Under the grant, REU's were conducted during summers of 2007 and 2008. During the school years 2007-2008 and 2008-2009 various undergraduate students conducted research activities under the grant. For the summers, the undergraduate students participating were chosen through a rigorous application and screening process. There were 2 groups during summer of 2007. Both groups met for 8 weeks, starting in early June and lasting until late July. They Were led by Dr. Jason Cantarella and Dr. Joseph Fu. Dr. Jason Cantarella's group was assisted by graduate Students Matt Mastin, and Aja JohnsonThey consisted of Ricky Biggs, Erik Forseth, Chris Green, Paul James, Drew Lupton, Jesse Rao, and Matt Wise. This group worked on the behavior of energies for plane curves under geometric evolution equations, studying results of Pavel Exner, Michael Gage, Gerhard Huisken, Chad Mullikan, and others. The essential questions in this area are simple, but hard to analyze. For instance, suppose that a plane curve is varied so as to decrease its length as fast as possible. What is the limiting shape of the curve? Will the curve ever develop a self-intersection, if it does not have self-intersections to start? Will the curve ever get rid any self-intersections present at the start of the flow? Some of these questions were settled in the 1980's, but many fascinating questions remain wide open. The group held daily meetings over an 8 week period, eventually resolving into a few working groups. Paul James worked with Ricky Biggs on computer models of the curves and their flow. Drew Lupton and Jesse Rao got interested in the question of finding curves with bounded distortion. Erik Forseth completed an ambitious project of reinterpreting the results of the important papers of Michael Gage for the curve-shortening flow to deal with a flow that continuously rescaled the curve to remain at constant length. Chris Green proved a theorem about the distortion of polygonal knots, improving a result of Chad Mullikan. And Matt Wise worked on a theorem about energy functionals for knotted curves in space. Ricky Biggs deserves special credit for helping with several of the working groups. In addition, Biggs worked with Jesse Rao on an approach to finding curves of given length and maximum average L^p chord length. Biggs and Rao derived a criticality condition for these curves. Biggs also analyzed preprints of Exner et. al., finding an important correction which leads to a restatement of the main results. Matt Wise (undergraduate) gave a talk at the Mid-Hudson Math Conference. Dr. Fu's group investigated some problems in geometric probability, in particular the idea of Rota to use ideas from integral geometry to study the combinatorics of finite lattices with a group action. This had been the subject of two papers published by Dan Klain in the 1990s. The idea was to explore the ramifications in this setting of developments in integral geometry occuring since Klain's work. The group consisted of six undergraduate students: Jon Weed (Princeton), Phil Hudelson (Notre Dame), Ken Knox (UGA), Seongyoon Chong (Stony Brook), Kelly Aman (UT Arlington), and Caitlin Phillips (St. Edwards College). Their preparation and ability varied enormously. Dr. Fu began by giving several lectures, with a lot of student input, in which the group carried out some calculations. Dr. Fu then presented a few conjectures. A core of three students (Weed, Hudelson, Knox) caught on very quickly, and by the end of the fifth week or so had a rather complete picture of the situation. It turned out that the subject is somewhat simpler than had been hoped, and that in principle everything seems to reduce to the simplest case, where the lattice is a Boolean algebra. Seongyoon Chong also contributed some, but his understanding of linear algebra was weak. He also developed a rather negative attitude and after a few weeks retreated from the group discussions. Dr. Fu tried to bring Seongyoon Chong, Aman and Phillips up to speed by meeting with them separately to go over the underlying background, with limited success. Eventually they were set to working out the details of the main example and its relation with the hypergeometric probability distribution. Aman and Phillips worked on this with some diligence, and with some assistance produced a short paper about it. After week 5 or so the group lost steam. Dr. Fu tried to push them to work out more examples, but for some reason they didn't really do it. Also there were some loose ends that they never worked out. He also tried to introduce them to some of the integral geometry lying in the background, but again this never caught on. In retrospect it may have helped to return to the initial approach at that point, resuming lectures and working out detailed examples with them. There is some hope that the group will be able to publish a paper with the results; although they are largely negative, they do seem to explain the surprising simplicity of Klain's results. All the students except for Seongyoon Chong applied to give presentations at the Young Mathematicians' Conference at Ohio State in mid-August. Weed and Knox applied as a pair, and were accepted; so did Hudelson and Phillips, but they were not. Aman's poster presentation was accepted too. Dr. Fu's group had the assistance of two graduate students, Yang Liu and Jennifer Belton. The groups conducted 2 Symposia to share their results with each other and other interested parties, one half-way through the program, and one at the end of it. Almost all the students gave presentations. In addition, 2 students who were participating in the University of Georgia's Summer Undergraduate Research Program (SURP) Summer Undergraduate Research Program (SURP) under the supervision of Dr. Robert Varley participated and gave presentations. At the end of the summer, each student participated in an exit interview. Their evaluation of their experience was generally enthusiastically positive, and they felt that they had gained valuable experience from the project. They indicated that they felt that the program was run very well, and that the level of the research was right. The REU's in summer of 2008 were led by Dr. Caner Kacanci and Dr. Ming-Jun Lai. Dr. Caner Kacanci ran his REU for 7 weeks during summer 2008 (May-June ) . Caner's REU group worked on systems ecology, investigating universal properties of ecosystem models. He had 7 undergraduate students in his group, with a graduate student (Leopold Matamba) assisting him. Caner at the start of the program gave a series of lectures on Mathematical Biology, modeling, ecosystems, ecological network analysis, differential equation based models, agent based simulations, and the particle tracking algorithm. Caner included guest lectures from faculty in Ecology and Engineering in the schedule, to make the program a true interdisciplinary experience for students. On a field trip to Lake Herrick, students built terrestrial and aquatic carbon flow models of Lake Herrick ecosystem. Some of the projects required computational work, so Caner gave lectures and assigned tutorial based exercises on Matlab, C++, Linux OS and shell scripting. He also lectured them on LaTeX, LyX, Powerdot and BibTeX. At the end of this period, Caner presented students several projects that they might be interested in. Students formed three groups based on their interests. One group with 2 students analyzed indirect effects in ecosystems. A second group of 3 students worked on ecological utility analysis and ecological temperature. The third group of 2 students generalized the current throughflow and storage analysis to a wider set of models. The first group worked on improving the current definition of indirect effect coefficient (IEC) for ecosystem models. IEC quantifies how much of the flow in ecosystems occur through indirect pathways, than direct links. The current IEC definition is based on matrix algebra, and over-estimates indirect effects in case of cyclic networks. The students proved to be extremely capable and they worked well as a team. They were successful in coming up with a totally new definition that decouples cycling from indirect effects. They tested their results on twenty different real-life ecosystem models, and wrote an extensive report. Caner believes that this work is definitely publishable, and intends to work on their initial draft himself and submit this work to a major journal. The second group worked on two projects. First, they focused on ecological utility analysis, which quantifies the utility relation among all compartments in an ecosystem. They investigated whether these relations solely depend on model structure, or on flow types as well. Two of the students used the symbolic toolbox of Matlab to reconstruct general utility relations for simple ecological networks and did an exhaustive search to test all possible cases for various classes of models. They got some unexpected results, and the work may be publishable with some additional effort. The third student in the group took a more high risk, high benefit approach by focusing more on the theoretical aspect of utility analysis. He tried to implement topics from algebraic graph theory of directed graphs. It turned out that this theory was not easily applicable to ecological networks, although he enjoyed it and learned a lot along the way. Later, he and one of the other students worked on a statistical approach for defining a temperature for an ecosystem through pathway history distributions of individual particles in the system. Some interesting results were obtained and Caner is planning additional work and to write a paper on this topic. Throughflow and Storage analysis provide information as to how environmental inputs are distributed as storages and fluxes among compartments. This analysis is restricted to donor-controlled flows and systems at steady state. Therefore it is not applicable to models involving predator-prey interactions, or temporal fluctuations (eg. seasonal variations). The third group was successful at generalizing Throughflow and Storage analysis to all interaction types. This was not an easy task, and kept them busy throughout. One member of this group is a minority student with a weak background. Although she had difficulties in keeping up with the rest of the REU group she had a positive attitude and worked well with the others as a group. While the other student concentrated on the theory, she worked on the report, presentation, figures and associated simulations. She wrote a report on Throughflow analysis, which Leopold is working on turning into a paper, with himself as main author, and Caner and the 2 students as co-authors. They plan to submit this paper to the journal Ecological Modeling. Another paper on Storage analysis is also planned. Leopold was out of the country for first 2 and 1/2 weeks of the project. Caner showed his amazing dedication by meeting with the REU group from 9am till 5pm every day during that time. Students presented their works halfway through the REU, and at the end. There were four final project presentations open to faculty and graduate students. The students were encouraged to attend Mathfest (MAA) next year, and present their work. At the end of the program exit interviews were conducted with the students. They expressed a high level of satisfaction with the program. They found the problems accessible and felt that they had made good progress on them. They were of the opinion that Dr. Kazanci had done an excellent job at preparing them for exactly what they needed to know to attack the problems. One student for example had had no previous experience in programming before, yet was able to successfully pursue her problem, which was done mainly through programming. The participants all agreed that they enjoyed the program immensely. They learned a lot and were challenged by it. Ming-Jun Lai's REU Group on Numerical Analysis had 7 undergraduate students: Katie Agle, Dustin Burns, Cooper Cumliffe, Grant Fiddyment, Max Mautner, and Tarik Trent, working on various research problems in the summer, 2008 for 7 weeks (June-July). They were introduced to bivariate splines and experienced doing research projects. Katie Agle, University of Tennessee, Knoxville works on bivariate splines for surface design. She studied how to use bivariate splines to construct interpolatory and/or fitting surfaces under various energy functionals. Dustin Burns, Georgia Institute of Technology worked on 3D data sampler to collect data from a truck model and then applied various surface design methods to generate truck body surface. Cooper Cunliffe, University of North Carolina, Ashville studied the convergence of data fitting methods for minimal triharmonic energy method and minimal surface area method. James Alexander, Rutgers University studies the bivariate splines for weak solution of Bousinesque equations and their approximation of the global weak solution. See his presentation for more detail. Max Mautner, Clairmount McKena College and Tarik Trent, Emory University worked on image segmentation together to obtain the boundary of local regions of an image. Max and Tarik invented an edge-crawling method to find edges. Grant Fiddyment, University of Georgia worked on image denoising to remove noises in local regions of an image. They all gave ptresentations of their work at the end of the program. Details and accounts of their presentations may be found at: http://www.math.uga.edu/~mjlai/REU08.html At the end of the program, exit interviews were conducted with the students. They expressed a high level of satisfaction with the program. In addition to the summer research groups, several undergraduate students participated in VIGRE research groups during the academic year. Four students were supported under the grant during the 2007-2008 academic year. Kyle Istvan worked with Dr. Izadi in her Vigre group on algebraic geometry. He worked on several projects with this group, such as understanding cages, and their uses in evaluating objects in projective space. An example of a project he worked on is trying to prove a claim the group found in a paper, that the fano variety of lines in the fermat quartic in P4 is 2-dimensional. The paper gave no proof or citation, so they decided to see if we could figure out how to show this. They believe that they have succeeded by using MAPLE to do the computation. Eric Cho worked with Dr. Alexeev's VIGRE group. The topic concerns the shape of a particular boundary contour related to a melted crystal. One of the main results from the literature recently set forth by Kenyon and Okounkov is that the projection of the outer contour to the plane is an algebraic curve, to which one naturally assigns a degree. The case when the plane curve is a cardioid of degree 4, has been considered by Kenyon-Okounkov. Cho's research involved exploring other cases of different degrees and visualizing the corresponding curves in 3-dimensional space. Meredith Perrie was a member of Dr. Hersonsky's geometry VIGRE group on circle packing (http://www.math.uga.edu/~saarh/CirclePacking.html). They worked on unsolved problems from Stephenson's book ,An Introduction to Circle Packing . Alex Rice worked with Dr Pete Clark's VIGRE group on Computations on CM elliptic curves (http://www.math.uga.edu/~pete/VIGRE.html). Ten undergraduate students were supported under the grant during the 2008-2009 academic year. Grant Fiddyment spent the fall semester continuing his summer REU research with Dr. Lai on multivariate spline applications. They worked to implement two summer students' region segmenting code into their existing methods for triangulation. They also worked to improve the old boundary fitting algorithm -- the part of the triangulation (Tdomain.m) code that, given a set of pixel values constituting the boundary of one of a picture's regions, determines a parametrized curve interpolating them. The old code used a simple linear fitting for this process, whereas they improved it to use a cubic spline instead. Kyle Istvan spent Fall semester 2008 inthe IVRG research group led by Dr. Varley and Dr. Burgoyne. They began by looking at tilings in the plane and how they could be expressed mathematically. He partnered with Max Pilzer during this period to investigate the properties of aperiodic tilings (such as the different Penrose tilings), and what sort of mathematical structure they embodied. That quickly led to the higher dimensional cases, where they tried using group representation theory to help them interpret the rigid symmetries of different shapes. Throughout fall semester, he helped to introduce group theory to the other undergraduates, many of whom had no background in the subject. WIth guidance from Dr. Varley, he took some of this work and put together a presentation that he gave at the MUURMAC conference at Mercer University in February. The gist of the project was that group representation theory could be used to interpret some of the symmetric groups (and alternating groups) as symmetries of the platonic solids, without having to explicitly find matrix representations for the transformations. They found arguments to prove that faithful, irreducible representations existed over R^3, and were able to look at orbits and stabilizers under action upon vectors in R^3 to deduce the symmetry groups of the solids. Briefly, near the end of the fall semester, he worked with Dr. Burgoyne on how character tables could be used to deduce much of this information quickly and efficiently. Eleanor Dannenberg worked with Dr. Cantarella's knot theory group. She has been drawing copies of kawauchi knots on the computer, using KnotPlot. She will later tighten these knots, using a different program.