on proof and progress in mathematics author s thurston william p abstract in response to jaffe and quinn math ho 9307227 the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems especially in his own work in the theory of foliations and geometrization of 3 manifolds and dynamical systems

degenerations of k3 surfaces of degree two we consider a semistable degeneration of k3 surfaces equipped with an effective divisor that defines a polarisation of degree two on a general fibre we show that the map to the relative log canonical model of the degeneration maps every fibre to either a sextic hypersurface in p 1 1 1 3 or a complete intersection of degree 2 6 in p 1 1 1 2 3 furthermore we find an explicit description of the hypersurfaces and complete intersections that can arise thereby giving a full classification of the possible singular fibres

exceptional collections on dolgachev surfaces associated with degenerations abstract dolgachev surfaces are simply connected minimal elliptic surfaces with p g q 0 and of kodaira dimension 1 these surfaces are constructed by logarithmic transformations of rational elliptic surfaces in this paper we explain the construction of dolgachev surfaces via q gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities this construction is based on the paper 25 also some exceptional bundles on dolgachev surfaces associated with q gorenstein smoothing have been constructed based on the idea of hacking 12 in the case if dolgachev surfaces were of type 2 3 we describe the picard group and present an exceptional collection of maximal length finally we prove that the presented exceptional collection is not full hence there exists a nontrivial phantom category in the derived category

log canonical models of elliptic surfaces abstract we classify the log canonical models of elliptic surface pairs f x c s f a where f x c is an elliptic fibration s is a section and f a is a weighted sum of marked fibers in particular we show how the log canonical models depend on the choice of the weights we describe a wall and chamber decomposition of the space of weights based on how the log canonical model changes in addition we give a generalized formula for the canonical bundle of an elliptic surface with section and marked fibers this is the first step in constructing compactifications of moduli spaces of elliptic surfaces using the minimal model program

on four dimensional terminal quotient singularities we report on an investigation of four dimensional terminal cyclic quotient singularities which are not gorenstein for simplicity we focus on quotients by cyclic groups of prime order an enumeration using a computer of all such singularities for primes 1600 led us to conjecture a structure theorem for these singularities which is rather more complicated than the known structure theorem in dimension three we discuss this conjecture and our evidence for it we also discuss properties of the anticanonical and antibicanonical linear systems of these singularities introduction the recent successes in understanding the birational geometry of algebraic varieties of dimension greater than two have focused attention on a class of singularities called terminal singularities which appear on the birational models which the theory selects in dimension three the structure of these terminal singularities is known in some detail the terminal quotient singularities were classified in what is now called the terminal lemma by several groups of people working independently cf 1 2 5 8 and all other three dimensional terminal singularities were subsequently classified by work of the first author 6 and of kollar and shepherd barron 4 both of these classifications were based on key results of reid 10 11 who reduced the problem to an analysis of quotients of smooth points and double points by cyclic group actions a detailed account of the classification can be found in 12 a consequence of this classification apparently first observed by reid 12 is that for any three dimensional terminal singularity t the general element of the anticanonical linear system i kt has only canonical singularities this in turn implies that if we form the double cover of t branched on the general antibicanonical divisor d e i 2kt i that double cover will also have only canonical singularities at first glance this second property appears to have no advantages over the first but looking at things in these terms proved decisive in a slightly more global context kawamata 3 showed that if t is an extremal neighborhood of a rational curve c on a threefold then the existence of such a divisor d globally on t is sufficient to conclude the existence of a certain kind of birational modification a directed flip centered on c the first author 7 showed that such divisors always exist completing the proof of the minimal model theorem for threefolds see 7 for more details received august 26 1987 revised december 17 1987 1980 mathematics subject classification 1985 revision primary 14b05 secondary 14 04 14j10 14l32 research partially supported by the national science foundation current addresses s m department of mathematics faculty of science nagoya university nagoya 464 japan d r m department of mathematics duke university durham north carolina 27706 i m department of mathematics fordham university bronx new york 10458 1988 american mathematical society 0025 5718 88 1 00 25 per page

equivalent representations of set functions this paper introduces four alternative representations of a set function the ma bius transformation the co ma bius transformation and the interactions between elements of any subset of a given set as extensions of shapley and banzhaf values the links between the five equivalent representations of a set function are emphasized in this presentation

simple and fast algorithms for linear and integer programs with two variables per inequality the authors present an o mn 2 log m algorithm for solving feasibility in linear programs with up to two variables per inequality which is derived directly from the fourier motzkin elimination method the number of variables and inequalities are denoted by n and m respectively the running time of the algorithm dominates that of the best known algorithm for the problem and is far simpler integer programming on monotone inequalities i e inequalities where the coefficients are of opposite sign is then considered this problem includes as a special case the simultaneous approximation of a rational vector with specified accuracy which is known to be np complete however it is shown that both a feasible solution and an optimal solution with respect to an arbitrary objective function can be computed in pseudo polynomial time

incremental closure for systems of two variables per inequality subclasses of linear inequalities where each inequality has at most two vari ables are popular in abstract interpretation and model checking because they strike a balance between what can be described and what can be efficiently computed this paper focuses on the tvpi class of inequalities for which each coefficient of each two variable inequality is unrestricted an implied tvpi in equality can be generated from a pair of tvpi inequalities by eliminating a given common variable echoing resolution on clauses this operation called result can be applied to derive tvpi inequalities which are entailed implied by a given tvpi system the key operation on tvpi is calculating closure satisfiability can be observed from a closed system and a closed system also simplifies the calculation of other operations a closed system can be derived by repeatedly applying the result operator the process of adding a single tvpi inequality to an already closed input tvpi system and then finding the closure of this augmented system is called incremental closure this too can be calcu lated by the repeated application of the result operator this paper studies the calculus defined by result the structure of result derivations and how deriva tions can be combined and controlled a series of lemmata on derivations are presented that collectively provide a pathway for synthesising an algorithm for incremental closure the complexity of the incremental closure algorithm is analysed and found to be o n 2 m 2 lg m where n is the number of variables and m the number of inequalities of the input tvpi system

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

compact moduli of plane curves we construct a compactication md of the moduli space of plane curves of degreed we regard a plane curvec p 2 as a surface divisor pair p 2 c and dene md as a moduli space of pairs x d where x is a degeneration of the plane we show that if d is not divisible by 3 the stackmd is smooth and the degenerate surfaces x can be described explicitly msc2000 14h10 14j10 14e30

a calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves any graph manifold can be obtained by plumbing according to some plumbing graph i a calculus for plumbing which includes normal forms for such graphs is developed this is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves for instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity in this paper we describe a calculus for plumbed manifolds which lets one algorithmically determine when the oriented 3 manifolds m tx and a ftj ob tained by plumbing according to two graphs tx and t2 are homeomorphic 3 mani folds are oriented 3 manifolds throughout this paper and homeomorphisms of 3 manifolds are orientation preserving we then apply the calculus to answer several questions about the topology of isolated singularities of complex surfaces and one parameter families of complex curves these results are described below since the class of 3 manifolds obtainable by plumbing is precisely the class of graph manifolds which were classified with minor exceptions by waldhausen 24 a calculus for plumbing is in some sense implicit in waldhausen s work moreover it has been known for some time that the calculus can be put in a form like the one given here but the details have never appeared in the literature a related calculus for plumbing trees has been worked by bonahon and siebenmann 1 in order to classify their algebraic knots we describe the calculus in greater generality than is needed for the present applications since this involves minimal extra work and the calculus is needed elsewhere 4 14 and 15 in particular in an appendix we describe two generalizations of it the calculus consists of a collection of moves one can do to a plumbing graph t without altering the plumbed manifold m t to see these moves are sufficient we describe how they can be used to reduce any graph to a normal form which is

centers of f purity in this paper we study a positive characteristic analogue of the centers of log canonicity of a pair r we call these analogues centers of f purity we prove positive characteristic analogues of subadjunction like results prove new stronger subadjunction like results and in some cases lift these new results to characteristic zero using a generalization of centers of f purity which we call uniformly f compatible ideals we give a characterization of the test ideal which unifies several previous characterizations finally in the case that 0 we show that uniformly f compatible ideals coincide with the annihilators of the mathcal f e r k submodules of e r k as defined by lyubeznik and smith

degeneration of surfaces with trivial canonical bundle the object of this note is to prove the theorem stated below the main idea of the proof and much of the detail are already to be found in v i kulikov s important paper k our contribution to the proof and the differences with kulikov s paper are discussed later in this introduction theorem let wl x d be a semistable degeneration of surfaces such that

automorphisms of smooth canonically polarized surfaces in positive characteristic abstract this paper investigates the geometry of a smooth canonically polarized surface x defined over an algebraically closed field of characteristic p 0 in the case when the automorphism scheme of x is not smooth in particular it is shown that a smooth canonically polarized surface x with 1 k x 2 2 and non smooth automorphism scheme tends to be uniruled and simply connected and is the purely inseparable quotient of a ruled or rational surface by a rational vector field moreover restrictions on certain numerical invariants of x are obtained in order for aut x to be smooth

the computational complexity of simultaneous diophantine approximation problems simultaneous diophantine approximation in d dimensions deals with the approximation of a vector bf alpha alpha 1 cdots alpha d of d real numbers by vectors of rational numbers all having the same denominator this paper considers the computational complexity of algorithms to find good simultaneous approximations to a given vector bf alpha of d rational numbers we measure the goodness of an approximation using the sup norm we show that a result of h w lenstra jr produces polynomial time algorithms to find sup norm best approximations to a given vector bf alpha when the dimension d is fixed we show that a recent algorithm of a k lenstra h w lenstra jr and l lovasz to find short vectors in an integral lattice can be used to find a good approximation to a given vector bf alpha in d dimensions with a denominator q satisfying 1 leqq q leqq 2 d 2 n which is within a factor sqrt 5d 2 d 1 2 of the best approximation with denominato

m resolutions and deformations of quotient singularities introduction due to the work in brieskorn tjurina artin wahl 2 and lipman one understands very well the notion of simultaneous resolution for rational surface singularities given a rational surface singularity x there exists a smooth parameter space res representing the functor of deformations of the minimal resolution the family y res contracts to x res the fibers yt x 1 are the minimal resolutions of xt there is a finite and galois map res defx the versa base space of x the image is the artin component which represents the functor of deformations of x with simultaneous resolution after finite base change and the covering group w is a reflection group which is also the monodromy group of the component in the case of rational double points rdps w is the weyl group of the corresponding root system ak dk e6 e1 ores in general much information about the deformation e g the discriminant and adjacencies may be read off the geometry of this covering the reflections in w are related to certain divisors called roots on y and their liftings the contraction map is induced by contracting rdp configurations on y in connection with his work on deformations of cyclic quotient singularities cqs the second author observed a natural galois covering of each component in the reduced versal base space of a cqs in the case of the artin component the observed covering group was the same as for res def see christophersen 1 and christophersen 2 the first author conjectured that this was the monodromy cover and we asked ourselves if there was a deformation theoretic explanation the purpose of this paper is to answer this question we will show that something similar to the res def picture actually happens for every non embedded component of the versa base space of a quotient singularity for us a quotient singularity is the singularity of c2 g where g c gl 2 c is a finite subgroup which we can assume to be without pseudo reflections for quotient singularities the construction of the artin component is a special case of a procedure involving deformations of certain modifications which we call m resolutions the application of threefold theory to deformations of rational singularities as found in kollfu shepherd barron was important both for discovering m resolutions and for the proofs of their properties in fact our results are anticipated in theorem 3 5 a of that paper still the definitions and statement of the main result may be made without reference to that work the relevant results from kollfu shepherd barron are postponed to 1 consider a smoothing of a normal surface singularity x with smooth generic fiber f the milnor number of this smoothing is p rk h2 f and depends upon the component of the versa base space of x on which the smoothing appears

using multiple reference levels in multi criteria decision aid the generalized additive independence model and the choquet integral approaches abstract in many multi criteria decision problems one can construct with the decision maker several reference levels on the attributes such that some decision strategies are conditional on the comparison with these reference levels the classical models such as the choquet integral cannot represent these preferences we are then interested in two models the first one is the choquet with respect to a p ary capacity combined with utility functions where the p ary capacity is obtained from the reference levels the second one is a specialization of the generalized additive independence gai model which is discretized to fit with the presence of reference levels these two models share common properties monotonicity continuity properly weighted but differ on the interpolation means lovasz extension for the choquet integral and multi linear extension for the gai model a drawback of the use of the choquet integral with respect to a p ary capacity is that it cannot satisfy decision strategies in each domain bounded by two successive reference levels that are completely independent of one another we show that this is not the case with the gai model

on importance indices in multicriteria decision making we address in this paper the problem of how to define an importance index in multicriteria decision problems when a numerical representation of preference is given we make no restrictive assumption on the model which could have discrete or continuous attributes and in particular it is not assumed that the model is monotonically increasing or decreasing with respect to w r t the attributes our analysis first considers discrete models which are seen to be equivalent to multichoice games we propose essentially two importance indices namely the signed importance index and the absolute importance index both based on the average variation of the value of the model induced by a given attribute we provide several axiomatizations for these importance indices extend them to the continuous case and finally illustrate them with examples classical simple models and a example of discomfort evaluation based on real data

special test configuration and k stability of fano varieties for any at projective family x l c such that the generic bre x is a klt q fano variety and ljx q kx we use the techniques from the minimal model program mmp to modify the total family the end product is a family such that every ber is a klt q fano variety moreover we can prove that the donaldson futaki invariants of the appearing models decrease when the family is a test conguration of a xed fano variety x kx this implies tian s conjecture given x a fano manifold to test its k semi poly stability we only need to test on the special test congurations

interaction indices for multichoice games abstract models in multicriteria decision analysis mcda can be analyzed by means of an importance index and an interaction index for every group of criteria we consider first discrete models in mcda without further restriction which amounts to considering multichoice games that is cooperative games with several levels of participation we propose and axiomatize two interaction indices for multichoice games the signed interaction index and the absolute interaction index in a second part we consider the continuous case supposing that the continuous model is obtained from a discrete one by means of the choquet integral we show that as in the case of classical games the interaction index defined for continuous aggregation functions coincides with the signed interaction index up to a normalizing coefficient

improved algorithms for linear inequalities with two variables per inequality the authors show that a system of m linear inequalities with n variables where each inequality involves at most two variables can be solved in tilde o mn 2 time we denote tilde o f o f polylog n polylog m deterministically and in tilde o n 3 mn expected time using randomization parallel implementations of these algorithms run in tilde o n time where the deterministic algorithm uses tilde o mn processors and the randomized algorithm uses tilde o n 2 m processors the bounds significantly improve over previous algorithms the randomized algorithm is based on novel insights into the structure of the problem

bipolarization of posets and natural interpolation the choquet integral w r t a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of 0 1 n we take this basic fact as a starting point to define the choquet integral in a very general way using the geometric realization of lattices and their natural triangulation as in the work of koshevoy a second aim of the paper is to define a general mechanism for the bipolarization of ordered structures bisets or signed sets as well as bisubmodular functions bicapacities bicooperative games as well as the choquet integral defined for them can be seen as particular instances of this scheme lastly an application to multicriteria aggregation with multiple reference levels illustrates all the results presented in the paper

on the k 2 of degenerations of surfaces and the multiple point formula in this paper we study some properties of reducible surfaces in particular of unions of planes when the surface is the central fibre of an embedded flat degeneration of surfaces in a projective space we deduce some properties of the smooth surface which is the general fibre of the degeneration from some combinatorial properties of the central fibre in particular we show that there are strong constraints on the invariants of a smooth surface which degenerates to configurations of planes with global normal crossings or other mild singularities our interest in these problems has been raised by a series of interesting articles by guido zappa in the 1950 s