Consortium Schools:

Summer 2007 Research Projects:
Number Theory: The Frobenius Problem I Johnson, C Starr at Willamette University. The Frobenius problem, also known as the ``postagestamp problem," is for a given set A of positive relatively prime integers the problem of finding the largest integer that is not a sum of elements of A. For example, if A ={4, 7}, then the largest integer which is not a sum of 4s and 7s is 17. Finding the Frobenius number of sets of integers has been a widely studied problem since the early 1900's, at which time Frobenius was reported to have posed the question frequently in lectures. In 1884, Sylvester showed that for a pair of relatively prime integers there is a nice formula for the Frobenius number. In 1990, Curtis showed that for an arbitrary relatively prime set of three positive integers a formula for the Frobenius number cannot be expressed in terms of a finite set of polynomials. The Frobenius problem for sets A of three or more elements has been shown to be NPhard. Despite the difficulty of the Frobenius problem there are algorithms to find the Frobenius number of a given set A and there are known formulae for special sets A. We will consider special infinite families of sets G_{i} that are recursively defined. We will study the Frobenius number for these sets, as well as additional properties and structures of the corresponding family of sets of integers which are not sums of elements of the sets G_{i}. Prerequisites: Students should have experience in at least one proofbased mathematics course. Differential Geometry: Listening to Orbifolds and Orbigraphs L Stanhope at Lewis and Clark College. There is a beautiful pairing of theorems between the smooth world of geometric shapes and the discrete world of graph theory. For example, geometers proved that you can hear the volume of a smooth object. Using natural definitions of the volume and 'sound' of a graph, graph theorists obtained the analogous result. This summer we ask 'What happens to this pairing if the geometric objects have sharp corners?' These pointy shapes are called orbifolds. Our challenge is to define what an orbigraph should be, and then see if geometric facts about orbifolds correspond to graph theoretic facts about orbigraphs. Prerequisites: Students should have experience in at least one proofbased mathematics course. Computer Science: Sensor Networks and Grid Computing J Mache at Lewis and Clark College. Wireless sensor networks and grid computing are both considered "Emerging Technologies That Will Change the World" according to MIT Technology Review. (Grid computing uses the Internet to allow sharing of computational and data resources among geographically distributed users. Wireless sensor networks are systems that combine potentially thousands of lowpowered, remotelydeployed minicomputers, embedded in the physical world.) New applications as well as security issues are rising. In order to achieve performance, scalability and robustness, many resource management problems have to be solved. This internship includes studying existing systems, writing software and experimentation with various designs and algorithms. Prerequisites: Coursework in Computer Architecture and Networking is desirable. Algebra: Algebraic Geometry and Finite Group Theory A Wootton, H Nordstrom at the University of Portland. An interesting problem in complex algebraic geometry is to determine all the groups of symmetries which can act on a surface. Surprisingly, despite its geometrical roots, this problem can be completely translated into a problem in group theory, and in many circumstances, finite group theory. This has led to tremendous progress in the last 20 years due to the great advances in computer algebra systems and computational finite group theory. This summer we shall ask ourselves what types of groups arise as symmetry groups of surfaces when we impose additional restrictions on the structure of the surfaces. To answer such questions, we shall use traditional arguments in finite group theory as well as making use of computer algebra systems such as GAP and magma. Prerequisites: Though we shall be developing much of the material in the first two weeks, some knowledge of finite group theory, complex variables and topology would be useful (though not necessary). Some knowledge of basic computer programming (for example, C or C+) would also be useful. 