This grant is to support 22 postdoctoral and mid-career mathematicians per year as Members of the School of Mathematics of the Institute for Advanced Study, where they will join a group of 45-55 other mathematicians (supported by funds from other sources) to form a community of mathematicians devoted to research at the highest level. This community is diverse in ages to promote mentoring of younger mathematicians by more experienced leaders. It is diverse in fields of mathematics to promote cross- fertilization of ideas from one mathematical field to another. The School of Mathematics provides this community of scholars with an academic program of rich mathematical content, an atmosphere that encourages mathematical interactions, academic facilities of the highest quality and a peaceful environment with a minimum of distractions.
The School of Mathematics program makes several contributions to science in the United States: 1. Former Members go on to have very successful mathematical careers. A large fraction of the country's mathematical leaders are former Members of the School. 2. Every year many new and significant mathematical results are produced by the community of mathematicians at the School. Historically, some of the most important mathematical developments of the last 75 years originated at the Institute. 3. By bringing mathematicians from different areas together, new connections between different fields of mathematics are found. Former Members leave with a broad understanding and appreciation of other fields of mathematics.
Over the five year period from 2007-2012, this NSF grant supported 207 mathematicians for appointments called "memberships" in the School of Mathematics at the Institute for Advanced Study. The duration of most of these memberships was one year, but 16 were supported for two or more years and 59 for one term only. The members suported by the grant were all fairly young scientists: at most 15 years past receiving their Ph.D. degrees. There were three primary project outcomes: 1. Producing new research results in mathematics. Every aspect of the Institute for Advanced Study is carefully tailored to encourage research productivity at the highest levels. The School of Mathematics provides an academic program of rich mathematical content, an atmosphere that encourages mathematical interactions, academic facilities of the highest quality, and a peacefull environment with a minimum of distractions. During their time at the Institute, the members produced many hundreds of mathematical papers in a number of widely diverse fields of mathematics. To give one outstanding example, Bao-Chau Ngo proved a result called "The Fundamental Lemma" during the period of this grant. Ngo's proof uses ideas from several different fields of mathematics. The statement of the Fundamental Lemma is too technical to be described to non-specialists, but the Fundamentall Lemma is, in turn, key to establishing deep results in several very diverse fields of mathematics. For this work done at the Institute , Ngo has already received the Fields Medal, which is widely regarded as the Nobel Prize for mathematics. The hundreds of mathematical results produced by the members supported by this grant are mostly in various branches of "pure" mathematics, where the quality of the work is measured in the depth, elegance and power of the result rather than its immediate applicability to real world problems. Experience has repeatedly shown that deep results in pure mathematics eventually become key to significant practical applications to problems in physics, chemistry, computer technology, engineering, biology, medicine, etc. 2. Training leaders of the next generation of mathematicians. The 207 Institute members supported by this grant were carefully selected by the faculty to have a background and previous accomplishments that showed that they could profitably benefit from the type of research and learning experience that the School provides. Once at the Institute, every effort is made to enhance their learning experience. They joined a group of 45-55 other mathematicians (supported by funds from other sources) to form a community of mathematicians devoted to research at the highest level. This community was diverse in ages to promote mentoring of younger mathematicians by more experienced leaders. It was diverse in fields of mathematics to promote cross-fertilization of ideas from one mathematical field to another. The mathematical progress of each member supported by the grant was monitored by the Faculty of the School of Mathematics, and they were provided with careful mentoring as needed. In the past, former members have gone on to have very successful mathematical careers. A large fraction of the country's mathematical leaders are former members of the School of Mathematics. Since the mathematical level and quality of members has only increased over time, there is every reason to believe that the 207 members supported by this grant will similarly be among tomorrow's mathematical leaders. 3. Promoting the heatlh and vitality of American mathematics as a subject. By bringing mathematicians from different subfields of mathematics together, new connections between different fields of mathematics have been found. Former members have left with a broad understanding and appreciation of other areas of mathematics. In an era of increasing fragmentation of science into smaller and smaller isolated subfields, the School of Mathematics of the Institute for Advanced Study has been a force to counter the trend.
