On Almost Randomizing Channels with a Short Kraus Decomposition

For large d, we study quantum channels on C ^d obtained by selecting randomly N independent Kraus operators according to a probability measure μ on the unitary group . When μ is the Haar measure, we show that for , such a channel is ε-randomizing with high probability, which means that it maps every state within distance ε/d (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discretization argument. Moreover, for general μ, we obtain an ε-randomizing channel provided . For d = 2 ^k (k qubits), this includes Kraus operators obtained by tensoring k random Pauli matrices. This leads to more efficient constructions of almost randomizing channels. The proof uses recent results on empirical processes in Banach spaces.     

Quantum Conjugacy Classes of Simple Matrix Groups

Let G be a simple complex classical group and its Lie algebra. Let be the Drinfeld-Jimbo quantization of the universal enveloping algebra . We construct an explicit -equivariant quantization of conjugacy classes of G with Levi subgroups as the stabilizers. Dedicated to the memory of Joseph Donin This research is partially supported by the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center “Group Theoretic Methods in the study of Algebraic Varieties” of the Israel Science foundation, by the EPSRC grant C511166, and by the RFBR grant no. 06-01-00451.     

Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation

We consider here the 1 D semilinear wave equation with a power nonlinearity and with no restriction on initial data. We first prove a Liouville Theorem for that equation. Then, we consider a blow-up solution, its blow-up curve and the set of non-characteristic points. We show that I ₀ is open and that T(x) is C ¹ on I ₀. All these results fundamentally use our previous result in [19] showing the convergence in selfsimilar variables for . This work was supported by a grant from the french Agence Nationale de la Recherche, project ONDENONLIN, reference ANR-06-BLAN-0185.     

A Class of Integrable Flows on the Space of Symmetric Matrices

For a given skew symmetric real n × n matrix N, the bracket [X, Y] _N = XNY − YNX defines a Lie algebra structure on the space Sym(n, N) of symmetric n × n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry, integrability, and linearizability of the Hamiltonian system , or equivalently in Lax form, the equation on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure of Sym(n, N). This is established by finding another compatible Poisson structure. If N is invertible, several remarkable identifications can be implemented. First, (Sym(n, N), [·, ·]) is Lie algebra isomorphic with the symplectic Lie algebra associated to the symplectic form on given by N ⁻¹. In this case, the system is the reduction of the geodesic flow of the left invariant Frobenius metric on the underlying symplectic group Sp(n, N ⁻¹). Second, the trace of the product of matrices defines a non-invariant non-degenerate inner product on Sym(n, N) which identifies it with its dual. Therefore Sym(n, N) carries a natural Lie-Poisson structure as well as a compatible “frozen bracket” structure. The Poisson diffeomorphism from Sym(n, N) to maps our system to a Mischenko-Fomenko system, thereby providing another proof of its integrability if N is invertible with distinct eigenvalues. Third, there is a second ad-invariant inner product on Sym(n, N); using it to identify Sym(n, N) with itself and composing it with the dual of the Lie algebra isomorphism with , our system becomes a Mischenko- Fomenko system directly on Sym(n, N). If N is invertible and has distinct eigenvalues, it is shown that this geodesic flow on Sym(n, N) is linearized on the Prym subvariety of the Jacobian of the spectral curve associated to a Lax pair formulation with parameter of the system. If, on the other hand, N has nullity one and distinct eigenvalues, in spite of the fact that the system is completely integrable, it is shown that the flow does not linearize on the Jacobian of the spectral curve. Research partially supported by NSF grants CMS-0408542 and DMS-0604307. Research partially supported by the Swiss SCOPES grant IB7320-110721/1, 2005-2008, and MEdC Contract 2-CEx 06-11-22/25.07.2006. Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932. Research partially supported by the Swiss NSF and the Swiss SCOPES grant IB7320-110721/1.     

The <Emphasis Type="Italic">N</Emphasis> = 1 Triplet Vertex Operator Superalgebras

We introduce a newfamily of C ₂-cofinite N = 1 vertex operator superalgebras , m ≥ 1, which are natural super analogs of the triplet vertex algebra family , p ≥ 2, important in logarithmic conformal field theory. We classify irreducible -modules and discuss logarithmic modules. We also compute bosonic and fermionic formulas of irreducible characters. Finally, we contemplate possible connections between the category of -modules and the category of modules for the quantum group , , by focusing primarily on properties of characters and the Zhu’s algebra . This paper is a continuation of our paper Adv. Math. 217, no.6, 2664–2699 (2008). The second author was partially supported by NSF grant DMS-0802962.     

Blowup Solutions of Some Nonlinear Elliptic Equations Involving Exponential Nonlinearities

In conformal geometry and several fields of physics, the blowup analysis of the equation Δu + V(x)e ^u = 0 in has led to interesting results. In 1999 Li [11] gave a uniform asymptotic estimate of a sequence of blowup solutions near an isolated blowup point. In this paper we improve Li’s result to the sharp form by the moving sphere method.Lei Zhang is partially supported by grant NSF-DMS-0600275     

Decay of Weak Solutions to the 2D Dissipative Quasi-Geostrophic Equation

We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data θ₀ is in L ² only, we prove that the L ² norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For θ₀ in L ^p ∩ L ², with 1 ≤ p < 2, we are able to obtain a uniform decay rate in L ². We also prove that when the norm of θ₀ is small enough, the L ^q norms, for , have uniform decay rates. This result allows us to prove decay for the L ^q norms, for , when θ₀ is in . The second author was partially supported by NSF grant DMS-0600692.     

Local Critical Perturbations of Unimodal Maps

We introduce a new complete metric in the space of unimodal C ²-maps of the interval, with two maps close if they are close in the C ²-metric and differ only on a small interval containing their critical points. We identify all structurally stable maps in the sense of this metric. They are maps for which either (1) the trajectory of the critical point is attracted to a topologically attracting (at least from one side) periodic orbit, but never falls into this orbit, or (2) the critical point is mapped by some iterate to the interior of an interval consisting entirely of periodic points of the same (minimal) period. We verify the generalized Fatou conjecture for and show that structurally stable maps form an open dense subset of . Partially supported by NSF grant DMS 0456748. Partially supported by NSF grant DMS 0456526.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Restriction Bounds for Eigenfunctions Along Curves in the Quantum Completely Integrable Case

We show that for a quantum completely integrable system in two dimensions, the L ²-normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds of the form for generic curves γ on the surface. We also prove that the maximal restriction bounds of Burq-Gerard-Tzvetkov [BGT] are generically attained for certain exceptional subsequences of eigenfunctions. The author was supported by a William Dawson Fellowship and NSERC Grant OGP0170280.     

An Infinite Genus Mapping Class Group and Stable Cohomology

We exhibit a finitely generated group whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of into the restricted symplectic group of the real Hilbert space generated by the homology classes of non-separating circles on , which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in is the pull-back of the Pressley-Segal class on the restricted linear group via the inclusion . L. F. was partially supported by the ANR Repsurf:ANR-06-BLAN-0311.     

Optimal Concentration for SU(1, 1) Coherent State Transforms and An Analogue of the Lieb-Wehrl Conjecture for SU(1, 1)

We derive a lower bound for the Wehrl entropy in the setting of SU(1, 1). For asymptotically high values of the quantum number k, this bound coincides with the analogue of the Lieb-Wehrl conjecture for SU(1, 1) coherent states. The bound on the entropy is proved via a sharp norm bound. The norm bound is deduced by using an interesting identity for Fisher information of SU(1, 1) coherent state transforms on the hyperbolic plane and a new family of sharp Sobolev inequalities on . To prove the sharpness of our Sobolev inequality, we need to first prove a uniqueness theorem for solutions of a semi-linear Poisson equation (which is actually the Euler-Lagrange equation for the variational problem associated with our sharp Sobolev inequality) on . Uniqueness theorems proved for similar semi-linear equations in the past do not apply here and the new features of our proof are of independent interest, as are some of the consequences we derive from the new family of Sobolev inequalities. Work partially supported by U.S. National Science Foundation grant DMS 06-00037.     

Maximum Solutions of Normalized Ricci Flow on 4-Manifolds

We consider the maximum solution g(t), t ∈ [0,  + ∞), to the normalized Ricci flow. Among other things, we prove that, if (M, ω) is a smooth compact symplectic 4-manifold such that and let g(t), t ∈ [0, ∞), be a solution to (1.3) on M whose Ricci curvature satisfies that |Ric(g(t))| ≤ 3 and additionally χ(M) = 3τ (M) > 0, then there exists an , and a sequence of points {x _j,k ∈ M}, j = 1, . . . , m, satisfying that, by passing to a subsequence,

Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations

We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations if the vorticity decays sufficiently fast near infinity in . By a similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in . This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions. The work was supported partially by the KOSEF Grant no. R01-2005-000-10077-0, and KRF Grant (MOEHRD, Basic Research Promotion Fund).     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Isolation Phenomenon for Complete Surfaces Arising from Yang–Mills Theory

Let M be a complete surface with parallel mean curvature in a complete simply connected space form F ^2+p (c) of constant curvature c. Denote by H and S the mean curvature and the squared length of the second fundamental form of M respectively. Motivated by L ²-isolation phenomenon in Yang–Mills theory, we prove that if , where c + H ² > 0, D(H,c) is an explicit positive constant depending on H and c, then , i.e., M is a totally umbilical sphere . Research supported by the Chinese NSF, Grant No. 10231010; Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China.     

On the Rate of Explosion for Infinite Energy Solutions of the Spatially Homogeneous Boltzmann Equation

Let μ ₀ be a probability measure on ℝ³ representing an initial velocity distribution for the spatially homogeneous Boltzmann equation for pseudo Maxwellian molecules. As long as the initial energy is finite, the solution μ _t will tend to a Maxwellian limit. We show here that if , then instead, all of the mass “explodes to infinity” at a rate governed by the tail behavior of μ ₀. Specifically, for L0, define

A Criterion for Uniqueness of Lagrangian Trajectories for Weak Solutions of the 3D Navier-Stokes Equations

Foias, Guillopé, & Temam showed in 1985 that for a given weak solution of the three-dimensional Navier-Stokes equations on a domain Ω, one can define a ‘trajectory mapping’ that gives a consistent choice of trajectory through each initial condition , and that respects the volume-preserving property one would expect for smooth flows. The uniqueness of this mapping is guaranteed by the theory of renormalised solutions of non-smooth ODEs due to DiPerna & Lions. However, this is a distinct question from the uniqueness of individual particle trajectories. We show here that if one assumes a little more regularity for u than is known to be the case, namely that , then the particle trajectories are unique and C ¹ in time for almost every choice of initial condition in Ω. This degree of regularity is more than can currently be guaranteed for weak solutions () but significantly less than that known to ensure that u is regular ( . We rely heavily on partial regularity results due to Caffarelli, Kohn, & Nirenberg and Ladyzhenskaya & Seregin.     

Double Standard Maps

We investigate the family of double standard maps of the circle onto itself, given by (mod 1), where the parameters a,b are real and 0 ≤ b ≤ 1. Similarly to the well known family of (Arnold) standard maps of the circle, (mod 1), any such map has at most one attracting periodic orbit and the set of parameters (a,b) for which such orbit exists is divided into tongues. However, unlike the classical Arnold tongues that begin at the level b = 0, for double standard maps the tongues begin at higher levels, depending on the tongue. Moreover, the order of the tongues is different. For the standard maps it is governed by the continued fraction expansions of rational numbers; for the double standard maps it is governed by their binary expansions. We investigate closer two families of tongues with different behavior. The first author was partially supported by NSF grant DMS 0456526. The second author was supported by FCT Grant SFRH/BD/18631/2004. CMUP is supported by FCT through POCTI and POSI of Quadro Comunitário de apoio III (2000-2006) with FEDER and national funding.     

Regularity Criteria for the Dissipative Quasi-Geostrophic Equations in Hölder Spaces

We study regularity criteria for weak solutions of the dissipative quasi-geostrophic equation (with dissipation (−Δ) ^γ/2, 0 < γ ≤ 1). We show in this paper that if , or with is a weak solution of the 2D quasi-geostrophic equation, then θ is a classical solution in . This result improves our previous result in [18]. Partially supported by a start-up funding from the Division of Applied Mathematics of Brown University and NSF grant number DMS 0800129. Partially supported by a start-up funding from the College of Natural Sciences of the University of Texas at Austin, NSF grant number DMS 0758247 and an Alfred P. Sloan Research Fellowship.     

Modular Conjugation and the Implementation of Supersymmetry

Any -graded C ^*-dynamical system with a self-adjoint graded-Kubo-Martin-Schwinger (KMS) functional on it can be represented (canonically) as a -graded algebra of bounded operators on a -graded Hilbert space, so that the grading of the latter is compatible with the functional. The modular conjugation operator plays a crucial role in this reconstruction. The results are generalized to the case of an unbounded graded-KMS functional having as dense domain the union of a net of C ^*-subalgebras. It is shown that the modulus of such an unbounded graded-KMS functional is KMS.      

Quantum Groups and Double Quiver Algebras

For a finite dimensional semisimple Lie algebra and a root q of unity in a field k, we associate to these data a double quiver . It is shown that a restricted version of the quantized enveloping algebras is a quotient of the double quiver algebra .*The author is partially supported by the National Science Foundation of China (Grant. 10271014) and Natural Science Foundation of Beijing City (grant. 1042001)