# Fall 2014 Spring 2015 Program Details

Organized by Mark Haskins, Dietmar Salamon, and Simon Donaldson

*August 18 – October 3, 2014*

This program seeks to connect recent developments and open questions in the theory of compact manifolds with special or exceptional holonomy (especially G_2 manifolds) with other areas of mathematics and theoretical physics: differential topology, algebraic geometry, (non compact) Calabi-Yau 3-folds, K3 surfaces, submanifold theory, J-holomorphic curves, gauge theory and M-theory in particular. One of the aims of the program is to bring to wider attention some of the problems in other parts of mathematics that have arisen recently in the theory of compact G_2 manifolds and to bring together experts from these disparate areas.

Attendee ListApplication for program is now closed.

**Gauge Theory, Integrability, and Novel Symmetries of Quantum Field Theory**

Organized by Anton Kapustin, Nikita Nekrasov, Samson Shatashvili, Volker Schomerus, and Konstantin Zarembo

* September 2 – December 19, 2014*

The interplay between the supersymmetric gauge theories and (non-supersymmetric) integrable theories in various dimensions is a puzzling development of several decades of research. In recent years the progress has been achieved in the understanding the non-perturbative dynamics of supersymmetric gauge theories subject to various supergravity backgrounds, and relating it to conformal and integrable field theories and lattice models in 1+1 dimensions. Quite independently, the scattering amplitudes and scaling dimensions in the maximally supersymmetric Yang-Mills theory are also related to the conformal and integrable field theories and lattice models in 1+1 dimensions. Trying to take these coincidences seriously, understanding the dictionary, deriving the consequences, and discussing the applications, will be the main theme of the program.

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**Interactions of Homotopy Theory and Algebraic Topology with Physics through Algebra and Geometry**

Organized by John Morgan and Dennis Sullivan

*October 1, 2014 – June 30, 2015*

While activities will depend on the visitors for their specific focus, we expect them to be organized around several general themes: (i) rigorous approaches to perturbative quantum field theories, and especially to gauge theories using homological and homotopy-theoretic techniques, (ii) formal quantization, and (iii) TQFTs and infinity structures. Within those themes, a partial list of topics would include: BV algebras; operads; the Fukaya category; various compactifications of moduli spaces of stable curves and stable maps, as in string topology and contact homology and twisted K-theory. Our plan is to have activity spread over the entire academic year, rather than a more concentrated activity during one semester, with between 4 and 6 visitors in residence at any one time. In addition to Fukaya, Sullivan, and Morgan, who are permanently in residence at Stony Brook, those who have expressed interest in attending the program (for 3 weeks to a month, with a few longer visits) include Kevin Costello, Jacob Lurie, Dan Freed, Constantin Teleman, and Alberto Cattaneo.

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**Geometric Flows**

Organized by Simon Brendle, Xiuxiong Chen, Simon Donaldson, and Yuanqi Wang

*October 13 – December 19, 2014*

Since its invention in 1982, Hamilton’s Ricci flow has become a central tool in global differential geometry. In particular, the Ricci flow has played a central role in Perelman’s proof of the Poincare conjecture, as well as in the proof of the Differentiable Sphere Theorem. Other geometric flows like mean curvature flow, inverse mean curvature flow have shed light on deep problems in the geometry of hypersurfaces.

In recent years, there has been exciting progress in many directions. In particular, there have been major advances in our understanding of singularity formation and soliton solutions. Furthermore, geometric flows in the Kahler setting, including Kahler-Ricci flow on manifolds of general type and the Calabi flow, have led to substantial progress on core problems in complex geometry.

In this program, we will bring together researchers in the area of elliptic and parabolic partial differential equations in Riemannian geometry, Kahler Geometry and Topology.

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**Mathematical Problems in General Relativity**

Organized by Mike Anderson, Sergiu Klainerman, Philippe LeFloch, and Jared Speck

*January 5 – February 6, 2015*

Einsteins field equation of general relativity is one of the most important geometric partial differential equations. Over the past decade, the mathematical research on Einstein equation has made spectacular progress on many fronts, including the Cauchy problem, cosmic censorship, and asymptotic behavior. These developments have brought into focus the deep connections between the Einstein equation and other important geometric partial differential equations, including the wave map equation, Yang-Mills equation, Yamabe equation, as well as Hamiltons Ricci flow. The field is of growing interest for mathematicians and of intense current activity, as is illustrated by major recent breakthroughs concerning the uniqueness and stability of black hole models, the formation of trapped surfaces, and the bounded L2 curvature problem. The themes of mathematical interest that will be particularly developed in the present Program include the formation of trapped surfaces and the nonlinear interaction of gravitational waves. The new results are based on a vast extension of the earlier technique by Christodoulou and Klainerman establishing the nonlinear stability of the Minkowski space. This Program will be an excellent place in order to present the recent breakthrough on the bounded L2 curvature problem for the Einstein equation, which currently provides the lower regularity theory for the initial value problem, as well as the recently developed theory of weakly regular Einstein spacetimes with distributional curvature.

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**Knot homologies, BPS states, and SUSY gauge theories**

Organized by Sergei Gukov, Mikhail Khovanov, and Piotr Sukowski

*March 16 – June 12, 2015*

The aim of this program is to understand new relations between knot theory, supersymmetric field theories, and string theory. Tremendous development in knot theory in recent decades led to the formulation of polynomial knot invariants, such as the Jones polynomial and its generalizations. Intimate relations between knots and physics have been found around 20 years ago in the seminal work by Witten in which he reformulated and vastly generalized the Jones polynomial in terms of Chern-Simons quantum field theory. Moreover, following the ideas of Khovanov within the last decade it has been found that knot polynomials can be categorified, meaning that they arise as the Euler characteristics of much richer homological spaces. This perspective immediately provides a new, large family of polynomial knot invariants, more generally called (colored) superpolynomials, which arise as the Poincare characteristics of these homological spaces.

Very recently it has been realized that the above mentioned results arise naturally from multiple new physics viewpoints, which provide a fascinating interpretation of mathematical phenomena and often provide

a very effective computational framework. These new physics perspectives include topics such as BPS states, topological strings, differentials in homological spaces, super-A-polynomials, 3d-3d correspondence, 3-dimensional holomorphic blocks, refined Chern-Simons theory, etc.

The aim of this program is to understand relations between all these topics, and to use the language and tools they provide to find a natural interpretation of abstract mathematical formulations of knot homologies. We also plan to develop new computational tools that allow to derive an explicit form of superpolynomials, super-A-polynomial, holomorphic blocks, etc.

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**Large N limit problems in Kahler Geometry**

Organized by Robert Berman, Semyon Klevtsov, Paul Wiegmann, and Steve Zelditch

*April 20 – June 19, 2015*

This program centers on the use of holomorphic sections of high powers of positive Hermitian holomorphic line bundles over a Kahler manifold to construct projective embeddings, Bergman Kahler metrics, and Gaussian random fields. The main tool is the Bergman kernel and its large N asymptotics, on and off the diagonal. Its asymptotics has many applications in geometry, probability theory and mathematical physics.

In geometry one often uses the Kodaira-type embedding of the manifold to the projective space of the holomorphic sections, in order to pull back the Fubini-Study metric to define a space of algebro-geometric metrics, known as Bergman metrics. For large N, they approx- imate all Kahler metrics, and in fact the finite dimensional symmetric space geometry of the space of Bergman metrics approximates the full infinite dimensional symmetric space geometry of the space of all Kahler metrics. In particular, one may study the geodesics on the latter, approximating them by the geodesics on the space of Bergman metrics. One can also study integral geometry on the space of metrics and use the finite dimensional approximations to define random Kahler metrics, with potential applications in physics.

In another directions Gaussian random holomorphic sections of the line bundle induce random zero sets and point processes. The two point function of the process equals to the Bergman kernel and, therefore, its asymptotics control the geometry of random zero sets. One may also define a canonical point process from the Bergman kernel and expect that as the number of points tends to infinity it approximates important metrics. The Bergman kernel has further uses in dimension one to study the geometric properties of random normal matrix ensembles.

All these applications are partly inspired by the Yau-Tian-Donaldson program of relating GIT stability and Kahler-Einstein metrics, but the focus of the program is on holomorphic stochastic geometry and mathematical physics.

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