Topology and Physics

摘要奖：DMS-1207817主要研究者：丹尼尔S.自由在这个建议的研究适合几何和高能理论物理之间的广泛的相互作用，重点是代数拓扑。 既有受物理学影响的纯数学项目，也有将拓扑学应用于量子场论、弦理论和凝聚态理论中的具体当前问题的项目。 在数学方面，我们寻求Atiyah-Singer指数定理的分类。 这就要求我们进一步发展广义微分上同调理论。 我们也将研究拓扑场论中的几个问题，特别是关于可逆理论，这对代数和低维拓扑有影响。 在物理方面，我们开始研究方面的最大超共形场论在六个方面，这是发挥越来越核心的作用，在应用数学。 我们也继续我们的工作拓扑问题的第二型超弦理论。 PI将利用他在这一领域的经验来攻击凝聚态物理学中的拓扑问题，这是他研究的一个新领域。 此外，还有几何和拓扑学各个方向的研究生项目。该资助支持的研究发展了与当前理论物理工作密切相关的几何学现代思想。 量子场论是在上个世纪发展起来的，用来解释基本粒子物理学，它对数学也有着惊人的影响。从物理学的角度来看，弦论更具思辨性，在数学上甚至更神秘，但它也提供了深刻的几何见解。许多项目要追求在这里躺在几何和这些物理理论之间的接口。 他们是更广泛的社区努力的一部分，将物理学的新思想带入数学。 一些项目研究物理学提出的数学结构和问题。在其他PI与物理学家合作，直接应用现代技术在几何和拓扑物理问题。 这种基础研究对社会的长期价值，超越其内在的知识价值，现在只能通过理解我们当前的技术和经济在根本上依赖于过去几十年和几个世纪的基础研究来衡量。 这笔赠款还支持奥斯汀的一个长期推广计划，该计划让初中和高中学生参与数学。

AbstractAward: DMS-1207817 Principal Investigator: Daniel S. FreedThe research in this proposal fits into the broad interaction between geometry and high energy theoretical physics, with a focus on algebraic topology. There are both purely mathematical projects influenced by the physics and projects which apply topology to concrete current problems in quantum field theory, string theory, and condensed matter theory. On the mathematics side we seek a categorification of the Atiyah-Singer index theorem. This requires us to develop generalized differential cohomology theory further. We will also investigate several problems in topological field theory, particularly concerning invertible theories, which have implications for algebra and low-dimensional topology. On the physics side we begin to study aspects of the maximal superconformal field theory in six dimensions, which is playing an increasingly central role in applications to mathematics. We also continue our work on topological questions in Type II superstring theory. The PI will use his experience in this area to attack topological questions in condensed matter physics, a new area for his research. There are also graduate student projects in various directions in geometry and topology.The research supported by this grant develops modern ideas in geometry which are closely connected with current work in theoretical physics. Quantum field theory was developed in the last century to explain elementary particle physics, and it has had spectacular implications for mathematics as well. String theory is more speculative from the physics point of view, even more mysterious in mathematics, yet it too provides profound geometric insights. Many of the projects to be pursued here lie at the interface between geometry and these physical theories. They are part of a much broader community effort to bring new ideas from physics into mathematics. Some projects investigate mathematical structures and problems suggested by the physics. In others the PI collaborates with physicists to apply modern techniques in geometry and topology directly to problems in physics. The longterm value of such basic research for society, above and beyond its intrinsic intellectual value, can be only be measured now by understanding how fundamentally our current technology and economy rely on basic research of past decades and centuries. This grant also supports a longstanding outreach program in Austin which engages middle and high school students in mathematics.