variations on the boman covering lemma abstract we explore some variants of the boman covering lemma and their relationship to the boundedness properties of the maximal operator let 1 p and let q be its conjugate exponent we prove that the strong type q q of the uncentered maximal operator by itself implies certain generalizations of the boman covering lemma for the exponent p and in turn these generalizations entail the weak type q q of the centered maximal operator we show by example that it is possible for the uncentered maximal operator to be unbounded for all 1 s while the conclusion of the lemma holds for every 1 p thus the latter condition is much weaker also the boundedness of the centered maximal operator entails weak versions of the lemma

on the variety of projectors in this paper we are going to give an answer to the following two questions let k be a field and a4 the space of n x n square matrices with coefficients in k the first question we want to investigate is what are the equations of the subvariety a of ikf consisting of rank h matrices a such that a2 a by equations we mean of course a set of explicit generators of the prime ideal of x the second question is what are the equations of the varieties yh h n 2 which are the closures of the nilpotent conjugacy classes in h4 e consisting of matrices of rank h whose square is the matrix 0 in this case we shall also prove that these varieties y are normal and cohen macaulay in arbitrary characteristic thus generalizing a result of hesselink 3 to arbitrary characteristic note that in characteristic 0 all closures of nilpotent conjugacy classes have been proved to be normal and cohen macaulay by kraft and procesi 4 but our method is not suitable for generalizing their result the method used in this paper is the one of hodge algebras 11 namely we shall exhibit an explicit linear basis for the coordinate rings of the varieties x and y

conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

the structure of time optimal trajectories for single input systems in the plane the c nonsingular case for single input c infty systems in the plane in which the control enters linearly we prove if the system is suitably nondegenerate that the time optimal trajectories are finite concatenations of bang bang and singular arcs with local bounds on the number of switchings

optimal control in laser induced population transfer for two and three level quantum systems we apply the techniques of control theory and of sub riemannian geometry to laser induced population transfer in two and three level quantum systems the aim is to induce complete population transfer by one or two laser pulses minimizing the pulse fluences sub riemannian geometry and singular riemannian geometry provide a natural framework for this minimization where the optimal control is expressed in terms of geodesics we first show that in two level systems the well known technique of pulse transfer in the rotating wave approximation emerges naturally from this minimization in three level systems driven by two resonant fields we also find the counterpart of the pulse transfer this geometrical picture also allows one to analyze the population transfer by adiabatic passage

ensemble control of bloch equations in this article we study a class of control problems which involves controlling a continuum of dynamical systems with different values of parameters characterizing the system dynamics by using the same control signal we call such problems control of ensembles the motivation for looking into these problems comes from the manipulation of an ensemble of nuclear spins in nuclear magnetic resonance nmr spectroscopy and imaging mri with dispersions in natural frequencies and the strengths of the applied radio frequency rf field from the standpoint of mathematical control theory the challenge is to simultaneously steer a continuum of systems between points of interest with the same control input this raises some new and unexplored questions about controllability of such systems we show that controllability of an ensemble can be understood by the study of the algebra of polynomials defined by the noncommuting vector fields that govern the system dynamics a systematic study of these systems has immediate applications to broad areas of control of ensembles of quantum systems as arising in coherent spectroscopy and quantum information processing the new mathematical structures appearing in such problems are excellent motivation for new developments in control theory

minimum time frictionless atom cooling in harmonic traps frictionless atom cooling in harmonic traps is formulated as a time optimal control problem and a synthesis of optimal controlled trajectories is obtained

galois representations and torsion in the coherent cohomology of hilbert modular varieties let f be a totally real number eld and let p be a prime unramied in f we prove the existence of galois pseudo representations attached to modp m hecke eigenclasses of paritious weight occurring in the coherent cohomology of hilbert modular varieties forf of level prime to p

twisted loop groups and their affine flag varieties we develop a theory of affine flag varieties and of their schubert varieties for reductive groups over a laurent power series local field k t with k a perfect field this can be viewed as a generalization of the theory of affine flag varieties for loop groups to a twisted case a consequence of our results is that our construction also includes the flag varieties for kac moody lie algebras of affine type we also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k t and some applications to local models of shimura varieties

construction of central elements in the affine hecke algebra via nearby cycles 0 1 overview let g be a connected reductive group over a finite field fq and let g k be the corresponding group over the local field k fq t let g o g k be a maximal compact subgroup of g k here o fq t and let hsph denote the hecke algebra of g k with respect to g o in other words hsph as a vector space consists of compactly supported bi g o invariant functions g k q and the product is defined by

finite controllability of infinite dimensional quantum systems quantum phenomena of interest in connection with applications to computation and communication often involve generating specific transfers between eigenstates and their linear superpositions for some quantum systems such as spin systems the quantum evolution equation the schrodinger equation is finite dimensional and old results on controllability of systems defined on on lie groups and quotient spaces provide most of what is needed insofar as controllability of non dissipative systems is concerned however in an infinite dimensional setting controlling the evolution of quantum systems often presents difficulties both conceptual and technical in this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues in particular we analyze controllability for infinite dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre defined subspaces this result which we call the finite controllability theorem provides a set of sufficient conditions for controllability in an infinite dimensional setting we consider specific physical systems that are of interest for quantum computing and provide insights into the types of quantum operations gates that may be developed

on the flatness of local models for the symplectic group abstract we investigate the bad reduction of certain shimura varieties associated to the symplectic group more precisely we look at a model of the shimura variety at a prime p with parahoric level structure at p we show that this model is flat as conjectured by rapoport and zink ann of math stud 141 1996 and that its special fibre is reduced a crucial ingredient is faltings theorem on the normality of schubert varieties in the affine flag variety

control and synchronization of neuron ensembles synchronization of oscillations is a phenomenon prevalent in natural social and engineering systems controlling synchronization of oscillating systems is motivated by a wide range of applications from surgical treatment of neurological diseases to the design of neurocomputers in this paper we study the control of an ensemble of uncoupled neuron oscillators described by phase models we examine controllability of such a neuron ensemble for various phase models and furthermore study the related optimal control problems in particular by employing pontryagin s maximum principle we analytically derive optimal controls for spiking single and two neuron systems and analyze the applicability of the latter to an ensemble system finally we present a robust computational method for optimal control of spiking neurons based on pseudospectral approximations the methodology developed here is universal to the control of general nonlinear phase oscillators

on the coherence conjecture of pappas and rapoport we prove the generalized coherence conjecture proposed by pappas and rapoport as a corollary one of their theorems which describes the geometry of the special bers of the local models for ramied unitary groups holds unconditionally our proof is based on the study of the geometry in particular certain line bundles and adic sheaves of the global schubert varieties which are the equal characteristic counterparts of the local models

quantum pontryagin principle under continuous measurements in this paper we develop the theory of the quantum pontryagin principle for continuous measurements and feedback the analysis is carried out under the assumption of compatible events in the output channel the plant is a quantum system which generally is in a mixed state coupled to a continuous measurement channel the pontryagin maximum principle is derived in both the schrodinger picture and heisenberg picture in particular in statistical moment coordinates to avoid solving stochastic equations we derive a linear quadratic gaussian scheme which is more suitable for control purposes finally we use the quantum harmonic oscillator as a concrete example to illustrate the performance of the controller

special cycles on unitary shimura varieties i unramified local theory the supersingular locus in the fiber at p of a shimura variety attached to a unitary similitude group gu 1 n 1 over is uniformized by a formal scheme mathcal n in the case when p is an inert prime we define special cycles mathcal z bold x in mathcal n associated to collections bold x of m special homomorphisms with fundamental matrix t hermm ok when m n and t is nonsingular we show that the cycle mathcal z bold x is either empty or is a union of components of the ekedahl oort stratification and we give a necessary and sufficient condition in terms of t for mathcal z bold x to be irreducible when mathcal z bold x is zero dimensional in which case it reduces to a single point we determine the length of the corresponding local ring by using a variant of the theory of quasi canonical liftings we show that this length coincides with the derivative of a representation density for hermitian forms

on the differentiability of lipschitz besov functions lr and ordinary differentiability is proved for functions in the lipschitz besov spaces b q 1 0 using certain maximal operators measuring smoothness these techniques allow also the study of lacunary directional differentiability and of tangential convergence of poisson integrals

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

on certain maximal functions and approach regions if f e l ir and u x y is the poisson integral of j a classical theorem of fatou 2 asserts that u has nontangential limits almost everywhere on r littlewood 5 and later zygmund 9 showed that this conclusion is false if nontangential approach is replaced by approach along translates of a curve which approaches the boundary tangentially in this paper we find a necessary and sufficient condition on an approach region for the associated maximal function to be appropriately bounded it then turns out that there are many approach regions which are not contained in any nontangential region but for which the conclusions of fatou s theorem remain true suppose q c ir has the property that whenever xi u e b and whenever x x y y it follows that x v e q then the maximal function associated to such a set is bounded if and only if the cross sectional measure at height y is bounded by a constant times y we also study regions where the cross sectional area is larger than y we introduce a modified maximal function which is again bounded this result leads to a simple proof of the boundedness of certain tangential maximal functions of poisson integrals of potentials studied in 161 the plan of this paper is as follows in section 1 we motivate our main ideas by discussing two particular examples the method of proof here is somewhat different from the proofs of the more general results in section 2 we obtain the necessary and sufficient conditions for fatou s theorem in section 3 we study more general approach regions and the appropriately modified maximal functions in section 4 we show how this maximal function controls the tangential boundary behavior of poisson integrals of potentials

topological flatness of local models in the ramified case local models are schemes defined in terms of linear algebra which can be used to study the local structure of integral models of certain shimura varieties with parahoric level structure we investigate the local models for groups of the form open image in new window is a totally ramified extension as defined by pappas and rapoport and show that they are topologically flat in the linear case flatness can be deduced from this

uber die lokale zetafunktion von shimuravarietaten monodromiefiltration und verschwindende zyklen in ungleicher charakteristik

cells in affine weyl groups and tensor categories abstract in this paper we construct a tensor or monoidal category for any two sided cell in a finite or affine weyl group the isomorphism classes of simple objects of this category correspond to the elements of the two sided cell the tensor product arises by a certain truncation procedure from the standard convolution of perverse sheaves on a flag manifold which imitates geometrically the definition of the ring j given in g lusztig j algebra 109 1987 536 548 similarly we construct for any left cell a tensor category in which the isomorphism classes of simple objects correspond to the elements in the left cell intersected with its inverse a right cell this contains as a special case the category of equivariant perverse sheaves on an affine grassmannian which as a consequence of g lusztig asterisque 101 102 1983 208 209 is a tensor category for convolution without truncation closely related to the tensor category of representations of the group dual in the sense of langlands to the group which gives rise to the affine grassmannian but our construction can be specialized in other ways and it gives a conjectural realization of the tensor category of representations of the reductive quotient of the centralizer of any unipotent element of the langlands dual group

topological flatness of orthogonal local models in the split even case i local models are schemes defined in terms of linear algebra that were introduced by rapoport and zink to study the etale local structure of integral models of certain pel shimura varieties over p adic fields a basic requirement for the integral models or equivalently for the local models is that they be flat in the case of local models for even orthogonal groups genestier observed that the original definition of the local model does not yield a flat scheme in a recent article pappas and rapoport introduced a new condition to the moduli problem defining the local model the so called spin condition and conjectured that the resulting spin local model is flat we prove a preliminary form of their conjecture in the split iwahori case namely that the spin local model is topologically flat an essential combinatorial ingredient is the equivalence of admissibility and permissibility for two minuscule cocharacters in root systems of type d

schubert varieties in twisted affine flag varieties and local models abstract our concern in this paper is the dimension and inclusion relations of schubert varieties in twisted partial affine flag varieties in the end we apply our results to some local models of certain shimura varieties

controllability issues for continuous spectrum systems and ensemble controllability of bloch equations we study the controllability of the bloch equation for an ensemble of non interactinghalf spins inastaticmagneticfield withdispersioninthelarmorfrequency this system may be seen as a prototype for infinite dimensional bilinear systems with continuous spectrum whose controllability is not well understood we provide several mathematical answers with discrimination between approximate and exact controlla bility and between finite time or infinite time controllability this system is not exactly controllable in finite time t with bounded controls in l 2 0 t but it is approximately controllable in l in finite time with unbounded controls in l 0 moreover we propose explicit controls realizing the asymptotic exact controllability to a uniform state of spin 1 2o r 1 2

optimal control of two level quantum systems we study the manipulation of two level quantum systems this research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two level quantum system a quantum bit we consider the problem of driving the evolution operator to a desired state while minimizing an energy type cost mathematically this problem translates into an optimal control problem for systems varying on the lie group of special unitary matrices of dimension two with cost that is quadratic in the control we develop a comprehensive theory of optimal control for two level quantum systems this includes in particular a classification of normal and abnormal extremals and a proof of regularity of the optimal control functions the impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

the energy minimization problem for two level dissipative quantum systems in this article we study the energy minimization problem of dissipative two level quantum systems whose dynamics is governed by the kossakowski lindblad equations in the first part we classify the extremal curve solutions of the pontryagin maximum principle the optimality properties are analyzed using the concept of conjugate points and the hamilton jacobi bellman equation this analysis completed by numerical simulations based on adapted algorithms allows a computation of the optimal control law whose robustness with respect to the initial conditions and dissipative parameters is also detailed in the final section an application in nuclear magnetic resonance is presented

alcoves associated to special fibers of local models the special fiber of the local model of a pel shimura variety with iwahori type level structure admits a cellular decomposition the set of strata is in a natural way a finite subset of the affine weyl group determined by the shimura data in this paper we generalize and give a new proof of a theorem of kottwitz and rapoport concerning the description of this set for the case of linear and symplectic groups for higher rank groups not of type an we produce counterexamples showing the situation is not as nice in general we also recover a theorem of deodhar characterizing the bruhat order on sn the local geometry can be approached using the rapoport zink local model m j 13 this protective op scheme is a local model for the singularities in the special fiber of sh g x9 k 9 but it is defined in terms of the pair g p using linear algebra and is somewhat easier to deal with than the shimura variety itself in this paper we address certain combinatorial questions which arise in the study of the special fiber m f of m to fix ideas consider the case where g is split it turns out that m f can be regarded as a finite dimensional union of iwahori orbits in the affine flag variety for g fp t see 5 and 8 it follows that m f has a stratification indexed by a finite set perm p of the extended affine weyl group w g for g equivalently perm p can be regarded as a finite set of alcoves in the affine coxeter complex determined by a splitting for g let wo denote the finite weyl group of g for each translation in the wo orbit

topological flatness of local models for ramified unitary groups i the odd dimensional case abstract local models are certain schemes defined in terms of linear algebraic moduli problems which give etale local neighborhoods of integral models of certain p adic pel shimura varieties defined by rapoport and zink when the group defining the shimura variety ramifies at p the local models and hence the shimura models as originally defined can fail to be flat and it becomes desirable to modify their definition so as to obtain a flat scheme in the case of unitary similitude groups whose localizations at q p are ramified quasi split gu n pappas and rapoport have added new conditions the so called wedge and spin conditions to the moduli problem defining the original local models and conjectured that their new local models are flat we prove a preliminary form of their conjecture namely that their new models are topologically flat in the case n is odd

local models of shimura varieties and a conjecture of kottwitz we give a group theoretic definition of local models as sought after in the theory of shimura varieties these are projective schemes over the integers of a p adic local field that are expected to model the singularities of integral models of shimura varieties with parahoric level structure our local models are certain mixed characteristic degenerations of grassmannian varieties they are obtained by extending constructions of beilinson drinfeld gaitsgory and the second named author to mixed characteristics and to the case of general tamely ramified reductive groups we study the singularities of local models and hence also of the corresponding integral models of shimura varieties in particular we study the monodromy inertia action and show a commutativity property for the sheaves of nearby cycles as a result we prove a conjecture of kottwitz which asserts that the semi simple trace of frobenius on the nearby cycles gives a function which is central in the parahoric hecke algebra

optimal shortcuts to adiabaticity for a quantum piston abstract in this paper we use optimal control to design minimum time adiabatic like paths for the expansion of a quantum piston under realistic experimental constraints we calculate the minimum expansion time and compare it with that obtained from a state of the art inverse engineering method we use this result to rederive the known upper bound for the cooling rate of a refrigerator which provides a quantitative description for the unattainability of absolute zero the third law of thermodynamics we finally point out the relation of the present work to the fast adiabatic like expansion of an accordion optical lattice a system which can be used to magnify the initial quantum state quantum microscope

poisson integrals of regular functions tangential convergence of poisson integrals is proved for certain spaces of regular functions which contain the spaces of bessel potentials of lp functions 1 1

on the controllability of quantum mechanical systems the systems theoretic concept of controllability is elaborated for quantum mechanical systems sufficient conditions being sought under which the state vector can be guided in time to a chosen point in the hilbert space h of the system the schrodinger equation for a quantum object influenced by adjustable external fields provides a state evolution equation which is linear in and linear in the external controls thus a bilinear control system for such systems the existence of a dense analytic domain d in the sense of nelson together with the assumption that the lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension permits the adaptation of the geometric approach developed for finite dimensional bilinear and nonlinear control systems conditions are derived for global controllability on the intersection of d with a suitably defined finite dimensional submanifold of the unit sphere sh in h several soluble examples are presented to illuminate th

integral models of hilbert modular varieties in the ramified case deformations of modular galois representations and weight one forms we prove the strong artin conjecture for continuous totally odd two dimensional representations of the absolute galois group of a totally real field f

time minimal control of dissipative two level quantum systems the integrable case the objective of this article is to apply recent developments in geometric optimal control to analyze the time minimum control problem of dissipative two level quantum systems whose dynamics is governed by the lindblad equation we focus our analysis on the case where the extremal hamiltonian is integrable

characterization of traces of smooth functions on ahlfors regular sets abstract we extend the results of p shvartsman on characterizing the traces of besov and triebel lizorkin spaces on ahlfors n regular sets to the case of d regular sets n 1 d n the characterizations of trace spaces are given in terms of local polynomial approximations

constrained minimum energy optimal control of the dissipative bloch equations abstract in this letter we apply optimal control theory to design minimum energy 2 and pulses for the bloch system in the presence of relaxation with constrained control amplitude we consider a commonly encountered case in which the transverse relaxation rate is much larger than the longitudinal one so that the latter can be neglected using pontryagin s maximum principle we derive optimal feedback laws which are characterized by the number of switches depending on the control bound and the coordinates of the desired final state

controllability of quantum mechanical systems by root space decomposition of su n the controllability property of the unitary propagator of an n level quantum mechanical system subject to a single control field is described using the structure theory of semisimple lie algebras sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases

adiabatic quantum computation is equivalent to standard quantum computation adiabatic quantum computation has recently attracted attention in the physics and computer science communities but its computational power was unknown we describe an efficient adiabatic simulation of any given quantum algorithm which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent our result can be extended to the physically realistic setting of particles arranged on a two dimensional grid with nearest neighbor interactions the equivalence between the models allows stating the main open problems in quantum computation using well studied mathematical objects such as eigenvectors and spectral gaps of sparse matrices