Principal Investigator: Ralf Spatzier, Mario Bonk, Richard D. Canary, John E. Fornaess, Juha M. Heinonen
This proposal calls for a research training program in geometry, topology and dynamics at the University of Michigan. Recent Ph.D.'s and advanced graduate students will be the main beneficiaries. The Mathematics Department at UM has one of the largest and most vigorous post-doctoral and graduate programs in the country with an excellent record of producing high-quality researchers in geometry, topology and dynamics. This proposal calls to bolster the training of post-docs and graduate students in all of these areas by deepening and broadening it and providing them with ample opportunities to excel in their research. Five faculty members (Mario Bonk, Richard Canary, John Erik Fornaess, Juha Heinonen and Ralf Spatzier) will lead this project in collaboration with eleven other senior faculty. The proposal calls for several specific innovations in the training program by: providing intensive exploratory seminars and workshops, travel semesters to deepen the scientific training at other insitutions at the forefront of research, opportunities to develop lecturing skills, intensive mentoring and exposure to research in the three areas to undergraduates.
Geometry, topology and dynamical systems are core areas of mathematics. Geometry investigates the shape of spaces, through invariants such as curvature. Topology explores the properties of spaces which remain invariant under deformations such as the number of holes in a surface. Dynamical systems concern the evolution of a physical or mathematical system over time. Especially in recent years, they have developed in mutually beneficial interaction. Case in point are topology, geometry and complex dynamics in low dimension which in many aspects mirror each other. Many of the most exciting developments in these areas are truly interrelated, and benefit from each other either by idea, analogy or actual tool. At the same time, connections with other fields such as algebraic geometry and mathematical physics have strengthened dramatically. These developments have been amazing in their breadth and depth, and demonstrate the vitality of these areas. This project will train young researchers in these exciting and interconnected fields, and will help insure their future health and continued interaction.
This Research Training Grant focussed on the education and professional preparation of young mathematicians working in geometry, topology and dynamical systems. Here are brief descriptions of these areas. Geometry studies how spaces are shaped. The central invariant is curvature which measures how the space bends near a point. Topology studies aspects of the shape of spaces that are invariant under continuous deformation, e.g., the number of holes in a surface (consider a bagel versus a pretzel). Dynamics finally studies the evolution of systems that change over time. In particular, one wants to know if a typical orbit visits every part of the underlying space. This is guaranteed by asymptotic properties such as ergodicity or mixing, or their quantifications, entropy in particular. It has been known for about a century that geometry, topology and dynamics are closely linked. These three fields have developed in mutually beneficial interaction, especially in recent years. Many of the most exciting developments in these areas are truly interrelated, and benefit from each other either by idea, analogy or actual tool. For example, studying the topology and geometry of 3-manifolds is closely related to the arithmetic and dynamics of the fundamental group and the analysis of geometric PDE. Old important conjectures, including the Poincare and geometrization conjectures, Ahlfors’ measure conjecture, Thurston’s ending lamination conjecture and surface subgroup problem, have been successfully resolved in this fashion. In the latter case for example, all three areas come together and combine for the ultimate proof. At the same time, connections with other fields have strengthened dramatically. Algebraic geometry and mathematical physics have close connections with symplectic geometry and algebraic topology. Some recent advances in number theory draw on ideas from dynamics and ergodic theory. These developments have been amazing in their breadth and depth, and demonstrate the vitality of these fields. Completely new areas have arisen in the last two decades. Examples par excellence are symplectic topology, the recent geometrization of combinatorial group theory, and especially the geometric analysis of arithmetic, moduli and outer automorphism groups. These areas abound with new ideas and offer tremendous opportunities for interaction between each other and areas outside. While these advances have been tremendous, young researchers need to understand and master materials and techniques from several areas to be successful. This requires intensive training broadly between several areas. This RTG grant has been very successful in this regard. It was set up to be interdisciplinary between these three major fields which are closely connected and are all represented strongly at Michigan. This led to successful interaction between advisors, postdocs and graduate students. Most importantly, the grant freed up time for the participants by reducing their teaching loads so that these young investigators could concentrate on their learning and their research. The department offers many opportunities for learning, e.g., advanced classes and many research and learning seminars. In addition, we held thirteen RTG workshops, six RTG lecture series, and partially supported three graduate student conferences. Furthermore, we supported travel of trainees to conferences and to consult with specialists at other institutions. 370 students and post-docs learned from expository lecture series given by thirty-two experts at the RTG workshops. Of those, 221 received financial support to come to the meetings. The workshops drew attendees both locally, from the Midwest and even both Coasts. This RTG grant has trained 16 postdocs, 28 graduate students and 19 undergraduates, the latter via Summer research opportunities. Of the 28 graduate students, 17 finished their Ph.D.s, another five are expected to finish in 2012/13 and the remaining the following year. The grant provided REU support to 19 undergraduate students who worked directly with faculty members on summer research projects. 82 research papers were written in topology, algebraic topology, geometry, geometric group theory, dynamical systems, complex and geometric analysis. This included a broad range of special topics. We mention a few: Riemann surfaces, moduli spaces, Out(Fn), limit groups, CAT(0) geometry, asymptotic properties of metric spaces, manifolds of non-positive curvature and generalizations, arithmetic groups, geodesic flows, closed geodesics, frame flows, horocycle flows, homogeneous dynamics, cocycles in dynamics, Gromov-Witten theory, Kahler geometry, mirror geometry, conformal and quasi-coformal maps, complex analysis and dynamics. We realize the importance of teaching in the careers of most of our young researchers, and provide extensive teaching training. Many of our postdocs then either teach one of our calculus classes that are renowned and have proved very effective. A few of them are offered the opportunity to teach inquiry based classes. This allows them to expand on their teaching techniques beyond the ordinary lecture centric teaching. Finally, the grant offered professional training opportunities, for applying for grants, jobs and about ethics in the research environment.
