on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

crossed products of the cantor set by free minimal actions of ℤd let d be a positive integer let x be the cantor set and let d act freely and minimally on x we prove that the crossed product c d x has stable rank one real rank zero and cancellation of projections and that the order on k0 c d x is determined by traces we obtain the same conclusion for the c algebras of various kinds of aperiodic tilings

sobolev inequalities the poisson semigroup and analysis on the sphere sn abstract hypercontractive estimates are obtained for the poisson semigroup on lp sn such estimates are determined by a sharp logarithmic sobolev inequality that relates the entropy of a function on sn to its smoothness

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

critical metrics for the determinant of the laplacian in odd dimensions let m be a closed compact n dimensional manifold with n odd we calculate the first and second variations of the zeta regularized determinants det a and det l as the metric on m varies where a denotes the laplacian on functions and l denotes the conformal laplacian we see that the behavior of these functionals depends on the dimension indeed every critical metric for 1 n 1 2det a or 1 n 1 21 det l has finite index consequently there are no local maxima if n 4rm 1 and no local minima if n 4m 3 we show that the standard 3 sphere is a local maximum for det a while the standard 4m 3 sphere with m 1 2 is a saddle point by contrast for all odd n the standard n sphere is a local extremal for det l an important tool in our work is the canonical trace on odd class operators in odd dimensions this trace is related to the determinant by the formula det q tr log q and we prove some basic results on how to calculate this trace

an uncertainty principle for fermions with generalized kinetic energy we derive semiclassical upper bounds for the number of bound states and the sum of negative eigenvalues of the one particle hamiltoniansh f i v x acting onl2 n these bounds are then used to derive a lower bound on the kinetic energy sum limits j 1 n left langle psi f i nabla j psi right rangle for ann fermion wavefunction we discuss two examples in more detail f p p andf p p2 m2 1 2 m both in three dimensions

every classifiable simple c algebra has a cartan subalgebra we construct cartan subalgebras in all classifiable stably finite c algebras together with known constructions of cartan subalgebras in all uct kirchberg algebras this shows that every classifiable simple c algebra has a cartan subalgebra

extremals of functionals with competing symmetries abstract we present a new method of producing optimizing sequences for highly symmetric functionals the sequences have good convergence properties built in we apply the method in different settings to give elementary proofs of some classical inequalities such as the hardy littlewood sobolev and the logarithmic sobolev inequality in their sharp form

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

cartan subalgebras in c algebras existence and uniqueness we initiate the study of cartan subalgebras in c algebras with a particular focus on existence and uniqueness questions for homogeneous c algebras these questions can be analyzed systematically using the theory of fiber bundles for group c algebras while we are able to find cartan subalgebras in c algebras of many connected lie groups there are classes of discrete groups for instance non abelian free groups whose reduced group c algebras do not have any cartan subalgebras moreover we show that uniqueness of cartan subalgebras usually fails for classifiable c algebras however distinguished cartan subalgebras exist in some cases for instance in nuclear uniform roe algebras

intrinsic ultracontractivity and conditional gauge for symmetric stable processes abstract it is shown in this paper that the conditional gauge theorem holds for symmetric stable processes on bounded c 1 1 domains in r n where 0 n 2 two of the major tools used to prove this conditional gauge theorem are logarithmic sobolev inequality and intrinsic ultracontractivity

extensional channel flow revisited a dynamical systems perspective extensional self similar flows in a channel are explored numerically for arbitrary stretching shrinking rates of the confining parallel walls the present analysis embraces time integrations and continuations of steady and periodic solutions unfolded in the parameter space previous studies focused on the analysis of branches of steady solutions for particular stretching shrinking rates although recent studies focused also on the dynamical aspects of the problems we have adopted a dynamical systems perspective analysing the instabilities and bifurcations the base state undergoes when increasing the reynolds number it has been found that the base state becomes unstable for small reynolds numbers and a transitional region including complex dynamics takes place at intermediate reynolds numbers depending on the wall acceleration values the base flow instabilities are constitutive parts of different codimension two bifurcations that control the dynamics in parameter space for large reynolds numbers the restriction to self similarity results in simple flows with no realistic behaviour but the flows obtained in the transition region can be a valuable tool for the understanding of the dynamics of realistic navier stokes solutions

a generalized cuntz krieger uniqueness theorem for higher rank graphs we present a uniqueness theorem for k graph c algebras that requires neither an aperiodicity nor a gauge invariance assumption specifically we prove that for the injectivity of a representation of a k graph c algebra it is sufficient that the representation be injective on a distinguished abelian c subalgebra a crucial part of the proof is the application of an abstract uniqueness theorem which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient c algebra each of which is the unique extension of a state on the distinguished abelian c subalgebra

constructing minimal homeomorphisms on point like spaces and a dynamical presentation of the jiang su algebra the principal aim of this paper is to give a dynamical presentation of the jiang su algebra originally constructed as an inductive limit of prime dimension drop algebras the jiang su algebra has gone from being a poorly understood oddity to having a prominent positive role in george elliott s classification programme for separable nuclear c algebras here we exhibit an etale equivalence relation whose groupoid c algebra is isomorphic to the jiang su algebra the main ingredient is the construction of minimal homeomorphisms on infinite compact metric spaces each having the same cohomology as a point this construction is also of interest in dynamical systems any self map of an infinite compact space with the same cohomology as a point has lefschetz number one thus if such a space were also to satisfy some regularity hypothesis which our examples do not then the lefschetz hopf theorem would imply that it does not admit a minimal homeomorphism

cartan subalgebras for non principal twisted groupoid c algebras abstract renault proved in 2008 22 theorem 5 2 that if g is a topologically principal groupoid then c 0 g 0 is a cartan subalgebra in c r g for any twist over g however there are many groupoids which are not topologically principal yet their twisted c algebras admit cartan subalgebras this paper gives a dynamical description of a class of such cartan subalgebras by identifying conditions on a 2 cocycle c on g and a subgroupoid s g under which c r s c is cartan in c r g c when g is a discrete group we also describe the weyl groupoid and twist associated to these cartan pairs under mild additional hypotheses

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part