XVIIth International Congress on Mathematical Physics

The following sections are included:

• PREFACE

• CONGRESS COMMITTEES

• SPONSORS

• CONTENTS

The following sections are included:

• PRIZE CEREMONIES
  • The Henri Poincaré Prize sponsored by the Daniel Iagolnitzer Foundation
  • The IAMP Early Career Award
  • IUPAP Young Scientist Prize
  • TALKS BY PRIZE RECIPIENTS

We review various combinatorial problems with underlying classical or quantum integrable structures.

We review several recent results showing that small piecewise smooth perturbations of integrable systems may exhibit unstable behavior on the set of initial condition of large measure. We also present open questions related to this subject.

Since Wigner’s pioneering work from 1955 random matrices have been an important tool in Mathematical Physics. After Voiculescu in 1991–95 used random matrices to solve some deep open problems about von Neumann algebras, random matrices have also played a key role in operator algebra theory. In 2005 Steen Thorbjørnsen and the speaker were able to solve an old problem on C*-algebras, by making careful estimates of the largest and smallest eigenvalues in random ensembles, which can be expressed as (non-commutative) polynomials in two or more independent GUE-random matrices, [1]. Shortly after we obtained (in collaboration with Hanne Schultz) similar estimates for polynomials in GOE- and GSE- matrices [2], but the corresponding problem for polynomials in two or more non-Gaussian random matrices with independent entries was solved only recently by Greg Anderson (2011).

This article contains the abstract only.

Reading is a highly complex task involving precise integration of vision, attention, rapid eye movements, and high-level language processing. In the past my colleagues and I [6, 7] have constructed a biologically realistic model of the frontal eye fields that simulates the control of eye movements in human readers. The model couples processes of oculomotor control and cognition in a microcircuit of spiking neurons. In this talk I will use this model to give an introduction to neuro-linguistics with special emphasis on reading and understanding in geometry [10].

We give a summary of a talk delivered at the ICMP in Aalborg, Denmark, August, 2012. We review d = 4, N = 2 quantum field theory and some of the exact statements which can be made about it. We discuss the wall-crossing phenomenon. An interesting application is a new construction of hyperkähler metrics on certain manifolds. Then we discuss recent geometric constructions which lead to exact results on the BPS spectra for some d = 4, N = 2 field theories and on expectation values of – for example – Wilson line operators. These new constructions have interesting relations to a number of other areas of physical mathematics.

Here we review several recent results on the propagation of microlocal singularities for (1) the solutions to Schrödinger equations; and (2) scattering matrices for Schrödinger operators on manifolds. These results are both closely related to a construction of classical mechanical scattering theory on manifolds, and scattering type time evolutions.

We first recall the basic ideas of scattering theories, both classical mechanical and quantum mechanical ones. Then we construct a classical mechanical scattering theory on asymptotically conic manifolds. By using different quantizations, we obtain two different sets of microlocal results described above.

Since the previous ICMP in 2009 in Prague, there has been considerable progress on the Kardar-Parisi-Zhang equation. Our goal here is to give a very brief discussion of some of the results. More comprehensive surveys are available [1–4].

We study an entropy measure for quantum systems that generalizes the von Neumann entropy as well as its classical counterpart, the Gibbs or Shannon entropy. The entropy measure is based on hypothesis testing and has an elegant formulation as a semidefinite program, a type of convex optimization. After establishing a few basic properties, we prove upper and lower bounds in terms of the smooth entropies, a family of entropy measures that is used to characterize a wide range of operational quantities. From the formulation as a semidefinite program, we also prove a result on decomposition of hypothesis tests, which leads to a chain rule for the entropy.

This is the summary of a plenary talk given at ICMP 2012, where I delineate the ideas and main results of the Associative Algebraic Approach to Logarithmic Conformal Field Theory we proposed with N. Read in 2007. I have tried to make this condensed and quickly accessible. This means that the language is not very mathematical, and some of the statements are qualitative. Also, this is not a review, and only a subset of the rather large relevant literature is mentioned. Details and many more references can be found in the papers mentioned below.

The following sections are included:

• Introduction

• Calculus of variations

• Wave equations

• The cubic Klein-Gordon equation

• Focusing cubic Klein-Gordon equation

• Wave Maps

• Acknowledgement

• References

Systems of interest in physics are usually composed by a very large number of interacting particles. At equilibrium, these systems are described by stationary states of the many-body Hamiltonian (at zero temperature, by the ground state). The reaction to perturbations, for example to a change of the external fields, is governed by the time-dependent many-body Schrödinger equation. Since it is typically very difficult to extract useful information from the Schrödinger equation, one of the main goals of nonequilibrium statistical mechanics is the derivation of effective evolution equations which can be used to predict the macroscopic behavior of the system. In these notes, we are going to consider systems of interacting bosons in the so called Gross-Pitaevskii regime, and we are going to show how coherent states and Bogoliubov transformations can be used to approximate the many body dynamics.

This is a summary of a talk given at ICMP 2012. It discusses some recent results in spectral theory through the prism of a new-found synergy between the spectral theory and OP communities.

We discuss the concepts of energy and mass in relativity. On a finitely extended spatial region, they lead to the notion of quasilocal energy/mass for the boundary 2-surface in spacetime. A new definition was found in [27] that satisfies the positivity, rigidity, and asymptotics properties. The definition makes use of the surface Hamiltonian term which arises from Hamilton-Jacobi analysis of the gravitation action. The reference surface Hamiltonian is associated with an isometric embedding of the 2-surface into the Minkowski space. We discuss this new definition of mass as well as the reference surface Hamiltonian. Most of the discussion is based on joint work with PoNing Chen and Shing-Tung Yau.

The Anderson model on the Bethe lattice is historically among the first for which an energy regime of extended states and a separate regime of localized states could be established. In this paper, we review recently discovered surprises in the phase diagram. Among them is that even at weak disorder, the regime of diffusive transport extends well beyond energies of the unperturbed model into the Lifshitz tails. As will be explained, the mechanism for the appearance of extended states in this non-perturbative regime are disorder-induced resonances. We also present remaining questions concerning the structure of the eigenfunctions and the associated spectral statistics problem on the Bethe lattice.

Man has grappled with the meaning and utility of randomness for millennia. Randomness is paramount to many fields of science, and probabilistic algorithms play a key role is solving important problems in many disciplines. Computational complexity theory offers new insights about the power and limit of randomness in such settings, and a concrete computational notion of pseudo-randomness. Surprisingly, under standard believable assumptions about computational difficulty, it turns out that randomness is not essential for any of these applications!

The closure of periodic orbits in the phase space of the spatial, planetary N-body problem (with well separated semimajor axes) has full measure in the limit of small planetary masses and small eccentricities and mutual inclinations.

In this talk, we present an answer to the long standing problem on the implication of positive entropy of a random dynamical system. We study C⁰ infinite dimensional random dynamical systems in a Polish space, do not assume any hyperbolicity, and prove that chaos and weak horseshoe exist inside the random invariant set when its entropy is positive. This result is new even for finite dimensional random dynamical systems and infinite dimensional deterministic dynamical systems generated by either parabolic PDEs or hyperbolic PDEs. We mention that in general one does not expect to have a horseshoe without assuming hyperbolicity. For example, consider the product system of a circle diffeomorphism with an irrational rotation number and a system with positive entropy. This product system has positive entropy and a weak horseshoe, but has no horseshoe.

Note from Publisher: This article contains the abstract only.

There are many systems that appear in applications that have negligible friction, like the models of celestial mechanics and Astrodynamics, motion of charged particles in magnetic fields chemical reactions, etc.

A general model for this kind of systems is to consider time periodic pertubations of integrable Hamiltonian systems with 2 or more degrees of freedom.

One problem that has attracted attention for a long time since the example of Arnold in 1964 [1] is whether the effect of perturbations accumulate over time and lead to large effects (instability) or whether these effects average out (stability).

In this talk we present some mechanisms that cause instabilitites for general perturbations. We use the so called geometric methods, which work for a priori-unstable systems, where the unperturbed system has some (possibly weakly) hyperbolic object with stable and unstable manifolds.

The main technique is to develop a toolkit to study, in a unified way, tori of different topologies and their invariant manifolds, their intersections as well as shadowing properties of these bi-asymptotic orbits. Part of this toolkit is to unify standard techniques (normally hyperbolic manifolds, KAM theory, averaging theory) so that they can work together. A fundamental tool used here is the scattering map of normally hyperbolic invariant manifolds.

The conditions needed are explicit and are based in the computation of a general Melnikov function. Therefore, they can be checked in specific examples. When the hyperbolic structure of the system is weakly hyperbolic, this Melnikov function is exponentially small as happens in some problems of celestial mechanics, as the restricted three body problem.

This article contains the abstract only.

We consider the nonlinear instability of a steady state υ₀ of the Euler equation in a fixed bounded domain in Rⁿ. When considered in H^s, s > 1, at the linear level, the stretching of the steady fluid trajectories induces unstable essential spectrum which corresponds to linear instability at small spatial scales and the corresponding growth rate depends on the choice of the space H^s. Therefore, more physically interesting linear instability relies on the unstable eigenvalues which correspond to large spatial scales. In the case when the linearized Euler equation at υ₀ has an exponential dichotomy of unstable (from eigenvalues) and center-stable directions, most of the previous results obtaining the expected nonlinear instability in L² (the energy space) were based on the vorticity formulation and therefore only work in 2-dim. In this talk, we prove, in any dimensions, the existence of the unique local unstable manifold of υ₀, under certain co nditions, and thus its nonlinear instability. Our approach is based on the observation that the Euler equation on a fixed domain is an ODE on an infinite dimensional manifold of volume preserving maps in function spaces.

Note from Publisher: This article contains the abstract only.

The following sections are included:

• CONTRIBUTED TALKS

• POSTERS

Some invariances under perturbations of the spin glass phase are introduced, their proofs outlined and their consequences illustrated as factorization rules for the overlap distribution. A comparison between the state of the art for mean field and finite dimensional models is shortly discussed.

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters q, t ∈ [0; 1). We recall this theory and record several results about these processes as developed and proved in [9]. The contributions of [9] include the following. (1) We explicitly evaluate expectations of a rich family of observables for these processes. (2) In the case t = 0, we find a Fredholm determinant formula for a q-Laplace transform of the distribution of the last part of the Macdonald-random partition. (3) We introduce Markov dynamics that preserve the class of Macdonald processes and lead to new “integrable” 2d and 1d interacting particle systems. (4) In a large time limit transition, and as q goes to 1, the particles of these systems crystallize on a lattice, and fluctuations around the lattice converge to O’Connell’s Whittaker process that describe semi-discrete Brownian directed polymers. (5) This yields a Fredholm determinant for the Laplace transform of the polymer partition function, and taking its asymptotics we prove KPZ universality for the polymer (free energy fluctuation exponent 1=3 and Tracy-Widom GUE limit law). (6) Under intermediate disorder scaling, we recover the Laplace transform of the solution of the KPZ equation with narrow wedge initial data. (7) We provide contour integral formulas for a wide array of polymer moments. (8) This results in a new ansatz for solving quantum many body systems such as the delta Bose gas.

This is an extended abstract of my talk at the ICMP in Aalborg, focusing primarily on background rather than technical details.

We review recent results with D. Chelkak and K. Izyurov [10], where we rigorously prove existence and conformal invariance of scaling limits of magnetization and multi-point spin correlations in the critical Ising model on an arbitrary simply connected planar domain. This solves a number of conjectures coming from physical and mathematical literatures. The proof is based on convergence results for discrete holomorphic spinor observables.

We describe quantum entropic functionals and outline a research program dealing with entropic fluctuations in non-equilibrium quantum statistical mechanics.

We announce our recent result on the study of the spectral gap estimate for a class of stochastic energy exchange models. We give a lower bound estimate of the spectral gap for the case where a rate function does not have a uniform lower bound. The result can be applied for the mesoscopic dynamics obtained from purely deterministic billiard lattice models. We also present an interesting observation on the hydrodynamic equation of a nice class of stochastic energy exchange models.

We review random loop representations for the spin-½ quantum Heisenberg models, that are due to Tóth (ferromagnet) and Aizenman–Nachtergaele (antiferromagnet). These representations can be extended to models that interpolate between the two Heisenberg models, such as the quantum XY model. We discuss the relations between long-range order of the quantum spins and the size of the loops. Finally, we describe conjectures about the joint distribution of the lengths of macroscopic loops, and of symmetry breaking.

The following sections are included:

• CONTRIBUTED TALKS

• POSTERS

A natural question in general relativity is to find initial data for the Einstein equations whose past evolution is regular and whose future evolution contains a black hole. In [1] initial data of this kind is constructed for the spherically symmetric Einstein-Vlasov system. One consequence of the result is that there exists a class of initial data for which the ratio of the Hawking mass and the area radius r is arbitrarily small everywhere, such that a black hole forms in the evolution. Another consequence is that there exist black hole initial data such that the solutions exist for all Schwarzschild time t ∈ (−∞, ∞). In the present article we review the results in [1].

This note surveys how energy generation and strengthening has been used to prove Morawetz estimates for various field equations in Minkowski space, the exterior of the Schwarzschild spacetime, and the exterior of the Kerr spacetime. It briefly outlines an approach to proving a decay estimate for the Maxwell equation outside a Kerr black hole.

We construct non-stationary vacuum black holes which converge asymptotically to a Schwarzschild spacetime. The (smooth) solutions are constructed by specifying suitable data on the horizon and null-infinity and solving backwards. This is joint work with M. Dafermos and I. Rodnianski.

Note from Publisher: This article contains the abstract only.

We discuss elliptic systems of Liouville type in presence of singular sources, as derived from the study of non-abelian (selfdual) Chern-Simons vortices. We shall focus on the so called non-topological vortex configurations, and present some recent results.

Let ϕ_ω, ω ∈ I, be a branch of unstable solitary waves (solitons) of a nonlinear Schrödinger equation (NLS) whose linearized operators have one pair of simple real eigenvalues ±e₊(ω) in addition to 0 eigenvalue. With localized perturbation to the initial data, the solution will locally either converge to the branch, or exit a neighborhood of the branch. This has implication to the blowup behavior of NLS with supercritical nonlinearity.

Joint work with Vianney Combet, Université Lille 1, and Ian Zwiers, University of British Columbia.

Note from Publisher: This article contains the abstract only.

I report on my work [12, 13] on a geometric criterion for the breakdown of Einstein vacuum space-times with constant mean curvature (CMC) foliation. In this work, the criterion is formulated in terms of time-integrability of the sup-norms of the second fundamental form and derivatives of the lapse function associated to the CMC foliation of the space-time.

The following sections are included:

• CONTRIBUTED TALKS

• POSTERS

Can one count the number of critical points for random smooth functions of many variables? How complex is a typical random smooth function? How complex is the topology of its level sets? We study here the simplest case of smooth Gaussian random functions defined on the sphere in high dimensions. We show that such a randomly chosen smooth function is very complex, i.e. that its number of critical points of given index is exponentially large. We also study the topology of the level sets of these functions, and give sharp estimates of their Euler characteristic. This study, which is a joint work with Tuca Auffinger (Chicago) and partly with Jiri Cerny (Vienna), relies rather surprisingly on Random Matrix Theory, through the use of the classical Kac-Rice formula. The main motivation comes from the study of energy landscapes for general spherical spin-glasses. I will detail the interesting picture we get for the complexity of these random Hamiltonians, for the bottom of the energy landscape, and in particular a strong correlation between the index and the critical value. We also propose a new invariant for the possible transition between the 1-step replica symmetry breaking and a Full Replica symmetry breaking scheme.

Note from Publisher: This article contains the abstract only.

In this note we consider β-ensembles with real analytic potential and arbitrary inverse temperature β, and review some recent universality results for these measures, obtained in joint works with L. Erdős and H.-T. Yau. In the limit of a large number of particles, the local eigenvalues statistics in the bulk are universal: they coincide with the spacing statistics for the Gaussian β-ensembles. We also discuss the proof of the rigidity of the particles up to the optimal scale N^−1+ε.

Consider a random walk on a finite graph, like e.g. a discrete torus. We investigate percolative properties of the vacant set of this walk, that is of the set of vertices not visited by this walk before certain fixed time. It is conjectured, and supported by simulations, that the vacant set exhibits a phase transition similar to usual Bernoulli percolation on finite graphs. For the torus, however, there is no proof of this fact at present.

In this talk, I explain that the phase transition can be proved on other graphs, like large-girth expanders, or random regular graphs, where it is even possible to investigate the critical window around the phase transition threshold. The results are based on the connection with the ‘infinite volume limit’ of the problem, the Random Interlacement percolation.

Note from Publisher: This article contains the abstract only.

The soliton resolution conjecture for the focusing nonlinear Schrödinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multi-soliton solution. Considered to be one of the fundamental problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation till date. I will present a theorem that proves a “statistical version” of this conjecture at mass-subcritical non-linearity. The proof involves a combination of techniques from large deviations, PDE, harmonic analysis, and bare hands probability theory.

Note from Publisher: This article contains the abstract only.

The KPZ equation is the stochastic PDE formally given by

where ξ denotes space-time white noise. It was originally introduced in the eighties as a model of surface growth, but it was soon realised that its solution is a much more universal object describing the crossover between the Gaussian universality class and the KPZ universality class. The mathematical proof of its universality however is still an open problem, in particular because of the lack of a good approximation theory for the equation. Indeed, the only known way so far to mathematically interpret solutions to the KPZ equation is to reduce it to a linear stochastic PDE via a non-linear transformation called the Cole-Hopf transform. Unfortunately, the resulting linear equation does itself lack a good approximation theory and many microscopic models do not behave well under the Cole-Hopf transform.

In this talk, we present a new notion of solution to the KPZ equation that bypasses the use of the Cole-Hopf transform. Our approach also allows to factorise the solution map into a “universal” (i.e. independent of initial condition) measurable map, composed with a solution map with good continuity properties. This lays the foundations for a robust approximation theory to the KPZ equation, which is needed to prove its universality. As a byproduct of the construction, we obtain very detailed regularity estimates on the solutions, as well as a new homogenisation result.

Note from Publisher: This article contains the abstract only.

We present recent work about the scaling limit of random planar maps, which are random graphs embedded in the two-dimensional sphere. We consider a random planar map M_n which is uniformly distributed over the class of all rooted q-angulations with n faces. We let m_n be the vertex set of M_n, which is equipped with the graph distance d_gr. Both when q ≥ 4 is an even integer and when q = 3, there exists a positive constant c_q such that the rescaled metric spaces (m_n; c_qn^−1/4d_gr) converge in distribution in the Gromov-Hausdorff sense, towards a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.

The following sections are included:

• CONTRIBUTED TALKS

It is well known that the configurations of the 6-vertex model on a square grid with Domain Wall boundary conditions are in bijections with Alternating Sign Matrices and with Fully-Packed Loop (FPL) configurations on the square with alternating boundary conditions.

In 2001 Razumov and Stroganov conjectured that the enumerations of FPL configurations on the square refined according to the link pattern for the boundary points coincide with the (properly normalized) components of the ground state of the dense O(1) loop model on a semi-infinite cylinder. In [1] we have provided a proof of the Razumov Stroganov conjecture. Recently [2] we have found and proven a generalization of this result by identifying certain weighted enumerations of FPLs with the components of the ground state of an inhomogeneous version of the O(1) loop model.

Joint work with Andrea Sportiello, University of Milan.

Note from Publisher: This article contains the abstract only.

The Weyl algebra is the standard C*-algebraic version of the algebra of canonical commutation relations, but in applications it often causes difficulties. These stem from its failure to admit the formulation of physically interesting dynamical laws as automorphism groups, and that it does not contain important (bounded) physical observables. We consider a new C*-algebra of the canonical commutation relations which circumvents such problems. It is based on the resolvents of the canonical operators and their algebraic relations. The resulting C*-algebra, the resolvent algebra, has many desirable analytic properties. In particular, the resolvent algebra has one–parameter automorphism groups corresponding to a large class of physically relevant dynamics, and it contains the resolvents of many interesting Hamiltonians. It has a rich ideal structure, and in fact its primitive ideal space can detect the dimension of the underlying symplectic space. However, all regular representations are faithful. In applications to canonical quantum systems it has been a substantial improvement on the Weyl algebra, already in the areas of C*-supersymmetry, dynamics of infinite lattice quantum systems and BRST-constraints.

For the integrable 6 vertex model, the expectation values of local operators are known to be given by complicated multiple integrals. We show that there exists a basis of (quasi) local operators for which the expectation values simplify drastically. Such a basis is constructed out of a simple ‘tail’ operator (analogous to the disorder field in the Ising model) by acting with integrals of motion and a newly introduced set of fermions. The expectation values for their generating functions are given by determinants with explicit entries. This fermionic structure is present at a generic coupling, away from the usual ‘free fermion point’.

Taking the continuum limit to CFT and the sine-Gordon model, we formulate conjectural explicit formulas for the one-point functions of all descendant fields in both cases, generalizing the remarkable formulas due to Lukyanov, Zamolodchikov and others. We argue also that at the level of form factors our fermions coincide with yet another fermions which have been introduced some time ago by Babelon, Bernard and Smirnov.

This talk is based on a series of joint works with H. Boos, T. Miwa, F. Smirnov and T. Takeyama.

Note from Publisher: This article contains the abstract only.

We review the key steps of the construction of Levin-Wen type of models enriched by gapped boundaries and defects of codimension 1,2,3 in a joint work with Alexei Kitaev [3]. We emphasize some universal properties, such as boundary-bulk duality and duality-defect correspondence, shared by all these models.

We present a microscopic approach in the framework of Sklyanin's quantum separation of variables (SOV) for the exact solution of 1 + 1-dimensional quantum field theories by integrable lattice regularizations. Sklyanin's SOV is the natural quantum analogue of the classical method of separation of variables and it allows a more symmetric description of classical and quantum integrability w.r.t. traditional Bethe ansatz methods. Moreover, it has the advantage to be applicable to a more general class of models for which its implementation gives a characterization of the spectrum complete by construction. Our aim is to introduce a method in this framework which allows at once to derive the spectrum (eigenvalues and eigenvectors) and the dynamics (correlation functions) of IQFTs. This approach is presented for a paradigmatic example of relativistic IQFT, the sine-Gordon model.

We review our recent construction of operator-algebraic quantum field models with a weak localization property. Chiral components of two-dimensional conformal fields and certain endomorphisms of their observable algebras play a crucial role. In one case, this construction leads to a family of strictly local (Haag-Kastler) nets.

The following sections are included:

• CONTRIBUTED TALKS

• POSTERS

We summarize our recent results on the ground state energy of multi-polaron systems. In particular, we discuss stability and existence of the thermodynamic limit, and we discuss the absence of binding in the case of large Coulomb repulsion and the corresponding binding-unbinding transition. We also consider the Pekar-Tomasevich approximation to the ground state energy and we study radial symmetry of the ground state density.

This note describes recent results on the localization properties of Random Quantum Walks on the d–dimensional lattice in a regime analogous to the large disorder regime by means of the Fractional Moments Method adapted to the unitary framework.

We study Schrödinger operators with finitely supported potentials on the square lattice. We show that the potential is uniquely reconstructed from the S-matrix of all energies. We also derive new trace formulas, and estimate the discrete spectrum in terms of the potentials.

We review some results on the ionization conjecture, which says that a neutral atom can bind at most one or two extra electrons.

We consider the Landau Hamiltonian perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of V.

We give an account of a recent work on absence of positive eigenvalues for generalized 2-body hard-core Schrödinger operators [12]. We show absence of such eigenvalues under the condition of bounded strictly convex obstacles. A scheme for showing absence of positive eigenvalues for generalized N-body hard-core Schrödinger operators, N ≥ 2, is presented. This scheme involves high energy resolvent estimates, and for N = 2 it is implemented by a Mourre commutator type method. A particular example is the Helium atom with the assumption of infinite mass and finite extent nucleus.

The following sections are included:

• CONTRIBUTED TALKS

• POSTERS

Originating from the works of Bekenstein and Hawking on the entropy of black holes, area laws constitute a central tool for understanding entanglement and locality properties in quantum systems. Essentially, in a system that obeys an area law, the entanglement entropy of a bounded region scales like its boundary area, rather than its volume.

In 2007, in a seminal paper, Hastings proved that all 1D quantum spin systems with a constant spectral gap obey an area law in their ground state. The proof was based on the analytical tool of Lieb-Robinson velocity. A major open problem is whether an area law holds also in 2 or more dimensions.

In this talk I will present a line of research of the past couple of years culminating in an alternative, entirely combinatorial proof for the 1D area law. The proof uses the Chebyshev polynomial to describe the structure of entanglement in the ground state, yielding an exponentially better bound on the entanglement entropy compared to Hastings' bound. Just a slight improvement of our parameters would give a sub-volume law for the 2D case; the combinatorial approach raises hopes that such improvements might be doable.

Joint work with Alexei Kitaev, Caltech, Zeph Landau and Umesh Vazirani, both at University of California at Berkeley.

Note from Publisher: This article contains the abstract only.

For a random quantum state on obtained by partial tracing a random pure state on , we consider the question whether it is typically separable or typically entangled. We show the existence of a sharp threshold s₀ = s₀(d) of order roughly d³. More precisely, for any ε > 0 and for d large enough, such a random state is entangled with very large probability when s ≤ (1 − ε)s₀, and separable with very large probability when s ≥ (1 + ε)s₀.

Frustration-free (FF) spin chains have a property that their ground state minimizes all individual terms in the chain Hamiltonian. We ask how entangled can the ground state of a FF quantum spin-s chain with nearest-neighbor interactions be for small values of s. While FF spin-1/2 chains are known to have unentangled ground states, the case s = 1 remains less explored. We propose the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits some signatures of a critical behavior. The ground state can be viewed as the uniform superposition of balanced strings of left and right parentheses separated by empty spaces. Entanglement entropy of one half of the chain scales as ½ log (n) + O(1), where n is the number of spins. We prove that the energy gap above the ground state is polynomial in 1/n. The proof relies on a new result concerning statistics of Dyck paths which might be of independent interest.

Note from Publisher: This article contains the abstract only.

The one-body quantum marginal problem (1-QMP) asks which single-site reduced density matrices arise from globally pure states. We show that this set allows for an explicit inner approximation, which becomes increasingly accurate as the number of particles grows.

The 1-QMP had been a long-standing open problem in quantum many-body theory. Recently, Klyachko recognized the set in question as a moment polytope in the sense of symplectic geometry and gave a recipe for computing its facets. However, both the algorithm and the polytope itself seem to be highly non-trivial and only a few cases have been explored. We present first systematic results on the properties of the polytope for large quantum systems.

More precisely: the set of eigenvalues of the one-body reduced density matrices of globally pure states allow for a simple outer approximation. It is obtained by imposing only positivity and normalization constraints. We show that for an n-body system, there is an explicit inner approximation, whose distance to the outer one is bounded by (log n)/n. For reasons that will become clear (and are by no means connected to the host nation of the present conference), we refer to the inner polytope as a lego simplex. There is a sense in which the construction is tight, up to, possibly, the log-factor.

Perhaps surprisingly, the technical analysis rests on an analytic diagonalization of a certain random walk on the representations of the symmetric group.

Joint work with Matthias Christandl and Michael Walter, ETH Zürich.

Note from Publisher: This article contains the abstract only.

Many proposed quantum mechanical models of black holes include highly nonlocal interactions. The time required for thermalization to occur in such models should reflect the relaxation times associated with classical black holes in general relativity. Moreover, the time required for a particularly strong form of thermalization to occur, sometimes known as scrambling, determines the time scale on which black holes should start to release information. It has been conjectured that black holes scramble in a time logarithmic in their entropy, and that no system in nature can scramble faster. We address the conjecture from two directions. First, we exhibit examples of systems that do indeed scramble in logarithmic time: Brownian quantum circuits and the antiferromagnetic Ising model on a sparse random graph. Unfortunately, both fail to be truly ideal fast scramblers for reasons we discuss. Second, we use Lieb-Robinson techniques to prove a logarithmic lower bound on the scrambling time of systems with finite norm terms in their Hamiltonian. The bound holds in spite of any nonlocal structure in the Hamiltonian, which might permit every degree of freedom to interact directly with every other one.

Joint work with Nima Lashkari, McGill University, Douglas Stanford, Stanford University, Matthew Hastings, Duke University, and Tobias Osborne, Institut für Theoretische Physik, Hannover.

Note from Publisher: This article contains the abstract only.

We study the requirements for and implications of the existence of finite-dimensional approximations of quantum systems with an infinite number of degrees of freedom. We find a close relationship between these physical questions and the embedding problem of Alain Connes, appearing in the theory of von Neumann algebras. We are mainly interested in the bipartite scenario known from quantum information theory, where two independent parties act on a joint physical system, i.e. by performing measurements. This latter example leads to Tsirelson's problem, which asks whether the modeling of bipartite situations using the usual approach of tensor product of Hilbert spaces compared to the situation of only commuting observables leads to the same correlation tables. We model this situation using the theory of operator systems and explain the connection to Connes' embedding problem. We go on by employing the language of operator systems to elaborate on the requirements for the existence of finite-dimensional approximations for general quantum systems. We furthermore introduce the concept of ultraproducts of operator systems and use it to study the implications of two different kinds of finite-dimensional approximations.

Note from Publisher: This article contains the abstract only.

The following sections are included:

• CONTRIBUTED TALKS

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We set up a functional setting for mean-field electronic structure models of Hartree-Fock or Kohn-Sham types for disordered quantum systems. We then use these tools to study a specific mean-field model (reduced Hartree-Fock, rHF) for a disordered crystal where the nuclei are classical particles whose positions and charges are random. We prove the existence of a minimizer of the energy per unit volume and the uniqueness of the ground state density. For (short-range) Yukawa interactions, we prove in addition that the rHF ground state density matrix satisfies a self-consistent equation, and that our model is the thermodynamic limit of the supercell model.

In this talk I consider a two-dimensional lattice model for liquid crystals consisting of long rods interacting via purely hard core interactions, with two allowed orientations defined by the underlying lattice. I report a rigorous proof of the existence of a nematic phase: by this I mean that at intermediate densities the system exhibits orientational order, either horizontal or vertical, but no positional order. The proof is based on a two-scales cluster expansion: first the system is coarse-grained on a scale comparable with the rods' length; then the resulting effective theory is re-expressed as a contours' model, which can be treated by Pirogov-Sinai methods. The talk is based on joint work with Margherita Disertori.

The semiclassical Schrödinger equation, i.e. the Schrödinger equation coupled to a classical electromagnetic field, is a good model to describe many physical effects in quantum mechanics. In the manuscript, we will show, for the special case of a Bose-Einstein condensate, how one can derive the semiclassical Schrödinger equation from QED. Consider a condensate of N interacting bosons. It is well understood that, for certain scalings of the interaction, the bosons remain in a condensate as time evolves. Now, turn on the coupling to a radiation field, scaling the coupling constants such that the interaction with the radiation field is of order one. We will show that, for large N, the created photons are (in some sense) close to a coherent state and that the system is well described by the Hartree equation coupled to the Maxwell's equations.

We present a summary of our recent rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macro-scopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.

In joint work with Etienne Sandier, we studied the statistical mechanics of a classical two-dimensional Coulomb gas, particular cases of which also correspond to random matrix ensembles. We connect the problem to the “renormalized energy” W, a Coulombian interaction for an infinite set of points in the plane that we introduced in connection to the Ginzburg-Landau model, and whose minimum is expected to be achieved by the “Abrikosov” triangular lattice. Results include a next order asymptotic expansion of the partition function, and various characterizations of the behavior of the system at the microscopic scale. When the temperature tends to zero we show that the system tends to “crystallize” to a minimizer of W.

We prove analyticity of solutions in ℝⁿ, n ≥ 1, to certain nonlocal linear Schrödinger equations with analytic potentials.

We study the effects of random scatterers on the ground state of the one-dimensional Lieb-Liniger model of interacting bosons on the unit interval in the Gross-Pitaevskii regime. We prove that Bose Einstein condensation survives even a strong random potential with a high density of scatterers. The character of the wave function of the condensate, however, depends in an essential way on the interplay between randomness and the strength of the two-body interaction. For low density of scatterers or strong interactions the wave function extends over the whole interval. High density of scatterers and weak interaction, on the other hand, leads to localization of the wave function in a fragmented subset of the interval.

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Within the general setting of algebraic quantum field theory, a new approach to the analysis of the physical state space of a theory is presented; it covers theories with long range forces, such as quantum electrodynamics. Making use of the notion of charge class, which generalizes the concept of superselection sector, infrared problems are avoided. In fact, on this basis one can determine and classify in a systematic manner the proper charge content of a theory, the statistics of the corresponding states and their spectral properties. A key ingredient in this approach is the fact that in real experiments the arrow of time gives rise to a Lorentz invariant infrared cutoff of a purely geometric nature.

Two-dimensional toy models display, in a gentler setting, many salient aspects of Quantum Field Theory. Here I discuss a concrete two-dimensional case, the Thirring model, which illustrates several important concepts of this theory: the anomalous dimension of the fields; the exact solvability; the anomalies of the Ward-Takahashi identities. Besides, I give a glimpse of the decisive role that this model plays in the study of an apparently unrelated topic: correlation critical exponents of two dimensional lattice systems of Statistical Mechanics.

We review our recent successful attempt to construct the planar sector of a non-local scalar field model in four-dimensional Euclidean deformed space-time, which needs 4 (instead of 3) relevant/marginal operators in the defining Lagrangian. As we have shown earlier, this model is renormalizable up to all orders in perturbation theory. In addition a new fixed point appears, at which the beta function for the coupling constant vanishes. This way, we were able to tame the Landau ghost.

We next discuss Ward identities and Schwinger-Dyson equations and derive integral equations for the renormalized N-point functions. They are the starting point of an exact non-perturbative solution of the model.

Matrix models are a highly successful framework for the analytic study of random two-dimensional surfaces with applications to quantum gravity in two dimensions, string theory, conformal field theory, statistical physics in random geometry, etc. Their success relies crucially on the so called 1/N expansion introduced by 't Hooft.

In higher dimensions matrix models generalize to tensor models. In the absence of a viable 1/N expansion tensor models have for a long time been less successful in providing an analytically controlled theory of random higher dimensional topological spaces. This situation has drastically changed recently. Models for a generic complex tensor have been shown to admit a 1/N expansion dominated by graphs of spherical topology in arbitrary dimensions and to undergo a phase transition to a continuum theory.

Most quantum field theories (QFT's) have classical counterparts that are described by Lagrangians, and these are in fact often actually taken as the starting point for defining/constructing the theory. One possible viewpoint on the “quantization” of a classical field theory of an algebraic nature is to think of QFT as a deformation (in the sense of “deformation quantization”) of the Poisson-algebra describing the classical field theory. This viewpoint has been worked out, including the treatment of “renormalization”, in the context of perturbative quantum field theory by Fredenhagen et al., and also by the author in recent years. In this talk I want to indicate another approach to this problem which relates it to a prescription due to Fedosov, which was developed originially in the context of the deformation quantization of finite dimensional symplectic manifolds (and their associated Poisson algebras).

Note from Publisher: This article contains the abstract only.

We shall present here a series of recent articles [1], [2] dedicated to the definition of iterated integrals for stochastic processes with very low Hölder regularity index α. According to the general principles of Lyons' theory of rough paths, iterated integrals, a priori ill-defined, should either be seen as some limit of actual iterated integrals, or equivalently as a stack of data satisfying some Hölder regularity and algebraic axioms; such data define unambiguously a stochastic calculus and ‘rough path solutions’ to stochastic differential equations.

For many processes with α ≤ 1/4, in particular for Gaussian processes (the main example being fractional Brownian motion), a proper definition of iterated integrals was not available with standard tools of stochastic calculus. Through an approach (Fourier normal ordering) combining Hopf algebraic combinatorics, multi-scale expansions, Feynman diagram renormalization and finally constructive field theory, we gave a satisfactory answer to this problem. Ultimately the underlying structure is provided by the operator product expansions of ‘composite operators’ built out of the original process, and should be the key to the probabilistic study of solutions of stochastic differential equations driven by it.

Note from Publisher: This article contains the abstract and references only.

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I will describe the recent and profound advances in our understanding of quantum field theory and the connections between its analytic structure and the geometry of the positive part of Grassmannian manifolds. I will briefly review the recursive tools recently developed to understand the Feynman expansion more efficiently in terms of on-shell graphs, and describe how these tools extend to all-loop orders; in the case of planar, , the all-loop version of the BCFW recursion relations, expressed in terms of on-shell graphs becomes:

I will explain the deep connection between on-shell graphs, the positive Grassmannian, and combinatorics. A simple consequence of this connection will be a complete classification of on-shell functions and all their relations for planar as well as for pure Yang-Mills . Time permitting, the application of these techniques to the case of non-planar scattering amplitudes will be described.

Joint work with Nima Arkani-Hamed, Institute for Advanced Study, Freddy Cachazo, Perimeter Institute, and Jaroslav Trnka, Princeton University.

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This talk is concerned with the question, whether it is in principle possible to detect individual gravitons. The question is not whether quantum gravity is true, but whether quantum gravity is observable. I do not claim to have answered the question. I can prove that detectors with the LIGO design, detecting gravitational waves by measuring their effects on the distance between two mirrors, cannot detect single gravitons. To reduce quantum fluctuations in the measurement of distance, the mirrors must be heavy. To make the quantum noise small enough to observe the signal from a single graviton, the mirrors must be so heavy that they collapse together into a black hole. The laws of general relativity and quantum mechanics conspire to make the measurement impossible. I examine two other kinds of graviton detector that avoid this difficulty. The question whether any of them can detect single gravitons remains open.

The spectrum of super Yang-Mills theory can be studied using methods of integrability in the planar limit. We show that the exact spectrum is governed by a set of functional equations (Hirota equations) [1]

The set of functions T_{a, s} of the spectral parameter u belongs to an infinite lattice of a very particular shape called T-hook

The Hirota equations by itself describe a classical integrable system. This allows further simplification of the solution. We describe how the infinite set of functional equations (75) can be recast into a finite set of nonlinear integral equations [2] (FiNLIE) which can be solved numerically or analyzed analytically in various limits. This new FiNLIE is in the perfect agreement with the previously obtained numerical results [2] based on the Thermodynamic Bethe Ansatz (TBA) approach.

The presented solution of the spectral problem passes various very nontrivial tests. It agrees with extremely involved perturbative calculations in the gauge theory (up to five loops) as well as with the predictions of the string theory for the strong coupling limit (up two two loops).

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By using quantum Teichmüller theory, a new type of three-dimensional TQFT has been constructed with the following distinguishing features: it uses the combinatorial framework of shaped triangulations; it takes its values in the space of tempered distributions; the fundamental building block of the theory is given by Faddeev's quantum dilogarithm. The semi-classical behavior and the geometrical ingredients suggest that the constructed TQFT is related to exact solution of quantum Chern–Simons theory with gauge group SL(2, ℂ). We also remark that quantum Teichmüller theory itself admits an additional real parameter which preserves unitarity but affects the projective factor in the corresponding mapping class group representation.

I will review the recent work on the equivariant partition functions of quiver gauge theories in four dimensions and their relation with Seiberg-Witten theories and algebraic integrable systems.

Joint work with Nikita Nekrasov, Simons Center for Geometry and Physics.

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I will survey the formalism and main results of loop quantum gravity [1], [2] from a mathematical perspective. Then I take a closer look at the way black hole horizons are treated in the theory, by coupling a Chern-Simons theory on the horizon to the bulk degrees of freedom [3]. I will present some recent results on a new way to solve the self-duality equation involved directly in the quantum theory [4].

Note from Publisher: This article contains the abstract and references only.

Recently there was a substantial progress in understanding of supersymmetric theories (in particular, their BPS spectrum) in space-times of different dimensions due to the exact computation of superconformal indices and partition functions using localization method. Here we discuss a connection of 4d superconformal indices and 3d partition functions using a particular example of supersymmetric theories with matter in antisymmetric representation.

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