Fall 2013 – Spring 2014 Program Details
Starting date: August 26, 2013VISIT THE PROGRAM WEBPAGE
Modular functions, Jacobi forms and mock modular forms appear naturally in various contexts in string theory and conformal field theory. In particular, characters of conformal field theories (CFTs) define (vector-valued) modular functions, while Jacobi forms and mock modular forms arise from the elliptic genus of superstring compactifications. For example, the J-function is the character of the famous Monster CFT, while the unique weak Jacobi form of weight zero and index one, φ0,1, is the elliptic genus of the superstring compactification on a K3 surface.
About 30 years ago, it was noted that the Fourier coefficients of the J-function can be interpreted in terms of representations of the Monster group, the largest simple sproadic finite group, and this gave rise to a development that is now usually referred to as `Monstrous Moonshine’. A few years ago, Eguchi, Ooguri and Tachikawa (EOT) made a similar observation regarding φ0,1: they noted that its Fourier coefficients can be interpreted in terms of representations of M24-representations, where M24 is the largest Mathieu group, another simple sporadic finite group. Moreover, this observation has a nice formulation in terms of the mock modular form that is naturally associated to φ0,1.
While the conjecture of EOT has by now been largely established, there are many intriguing open questions that remain. For example, while the Monster CFT provided a microscopic explanation of at least some aspects of Monstrous Moonshine, none of the superstring theories on K3 actually possess M24 as their automorphism group. Moreover, intriguing extensions of the EOT conjecture to higher index weak Jacobi forms (umbral moonshine) and generalisations to situations with less supersymmetry have recently been found, but the proper context in which all of these observations fit remains to be understood. In this program, we hope to bring together experts from different areas, including vertex operator algebra, string theory, algebraic geometry and number theory, in order to make progress with these very topical problems.VISIT THE PROGRAM WEBPAGE
Moduli Spaces of Pseudo-holomorphic curves and their applications to Symplectic Topology
Organized by Kenji Fukaya, Dusa McDuff, and John Morgan
January 2 – June 30, 2014
The Simons Center is hosting a semester-long program entitled Moduli Spaces of Pseudo-holomorphic curves and their applications to Symplectic Topology from January 2, 2014 through June 30, 2014.
Gromov-Witten theory, Lagrangian-Floer homology and symplectic field theory arise from the notion of pseudo-holomorphic curves, possibly with boundary conditions, in symplectic manifolds. All these theories rely in a fundamental way on Gromov’s compactness result for moduli spaces of pseudo-holomorphic curves, as well as on results about gluing and transversality. A central issue is how to use these analytic results in various contexts to produce appropriate structures on the moduli spaces, structures that allow one to define a virtual fundamental cycle that can then be used to produce algebraic invariants.
The program will focus on two basic approaches to understanding the structure of these moduli spaces:
(1) local finite-dimensional models derived from the local Kuranishi obstruction picture, and
(2) infinite dimensional analysis — the polyfold approach.
The purpose of this program is to examine foundational analytic and topological questions associated with these approaches, especially in key examples such as the closed case (Gromov-Witten theory), Hamiltonian- and Lagrangian-Floer theory and contact homology. The aim is to compare and contrast the approaches with each other and with alternative methods, such as those in algebraic geometry or those provided by Joyce’s derived smooth manifold theory.
- Workshop on Moduli Spaces of Pseudo-holomorphic Curves I occurs from March 17 – 21, 2014. Please see the workshop webpage here: http://scgp.stonybrook.edu/archives/7149
- There will be an informal workshop on polyfolds May 5 – 9, 2014.
- A workshop on Equivariant Gromov-Witten Theory and Applications occurs from May 12 – 16, 2014. Please see the workshop webpage here: http://scgp.stonybrook.edu/archives/7153
- Workshop on Moduli Spaces of Pseudo-holomorphic Curves II occurs from June 2 – 6, 2014. Please see the workshop webpage here: http://scgp.stonybrook.edu/archives/7151
- There will be an informal workshop on the foundations of contact homology from June 16 – 20, 2014.
Recent developments in relativistic hydrodynamics place it at the crossroads of nuclear physics, condensed matter physics and string theory. Hydrodynamics is known to be very effective in describing the long-wavelength behavior of many body systems regardless of the strength of inter-particle interactions. It encapsulates general conservation laws and symmetries of the system.
Quantum anomalies can break some of the symmetries of the underlying theory. While it was known for a long time that anomalies induce observable effects in quantum field theories, only quite recently it has become clear that anomalies also have important macroscopic manifestations and affect transport and hydrodynamics — in particular, anomalies make possible non-dissipative transport and bring to the existence novel collective excitations. The effects of anomalies are especially important in the systems that possess chiral fermions (e.g. quantum Hall systems, graphene or quark-gluon plasma) and where topologically non-trivial configurations are present (e.g. vortices, skyrmions or sphalerons).
An important advantage of hydrodynamics is that it is formulated explicitly in terms of physical observables. Therefore, the hydrodynamical approach usually leads directly to the predictions for the experiment, both in condensed matter and nuclear physics. Furthermore, topological methods in hydrodynamics could shed some light on the search for new symmetries related to quantum anomalies. Topological fluid dynamics is a young branch of mathematics which studies group symmetries of various equations of hydrodynamical origin, as well as geometric and topological properties of their solutions and of the corresponding magnetic and vortex fields. The interaction of the relativistic and topological approaches in hydrodynamics might also lead to new insights into the turbulence and singularity problems.
The goal of the program is to develop hydrodynamic descriptions in condensed matter physics and QCD at finite temperature and density with an emphasis on the effects of quantum anomalies and topology. Among these effects is the non-dissipative transport of charges and energy — with a wide range of applications in science and technology, from quantum computing to the detection of topological fluctuations of QCD in heavy ion collisions. We aim at advancing quantum hydrodynamics through the use of topological and geometric methods.