singularities of energy minimizing maps from the ball to the sphere we study maps from the unit ball b in r3 to the unit sphere s2 in r3 which minimize dirichlet s energy integral varepsilon varphi int b nabla varphi 2 dv

regularity of area minimizing currents iii blow up this is the last of a series of three papers in which we give a new shorter proof of a slightly improved version of almgren s partial regularity of area minimizing currents in riemannian manifolds here we perform a blow up analysis deducing the regularity of area minimizing currents from that of dir minimizing multiple valued functions

a boundary value problem for nematic liquid crystals with a variable degree of orientation in this paper we represent the degree of microscopic order through a scalar valued function the degree of orientation which vanishes where the fluid becomes isotropic

nonlinear theory of defects in nematic liquid crystals phase transition and flow phenomena this is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals it also includes some new results concerning disclinations line defects which have not been published elsewhere

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

homotopy classes in sobolev spaces and the existence of energy minimizing maps consider the problem of finding maps f m n that are stationary for an energy functional such as fm idfl p where m and n are compact riemannian manifolds and p l such maps may be found by minimizing the functional but if we minimize among all maps from m to n then the minimum is 0 and is attained only by constant maps thus in order to find nontrivial stationary maps we would like to use the topology of m and n to define classes of maps from m to n in which we can minimize the functional for instance one could try to minimize among maps in a given homotopy class but this is not possible in general unless p dimm since a minimizing sequence of mappings in one homotopy class can converge in the appropriate weak topology to a map in another homotopy class however in this paper we show that it is possible to minimize among maps fwhose restrictions to a lower dimensional skeleton of a triangulation of m belong to a given homotopy class to state the results precisely we need to refer to certain sobolev spaces and norms we will assume without loss of generality that m and n are submanifolds of euclidean spaces r m and r respectively and we let lip m n denote the space of lipschitz maps from m to n we define li p m r to be the space of all functions fe lp m r n such that there exist functions

dirichlet and neumann boundary value problems for yang mills connections the yang mills equations have been extensively studied on compact riemannian manifolds without boundary a substantial amount of work has also been done on non compact examples with a boundary component at infinity see 5 10 11 14 15 here we define and describe a solution of dirichlet and neumann boundary value problems for yang mills connections on a compact four dimensional manifold with smooth boundary the equations in the interior are the full yang mills equations we do not restrict ourselves to self dual or anti self dual solutions in the case where the structure group in the gauge theory is abelian our results are contained in the hodge de rham theory for manifolds with boundary see 6 9 4 3 in the general case our results describe a non linear version of this theory the base manifold m of the theory is a compact orientable four dimensional manifold with smooth boundary m we assume that m is endowed with a riemannian metric we consider a principal fiber bundle p over m with structure group g a compact lie group which will have a metric invariant under the adjoint action in the usual example this comes from trace the unknown in the problem is the connection a i e a differential form in p ad g where g is the lie algebra of g there are two natural boundary value problems one of them requires fixing the tangential part of the connection a on the bundle restricted to m if m is locally described as x 0 this amounts to prescribing a1 x dx a2 x dx 2 a3 x dx 3 on m the connection is fixed in the sense that a particular choice is fixed in the gauge equivalence class of a on the boundary the yang mills equations are satisfied in the interior this corresponds to the dirichlet problem in the other boundary value problem we consider a is free at the boundary but the normal part of the curvature is set equal to zero we call this the neumann problem for the dirichlet problem we consider a sequence of l1 connections that minimizes the yang mills functional with prescribed admissible tangential boundary data we show that there exists a weak limiting connection a by finding local good gauges the limiting connection is only locally defined we show that a solves the yang mills equations and satisfies the boundary conditions up to

smoothness of harmonic maps for hypoelliptic diffusions harmonic maps are viewed as maps sending a fixed diffusion to manifold valued martingales under a convexity condition we prove that the continuity of real valued harmonic functions implies the continuity of harmonic maps then we prove with a probabilistic method that continuous harmonic maps are smooth under hormander s condition the proof relies on the study of martingales with values in the tangent bundle

space efficient preprocessing schemes for range minimum queries on static arrays given a static array of n totally ordered objects the range minimum query problem is to build a data structure that allows us to answer efficiently subsequent on line queries of the form what is the position of a minimum element in the subarray ranging from i to j we focus on two settings where 1 the input array is available at query time and 2 the input array is available only at construction time in setting 1 we show new data structures a of size frac 2n c n theta bigl frac n lg lg n c n lg n bigr bits and query time o c n for any positive integer function c n in o bigl n varepsilon bigr for an arbitrary constant 0 varepsilon 1 or b with o nh k o n bits and o 1 query time where h k denotes the empirical entropy of k th order of the input array in setting 2 we give a data structure of size 2n o n bits and query time o 1 all data structures can be constructed in linear time and almost in place

on nematic liquid crystals with variable degree of orientation energy minimizing static configurations of nematic liquid crystals with variable degree of orientation can be described by a map which minimizes certain degenerate variational integrals this mathematical model was recently posed by j l erickson in this paper we shall prove the existence local uniqueness and global regularity of minimizers we shall also estimate the size of possible defects

derived categories of quadric fibrations and intersections of quadrics abstract we construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the clifford algebras on the base corresponding to this quadric fibration generalizing the kapranov s description of the derived category of a single quadric as an application we verify that the noncommutative algebraic variety p s 2 w b 0 where b 0 is the universal sheaf of even parts of clifford algebras is homologically projectively dual to the projective space p w in the double veronese embedding p w p s 2 w using the properties of the homological projective duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics

space efficient frameworks for top k string retrieval the inverted index is the backbone of modern web search engines for each word in a collection of web documents the index records the list of documents where this word occurs given a set of query words the job of a search engine is to output a ranked list of the most relevant documents containing the query however if the query consists of an arbitrary string which can be a partial word multiword phrase or more generally any sequence of characters then word boundaries are no longer relevant and we need a different approach in string retrieval settings we are given a set de d1 d2 d3 m dd of d strings with n characters in total taken from an alphabet set l r and the task of the search engine for a given query pattern p of length p is to report the most relevant strings in d containing p the query may also consist of two or more patterns the notion of relevance can be captured by a function score p dr which indicates how relevant document dr is to the pattern p some example score functions are the frequency of pattern occurrences proximity between pattern occurrences or pattern independent pagerank of the document the first formal framework to study such kinds of retrieval problems was given by muthukrishnan lsoda 2002r he considered two metrics for relevance frequency and proximity he took a threshold based approach on these metrics and gave data structures that use o n log n words of space we study this problem in a somewhat more natural top k framework here k is a part of the query and the top k most relevant highest scoring documents are to be reported in sorted order of score we present the first linear space framework i e using o n words of space that is capable of handling arbitrary score functions with near optimal o p p klog k query time the query time can be made optimal o ppk if sorted order is not necessary further we derive compact space and succinct space indexes for some specific score functions this space compression comes at the cost of higher query time at last we extend our framework to handle the case of multiple patterns apart from providing a robust framework our results also improve many earlier results in index space or query time or both

liquid crystals and the monotonicity formula abstract we give numerical evidence that the monotonicity formula holds for the liquid crystal functional and study part of the associated bifurcation diagram

towards homological projective duality for s2p3 and s2p4 abstract we provide homological foundations to establish conjectural homological projective dualities in two cases 1 the duality between s 2 p 3 and the double cover of the projective 9 space branched along the symmetric determinantal quartic and 2 the duality between s 2 p 4 and the double cover of the symmetric determinantal quintic in p 14 branched along the symmetric determinantal locus of rank at most 3

on the regularity of harmonic functions and spherical harmonic maps defined on lattices abstract a growth lemma for certain discrete symmetric laplacians defined on a lattice z d zd rd with spacing is proved the lemma implies a de giorgi theorem that the harmonic functions for these laplacians are equi holder continuous 0 these results are then applied to establish regularity properties for the harmonic maps defined on zd and taking values in an n dimensional sphere sn uniform in questions of the convergence 0 and the dirichlet problem for these discrete harmonic maps are also addressed

exceptional collections for grassmannians of isotropic lines we construct a full exceptional collection of vector bundles in the derived categories of coherent sheaves on the grassmannian of isotropic two dimensional subspaces in a symplectic vector space of dimension 2n and in an orthogonal vector space of dimension 2n 1 for all n

succinct indexes for reporting discriminating and generic words we consider the problem of indexing a collection d of d strings documents of total n characters from an alphabet set of size such that whenever a pattern p of p characters and an integer 1 d come as a query we can efficiently report all i maximal generic words and ii minimal discriminating words as defined below maximal generic word is a maximal extension of p occurring in at least documents minimal discriminating word is a minimal extension of p occurring in at most documents these problems were introduced by kucherov et al spire 8 they proposed indexes occupying o n log n bits with query times o p output and o p log log n output for problem i and problem ii respectively the query time for problem ii is later improved to optimal o p output by gawrychowski et al spire 6 in this paper we describe succinct indexes of n log o n log o n bits space with near optimal query times i e o p log log n output for both these problems

harmonic maps from noncompact riemannian manifolds with non negative ricci curvature outside a compact set in this paper we prove the uniqueness and existence of harmonic maps of finite energy from a complete noncompact riemannian manifold m g with sobolev constant s2 m 0 and ricci curvature ric m 0 outside some compact subset into a complete manifold of nonpositive curvature or a regular ball in particular we prove the uniqueness and existence of bounded harmonic functions on m g

rectifiability of line defects in liquid crystals with variable degree of orientation in 2 hardt lin and the author proved that the defect set of minimizers of the modified ericksen energy for nematic liquid crystals consists locally of a finite union of isolated points and holder continuous curves with finitely many crossings in this article we show that each holder continuous curve in the defect set is of finite length hence locally the defect set is rectifiable for the most part the proof closely follows the work of de lellis et al rectifiability and upper minkowski bounds for singularities of harmonic q valued maps arxiv 1612 01813 2016 on harmonic mathcal q valued maps the blow up analysis in alper et al calc var partial differ equ 56 5 128 2017 allows us to simplify the covering arguments in 11 and locally estimate the length of line defects in a geometric fashion

determinantal quintics and mirror symmetry of reye congruences we study a certain family of determinantal quintic hypersurfaces in mathbb p 4 whose singularities are similar to the well studied barth nieto quintic smooth calabi yau threefolds with hodge numbers h1 1 h2 1 52 2 are obtained by taking crepant resolutions of the singularities it turns out that these smooth calabi yau threefolds are in a two dimensional mirror family to the complete intersection calabi yau threefolds in mathbb p 4 times mathbb p 4 which have appeared in our previous study of reye congruences in dimension three we compactify the two dimensional family over mathbb p 2 and reproduce the mirror family to the reye congruences we also determine the monodromy of the family over mathbb p 2 completely our calculation shows an example of the orbifold mirror construction with a trivial orbifold group

regularity of harmonic maps with prescribed singularities in this paper we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds this is motivated from one of hawking s conjectures on the uniqueness of kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum einstein equation in general relativity

partially constrained boundary conditions with energy minimizing mappings on considere la regularite a la frontiere pour une application minimisant l energie u m n pour deux types de conditions aux limites frontiere libre et condition limite partiellement contrainte

biaxial escape in nematics at low temperature abstract in the present work we study minimizers of the landau de gennes free energy in a bounded domain r 3 we prove that at low temperature minimizers do not vanish even for topologically non trivial boundary conditions this is in contrast with a simplified ginzburg landau model for superconductivity studied by bethuel brezis and helein merging this with an observation of canevari we obtain as a corollary the occurrence of biaxial escape the tensorial order parameter must become strongly biaxial at some point in in particular while it is known that minimizers cannot be purely uniaxial we prove the much stronger and physically relevant fact that they lie in a different homotopy class

compactness theorems for coupled yang mills fields in this paper we consider coupled yang mills fields on vector bundle e over compact riemannian manifold m under appropriate conditions on the curvature and the higgs field two compactness theorems are proved

singularly perturbed elliptic systems and multi valued harmonic functions with free boundaries here we study the asymptotic limits of solutions of some singularly perturbed elliptic systems the limiting problems involve multiple valued harmonic functions or in general harmonic maps to singular spaces and free interfaces between supports of various components of the maps the main results of the paper are the uniform lipschitz regularity of solutions as well as the regularity of free interfaces

an l p regularity theory for harmonic maps motivated by the harmonic map heat flow we consider maps between riemannian manifolds such that the tension field belongs to an lp space under an appropriate smallness condition a certain degree of regularity follows for suitable solutions of the harmonic map heat flow we have a partial regularity result as a consequence

parabolic theory as a high dimensional limit of elliptic theory the aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts this technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows specifically we give new proofs of an l 2 carleman estimate for the heat operator and the monotonicity formulas for the frequency function associated to the heat operator the two phase free boundary problem the flow of harmonic maps and the mean curvature flow the proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory in particular each parabolic theorem is proved by taking a high dimensional limit of the related elliptic result

rectifiable sets and the traveling salesman problem let k c c be a bounded set in this paper we shall give a simple necessary and sufficient condit ion for k to lie in a rectifiable curve we say that a set is a rectifiable curve if it is the image of a finite interval under a lipschitz mapping recall that for a connected set f c c f is a rectifiable curve not necessarily simple if and only if l f where l denotes one dimensional hausdorff measure this classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge but not necessarily each vertex at most twice if k is a finite set then we are essentially reduced to the classical traveling salesman problem tsp compute the length of the shortest hami l ton ian cycle which hits all points of k this is the same up to a constant multiple as asking for the inf imum of l f where f is a curve k c f such a f is called a spanning tree in tsp theory for infinite sets k we cannot hope in general to have k be a subset of a jordan curve what we should therefore look at is connected sets which conta in k let fmi n be the shortest minimal spanning tree then we cannot possibly solve our problem for sets k of infinite cardinality if we cannot find f i f c o fmin for any finite set k here and throughout the paper c co c1 c o etc denote various universal constants while there are several algorithms for computing l fml these algorithms work for finite graphs satisfying the triangle inequality and do not use the euclidean properties of k see 13 for an excellent discussion of some of these algorithms therefore these methods cannot solve our problem for general infinite k we present a method which is a minor modification of a well known algorithm farthest insert ion see 13 which yields a f with i f c o l fmi the farthest insert ion algorithm has been extensively studied with large numerical calculations on computers and is experimentally good in the sense that the f produced satisfy i f c o l f i for all examples which have