Applicable Algebraic Geometry: Real Solutions, Applications, and Combinatorics

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The main goals of this project are basic research in real algebraic geometry and the Schubert calculus, training the next generation of mathematical scientists, and deepening ties between mathematics and the applied sciences. It will support Sottile''''s basic research in real algebraic geometry and Schubert calculus and his interdisciplinary work on applications of algebraic geometry. These all have a substantial combinatorial component and an essential omputational/experimental core. This activity will result in a better understanding of real solutions to systems of polynomial equations through experimentation and conjectures by obtaining tighter fewnomial upper bounds and by developing a theory of lower bounds for the number of real solutions to structured polynomial systems. It will also result in the dissemination of ideas and techniques from algebraic geometry to other fields and the development of new directions in mathematics inspired by problems from these fields. Some of this work will be carried out by Sottile''''s research team at Texas A&M consisting of students and postdocs, and will include substantial experimentation. This team approach to research and training is consciously modeled on work patterns in the other sciences. It will also support Sottile''''s efforts organizing large-scale international and interdisciplinary scientific meetings.While algebraic geometry is concerned with basic questions about solutions to equations, its value to other disciplines is through concrete objects and computational tools, as applications require knowledge of specific geometric objects and explicit, often real-number, solutions. Modern tools from computational algebraic geometry have great potential in applications, but their use requires a concerted effort to transfer this technology into the hands of applied scientists. This project is intended to facilitate this technology transfer.