new characterizations of morrey spaces and their preduals with applications to fractional laplace equations abstract in this article the authors characterize the morrey spaces as well as their preduals via quadratic functions related to the taylor remainder of the kernel of the riesz potential as applications the authors obtain some strong capacitary inequalities which are then used to study the regularity of the duality weak solution to the fractional laplace equation with measure data

fake projective planes a fake projective plane is a complex surface different from but has the same betti numbers as the complex projective plane it is a complex hyperbolic space form and has the smallest euler poincare characteristic among smooth surfaces of general type the first example was constructed by mumford later on two more examples were found by ishida and kato a fourth possible one was recently proposed by keum in a recent joint work with gopal prasad we showed that there are seventeen non empty classes of fake projective planes and there can at most be four more specific classes higher dimensional analogues and examples were also obtained the main purpose of the talk is to explain the joint work with prasad and other related results such as arithmeticity of the lattices involved obtained earlier by klingler and yeung independently

factorization homology i higher categories abstract we construct a pairing which we call factorization homology between framed manifolds and higher categories the essential geometric notion is that of a vari framing of a stratified manifold which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata our main result constructs labeling systems on disk stratified vari framed n manifolds from n categories these n categories in contrast with the literature to date are not required to have adjoints this allows the following conceptual definition the factorization homology m c of a framed n manifold m with coefficients in an n category c is the classifying space of c labeled disk stratifications over m the core calculation underlying our main result is the following for any disk stratified manifold the space of conically smooth diffeomorphisms which preserve a vari framing is discrete

determination of the algebraic relations among special γ values in positive characteristic we devise a new criterion for linear independence over function fields using this tool in the setting of dual t motives we find that all algebraic relations among special values of the geometric r function over f q t are explained by the standard functional equations

a vanishing result for a casson type instanton invariant casson type invariants emerging from donaldson theory over certain negative definite 4 manifolds were recently suggested by andrei teleman these are defined by a count of a zero dimensional moduli space of flat instantons motivated by the cobordism program of proving witten s conjecture we use a moduli space of pu 2 seiberg witten monopoles to exhibit an oriented one dimensional cobordism of the instanton moduli space to the empty space the casson type invariant must therefore vanish

knots links and 4 manifolds in this paper we investigate the relationship between isotopy classes of knots and links in s and the di eomorphism types of homeomorphic smooth 4 manifolds as a corollary of this initial investigation we begin to uncover the surprisingly rich structure of di eomorphism types of manifolds homeomorphic to the k3 surface in order to state our theorems we need to view the seiberg witten invariant of a smooth 4 manifold as a multivariable laurent polynomial to do this recall that the seiberg witten invariant of a smooth closed oriented 4 manifold x with b2 x 1 is an integer valued function which is de ned on the set of spinc structures over x cf w km ko1 t1 in case h1 x z has no 2 torsion which will be the situation in this paper there is a natural identi cation of the spinc structures of x with the characteristic elements of h2 x z in this case we view the seiberg witten invariant as

characterizations of some operators on the vanishing generalized morrey spaces with variable exponent in this paper we will give some characterizations for the boundedness of the maximal operators potential operators and singular integral operators on the vanishing generalized morrey spaces vl p with variable exponent p and bounded set moreover we also obtain some corresponding results for unbounded set on such these spaces

algebraic relations among periods and logarithms of rank 2 drinfeld modules for any rank 2 drinfeld module rho defined over an algebraic function field we consider its period matrix p rho which is analogous to the period matrix of an elliptic curve defined over a number field suppose that the characteristic of the finite field bbb f q is odd and that rho does not have complex multiplication we show that the transcendence degree of the field generated by the entries of p rho over bbb f q theta is 4 as a consequence we show also the algebraic independence of drinfeld logarithms of algebraic functions which are linearly independent over bbb f q theta

a trace law for the hardy morrey sobolev space abstract this article shows that if h l p and h l q are p hardy morrey space based on the standard lebesgue measure and q hardy morrey space induced by the non negative radon measure respectively then one has the following trace law for the hardy morrey sobolev space i h l p of riesz potentials of order of h l p functions sup 0 f h l p i f h l q f h l p 1 sup x r r n 0 r b x r under 0 n 0 n 0 p n p n 0 q p n p

the donaldson witten function for gauge groups of rank larger than one we study correlation functions in topologically twisted d 4 supersymmetric yang mills theory for gauge groups of rank larger than one on compact four manifolds x we find that the topological invariance of the generator of correlation functions of brst invariant observables is not spoiled by noncompactness of field space we show how to express the correlators on simply connected manifolds of b 2 x 0 in terms of seiberg witten invariants and the classical cohomology ring of x for manifolds x of simple type and gauge group su n we give explicit expressions of the correlators as a sum over 1 vacua we describe two applications of our expressions one to superconformal field theory and one to large n expansions of su n d 4 supersymmetric yang mills theory

chern simons functional and the homology cobordism group for each integral homology sphere y we construct a function y on the set of integers we establish that y depends only on the homology cobordism class of y and that it recovers the froyshov invariant we state a relation between y and fintushel and stern s r invariant we show that the value of y at each integer is related to the critical values of the chern simons functional and we give some topological applications of y in particular we show that if y is trivial then there is no simply connected homology cobordism from y to itself

values of certain l series in positive characteristic we introduce a class of deformations of the values of the goss zeta function we prove with the use of the theory of deformations of vectorial modular forms as well as with other techniques a formula for their value at 1 and some arithmetic properties of values at other positive integers our formulas involve anderson and thakur s function we discuss how our formulas may be used to investigate the existence of a kind of functional equation for the goss zeta function

tannakian duality for anderson drinfeld motives and algebraic independence of carlitz logarithms we develop a theory of tannakian galois groups for t motives and relate this to the theory of frobenius semilinear difference equations we show that the transcendence degree of the period matrix associated to a given t motive is equal to the dimension of its galois group using this result we prove that carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent

an instanton invariant for 3 manifolds to an oriented closed 3 dimensional manifoldm withh1 m 0 we assign a 8 graded homology groupi m whose euler characteristic is twice casson s invariant the definition uses a construction on the space of instantons onm

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

donaldson theory on non kahlerian surfaces and class vii surfaces with b2 1 we prove that any class vii surface with b2 1 has curves this implies the global spherical shell conjecture in the case b2 1

the orientation of yang mills moduli spaces and 4 manifold topology si x est une 4 variete lisse orientee fermee dont la forme intersection q h 2 x z torsion z est definie negative alors la forme est equivalente sur les entiers a la forme standard 1 1 1 soit x une 4 variete lisse orientee fermee avec les trois proprietes suivantes 1 h 1 x z n a pas de 2 torsion 2 la forme intersection q sur h 2 x torsion a une partie positive de rang 1 ou 2 3 la forme intersection est paire alors q est equivalente sur les entiers a une des formes 01 10 01 10 01 10

orientability for gauge theories on calabi yau manifolds abstract we study orientability issues of moduli spaces from gauge theories on calabi yau manifolds our results generalize and strengthen those for donaldson thomas theory on calabi yau manifolds of dimensions 3 and 4 we also prove a corresponding result in the relative situation which is relevant to the gluing formula in dt theory

restricting riesz morrey hardy potentials abstract to strengthen the weak restriction principle discovered in both 2 theorem 5 1 for and 32 lemma 2 1 for this paper shows that the riesz operator i of order maps the morrey hardy space l h p to the morrey radon space l q continuously if and only if sup x r r n 0 b x r r provided 0 n 0 n 1 p n p n 0 q p n p moreover this brand new restriction principle is optimal in the sense that if n 0 then the iff statement fails quite remarkably such an optimization confirms affirmatively the conjecture in 7 remark 3 3 if p n q p p then there is no continuity of i l p l q n for an arbitrary radon measure with

on the blow up set of the yang mills flow on kahler surfaces the yang mills flow on a kahler surface with holomorphic initial data converges smoothly away from a singular set determined by the harder narasimhan seshadri filtration of the initial holomorphic bundle

instanton counting and donaldson invariants for a smooth projective toric surface we determine the donaldson invariants and their wallcrossing in terms of the nekrasov partition function using the solution of the nekrasov conjecture 33 38 3 and its refinement 34 we apply this result to give a generating function for the wallcrossing of donaldson invariants of good walls of simply connected projective surfaces with b 1 in terms of modular forms this formula was proved earlier in 19 more generally for simply connected 4 manifolds with b 1 assuming the kotschick morgan conjecture and it was also derived by physical arguments in 31

weighted norm inequality for the singular integral with variable kernel and fractional differentiation abstract for 0 1 let d be the fractional differentiation operator and t be the singular integral operator with variable kernel which is defined by t f x r n x x y x y n f y d y let t and t denote the adjoint of t and the pseudo adjoint of t respectively let t 1 t 2 denote the product of t 1 and t 2 t 1 t 2 denote the pseudo product of t 1 and t 2 in this paper we give the weighted norm inequalities for commutators of these singular integral operators and the fractional differentiation operator d finally we extend all results to the morrey spaces

magnetic monopoles on three manifolds on etudie des monopoles magnetiques sur une 3 variete de riemann complete orientee m on associe a m une collection d espaces de modules de solutions des equations des monopoles pourvu que m soit non compacte et que la metrique de riemann sur m soit bonne pres de l infini

morrey potentials and harmonic maps this paper discusses trace estimates for morrey potentials i e riesz potential integrals of morrey functions leading to a consideration of the c smoothness of a class of generalized harmonic maps

estimating the order of vanishing at infinity of drinfeld quasi modular forms we introduce and study certain deformations of drinfeld quasi modular forms by using rigid analytic trivialisations of corresponding anderson s t motives we show that a sub algebra of these deformations has a natural graduation by the group z 2 x z q 1 z and an homogeneous automorphism and we deduce from this and other properties multiplicity estimates

new topological field theories in two dimensions it is shown that two 1 1 dimensional 2d free abelian and self interacting non abelian gauge theories without any interaction with matter fields belong to a new class of topological field theories tfts these new theories capture together some of the key features of witten and schwarz types of tft because they are endowed with symmetries that are reminiscent of the schwarz type theories but their lagrangian density has the appearance of the witten type theories the topological invariants for these theories are computed on a 2d compact manifold and their recursion relations are obtained these new theories are shown to provide a class of tractable field theoretical models for the hodge theory in two dimensions of flat minkowski spacetime where there are no propagating degrees of freedom associated with the 2d gauge boson

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

holomorphic triangles and invariants for smooth four manifolds abstract in earlier articles the authors introduced invariants for closed oriented three manifolds defined using a variant of lagrangian floer homology in the symmetric products of riemann surfaces the aim of this article is to introduce invariants of oriented smooth four manifolds built using these floer homology groups this four dimensional theory also endows the corresponding three dimensional theories with additional structure an absolute grading of certain of its floer homology groups

partial regularity for harmonic maps and related problems via gauge theory we give a new proof of partial regularity for harmonic maps in dimensions m 3 into arbitrary targets this proof avoids the use of adapted frames and permits to consider targets of minimal c 2 regularity the proof we present moreover extends to a large class of elliptic systems of quadratic growth

pu 2 monopoles ii top level seiberg witten moduli spaces and witten s conjecture in low degrees in this article a continuation of 10 we complete the proof for a broad class of four manifolds of witten s conjecture that the donaldson and seiberg witten series coincide at least through terms of degree less than or equal to c 2 where c 1 4 7 11 andandare the euler characteristic and signature of the four manifold we use our computations of chern classes for the virtual normal bundles for the seiberg witten strata from the companion article 10 a comparison of all the orientations and the pu 2 monopole cobordism to compute pairings with the links of level zero seiberg witten moduli subspaces of the moduli space of pu 2 monopoles these calculations then allow us to compute low degree donaldson invariants in terms of seiberg witten invariants and provide a partial verification of witten s conjecture

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2