classical solutions of the chiral model unitons and holomorphic vector bundles this paper deals with classical solutions of thesu 2 chiral model on 2 and of a generalized chiral model on 2 1 such solutions are shown to correspond to certain holomorphic vector bundles over minitwistor space with an appropriate boundary condition the solutions called 1 unitons in 9 correspond to bundles over a compact 2 dimensional complex manifold and the problem becomes one of algebraic geometry

harmonic maps into the orthogonal group and null curves we find algebraic parametrizations of extended solutions of harmonic maps of finite uniton number from a surface to the orthogonal group mathrm o n in terms of free holomorphic data which lead to formulae for all such harmonic maps our work reveals an interesting correspondence between certain harmonic maps and the free weierstrass representation of null curves and minimal surfaces in 3 and 4 space

null vectors spinors and strings it is shown how in the frame of the cartan conception of spinors the old theorems onminimal surfaces as generated from null curves formulated by enneper weierstrass 1864 1866 for 3 dimensional ordinary space and by eisenhart 1911 for 4 dimensional space time may be reformulated in terms ofcomplex 2 and 4 component projective spinors respectively for the correspondingreal majorana spinors instead the same procedure naturally leads tostrings in 3 dimensional and 4 dimensional space time 2 1 and 3 1 it is suggested that this close connection with cartan spinors and the corresponding projective null geometry may be the clue for understanding the fundamental nature of strings

a note on the index bundle over the moduli space of monopoles donaldson has shown that the moduli space of monopolesmk is diffeomorphic to the spaceratk of based rational maps from the two sphere to itself we use this diffeomorphism to give an explicit description of the bundle onratk obtained by pushing out the index bundle frommk this gives an alternative and more explicit proof of some earlier results of cohen and jones

generalized interference alignment part i theoretical framework interference alignment ia has attracted enormous research interest as it achieves optimal capacity scaling with respect to signal to noise ratio on interference networks ia has also recently emerged as an effective tool in engineering interference for secrecy protection on wireless wiretap networks however despite the numerous works dedicated to ia two of its fundamental issues i e feasibility conditions and transceiver design are not completely addressed in the literature in this two part paper a generalized interference alignment gia technique is proposed to enhance the ia s capability in secrecy protection a theoretical framework is established to analyze the two fundamental issues of gia in part i and then the performance of gia in large scale stochastic networks is characterized to illustrate how gia benefits secrecy protection in part ii the theoretical framework for gia adopts methodologies from algebraic geometry determines the necessary and sufficient feasibility conditions of gia and generates a set of algorithms for solving the gia problem this framework sets up a foundation for the development and implementation of gia

gaussian interference channel capacity to within one bit the capacity of the two user gaussian interference channel has been open for 30 years the understanding on this problem has been limited the best known achievable region is due to han and kobayashi but its characterization is very complicated it is also not known how tight the existing outer bounds are in this work we show that the existing outer bounds can in fact be arbitrarily loose in some parameter ranges and by deriving new outer bounds we show that a very simple and explicit han kobayashi type scheme can achieve to within a single bit per second per hertz bit s hz of the capacity for all values of the channel parameters we also show that the scheme is asymptotically optimal at certain high signal to noise ratio snr regimes using our results we provide a natural generalization of the point to point classical notion of degrees of freedom to interference limited scenarios

kac moody monopoles and periodic instantons it is shown that calorons or periodic instantons are the same as monopoles with the loop group as their structure group their twistor correspondences and spectral data are defined the spectral data is shown to determine the general caloron

a geometric approach to the linear penrose transform on montre que sous une certaine condition cohomologique le theoreme de witten iseuberg yasskin et green sur la transformee de penrose inverse d une connexion est vraie dans une situation generale

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

on twistors and conformal field theories from six dimensions we discuss chiral zero rest mass field equations on six dimensional space time from a twistorial point of view specifically we present a detailed cohomological analysis develop both penrose and penrose ward transforms and analyse the corresponding contour integral formulae we also give twistor space action principles we then dimensionally reduce the twistor space of six dimensional space time to obtain twistor formulations of various theories in lower dimensions besides well known twistor spaces we also find a novel twistor space amongst these reductions which turns out to be suitable for a twistorial description of self dual strings for these reduced twistor spaces we explain the penrose and penrose ward transforms as well as contour integral formulae

monopoles and rational maps a note on a theorem of donaldson in a recent paper donaldson gave a description of the moduli space of su 2 monopoles in terms of rational maps this was done indirectly via the associated solution of nahm s equations we give here an interpretation of these rational maps in terms of the monopole s spectral curve and then as scattering data for the monopole itself

metric in the moduli of su 2 monopoles from spectral curves and gauss manin connection in disguise we show here that from the metric of the manifold m20 i e the reduced moduli of su 2 2 monopoles in yang mills higgs theory one can recover the respective moduli of spectral curves using the method gauss manin connection in disguise this work is a step toward creating an inverse process of finding the metric of any mk0 from spectral curves this is a thirty year old problem that we hope to shed some light on

why instantons are monopoles it is shown that instantons are hyperbolic monopoles for the loop group with non maximal symmetry breaking at infinity

feasibility of interference alignment for the mimo interference channel we study vector space interference alignment for the multiple input multiple output interference channel with no time or frequency diversity and no symbol extensions we prove both necessary and sufficient conditions for alignment in particular we characterize the feasibility of alignment for the symmetric three user channel where all users transmit along d dimensions all transmitters have m antennas and all receivers have n antennas as well as feasibility of alignment for the fully symmetric m n channel with an arbitrary number of users an implication of our results is that the total degrees of freedom available in a k user interference channel using only spatial diversity from the multiple antennas is at most 2 this is in sharp contrast to the k 2 degrees of freedom shown to be possible by cadambe and jafar with arbitrarily large time or frequency diversity moving beyond the question of feasibility we additionally discuss computation of the number of solutions using schubert calculus in cases where there are a finite number of solutions

cyclic monopoles affine toda and spectral curves we show that any cyclically symmetric monopole is gauge equivalent to nahm data given by sutcliffe s ansatz and so obtained from the affine toda equations further the direction the ercolani sinha vector and base point of the linearising flow in the jacobian of the spectral curve associated to the nahm equations arise as pull backs of toda data a theorem of accola and fay then means that the theta functions arising in the solution of the monopole problem reduce to the theta functions of toda

monopoles on asymptotically euclidean 3 manifolds on considere une generalisation de la theorie de yang mills higgs sur r 3 euclidien a des sommes convexes de r 3 avec des 3 varietes closes compactes

stratifying monopoles and rational maps a stratification of the moduli space of monopoles and of the space of rational maps into a flag variety is presented it is shown that the map associating a rational map to a monopole preserves these strata these strata explain some problems in the intepretation of the parameters of the moduli space in terms of superpositions of fundamental monopoles this interpretation is not valid on the individual strata the space of fundamental monopoles is described and shown to be the same as the corresponding space of rational maps

monopoles and maps from s 2 to s 2 the topology of the configuration space the configuration space for the su 2 yang mills higgs equations on 3 is shown to be homotopic to the space of smooth maps froms 2 tos 2 this configuration space indexes a family of twisted dirac operators the dirac family is used to prove that the configuration space does not retract onto any subspace on which the su 2 yang mills higgs functional is bounded

pre hilbert spaces in twistor quantization in twistor theory the canonical quantization procedure called twistor quantization is performed with the twistor operators represented as za za c and za za however it has not been clarified what kind of function spaces this representation is valid in in the present paper we intend to find appropriate pre hilbert spaces in which the above representation is realized as an adjoint pair of operators to this end we define an inner product for the helicity eigenfunctions by an integral over the product space of the circular space s1 and the upper half of projective twistor space using this inner product we define a hilbert space in some particular case and indefinite metric pre hilbert spaces in other particular cases showing that the above mentioned representation is valid in these spaces it is also shown that only the penrose transform in the first particular case yields positive frequency massless fields without singularities while the penrose transforms in the other particular cases

some integrable hierarchies in 2 1 dimensions and their twistor description the twistor space construction due to mason and sparling for solutions of the nonlinear schrodinger and korteweg devries hierarchies is generalized resulting in families of integrable models in 2 1 dimensions soliton solutions to one such hierarchy associated with the gauge group su 2 are constructed by adapting the methods used to construct instanton and monopole solutions these integrable models have the feature that their solutions depend on a number of arbitrary functions as well as arbitrary constants in contrast to the closely related davey stewartson hierachy

min max theory for the yang mills higgs equations in each monopole sector there exist an infinite number of finite energy solutions to the prasad sommerfield limit of the su 2 yang mills higgs equations on 3 whose energy is greater than any finite number

linear transceiver design for interference alignment complexity and computation consider a multiple input multiple output mimo interference channel where each transmitter and receiver are equipped with multiple antennas an effective approach to practically achieving high system throughput is to deploy linear transceivers or beamformers that can optimally exploit the spatial characteristics of the channel the recent work of cadambe and jafar ieee trans inf theory vol 54 no 8 suggests that optimal beamformers should maximize the total degrees of freedom and achieve interference alignment in the high signal to noise ratio snr regime in this paper we first consider the interference alignment problem without channel extension and prove that the problem of maximizing the total achieved degrees of freedom for a given mimo interference channel is np hard furthermore we show that even checking the achievability of a given tuple of degrees of freedom for all receivers is np hard when each receiver is equipped with at least three antennas interestingly the same problem becomes polynomial time solvable when each transmit receive node is equipped with no more than two antennas we also propose a distributed algorithm for transmit covariance matrix design that does not require the dof tuple preassignment under the assumption that each receiver uses a linear minimum mean square error mmse beamformer the simulation results show that the proposed algorithm outperforms the existing interference alignment algorithms in terms of system throughput

communication over mimo x channels interference alignment decomposition and performance analysis in a multiple antenna system with two transmitters and two receivers a scenario of data communication known as the x channel is studied in which each receiver receives data from both transmitters in this scenario it is assumed that each transmitter is unaware of the other transmitter s data noncooperative scenario this system can be considered as a combination of two broadcast channels from the transmitters points of view and two multiple access channels from the receivers points of view taking advantage of both perspectives two signaling schemes for such a scenario are developed in these schemes some linear filters are employed at the transmitters and at the receivers which decompose the system into either two noninterfering multiple antenna broadcast subchannels or two noninterfering multiple antenna multiple access subchannels the main objective in the design of the filters is to exploit the structure of the channel matrices to achieve the highest multiplexing gain mg it is shown that the proposed noncooperative signaling schemes outperform other known noncooperative schemes in terms of the achievable mg in particular it is shown that in some specific cases the achieved mg is the same as the mg of the system if full cooperation is provided either between the transmitters or between the receivers in the second part of the paper it is shown that by using mixed design schemes rather than decomposition schemes and taking the statistical properties of the interference terms into account the power offset of the system can be improved the power offset represents the horizontal shift in the curve of the sum rate versus the total power in decibels

aligned image sets under channel uncertainty settling conjectures on the collapse of degrees of freedom under finite precision csit a conjecture made by lapidoth et al at allerton 2005 also an open problem presented at ita 2006 states that the degrees of freedom dof of a two user broadcast channel where the transmitter is equipped with two antennas and each user is equipped with one antenna must collapse under finite precision channel state information at the transmitter csit that this conjecture which predates interference alignment has remained unresolved is emblematic of a pervasive lack of understanding of the dof of wireless networks including interference and x networks under channel uncertainty at the transmitter s in this paper we prove that the conjecture is true in all non degenerate settings e g where the probability density function of unknown channel coefficients exists and is bounded the dof collapse even when perfect channel knowledge for one user is available to the transmitter this also settles a related recent conjecture by tandon et al the key to our proof is a bound on the number of codewords that can cast the same image within noise distortion at the undesired receiver whose channel is subject to finite precision csit while remaining resolvable at the desired receiver whose channel is precisely known by the transmitter we are also able to generalize the result along two directions first if the peak of the probability density function is allowed to scale as o sqrt p alpha representing the concentration of probability density improving csit due to e g quantized feedback at rate alpha 2 log p then the dof is bounded above by 1 alpha which is also achievable under quantized feedback second we generalize the result to arbitrary number of antennas at the transmitter arbitrary number of single antenna users and complex channels the generalization directly implies a collapse of dof to unity under non degenerate channel uncertainty for the general k user interference and m times n user x networks as well

solutions of the yang mills higgs equations in 2 1 dimensional anti de sitter space time the solutions of the bogomolny equation in 2 1 dimensional anti de sitter space time are obtained by using darboux transformations with both constant spectral parameters and variable spectral parameters these solutions give the yang mills higgs fields in 2 1 dimensional anti de sitter space time some examples in the su 2 case are considered and qualitative asymptotic behaviors of the solutions as t are discussed in detail

singular hyperbolic monopoles we study the twistor theory of singular hyperbolic su 2 monopoles following the approach taken by kronheimer 9 in the euclidean case we use our results to show that the moduli space of charge 1 monopoles possesses a natural 2 sphere of scalar flat kahler metrics in the zero mass limit the metrics reduce to a class of metrics first studied by lebrun in 10

automated graph based analysis and correction of cortical volume topology the human cerebral cortex is topologically equivalent to a sheet and can be considered topologically spherical if it is closed at the brainstem low level segmentation of magnetic resonance mr imagery typically produces cerebral volumes whose tessellations are not topologically spherical the authors present a novel algorithm that analyzes and constrains the topology of a volumetric object graphs are formed that represent the connectivity of voxel segments in the foreground and background of the image these graphs are analyzed and minimal corrections to the volume are made prior to tessellation the authors apply the algorithm to a simple test object and to cerebral white matter masks generated by a low level tissue identification sequence the authors tessellate the resulting objects using the marching cubes algorithm and verify their topology by computing their euler characteristics a key benefit of the algorithm is that it localizes the change to a volume to the specific areas of its topological defects

the geodesic approximation for the yang mills higgs equations in this paper we consider the dynamics of the monopole solutions of yang mills higgs theory on minkowski space the monopoles are solutions of the yang mills higgs equations on three dimensional euclidean space it is of interest to understand how they evolve in time when considered as solutions of the yang mills higgs equations on minkowski space i e the time dependent equations it was suggested by manton that in certain situations the monopole dynamics could be understood in terms of geodesics with respect to a certain metric on the space of guage equivalence classes of monopoles the moduli space the metric is defined by taking thel 2 inner product of tangent vectors to this space in this paper we will prove that manton s approximation is indeed valid in the right circumstances which correspond to the slow motion of monopoles the metric on the moduli space of monopoles was analysed in a book by atiyah and hitchin so together with the results of this paper a detailed and rigorous understanding of the low energy dynamics of monopoles in yang mills higgs theory is obtained the strategy of the proof is to develop asymptotic expansions using appropriate gauge conditions and then to use energy estimates to prove their validity for the case of monopoles to be considered here there is a technical obstacle to be overcome when the equations are linearised about the monopole the continuous spectrum extends all the way to the origin this is overcome by using a norm introduced by taubes in a discussion of index theory for the yang mills higgs functional

monopoles non linear σ models and two fold loop spaces in this paper we study the topology of hat m k the moduli spaces ofsu 2 monopoles associated with the yang mills higgs and bogomol nyi equations and m k non linear models from quantum field theory beautiful work of donaldson 18 19 hitchin 24 25 and taubes 37 39 40 shows that gauge equivalence classes of monopoles correspond to based rational self maps of the riemann sphere similarly the non linear models we consider here are based harmonic maps from the riemann sphere to complex projectivem space in seminal work segal 35 studied m k the space of based rational maps from the riemann sphere to complex projectivem space of a fixed degreek any element of m k is clearly an element of k2cp m the space of all based continuous maps from the riemann sphere to complex projectivem space of a fixed degreek and this assignment gives rise to the natural inclusion of m k in k2cp m segal showed that these natural inclusions are homotopy equivalences through dimensionk 2m 1 as the topology of the two fold loop space 2cp m is well understood segal s result gives a very efficient way to explicitly determine the low dimensional topology of m k thus iterated loop spaces have much to say about the topology of monopoles and non linear models

symmetric monopoles and finite gap lame potentials for each positive integer g we construct a one parameter family of spectral curves for d 2 symmetric charge 2g 1 su 2 bps monopoles each spectral curve is reducible and is the union of a line with g elliptic curves we show that such a monopole is related to a g gap lame potential other symmetric monopoles related to elliptic curves are also shown to have a similar correspondence a suggestion is made on how this observation may be of use in the construction of new spectral curves

on feasibility of interference alignment in mimo interference networks we explore the feasibility of interference alignment in signal vector space based only on beamforming for k user mimo interference channels our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system which is considered extensively in algebraic geometry it is well known e g from bezout s theorem that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables following this intuition we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables rigorous connections between feasible and proper systems are made through bernshtein s theorem for the case where each transmitter uses only one beamforming vector the multibeam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems in this case we show that the connection between feasible and proper systems can be further strengthened since the equivalency between feasible and proper systems does not always hold by including standard information theoretic outer bounds in the feasibility analysis

monopoles and their spectral data a new definition of spectral data of a monopole is given for any compact lie or kac moody group it is shown that the spectral data determines the irreducible monopole in the case of maximal symmetry breaking the spectral data is shown to reduce to an earlier definition in terms of algebraic curves indexed by the nodes of the dynkin diagram of the group the structure of solutions to nahm s equations corresponding to the monopole is discussed

minitwistors and 3d yang mills higgs theory we construct a minitwistor action for yang mills higgs ymh theory in three dimensions the feynman diagrams of this action will construct perturbation theory around solutions of the bogomolny equations in much the same way that mhv maximally helicity violating diagrams describe perturbation theory around the self dual yang mills equations in four dimensions we also provide a new formula for all tree amplitudes in ymh theory and its maximally supersymmetric extension in terms of degree d maps to minitwistor space we demonstrate its relationship to the roiban spradlin volovich witten rsvw formula in four dimensions and show that it generates the correct mhv amplitudes at d 1 and factorizes correctly in all channels for all degrees we construct a minitwistor action for yang mills higgs ymh theory in three dimensions the feynman diagrams of this action will construct perturbation theory around solutions of the bogomolny equations in much the same way that mhv maximally helicity violating diagrams describe perturbation theory around the self dual yang mills equations in four dimensions we also provide a new formula for all tree amplitudes in ymh theory and its maximally supersymmetric extension in terms of degree d maps to minitwistor space we demonstrate its relationship to the roiban spradlin volovich witten rsvw formula in four dimensions and show that it generates the correct mhv amplitudes at d 1 and factorizes correctly in all channels for all degrees

cohomologie l 2 des varietes qale we give a topological interpretation of the space of l 2 harmonis forms of some qale manifolds introduced by d joyce we introduce a analytical criterium which make possible the used of mayer vietoris sequence

double solid twistor spaces the case of arbitrary signature in a recent paper 9 we constructed a series of new moishezon twistor spaces which are a kind of variant of the famous lebrun twistor spaces in this paper we explicitly give projective models of another series of moishezon twistor spaces on ncp2 for arbitrary n 3 which can be regarded as a generalization of the twistor spaces of double solid type on 3cp2 studied by kreusler kurke poon and the author similarly to the twistor spaces of double solid type on 3cp2 projective models of the present twistor spaces have a natural structure of double covering of a cp2 bundle over cp1 we explicitly give a defining polynomial of the branch divisor of the double covering whose restriction to fibers is degree four if n 4 these are new twistor spaces to the best of the author s knowledge we also compute the dimension of the moduli space of these twistor spaces differently from 9 the present investigation is based on analysis of pluri half anticanonical systems of the twistor spaces

strongly normal sets of contractible tiles in n dimensions the second and third authors and others have studied collections of usually convex tiles a generalization of pixels or voxels in r 2 and r 3 that have a property called strong normality sn for any tile p only finitely many tiles intersect p and any nonempty intersection of those tiles also intersects p this paper extends basic results about strong normality to collections of contractible polyhedra in r n whose nonempty intersections are contractible we also give sufficient and trivially necessary conditions on a locally finite collection of contractible polyhedra in r 2 or r 3 for their nonempty intersections to be contractible

on the interference management for k user partially connected fading interference channels the multiplexing gain muxg of the k user interference channel ic with partially connected interfering links is analyzed the motivation for the partially connected ic comes from the fact that not all interferences are equally strong in practice we introduce the idea of the graphical representation of interference alignment ia condition we show that for a k user partially connected mimo ic with m antennas at each transmitter and receiver the transmit beamforming matrices achieving the muxg of km 2 can be generated sequentially almost surely if the interference channel has a graphical representation which follows certain criteria we also present a novel scheme where we use the idea of partially connected ic to obtain improvements in conventional distributed interference alignment algorithm for fully connected ic the improvements are obtained in terms of greater average sum rate and faster convergence

the generalized spherical homeomorphism theorem for digital images the spherical homeomorphism conjecture proposed by shattuck and leahy in 2001 serves as the backbone of their algorithm to correct the topology of magnetic resonance images of the human cerebral cortex using a canonical image thickening technique and the authors previously proven spherical homeomorphism theorem for surfaces we formulate and prove a spherical homeomorphism theorem which is valid for all digital images when utilizing the 26 6 connectivity rule

the cohomology of the space of magnetic monopoles denote byx q the reduced space ofsu 2 monopoles of chargeq in 3 in this paper the cohomology ofx q the cohomology with compact supports ofx q and the image of the latter in the former are all calculated as representations of q which acts onx 2 this provides a non trivial lower bound for thel 2 cohomology ofx q which is compatible with some conjectures of sen it is also shown that granted some assumptions about the metric onx q itsl 2 cohomology does not exceed this bound in the situation referred to in the paper as the coprime case

instantons in two and four dimensions it is shown that yang mills instantons in four dimensions can naturally be identified with the instantons of a two dimensional theory with values in the loop group

ergodic secret alignment in this paper we introduce two new achievable schemes for the fading multiple access wiretap channel mac wt in the model that we consider we assume that perfect knowledge of the state of all channels is available at all the nodes in a causal fashion our schemes use this knowledge together with the time varying nature of the channel model to align the interference from different users at the eavesdropper perfectly in a one dimensional space while creating a higher dimensionality space for the interfering signals at the legitimate receiver hence allowing for better chance of recovery while we achieve this alignment through signal scaling at the transmitters in our first scheme scaling based alignment we let nature provide this alignment through the ergodicity of the channel coefficients in the second scheme ergodic secret alignment esa for each scheme we obtain the resulting achievable secrecy rate region we show that the secrecy rates achieved by both schemes in the two user fading mac wt scale with signal to noise ratio snr as hence we show the suboptimality of the independent identically distributed i i d gaussian signaling based schemes with and without cooperative jamming by showing that the secrecy rates achieved using i i d gaussian signaling with cooperative jamming do not scale with snr in addition we introduce an improved version of our esa scheme where we incorporate cooperative jamming to achieve higher secrecy rates moreover we derive the necessary optimality conditions for the power control policy that maximizes the secrecy sum rate achievable by our esa scheme when used solely and with cooperative jamming finally we discuss the extension of the proposed schemes to the case where there are more than two users and show that for the user fading mac wt each of the two schemes achieves secrecy sum rate that scales with snr as

the asymptotic higgs field of a monopole a simple formula is computed for the asymptotic higgs field of an su 2 monopole this formula is derived from the twistor description of monopoles and is applied to the study of boundary behaviour it is found to be harmonic and to have as its natural domain of definition a branched covering of 3 explicit formulae are given in simple cases

on the construction of monopoles for the classical groups forg a classical group an equivalence is exhibited between a g monopoles over 3 with maximal symmetry breaking at infinity b families of rank g algebraic curves int 1 along with divisors on those curves satisfying certain constraints c solutions of nahm s equations over rank g intervals satisfying the appropriate boundary conditions a and b are linked by twistor techniques b and c via the krichever method for solving non linear differential equations and a and c via the adhmn construction providing a unified picture of techniques for solution amongst other things an asymptotic formula for the higgs field of the monopole is computed

rational points and cohomology of discriminant varieties abstract let g be a finite unitary reflection group acting in a complex vector space v c n the discriminant varietyxg of g is defined as the space of regular orbits of g on v classical examples include the varieties of complex polynomials of degree n with distinct resp non zero distinct roots the normaliser g of g in gl v acts on xg in this work we determine the action of g g on the cohomology of xg in the classical cases this amounts to computing the cohomology of xg with certain local coefficient systems our methods are to compute equivariant weight polynomials by means of explicit counting of the rational points of certain varieties over finite fields and then to exploit the weight purity of the relevant varieties we obtain some power series identities as a byproduct

on the construction of monopoles we show that any self dual su 2 monopole may be constructed either by ward s twistor method or nahm s use of the adhm construction the common factor in both approaches is an algebraic curve whose jacobian is used to linearize the non linear ordinary differential equations which arise in nahm s method we derive the non singularity condition for the monopole in terms of this curve and apply the result to prove the regularity of axially symmetric solutions

spectral curves and the adhm method for a general monopole the algebraic curves defined by nahm are shown to be the same as the spectral curves

symmetric calorons and the rotation map we study su 2 calorons also known as periodic instantons and consider invariance under isometries of s1 r3 coupled with a non spatial isometry called the rotation map in particular we investigate the fixed points under various cyclic symmetry groups our approach utilises a construction akin to the adhm construction of instantons what we call the monad matrix data for calorons derived from the work of charbonneau and hurtubise to conclude we present an example of how investigating these symmetry groups can help to construct new calorons by deriving nahm data in the case of charge 2

interference alignment as a rank constrained rank minimization we show that the maximization of the sum degrees of freedom for the static flat fading multiple input multiple output mimo interference channel ic is equivalent to a rank constrained rank minimization problem rcrm when the signal subspaces span all available dimensions the rank minimization corresponds to maximizing interference alignment ia so that interference spans the lowest dimensional subspace possible the rank constraints account for the useful signal subspaces spanning all available spatial dimensions that way we reformulate the ia requirements to requirements involving ranks then we present a convex relaxation of the rcrm problem inspired by recent results in compressed sensing and low rank matrix completion theory that rely on approximating rank with the nuclear norm we show that the convex envelope of the sum of ranks of the interference matrices is the normalized sum of their corresponding nuclear norms and replace the rank constraints with asymptotically equivalent and tractable ones we then tune our heuristic relaxation for the multicell interference channel we experimentally show that in many cases the proposed algorithm attains perfect interference alignment and in some cases outperforms previous approaches for finding precoding and receive matrices for interference alignment

the topology of rational functions and divisors of surfaces segal studied the homotopy types of the spaces of holomorphic functions of the 2 sphere s 2 of closed surfaces of higher genus and of the spaces of divisors of these surfaces we continue segal s program by describing the entire stable homotopy types of these spaces in terms of the homotopy types of more familiar spaces

spectral curves and the mass of hyperbolic monopoles the moduli spaces of hyperbolic monopoles are naturally fibred by the monopole mass and this leads to a nontrivial mass dependence of the holomorphic data spectral curves rational maps holomorphic spheres associated to hyperbolic multi monopoles in this paper we obtain an explicit description of this dependence for general hyperbolic monopoles of magnetic charge two in addition we show how to compute the monopole mass of higher charge spectral curves with tetrahedral and octahedral symmetries spectral curves of euclidean monopoles are recovered from our results via an infinite mass limit

einstein metrics spinning top motions and monopoles on considere la construction de solutions pour l equation d einstein autoduale on applique la construction de lebrun a la sphere de berger la 3 sphere de metrique 1 2 2 2 i 3 3 2 ou 1 2 3 est une base de 1 formes invariante a gauche satisfaisant d i j k e ijk j k et i 3 est une constante

twistors in 2 1 dimensions the geometry of twistors for 2 1 dimensional flat space time is described functions on twistor space generate solutions of various field equations in space time as an illustrative example it is shown what a sine gordon soliton looks like in this twistor description

mimo multiple access channel with an arbitrarily varying eavesdropper secrecy degrees of freedom a two transmitter gaussian multiple access wiretap channel with multiple antennas at each of the nodes is investigated the channel matrices of the legitimate users are fixed and revealed to all the terminals whereas the channel matrices of the eavesdropper are arbitrarily varying and only known to the eavesdropper the secrecy degrees of freedom s d o f region under a strong secrecy constraint is characterized a transmission scheme that orthogonalizes the transmit signals of the two users at the intended receiver and uses a single user wiretap code for each user is shown to achieve the s d o f region the converse involves establishing an upper bound on a weighted sum rate expression this is accomplished by using induction where at each step one combines the secrecy and multiple access constraints associated with an adversary eavesdropping a carefully selected group of sub channels

l 2 cohomology of hyperkahler quotients using an argument of jost and zuo we give a criterion which implies that the l 2 harmonic forms on a complete noncompact hyperkahler manifold lie in the middle dimension and are invariant under the isometry group this is applied to various examples and in particular gives a verification of some of the predictions of sen on monopole moduli spaces

platonic hyperbolic monopoles we construct a number of explicit examples of hyperbolic monopoles with various charges and often with some platonic symmetry the fields are obtained from instanton data in mathbb r 4 that are invariant under a circle action and in most cases the monopole charge is equal to the instanton charge a key ingredient is the identification of a new set of constraints on adhm instanton data that are sufficient to ensure the circle invariance unlike for euclidean monopoles the formulae for the squared higgs field magnitude in the examples we construct are rational functions of the coordinates using these formulae we compute and illustrate the energy density of the monopoles we also prove for particular monopoles that the number of zeros of the higgs field is greater than the monopole charge confirming numerical results established earlier for euclidean monopoles we also present some one parameter families of monopoles analogous to known scattering events for euclidean monopoles within the geodesic approximation

moduli space of su 2 monopoles and complex cyclic toda hierarchy we study the problem of complete parametrization of the moduli space ofsu 2 yang mills higgs monopoles in terms of a nonlinear integrable system it is shown that the moduli space is homeomorphic to the solution space of a new generalization of finite nonperiodic toda equation called the complex cyclic toda hierarchy

on the degrees of freedom achievable through interference alignment in a mimo interference channel consider a k user flat fading mimo interference channel where the kth transmitter or receiver is equipped with mk respectively nk antennas if an exponential in k number of generic channel extensions are used either across time or frequency cadambe and jafar 1 showed that the total achievable degrees of freedom dof can be maximized via interference alignment resulting in a total dof that grows linearly with a even if mk and nk are bounded in this work we consider the case where no channel extension is allowed and establish a general condition that must be satisfied by any degrees of freedom tuple d1 d2 dk achievable through linear interference alignment for a symmetric system with mk m nk n dk d for all k this condition implies that the total achievable dof cannot grow linearly with k and is in fact no more than k m n k 1 we also show that this bound is tight when the number of antennas at each transceiver is divisible by d the number of data streams per user

monopoles and geodesics using the holomorphic geometry of the space of straight lines in euclidean 3 space it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the atiyah ward ansatz a k

a proof of the spherical homeomorphism conjecture for surfaces the human cerebral cortex is topologically equivalent to a sphere when it is viewed as closed at the brain stem due to noise and or resolution issues magnetic resonance imaging may see handles that need to be eliminated to reflect the true spherical topology shattuck and leahy 2001 present an algorithm to correct such an image the basis for their correction strategy is a conjecture which they call the spherical homeomorphism conjecture stating that the boundary between the foreground region and the background region is topologically spherical if certain associated foreground and background multigraphs are both graph theoretic trees in this paper we prove the conjecture and its converse under the assumption that the foreground background boundary is a surface

a feasibility test for linear interference alignment in mimo channels with constant coefficients in this paper we consider the feasibility of linear interference alignment ia for multiple input multiple output mimo channels with constant coefficients for any number of users antennas and streams per user and propose a polynomial time test for this problem combining algebraic geometry techniques with differential topology ones we first prove a result that generalizes those previously published on this topic in particular we consider the input set complex projective space of mimo interference channels the output set precoder and decoder grassmannians and the solution set channels decoders and precoders satisfying the ia polynomial equations not only as algebraic sets but also as smooth compact manifolds using this mathematical framework we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective if that mapping is not surjective then the solution variety projects into the input space in a singular way and the projection is a zero measure set this result naturally yields a simple feasibility test which amounts to checking the rank of a matrix we also provide an exact arithmetic version of the test which proves that testing the feasibility of ia for generic mimo channels belongs to the bounded error probabilistic polynomial complexity class

on the curvature of vortex moduli spaces we use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed riemann surface after computing the homotopy type of the universal cover of the moduli spaces which are symmetric products of the surface we prove that for genus g 1 the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case and this property extends to all kahler metrics on certain symmetric products our result rules out an established and natural conjecture on the geometry of the moduli spaces

on the hyperkahler quaternion kahler correspondence a hyperkahler manifold with a circle action fixing just one complex structure admits a natural hyperholomorphic line bundle this observation forms the basis for the construction of a corresponding quaternionic kahler manifold in the work of a haydys in this paper the corresponding holomorphic line bundle on twistor space is described and many examples computed including monopole and higgs bundle moduli spaces finally a twistor version of the hyperkahler quaternion kahler correspondence is established

linear degrees of freedom of the x channel with delayed csit we establish the degrees of freedom dof of the two user x channel with delayed channel knowledge at transmitters i e delayed channel state information at the transmitters csit assuming linear coding strategies at the transmitters we derive a new upper bound and characterize the linear dof of this network to be 6 5 the converse builds upon our development of a general lemma that shows that if two distributed transmitters employ linear strategies the ratio of the dimensions of received linear subspaces at the two receivers cannot exceed 3 2 due to delayed csit as a byproduct we also apply this general lemma to the three user interference channel with delayed csit thereby deriving a new upper bound of 9 7 on its linear dof this is the first bound that captures the impact of delayed csit on the dof of this network under the assumption of linear encoding strategies

the feasibility conditions for interference alignment in mimo networks interference alignment ia has attracted great attention in the last few years for its breakthrough performance in interference networks however despite the numerous works dedicated to ia the feasibility conditions of ia remains unclear for most network topologies the ia feasibility analysis is challenging as the ia constraints are sets of high degree polynomials for which no systematic tool to analyze the solvability conditions exists in this work by developing a new mathematical framework that maps the solvability of sets of polynomial equations to the linear independence of their first order terms we propose a sufficient condition that applies to mimo interference networks with general configurations we have further proved that this sufficient condition coincides with the necessary conditions under a wide range of configurations these results further consolidate the theoretical basis of ia

relation between the solitons of yang mills higgs fields in 2 1 dimensional minkowski space time and anti de sitter space time the yang mills higgs bogomolny equations in both 2 1 dimensional minkowski space time and 2 1 dimensional anti de sitter space time are known to be integrable and their soliton solutions have already been obtained in this article we show that there is a natural relation between the lax pairs and soliton solutions in these two space times when the curvature changes from 0 to 1 the changes of the asymptotic behaviors of the solitons are also discussed

mimo broadcast channel with an unknown eavesdropper secrecy degrees of freedom we study a multi antenna broadcast channel with two legitimate receivers and an external eavesdropper we assume that the channel matrix of the eavesdropper is unknown to the legitimate terminals but satisfies a maximum rank constraint as our main result we characterize the associated secrecy degrees of freedom for the broadcast channel with common and private messages we show that a direct extension of the single user wiretap codebook does not achieve the secrecy degrees of freedom our proposed optimal scheme involves decomposing the signal space into a common subspace which can be observed by both receivers and private subspaces which can be observed by only one of the receivers and carefully transmitting a subset of messages in each subspace we also consider the case when each user s private message must additionally remain confidential from the other legitimate receiver and characterize the s d o f region in this case

secrecy in mimo networks with no eavesdropper csit we consider two fundamental multi user channel models the multiple input multiple output mimo wiretap channel with one helper wth and the mimo multiple access wiretap mac wt channel in each case the eavesdropper has k antennas while the remaining terminals have n antennas each we consider a fast fading channel where the channel state information csi of the legitimate receiver is available at the transmitters but no csi at the transmitters csit is available for the eavesdropper s channel we determine the optimal sum secure degrees of freedom s d o f for each channel model for the regime k leq n and show that in this regime the mac wt channel reduces to the wth in the absence of eavesdropper csit for the regime n leq k leq 2n we obtain the optimal linear s d o f and show that the mac wt channel and the wth have the same optimal s d o f when restricted to linear encoding strategies in the absence of any such restrictions we provide an upper bound for the sum s d o f of the mac wt channel in the regime n leq k leq 2n our results show that unlike in the single input single output case there is loss of s d o f for even the wth due to lack of eavesdropper csit when k geq n

some characterizations of dirac type singularity of monopoles we study singular monopoles on open subsets in the 3 dimensional euclidean space we give two characterizations of dirac type singularities one is given in terms of the growth order of the norms of sections which are invariant by the scattering map the other is given in terms of the growth order of the norms of the higgs fields

monopoles and baker functions the work in this paper pertains to the solutions of nahm s equations which arise in the atiyah drinfield hitchin manin nahm construction of solutions to the bogomol nyi equations for static monopoles this paper provides an explicit construction of the solution of nahm s equations which satisfy regularity and reality conditions the lax form of nahm s equations is reduced to a standard eigenvalue problem by a special gauge transformation these equations may then be solved by the method of baker krichever this leads to a compact representation of the solutions of nahm s equations the regularity condition is shown to be related to the monodromy of the gauge reduced linear operator hitchin showed that the solutions of nahm s equations can be characterized by an algebraic curve and some data on that curve here this characterization reduces to a transcendental equation involving certain loop integrals of a meromorphic differential donaldson coordinatized the moduli space ofk monopoles by a class of rational maps from the riemann sphere to itself the data of a baker function is equivalent to this map this method gives an apriori construction of the known two monopole solutions we also give a generalization of the two monopole solution to a class of elliptic solutions of arbitrary charge these solutions correspond to reducible curves with elliptic components and the associated donaldson rational function has a simple partial fraction expansion

hamiltonian systems admitting a runge lenz vector and an optimal extension of bertrand s theorem to curved manifolds bertrand s theorem asserts that any spherically symmetric natural hamiltonian system in euclidean 3 space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a kepler system in this paper we extend this classical result to curved spaces by proving that any hamiltonian on a spherically symmetric riemannian 3 manifold which satisfies the same conditions as in bertrand s theorem is superintegrable and given by an intrinsic oscillator or kepler system as a byproduct we obtain a wide panoply of new superintegrable hamiltonian systems the demonstration relies on perlick s classification of bertrand spacetimes and on the construction of a suitable globally defined generalization of the runge lenz vector

nahm s equations and the classification of monopoles solutions of nahm s system of ordinary differential equations are produced by variational methods this leads to an explicit parametrisation of the solutions to the bogomolny equation over 3

ricci flat metrics harmonic forms and brane resolutions we discuss the geometry and topology of the complete non compact ricci flat stenzel metric on the tangent bundle of s n 1 we obtain explicit results for all the metrics and show how they can be obtained from first order equations derivable from a superpotential we then provide an explicit construction for the harmonic self dual p q forms in the middle dimension p q n 1 for the stenzel metrics in 2 n 1 dimensions only the p p forms are l 2 normalisable while for p q forms the degree of divergence grows with we also construct a set of ricci flat metrics whose level surfaces are u 1 bundles over a product of n einstein kahler manifolds and we construct examples of harmonic forms there as an application we construct new examples of deformed supersymmetric non singular m2 branes with such 8 dimensional transverse ricci flat spaces we show explicitly that the fractional d3 branes on the 6 dimensional stenzel metric found by klebanov and strassler is supported by a pure 2 1 form and thus it is supersymmetric while the example of pando zayas tseytlin is supported by a mixture of 1 2 and 2 1 forms we comment on the implications for the corresponding dual field theories of our resolved brane solutions