The main goals of the Conference on Homological Mirror Symmetry, to be held on January 27-February 1, 2014 at the University of Miami, will be dissemination of results and encouragement a wave of young, early career researchers to join these projects. The wide range of topics appearing in HMS research necessitates venues for the open exchanges of ideas in order for graduate students and early career researchers to stay afloat on current topics. The subject has developed extremely rapidly in recent years, and this conference will really help engage young researchers.
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. 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 super-symmetric 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.
The conference website: http://math.berkeley.edu/~auroux/miami2014.html
Several new projects came out of this conference. To mention a few: 1) Kontsevich, Gross, Keel, Soibelman finished a project on canonical coordinates. 2) Kontsevich, Katzarkov and Haiden completed a project on moduli of stability conditions. 3) Katzarkov, Kontsevich, Pantev completed a work on moduli spaces of LG models. 4) Goncharov kept developing his work on cluster varieties. Numerous works by a range of authors have demonstrated the interaction of mirror symmetry and HMS with a wide range of new and subtle mathematical structures. We note in particular methods from Lagrangian intersection Floer theory, integrable systems and wall-crossing, derived and higher categories, and non-commutative Hodge structures, each of which are of immense independent mathematical interest, and each of which are intimately connected with HMS. Such connections have established HMS as a dominating force in modern geometry. The exchanges have been greatly facilitated and accelerated by the month long periods of activities that have recently been taking place every winter in Miami. Another benefit of these activity periods has been the involvement of talented young mathematicians in this field -- in particular people such as Sheridan, Preygel, Isik, Pandit, Favero, Kerr, Ballard, Diemer, Dyckerhoff, etc. have repeatedly attended and benefited from these activities. These achievements -- educating a new generation of mathematicians, fostering and deepening connections among the many areas of mathematics related to HMS and with the relevant areas of physics, are the main broader impact outcomes of our project. From a scientific viewpoint, the conferences have in particular helped ideas around wall crossing, algebraic cycles and spectra crystallize; more broadly, the new ideas emerging from the conferences have allowed us mathematicians to give back our dues to Physics, where this area of mathematics originated.
