## Announcement

During the Second World War, the mathematics underlying game theory and cryptography played an important role in military planning. Since the war, it has become clear that mathematics has an important role to play in securing victory in any global conflict; the current war on terror is no exception. It has been estimated that $2 billion were spent in the development of game theory since the onset of the Cold War. Today, we need a new kind of mathematics to fight a new kind of war.

Since 2001, tremendous amounts of information have been gathered regarding terrorist cells and individuals potentially planning future attacks. There is now a pressing need to develop new mathematical and computational techniques to assist in the analysis of this information, both to quantify future threats and to quantify the effectiveness of counterterrorism operations and strategies. Concepts and techniques from mathematics, specifically, from Lattice Theory and Reflexive Theory, have already been applied to counterterrorism and computer security problems. The following is a partial list of such problems.

- Strategies for disrupting terrorist cells
- Border penetration and security
- Terrorist cell formation and growth
- Data analysis of terrorist activity
- Terrorism deterrence strategies
- Information security

This conference will draw researchers who bring these two important branches of mathematics to bear in the understanding of terrorist groups and in the development of strategies for defense against threats to international security.

Terrorist cells are often modeled as graphs, yet they are composed of leaders and of followers. Hence, neglecting the fact that they have a hierarchy leaves out an important aspect of their structure. The proper framework is therefore that of Order Theory (Lattice Theory).

In the 1980's, Ivan Rival coordinated three NATO Advanced Study Institutes that explored applications of Order Theory to different settings. In our case, tools from Order Theory will be applied to help intelligence agencies determine whether they have disrupted a terrorist cell. The same tools (along with some extra-mathematical analysis) will help law enforcement agencies determine which individuals in a terrorist cell should be captured first, in order to maximize the chances of disrupting a cell by expending as few resources as possible.

Information security is also a vital component in issues related to homeland security. Lattices of antichains have been applied to provide new models for improved computer security. Lattices of submodules of finite rings have been applied to provide new models for secret sharing schemes, whereby secrets of different levels of importance can be divided up among individuals with different levels of authority in an organization. (This work was further developed at AT&T's Shannon Laboratory.) Matroids, which are correlated with geometric lattices, have also been used in this context by the Information Security Group of Royal Holloway, University of London. Concept lattices have been used to find hierarchical relationships in terrorist-related data sets. (See Los Alamos National Laboratory Technical Report LAUR 02-7867, "Advanced Knowledge Integration in Assessing Terrorist Threats".)

Reflexive Theory is a branch of mathematical psychology developed in the 1960's and used by the Soviet military establishment. The underlying principles are the following: Predicting the behavior of the adversary is extremely difficult, if not impossible. Reflexive Theory enables one to control the behavior of the adversary by limiting the range of his or her options. This is done through the selective release of information and disinformation. Reflexive Theory enables one to control the adversary's decision-making process.

The art of battlefield deception is as old as war; Reflexive Theory enables one to formalize and ''mathematize'' the process of forming ''tricks''. Reflexive Theory has been used to:

- Build prototype computer simulations of the penetration of our borders by terrorists. In this simulation, terrorists attempt to enter at one of three areas. We must decide how to set our troop strength level at each of these areas, and how to advertise the troop strength level, so that the terrorist attempts to enter at an area of our choosing.
- Build models of terrorist recruitment by modeling moral choices (e.g., one's willingness to sacrifice one's life) as opposed to purely rational choices (i.e., those based on maximizing a utility function).