The inaugural conference of the Simons Collaboration on Homological Mirror Symmetry took place November 5–8, 2015 on the campus of the University of Pennsylvania in Philadelphia. The event attracted over 90 mathematicians and physicists from around the world, at all career stages, ranging from established senior researchers to beginning postdoctoral fellows and graduate students. The meeting focused on the current state of the Homological Mirror Symmetry program and on the major new techniques that are expected to lead the project to completion and resolve several outstanding conjectures. The focus on methodology and strategy, and the vast variety of different mathematical fields represented by the participants, resulted in a uniquely productive scientific atmosphere.

The conference featured an intensive program of talks covering the various research areas relevant to homological mirror symmetry: symplectic topology, noncommutative geometry, higher categories, Hodge theory, enumerative and birational algebraic geometry, homological algebra, representation theory, quantum field theory, integrable systems, supersymmetric gauge theory and string theory. At the core of the schedule were six series of two lectures each presenting the latest state of the art in the different approaches to homological mirror symmetry. M. Kontsevich gave a global overview of the goals of the collaboration and explained the categorical geometry aspects of the problem, with a particular emphasis on categorical stabilities and Kahler structures. S.-T. Yau and S.-C. Lau presented the state of the art in the SYZ approach to mirror symmetry, focusing on mirror transformations and the noncommutative geometry of SYZ mirrors. M. Abouzaid and K. Fukaya introduced the foundations of family Floer theory — a powerful new tool for building mirrors of symplectic manifolds from local analytic patches and constructing the mirror functor. S. Ganatra and N. Sheridan described their joint work with T. Perutz, explaining how to recover enumerative predictions of mirror symmetry from categorical equivalences. B. Siebert’s series introduced the algebro-geometric perspective for toric degenerations based on log geometry and tropical curve counting. Finally, K. Costello discussed higher genus instanton corrections to the mirror correspondence from the point of view of BCOV quantization.

These core topics were enhanced and supplemented by a series of stand-alone talks. E. Witten explained how supersymmetric gauge theory categories knot invariants and pointed to connections with Fukaya categories. Y. Soibelman described a new approach to Fukaya categories of holomorphic symplectic manifolds using a Riemann-Hilbert correspondence for constructible sheaves of differential graded categories. L. Katzarkov’s lecture elaborated on the notion of Kähler stability suggested by HMS, and discussed applications to low dimensional topology. E. Zaslow described a construction of Fukaya categories for cotangent bundles of surfaces relative to a Legendrian knot in the contact boundary. D. Nadler explained his theory of arboreal singularities and its implications for constructing Fukaya categories of Liouville manifolds. B. Lian gave an overview of recent progress on the periods of Fano and general-type varieties based on a function theoretic analysis of mirror maps. Finally, A. Efimov presented a refined version of the degeneration conjecture for noncommutative Hodge filtrations and discussed proofs of this refinement for certain spaces given by quivers.

The conference was preceded by two days of mathematical discussions attended by most of the PIs. These discussions covered future joint research as well as organizational plans for the collaboration.

In order to make the conference material available to an even wider audience, videos of all talks have been posted online. Links can be found at: https://schms.math.berkeley.edu/events/penn2015/

The inaugural conference of the Simons Collaboration on HMS saw the participation of over 90 mathematicians and physicists. The core of the conference consisted of six series of talks on the different approaches to mirror symmetry:

**M. Kontsevich**gave a global overview of the goals of the collaboration and the categorical aspects of the problem.**S.-T. Yau**and**S.-C. Lau**presented the state of the art in the SYZ approach to mirror symmetry.**M. Abouzaid**and**K. Fukaya**introduced the foundations of family Floer theory — a powerful new tool for constructing the mirror functor.**S. Ganatra**and**N. Sheridan**described their joint work with T. Perutz, explaining how to recover enumerative predictions of mirror symmetry from categorical equivalences.**B. Siebert**‘s series introduced the algebro-geometric perspective for toric degenerations based on log geometry and tropical curve counting.**K. Costello**discussed higher genus instanton corrections to the mirror correspondence from the point of view of BCOV quantization.

There were complementary talks by mathematicians (Efimov, Katzarkov, Lian, Nadler, Soibelman) and physicists (Zaslow, Witten). Videos of all talks have been posted online.