Foundations and Applications of Random Matrix Theory in Mathematics and Physics
Organized by Alexei Borodin, Peter Forrester, Yan Fyodorov, Alice Guionnet, Jon Keating, Mario Kieburg, and Jacobus Verbaarschot
August 24 – December 18, 2015
Random Matrix theory has been applied to many areas in pure and applied mathematics and in physics, ranging from correlations among the zeros of the Riemann function and the distribution of the longest increasing subsequences of permutations to the spacing distribution of nuclear levels and correlations of the eigenvalues of the Dirac operator in Quantum Chromo Dynamics. In this program we will discuss recent developments of random matrix theories beyond the ten fold classification in terms of large symmetric spaces. In particular we will focus on the following four areas: 1) Chiral random matrix theories with applications to gauge theories and the Riemann function. Nonperturbative properties of strongly interacting quantum held theories can be understood by means of random matrix theories with the same chiral symmetry breaking pattern. Recent work indicates that the same is true if the symmetries are softly broken by discretization effects or a chemical potential. Applications to the zeros of the Riemann function, Dirac spectra in lower dimensions and relations with topological insulators will be discussed as well. 2) The relation between integrability and random matrix theory. Invariant random matrix theories can be solved because of underlying integrable structures such as for example the Toda lattice equation. More complicated ensembles still can be solved suggesting that an underlying integrable structure exists and we hope to explore such relations in this program. 3) Universal properties on non-invariant random matrix ensembles. Examples are Wigner matrices, ensembles of sparse matrices and band matrices. In particular, band matrices describe the transition between Poisson and Wigner-Dyson eigenvalue statistics and have been used to study scaling properties of localization. Based on recent work showing that these properties are universal we hope to make further progress in this area. 4) Random Matrix Theory and dynamics. This goes back as far as Dyson Brownian motion model for the eigenvalues of random matrices, but recently this topic received a great deal of attention in the context of non-equilibrium dynamics and the Kardar-Parisi-Zhang equation. This is also the topic of the workshop Random matrices, random growth processes and statistical physics from September 6-11 which is part of this program. Related issues such as the Langevin evolution of random matrix spectra and relations between the stochastic Loewner equation and random matrix theory will be discussed as well. A second workshop related to physics applications of random matrix theory may be organized later in the program.
Program Application is now closed.Attendee List
Moduli spaces and singularities in algebraic and Riemannian geometry
Organized by Simon Donaldson, Song Sun, Henry Guenancia, Radu Laza (Stony Brook), Valentino Tosatti (Northwestern), Hans-Joachim Hein (Maryland), Yuji Odaka (Kyoto)
August 17-November 20th, 2015
The theme of this program is the interaction between algebro-geometric and differential-geometric approaches. In algebraic geometry, one is interested in constructing compact moduli spaces of varieties. Even if one starts with manifolds, the compactification will almost always involve the inclusion of appropriate singular varieties. In a case when the manifolds have canonical metrics, such as Kahler-Einstein metrics, one can approach these questions differential-geometrically, studying the convergence of the metrics and the metric nature of the singular limits. Such ideas are well-established in the case of the moduli of curves and the Deligne-Mumford compactification. The purpose of this program is to make progress in higher dimensions, in the light of a number of recent developments coming from pluripotential theory, Riemannian convergence theory and algebraic geometry. The program will be an opportunity for specialists in the various different fields to interact and share expertise. Topics which will be discussed include:
1. The differential geometric interpretation of the moduli of varieties of general type constructed by Alexeev, Kollar, Shepherd-Barron.
2. Moduli spaces of Fano varieties, and connections with stability.
3. Kahler-Einstein metrics on singular varieties; their singularities and metric tangent cones.
4. Riemannian collapsing and large complex structure limits of Calabi-Yau manifolds.
Program Application is now closed.
Geometric representation theory
Organized by: David Ben-Zvi, Roman Bezrukavnikov and Alexander Braverman
January 4-29th, 2016
The program will focus on emerging trends in representation theory and their relation to the more traditional ideas of the subject. The celebrated success of the perverse sheaves methods in 1980’s has led to development of a direction which may be called geometric categorification, where the primary object of study is a category of sheaves on an appropriate geometric space (stack): this includes theories of character sheaves, geometric Langlands duality, various approaches to categorification based on quiver varieties etc. Recently significant progress has been achieved in developing and systematizing these constructions and new algebraic methods inspired by D-modulesand perverse sheaves have been discovered. Other recent works indicate relation of this area to ideas of physical origin such as topological quantum field theory and wall-crossing, and their mathematical manifestations including cluster algebras, Bridgeland stabilities and quantum cohomology. The program will bring together experts in these different directions creating an opportunity for a synthesis of the diverse approaches and further progress.
Program Application Deadline: November 4, 2015 (or when event is at maximum capacity). Applicants will be contacted soon after this date.
Statistical mechanics and combinatoricsApply to a Program Now
Organized by Pavel Bleher, Vladimir Korepin, and Bernard Nienhuis
February 15 – April 15, 2016.
The purpose of the program is to relate physics and mathematics, and more specifically, statistical mechanics, algebraic combinatorics, and random matrices. The program will focus on the six-vertex model of statistical mechanics and related models, such as the XXZ spin chain, and loop models. The six-vertex model was introduced by Pauling in 1935mas a two dimensional version of a model for the hydrogen bonds in ice.
The bulk free energy and entropy in the six-vertex model were explicitly calculated for periodic boundary conditions by Lieb. Then other boundary conditions were studied. A very interesting case is the domain wallboundary conditions (DWBC). The six-vertex model provides an important ‘counterexample’ in statistical mechanics: the bulk free energy in the thermodynamic limit depends on boundary conditions. In particular, it is different for periodic and domain wall boundary conditions.
The partition function of the six-vertex model with DWBC in a finite box has been expressed, via the Yang-Baxter equations, in terms of a Hankel determinant. It can be furthermore expressed as a partition function of an ensemble of random matrices with a non-polynomial interaction. This expression was used via the powerful Riemann-Hilbert approach, the asymptotic behavior of the partition function of the six-vertex model with DWBC in different phase regions.
A remarkable observation was made by Razumov and Stroganov that concerns the six-vertex model at a special value of the parameter Delta=1/2, on afinite square grid, transformed into a loop model, i.e. a model of paths on the lattice that can end only on the boundary. In this case the paths visit all vertices exactly once, and all possible configurations are equally probable. This simple measure induces a probability measure on how the edges on the boundary are pairwise connected to each other. This is compared with another loop model placed on a half-infinite cylinder. In this case it is a loop model where the paths pass every edge once, every vertex twice and do not intersect. An equivalent six-vertex model has Delta=-1/2. Again every configuration is equally probable, and again this induces a probability measure on a pairing of the boundary edges. The observation of Razumov and Stroganov is that both probability measures of pairings of boundary edges are the same.
This observation was later proven by Cantini and Sportiello. The original observation has many generalizations on different geometries, but these have not been proven.
The six-vertex model with DWBC relates statistical mechanics to various problems of combinatorics: the statistics of alternating sign matrices, domino tilings, limiting shapes, nonintersecting lattice paths, loop models, and others. Limiting shape formulae in the six-vertex model were proposed recently. The six-vertex model has various generalizations to the eight-vertex model, higher-spin systems, coloring of a lattice, and others, which are important for applications. We plan on inviting leading experts, both physicists and mathematicians, workingon the combinatorics of six-vertex model and its generalizations.
Program Application Deadline: December 15, 2015 (or when event is at maximum capacity). Applicants will be contacted soon after this date.
Complex, p-adic, and logarithmic Hodge theory and their applicationsApply to a Program Now
Organized by Mark de Cataldo, Radu Laza, Christian Schnell
March 6, 2016 – April 29, 2016
Hodge theory is a very powerful tool for understanding the geometry of complex algebraic varieties and it has a wide range of applications in complex and algebraic geometry, mirror symmetry, representation theory, combinatorics, etc. This program focuses on different aspects of Hodge theory, their applications in algebraic geometry and related areas and, very importantly, on their interactions. We plan to cover the following four themes:
(1) p-adic Hodge theory and arithmetic geometry
(2) mixed Hodge modules and their applications
(3) log geometry, with an emphasis on degenerations and moduli problems
(4) applications of Hodge theory to questions about algebraic cycles
There will be four lecture series, “p-adic Hodge theory” (B. Bhatt), “Log geometry and log Hodge structures” (M. Olsson), and “Mixed Hodge modules” (Ch. Schnell), a weekly seminar, and several mini-courses.
Program Application Deadline: January 6, 2016 (or when event is at maximum capacity). Applicants will be contacted soon after this date.
Geometry of Quantum Hall States
Organized by Sasha Abanov, Tankut Can, Anton Kapustin, and Paul WiegmannApply to a Program Now
April 18, 2016 – June 17, 2016
The quantum Hall effect (QHE) is a fascinating and important phenomenon. Since its experimental discovery in the early 80’s the QHE continues to fuel work in experimental physics, metrology, fundamental theoretical physics and mathematics.
At very low temperatures and strong magnetic fields, highly entangled collective electronic quantum states are formed on surfaces of ultra clean doped semiconductors. Despite varying microscopic details and imperfections of materials, these states possess universal properties demonstrating very precise (up to 9 significant digits) quantization of the Hall conductivity in units of fundamental constants. Other observable features are: an excitation gap determined by electron interactions, fractional charge and statistics of elementary excitations, and chiral massless boundary modes with universal dynamics.
Many universal features of the QHE are also captured at the microscopic scale by model electron wave functions. An outstanding theoretical task is to connect this microscopic description with the macroscopic picture discussed above.
Recently, the physics of low energy transport phenomena has been connected to the response of the ground state to variations of the spatial geometry. This observation has led to a geometric description of QHE states, which links the physics of QH states to problems of modern geometry (Kahler geometry). Another rapidly developing link is a relation between the FQHE and the theory of random geometry and quantum gravity. The synthesis of subjects and intriguing links to modern mathematics give the QHE a very special status.
The goal of the program is to bring together physicists and mathematicians working on topics related to the geometry of QH states, in order to develop a geometric approach to quantum Hall states. The program will also encourage a broader discussion of the role of geometry in quantum states of condensed matter systems. Some of the key topics of the program are:
QH wave functions in geometric backgrounds
Effective field theory and geometric responses of QH states
Transport properties in inhomogeneous backgrounds
Bulk-boundary correspondence of QH states
Entanglement and topological order in QH states
Hydrodynamics of fractional QH electronic fluid
Methods of Conformal Field Theory and QH states
Other topics and connections to other systems (anomalous hydrodynamics, quantum gravity, random matrices, Kahler geometry)
Program Application Deadline: February 18, 2016 (or when event is at maximum capacity). Applicants will be contacted soon after this date.