This project will train undergraduates, graduate students, and postdoctoral associates in an interdisciplinary program that integrates modeling, analysis, and computations with contemporary experimental research. Scientifically, the overarching theme is large multi-scale nonlinear systems and the mathematical and numerical methods needed to analyze them. We are interested in applications that are highly nonlinear, noisy, have a large number of components, and evolve over an extensive range of spatial and temporal scales. The models are physically realistic but are so difficult that direct numerical simulations are not able to faithfully reproduce the processes they are intended to describe. Our research objective is to provide a more in-depth analysis, and this invariably involves integrating modeling, computations, and analysis with contemporary experimental research.
The core of our program is mentored participation of students and postdocs in an interdisciplinary research team. At the same time, the activities we will develop will reach a wide audience and combine research, mentoring, and broad education experiences. Specific examples are Applied Math Days and mini-symposia. Also included in this effort will be innovative curricular changes that will provide mentored transitions that will both increase the number of students interested in, and prepared for, graduate school in mathematics and which will also prepare our graduate students and postdoctoral associates for careers in academia or industry. In addition we will develop innovative recruitment and retention components, as well as careful management and assessment of the program.
Our RTG program had the trainees involved in a rich mentoring and research program driven by a broad choice of engineering, physical, and biological applications. Common to all the individual projects was an overarching modeling-analysis-computation approach. In doing this, the trainees mastered an application discipline to the extent that made it possible to have a productive collaboration with experts in that discipline. It also meant that their research was considered a significant contribution to mathematics, as well as the application area to which it applied. Intellectual Merit With RTG we supported 11 PhD students, 2 postdocs, and 12 undergraduates. The PhD students and postdocs published 17 research papers, presented 36 talks and 16 posters (two of which won ``best poster" awards), and received 18 travel awards from major professional conferences. This included 2 invited workshop presentations and 2 presentations in France and Spain by one of the PhD students. The postdocs organized 3 minisymposia/sessions at SIAM and other conferences. PhD student Lisa Rogers spent 3 months training in neuroscience at Harvard, Katie Newhall spent a semester at SAMSI, Kajetan Sikorski visited MBI and Warwick Mathematics Institute, and Matthew Reyna visited ICERN, with the last three getting local support. Our PhD graduates have secured postdocs at Courant Institute (2), MBI (1), and UC Santa Barbara (1), as well as an NSF postdoc (Rogers) and a Courant Instructorship (Newhall). RTG supported undergraduates are currently in PhD programs at Arizona, Brown, Courant, Duke, Maryland, Northwestern, and Princeton. The first three have graduated from Brown, Courant, and Duke. Postdoc Liwei Xu (permanent resident) received a ``Thousand Young Talents" award (equivalent to a CAREER award) to spend as an Assistant Professor at Chongqing University (he also had a tenure-track offer from Shanghai Jiao Tong University). Broader Impacts Over the span of the grant we conducted five Applied Math Days workshops, which averaged 25 participants from Boston U, Brown, Cornell, Courant, NJIT, SUNY Buffalo, Penn State, and Princeton. The participants at these annual meetings were PhD students and postdocs at various stages of their careers. They presented short talks about their research. More importantly, the meeting fostered a sense of a community for these budding applied mathematicians, increasing the interactions among students and postdocs, and establishing relationships that have been long lasting. The core activity in our project was the weekly working seminar, in which all of our students and postdocs participated. Those with a research topic would regularly present their work, while those preparing for their PhD project would present results from the research literature (aka, a journal club). During the funding period we hosted 69 external speakers. We taught special topics courses on Advanced Probability Modeling and Techniques, Introduction to Stochastic Differential Equations, Introduction to Stochastic Processes, Stochastic Modeling in Physics and Microbiology, Mathematical Foundations of Fluid Mechanics, Finite Element Methods, and Wave Theory. Note that these courses were taken by RTG as well as non-RTG students. The RTG topics course Stochastic Processes and Modeling is now offered as a biennial (catalogued) graduate course. With RTG funding we significantly modified how we ran the Mathematical Contest in Modeling (MCM). With the advent of the RTG seminars, and related courses, we had a built-in recruitment mechanism for this contest. This resulted in a noticeable increase in student participation. We also had one of our RTG grad students run the practice sessions, and then have other grad students act as assistants. This worked so well that RTG PhDs L. Rogers and K. Newhall started a training program for the MCM at New York University. As an indication of our success, one of our teams won the SIAM Prize in 2010. This is significant because a total of 2,254 teams from 14 countries participated in the competition. In 2011, we had a team achieve an outstanding designation (4 out of 1483 submissions for their problem), while one team of sophomores ranked in the finalist category (top 18 out of 1483 submissions for their problem). In 2013, for the first time, one of our teams received an outstanding designation (top 5 out of 957 teams, the only non-Chinese team to achieve this distinction) in the Interdisciplinary Contest in Modeling, which requires a deeper connection between the mathematics and scientific discipline (this year, ecology and environment). The remarkable success of Rensselaer's mathematical modeling teams in recent years is a true testament to the design of and investment in the preparation work by the RTG graduate students as well as the undergraduate students in the competition.
