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

additive mappings decreasing rank one abstract let b x be the algebra of bounded operators on a real or complex banach space x and f x a subalgebra of finite rank operators a complete description of additive mappings f x f x which map rank one operators to operators of rank at most one is given

simultaneous diagonalization of matrices parametrized by a projective algebraic curve abstract if k is an algebraic function field in one variable over an algebraically closed field k then conditions are presented to insure that a matrix a mn k is diagonalizable by means of a similarity transformation t gl n k this result generalizes results of friedland 1 and motzkin taussky 4

local automorphisms of finitary incidence algebras abstract let r be a commutative indecomposable ring with identity and p a partially ordered set let f i p denote the finitary incidence algebra of p over r we will show that in most cases local automorphisms of f i p are actually r algebra automorphisms in fact the existence of local automorphisms which fail to be r algebra automorphisms will depend on the chosen model of set theory and will require the existence of measurable cardinals we will discuss local automorphisms of cartesian products as a special case in preparation of the general result

a new positivity condition for the curvature of hermitian manifolds in this note we introduce a new type of positivity condition for the curvature of a hermitian manifold which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non kahler case we derive a bochner formula for closed 1 1 forms from which this condition appears naturally and prove that if a hermitian manifold satisfies our positivity condition then any class alpha in h 1 1 bc x can be represented by a closed 1 1 form which is parallel with respect to the bismut connection lastly we show that such a curvature positivity condition holds on certain generalized hopf manifolds and on certain vaisman manifolds

on linear ordinary differential equations with functionally commutative coefficient matrices a system of linear ordinary differential equations x a t x t i is called semiproper if a t a a a t t i also known as functional commutativity of a t it is known that a semiproper system has a closed form fundamental solution matrix xa t exp ta d where the matrix exponential is defined by the power series exp k 1 kk therefore the problem of solving a semiproper system amounts to that of finding a finite form expression for the matrix exponential based on some recent results obtained by the authors for decomposing semiproper matrix functions a systematic approach is developed for finding a finite form analytical solution for the entire family of semiproper systems this solution is then used to derive a number of important and practical stability criteria for semiproper systems applications of the new results are also discussed

morphisms and faces of pseudo effective cones let x y be a morphism of projective varieties and suppose that is a pseudo effective numerical cycle class satisfying 0 a conjecture of debarre jiang and voisin predicts that is a limit of classes of effective cycles contracted by we establish new cases of the conjecture for higher codimension cycles in particular we prove a strong version when x is a fourfold and has relative dimension one

normal matrices over hermitian discrete valuation rings abstract a generalization of the classical spectral theorem for normal complex matrices is proved for matrices with entries from a class of discrete valuation rings which we have called hermitian the hermitian discrete valuation rings include the convergent complex power series in one variable and the complex formal power series in one variable thus one obtains an algebraic proof of rellich s theorem on diagnolizability of hermitian analytic matrices which is simultaneously valid for both the rings of convergent and formal power series

positivity of the diagonal abstract we study how the geometry of a projective variety x is reflected in the positivity properties of the diagonal x considered as a cycle on x x we analyze when the diagonal is big when it is nef and when it is rigid in each case we give several implications for the geometric properties of x for example when the cohomology class of x is big we prove that the hodge groups h k 0 x vanish for k 0 we also classify varieties of low dimension where the diagonal is nef and big

2 local automorphisms of finite dimensional simple lie algebras abstract let f be an algebraically closed field of characteristic 0 l be a finite dimensional simple lie algebra of type a l l 1 d l l 4 e k k 6 7 8 over f a not necessarily linear map l l is called a 2 local automorphism if for every x y l there is an automorphism x y l l depending on x and y such that x x y x y x y y in this paper we prove that any 2 local automorphism of l is an automorphism

quasianalytic multiparameter perturbation of polynomials and normal matrices we study the regularity of the roots of multiparameter families of complex univariate monic polynomials p x z zn p n 1 1 jaj x znij with fixed degree n whose coefficients belong to a certain subringc of c 1 functions we require that c includes polynomial but excludes flat functions quasianalyticity and is closed under composition derivation division by a coordinate and taking the inverse examples are quasianalytic denjoy carleman classes in particular the class of real analytic functions c we show that there exists a locally finite covering k of the parameter space where each k is a composite of finitely many c mappings each of which is either a local blow up with smooth center or a local power substitution in coordinates given by x 7 x 1 1 x q q i n 0 such that for each k the family of polynomials p k admits a c parameterization of its roots if p is hyperbolic all roots real then local blow ups suffice using this desingularization result we prove that the roots of p can be parameterized by sbvloc functions whose classical gradients exist almost ev erywhere and belong to l1 in general the roots cannot have gradients in l p for any 1 p neither can the roots be in w 1 1 loc or v mo we obtain the same regularity properties for the eigenvalues and the eigen vectors of c families of normal matrices a further consequence is that every continuous subanalytic function belongs to sbvloc

invariance and commutativity properties of some classes of solutions of the matrix differential equation x t x t x t x t abstract this paper presents some new results about the 47 years old study of the matrix differential equation xx x x first we prove that if for every t in an interval r a solution x t c n n is similar to diag jq 0 jq 0 then for every k 0 q the kernel and the image of x t k do not depend on t furthermore we show that in this case the obvious nilpotency of index q of x t x t q 0 t spreads on x t 1 x t q 0 n t 1 t q this implies that when q 2 the solution x is necessarily commutative x s x t 0n x t x s for every s t however we show that when q 3 there are analytic solutions that are not commutative also we completely describe the solutions of xx x x such that x2 0 finally we construct a class of analytic solutions whose kernel and image are not constant

perturbation theory for normal operators let e 3 x 7 a x be a c mapping with its values being un bounded normal operators with common domain of denition and compact resolvent here c stands for c1 c real analytic c m denjoy carleman of beurling or roumieu type c0 1 locally lipschitz or ck the param eter domain e is either r or rn or an innite dimensional convenient vector space we completely describe the c dependence on x of the eigenvalues and the eigenvectors of a x thereby we extend previously known results for self adjoint operators to normal operators partly improve them and show that they are best possible for normal matrices a x we obtain partly stronger results

2 local derivations and automorphisms on b h abstract this paper is devoted to 2 local derivations and 2 local automorphisms on the algebra b h of all bounded linear operators on a hilbert space h we prove that every 2 local derivation on b h is a derivation a similar result is obtained for automorphisms

on fano manifolds with nef tangent bundles admitting 1 dimensional varieties of minimal rational tangents let x be a fano manifold of picard number 1 with numerically effective tangent bundle according to the principal case of a conjecture of campana peternell s x should be biholomorphic to a rational homogeneous manifold g p where g is a simple lie group and p g is a maximal parabolic subgroup in our opinion there is no overriding evidence for the campana peternell conjecture for the case of picard number 1 to be valid in its full generality as part of a general programme that the author has undertaken with jun muk hwang to study uniruled projective manifolds via their varieties of minimal rational tangents a new geometric approach is adopted in the current article in a special case consisting of a recovering the generic variety of minimal rational tangents c x and b recovering the structure of a rational homogeneous manifold from c x the author proves that when b 4 x 1 and the generic variety of minimal rational tangents is 1 dimensional x is biholomorphic to the projective plane p 2 the 3 dimensional hyperquadric q 3 or the 5 dimensional fano homogeneous contact manifold of type g 2 to be denoted by k g 2 the principal difficulty is part a of the scheme we prove that c x c pt x x is a rational curve of degrees 3 and show that d 1 resp 2 resp 3 corresponds precisely to the cases of x p 2 resp q 3 resp k g 2 let be the normalization of a choice of a chow component of minimal rational curves on x nefness of the tangent bundle implies that is smooth furthermore it implies that at any point x x the normalization x of the corresponding chow space of minimal rational curves marked at is smooth after proving that x is a rational curve our principal object of study is the universal family u of giving a double fibration p u u x which gives p 1 bundles there is a rank 2 holomorphic vector bundle v on whose projectivization is isomorphic to p u we prove that v is stable and deduce the inequality d 4 from the inequality c 2 1 v 4c 2 v resulting from stability and the existence theorem on hermitian einstein metrics the case of d 4 is ruled out by studying the structure of the curvature tensor of the hermitian einstein metric on v in the special case where c 2 1 v 4c 2 v

fano 5 folds with nef tangent bundles and picard numbers greater than one we prove that smooth fano 5 folds with nef tangent bundles and picard numbers greater than one are rational homogeneous manifolds

conditions for a vector subspace e t and for a projector p t not to depend on t abstract a projector p t may depend on t when its range e t does not conditions for a vector subspace e t and for a projector p t not to depend on t are established useful models of applications are mentioned

projective manifolds whose tangent bundles are numerically effective in 1979 mori mo proved the so called hartshorne frankel conjecture every projective n dimensional manifold with ample tangent bundle is isomorphic to the complex projective space p a differential geometric analogon assuming the existence of a k ihler metric on x with positive holomorphic bisectional curvature is independently due to siu yau sy thus it seems natural to classify projective manifolds x whose tangent bundle tx satisfy a degenerate condition of ampleness numerical effectivity abbreviated by nef this means that the tautological quotient line bundle d 1 on f tx is numerically effective i e c 0

a generalization of the motzkin taussky theorem abstract in this paper we generalize the motzkin taussky theorem to matrices with polynomial entries

matrices that commute with their derivative on a letter from schur to wielandt we examine when a matrix whose elements are dierentiable functions in one variable commutes with its derivative this problem was discussed in a letter from issai schur to helmut wielandt written in 1934 which we found in wielandt s nachlass we present this letter and its translation into english the topic was rediscovered later and partial results were proved however there are many subtle observations in schur s letter which were not obtained in later years using an algebraic setting we put these into perspective and extend them in several directions we present in detail the relationship between several conditions mentioned in schur s letter and we focus in particular on the characterization of matrices called type 1 by schur we also present several examples that demonstrate schur s observations

simultaneous diagonalization of an algebraic family of matrices abstract let k be an algebraically closed field and let x be a projective variety over k let a be an n n matrix with entries in the function field k x of x we give a criterion for the existence of an invertible n n matrix t with entries in k such that the matrix t 1 at is diagonal this result generalizes a recent theorem of adkins and earlier theorems of friedland and of motzkin and taussky

on the approximation property of excellent rings where d is a smooth finite type a algebra a theorem which would imply the existence of an embedding of the above type yields the positive solution of the bass quillen conjecture so far this theorem is only known in special cases there is the following result by artin 3 4

the eigenvalues of matrices which commute with their derivative abstract let f be a differential field whose field of constants is algebraically closed and let a be a matrix with coefficients in f which commutes with its derivative da we show that all the eigenvalues of a lie in f answering open problem 22 of 1 we also give a simple proof of a theorem of schur characterizing matrices a with the property that the derivatives of a of all orders mutually commute

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

matrices over differential fields which commute with their derivative abstract a theorem is proved concerning the diagonalizability of a matrix over a differential field by means of a similarity transformation from the field of constants of the differential field this result contains as a special case known results concerning the diagonalizability over the complex numbers of a hermitian matrix of analytic functions under the hypothesis that the matrix commutes with its derivative

differentiable perturbation of unbounded operators if a t is a c 1 curve of unbounded self adjoint operators with compact resolvents and common domain of definition then the eigenvalues can be parameterized c 1 in t if a is c then the eigenvalues can be parameterized twice differentiably

on matrix functions which commute with their derivative abstract we improve several results published from 1950 up to 1982 on matrix functions commuting with their derivative and establish two results of general interest the first one gives a condition for a finite dimensional vector subspace e t of a normed space not to depend on t when t varies in a normed space the second one asserts that if a is a matrix function defined on a set of the form a t u diag b1 t bp t u 1 t and if each matrix function bk has the polynomial form b k t i 0 k f ki t c k i t k 1 p then a itself has the polynomial form a t i 0 d 1 f i t c i t where d k 1 p d k dk being the degree of the minimal polynomial of the matrix ck for every k 1 p

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

hermitian function matrices which commute with their derivative abstract spectral properties of analytic function matrices are studied the results obtained are then used to prove that analytic hermitian function matrices which commute with their derivative must be functionally commutative

many parameter holder perturbation of unbounded operators if u mapsto a u is a c0 mapping for 0 1 having as values unbounded self adjoint operators with compact resolvents and common domain of definition parameterized by u in an even infinite dimensional space then any continuous in u arrangement of the eigenvalues of a u is indeed c0 in u

perturbation of spectral subspaces and solution of linear operator equations abstract let a and b be normal operators on a hilbert space let k a and k b be subsets of the complex plane at distance at least from each other let e be the spectral projector for a belonging to k a and let f be the spectral projector for b belonging to k b our main results are estimates of the form ef c e a b f in some special situations the constant c is as low as 1 as an application we prove for an absolute constant d that if the space is finite dimensional and if a and b are normal with a b d then the spectrum of b can be obtained from that of a multiplicities counted by moving each eigenvalue by at most our main results have equivalent formulations as statements about the operator equation aq qb s let a and b be normal operators on perhaps different hilbert spaces assume a k a and b k b where k a k b and are as before then we give estimates of the forms q c aq qb

local automorphisms of standard operator algebras let x be an infinite dimensional separable real or complex banach space and a a closed standard operator algebra on x then every local automorphism of a is an automorphism the assumptions of infinite dimensionality separability and closeness are all indispensable

the convenient setting for non quasianalytic denjoy carleman differentiable mappings abstract for quasianalytic denjoy carleman differentiable function classes c q where the weight sequence q q k is log convex stable under derivations of moderate growth and also an l intersection see 1 6 we prove the following the category of c q mappings is cartesian closed in the sense that c q e c q f g c q e f g for convenient vector spaces applications to manifolds of mappings are given the group of c q diffeomorphisms is a regular c q lie group but not better

remarks on seshadri constants which is called the seshadri constant of l at x here the infimum is taken over all irreducible curves c passing through x and mx c is the multiplicity of c at x for example if l is very ample then e l x 1 there has been recant interest in trying to give lower bounds for this invariant at a general point ein and lazarsfeld el show that if x is a surface then e l x 1 for very general x x in higher dimension n 3 ein kuchle and lazarsfeld ekl prove that e l x 1 n for a very general point we say that a point x x is very general if x is in the complement x z of z a countably union of proper subvarieties examples of miranda show that e l x can take arbitrarily small values in codimension two i e codim z 2 even for an ample line bundle one may expect that this general bounds are not optimal an elementary observation see remark 1 below for the proof shows that e l x n ln a natural question is are there conditions which guarantee equality even in relative simple cases it turns out to be hard to give an answer recently xu xu improved the surface bound given by ein and lazarsfeld he showed that if l2 3 4 4 5 for a given integer 1

2 local derivations on matrix algebras over commutative regular algebras abstract the paper is devoted to 2 local derivations on matrix algebras over commutative regular algebras we give necessary and sufficient conditions on a commutative regular algebra to admit 2 local derivations which are not derivations we prove that every 2 local derivation on a matrix algebra over a commutative regular algebra is a derivation we apply these results to 2 local derivations on algebras of measurable and locally measurable operators affiliated with type i von neumann algebras

2 local triple homomorphisms on von neumann algebras and jbw triples abstract we prove that every not necessarily linear nor continuous 2 local triple homomorphism from a jbw triple into a jb triple is linear and a triple homomorphism consequently every 2 local triple homomorphism from a von neumann algebra respectively from a jbw algebra into a c algebra respectively into a jb algebra is linear and a triple homomorphism

mmp for moduli of sheaves on k3s via wall crossing nef and movable cones lagrangian fibrations we use wall crossing with respect to bridgeland stability conditions to systematically study the birational geometry of a moduli space m of stable sheaves on a k3 surface x a we describe the nef cone the movable cone and the effective cone of m in terms of the mukai lattice of x b we establish a long standing conjecture that predicts the existence of a birational lagrangian fibration on m whenever m admits an integral divisor class d of square zero with respect to the beauville bogomolov form these results are proved using a natural map from the space of bridgeland stability conditions mathop mathrm stab nolimits x to the cone mathop mathrm mov nolimits x of movable divisors on m this map relates wall crossing in mathop mathrm stab nolimits x to birational transformations of m in particular every minimal model of m appears as a moduli space of bridgeland stable objects on x

fake quadrics from irreducible lattices acting on the product of upper half planes in the present article we provide examples of fake quadrics that is minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in mathbb p 3 which are quotients of the bidisc by an irreducible lattice of automorphisms moreover we list classes of arithmetic lattices over a real quadratic number field which define a fake quadric and give general results towards a classification of all such fake quadrics

sharp differential estimates of li yau hamilton type for positive p p forms on kahler manifolds in this paper we study the heat equation of hodge laplacian deformation of p p forms on a kahler manifold after identifying the condition and establishing that the positivity of a p p form solution is preserved under such an invariant condition we prove the sharp differential harnack in the sense of liyau hamilton estimates for the positive solutions of the hodge laplacian heat equation we also prove a nonlinear version coupled with the kahler ricci flow and some interpolating matrix differential harnack type estimates for both the kahler ricci flow and the ricci flow 2011 wiley periodicals inc

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

the uniformization theorem for compact kahler manifolds of nonnegative holomorphic bisectional curvature soit x h une variete de kahler compacte a n dimensions de courbure bisectionnelle holomorphe non negative et soit x h son espace de recouvrement universel alors il existe des entiers non negatifs k n 1 n l et des espaces symetriques hermitiens compacts ineductibles m 1 m k de rang 2 tels que x h est isometriquement bi holomorphe a c k g 0 p nu 1 1 p nu l l m 1 g 1 m p g p ou g 0 est la metrique euclidienne sur c k g 1 g p sont des metriques canoniques sur m 1 m p et i 1 i l est une metrique de kahler sur p nu i portant une courbure bisectionnelle holomorphe non negative

extraction and prediction of coherent patterns in incompressible flows through space time koopman analysis abstract we develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible time dependent fluid flows driven by ergodic dynamical systems our approach is based on representations of the generators of the koopman and perron frobenius groups of operators governing the evolution of observables and probability measures on lagrangian tracers respectively in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm these operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place and as a result their eigenfunctions correspond to global space time coherent patterns under a skew product dynamical system moreover using this data driven representation of the generators in conjunction with leja interpolation for matrix exponentiation we construct model free prediction schemes for the evolution of observables and probability densities defined on the tracers we present applications to periodic gaussian vortex flows and aperiodic flows generated by lorenz 96 systems