that support multi-granular switching. This requires new analytic formulations and planning heuristics that consider factors such as: placement of wavelength/band switching resources, traffic grooming resources, optical signal strengths/dispersions, and equipment costs. By quantifying tradeoffs between wavelength, band, and multi-granular switching techniques we seek to develop models that enable network designers to optimally leverage wavelength and band switching resources, while minimizing total network cost and maximizing network performance.
Survivability and QoS path protection. We are developing an alarm correlation expert system to perform root cause analysis of network alarms, and to determine when the QoS of a given path is degraded and protection paths should be enabled. This alarm management system will also be used to construct dynamic failure-condition models for individual network elements, and the lightpath route selection process uses these models to proactively route around network elements that are nearing dysfunctional states.
QoS routing, signalling and management protocols for ATM virtual-circuit switched networks.
Past Students. Nicolai Zeldovich (B.Sc), Ron Shacham (B.Sc), Jennifer Trotta (B.Sc), Rudy Kellner (B.Sc), Max Berman (B.Sc), Chris Westmoreland (B.Sc), Rodney Walker (B.Sc), Daniel Russakoff (B.Sc), Tolentino Correia (B.Sc), Lionel Pagador (B.Sc), Sean Mountcastle (B.Sc).
Grants/Collaborations. NRL Center for Computational Sciences, Abdella Battou, Mohsen Guizani, Ala Al Fuqaha (University of Western Michigan).
Wireless Ad-hoc Networks and Applications to Battlespace/Disaster RecoveryA Mobile Ad Hoc Network (MANET) consists of mobile router/host platforms connected by wireless communications channels. Important examples of MANETs include communication networks for emergency rescue operations, and military networks. In dynamic settings such as these, centralised coordination of connectivity is not feasible, and MANET nodes must execute distributed topology discovery and message delivery.
Cooperative Networks. Traditional wireless MANETs assume complete autonomy of all nodes with regards to mobility. While this popular model is appropriate for consumer cellular networks, it does not adequately capture the environmental parameters of military and battlespace networks. In battlespace settings, it is unnecessarily restrictive to require constant and complete autonomy for all nodes--indeed, it is feasible and realistic to have a subset of nodes which are subordinate and/or cooperative (e.g. unmanned vehicles). Indeed, Comms Infrastructure optimization is a necessary function of command and control. This research task begins with the premise that future battlespace networks will not be homogenous in terms of node autonomy. We will consider a setting where wireless nodes exhibit a full range of cooperativeness with regards to mobility: i.e. different nodes exhibit a different level of “willingness to move when asked". We develop an integrated solution to complementary online problems of routing and network design, in a model where some of the wireless MANET nodes can be moved by the network layer itself (at a per-node specified cost). Our goal is to devise scalable distributed schemes in which cooperative nodes can be leveraged to improve the performance of existing network connections.
Current Students. Ghassen Ben Brahim (Ph.D candidate).
Grants/Collaborations. IBM-ITA Research Grant from the DoD/MoD (Fall 2007-Spring 2008)
Data Management in Peer-to-Peer Networks and Applications to Scalable Spatiotemporal Information Retrieval The successful transition of current centralized geographic information systems to the next-generation of scalable open distributed systems for universal situational awareness. Besides issues in decentralized data management and search, this requires a transition from data that is geospatially indexed to data that is spatiotemporally indexed. We consider the following central problems:
Time Lens. How can the user be guided through time in order to efficiently discover temporal regions of interest? This entails the development of a framework for temporal aggregation schemes, multi-resolution representations of time-sequence data, and mechanisms for iterative refinement of temporal queries.
WorldLine. How can the user be dynamically informed about the data that is presently available within their declared spatiotemporal region of interest, at the appropriate level of detail? This requires the development of a distributed geospatial/temporal directory service that serves as a dynamic matchmaker between users and relevant publishers of data.
Past Students. James Johnson (B.Sc).
Grants/Collaborations. NRL Center for Computational Sciences, Large Data JCTD.
Cybercrime mapping for P2P Networks. Can one detect, model, and predict illicit transactions over P2P networks in a way that facilitates effective intervention by law enforcement agencies?
Current Students. Matthew Cheng (M.Sc candidate).
Applications of Mathematics and Computer Science in the Social Sciences, the Arts, and Humanities Bishop Study. Statistical modelling and inference for a study undertaken by John Jay College of Criminal Justice in 2003, sponsored by the United States Conference of Catholic Bishops, initiated by
the National Review Board and the Office of Child and Youth Protection.
Grants/Collaborations. (PI: Karen Terry)
Child Prostitution Study. Statistical modelling and inference for a study undertaken by John Jay College of Criminal Justice, using respondent driven sampling to assess the extent and structure of child prostitution networks in New York City.
Grants/Collaborations. (PI: Karen Terry)
National Incident Based Reporting System. We are developing guided navigation tools for NIBRS, an incident-based reporting system for crimes known to the police. For each crime incident, a variety of data are collected about the incident, including the nature and types of specific offenses in the incident, characteristics of the victim(s) and offender(s), types and value of property stolen and recovered, and characteristics of persons arrested in connection with a crime incident.
Current Students. Petrit Duraku (B.Sc candidate), Aljona Necaj (B.Sc candidate).