Collaborative Research: Midwest Geometry Conference 2005-2007

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AbstractAward: DMS 0509068, 0508661, 0509714Principal Investigator: Thomas P. Branson, Shihshu Walter Wei, Andrzej J. DerdzinskiInstitutions: University of Iowa, University of Oklahoma,Ohio State UniversityThe Midwest Geometry Conference has been meeting annually since1991. Its goal is to present recent advances in differentialgeometry, geometric analysis, and integral geometry to geometersfrom the Midwest, including junior level researchers and somegraduate students. The main part of the program consists ofseveral plenary lectures given by experts in specific sub-fields(who need not be Midwest-based); an opportunity is also providedfor some of the participants to give contributed talks. Thecurrent project provides for conferences in the spring of 2005,2006, and 2007 at the Ohio State University, the University ofOklahoma and, respectively, the University of Iowa. The OhioState conference, planned for April 29 to May 1, 2005, will focuson three topics: positively curved manifolds; Ricci flow,solitons and Einstein metrics; geometric group theory. The listsof topics for the other two conferences are still tentative, andinclude: symmetric criticality; p-harmonic geometry; convexity,positively curved manifolds, and Ricci flow (Oklahoma, 2006), aswell as: symmetries and overdetermined systems of PDEs; theAdS/CFT correspondence; nonlinear hyperbolic evolution equations;geometric scattering theory (Iowa, 2007).The Midwest Geometry Conference is dedicated to the exchange ofideas among members of the geometry community in the Midwest, aswell as dissemination of important research advances and ideas,nationwide and worldwide, in differential geometry and relatedfields. A key objective of the conference is the attempt toencourage researchers and potential researchers in the Midwest,especially those who work some distance from the large researchcenters, as well as young people, women, and members ofunderrepresented groups. "Warm-up" sessions for graduatestudents and young researchers are planned for the days leadingup to the conference. At least one of the three or four focustopics at each conference is related to applications of geometry.Historically, such applications lie primarily in physics, withprominent examples provided by Einstein's general theory ofrelativity and the currently accepted Standard Model ofelementary particles and their interactions. Other well-knownapplications of differential geometry are in engineering, forexample robotics. A further example pertains to evolutionequations, which are currently a subject of keen interest todifferential geometers and, at the same time, have the appliedaspect of describing how some materials, or the space itself,develop in time. A very specific example of the latter kind isprovided by the mean curvature flow. The surface of a globule ofmolten plastic evolves, or adapts in time, to have the minimumpossible curvature; this evolution is completely determined bythe globule's curvature. This fundamental observation providesan understanding of a wide range of flow problems, and allows anapproach to predicting the flow. This has immediate relevance toinjection molding of materials, for example in the dashboards andbumpers of automobiles. But the fundamental principles behindmean curvature flow have also provided a paradigm for problemslike the Ricci flow, in which the metric properties of spaceitself are the quantities which are evolving. Questions of thelatter kind will be central to the coming meetings of the MidwestGeometry Conference.