This project supports a conference on Homological Mirror Symmetry to be held at the University of Miami, Florida, from January 28 - February 1, 2013. More information can be found on the conference website: http://math.berkeley.edu/~auroux/miami2013.html
While mirror symmetry initially arose from phenomena in string theory, this active area at the interface between mathematics and physics has been shown to be relevant to numerous mathematical structures. In particular, methods from Lagrangian intersection theory and Floer homology theory, integrable systems and wall-crossing, derived and higher categories, and non-commutative Hodge structures are all intimately connected with homological mirror symmetry. The wide range of topics in homological mirror symmetry research necessitates venues such as this, particularly for early-career researchers to be introduced to and stay abreast of current topics. The main topics of this conference will be wall crossing, stability Hodge structures and topological quantum field theories. There will be 3 lecture courses, by M. Kontsevich, by M. Abouzaid and by S. Keel. There will be two discussion sessions.
The conference will result in dissemination of results and getting a young wave of researchers to join these projects. All considerations on giving opportunities to underrepresented groups were taken. The conference will also serve to facilitate interactions between US-based researchers and international participants who will be attended the conference. This will be particularly beneficial for US-based graduate students and should lead to a fruitful exchange of ideas among representatives of many mathematical communities. In addition, the conference will be a means of "giving back to physics." Many of the ideas at the beginning of the subject come from String theory - Mirror Symmetry. It is time for the mathematical aspect to give back to physics. The singularities on the discriminant loci we study correspond to a new physics theory of 6d type. In this way the circle comes to a close.
Over the last several years, a variety of new categorical structures have been discovered by physicists. Furthermore, it has become transparently evident that the higher categorical language is beautifully suited to describing cornerstone concepts in modern theoretical physics. The goal of this project was to develop these structures even further. As we head into the second decade of the 21st century, modern geometry and theoretical physics are more intertwined than ever before. The convergence of ideas from mathematics and physics is accelerating at the same time as elementary particle physics is on the cusp of a profound revolution to be brought about by the new experimental results coming out of the Large Hadron Collider (LHC). These will serve to identify among the multitude of theoretical possibilities currently open, which ones best address quantum field theory at the high energy scale. At the same time, a lot of mathematical work remains to be done to provide a suitable framework for the new physical theories that are being proposed. The geometric objects which we investigate today are the foundations for such a framework: homological mirror symmetry is the mathematical realization of dualities among sypersymmetric theories, and higher categories are the mathematical foundation for quantum field theories. These new flavors of geometry, in which categorical structures play a primordial role, will certainly continue to play a fundamental role in 21st century theoretical physics. There three main directions we have developed: 1) Categories HMS and Dynamical Systems. 2) Noncommutative Hodge theory and stability Hodge Structures.
