subdivisions and local h vectors in part i a general theory of f vectors of simplicial subdivisions or triangulations of simplicial complexes is developed based on the concept of lo cal h vector as an application we prove that the h vector of a cohen macaulay complex increases under quasi geometric subdivision thus establishing a spe cial case of a conjecture of kalai and this author techniques include com mutative algebra homological algebra and the intersection homology of toric varieties in part ii we extend the work of part i to more general situations first a formal generalization of subdivision is given based on incidence alge bras special cases are then developed in particular one based on subdivisions of eulerian posets and involving generalized h vectors other cases deal with kazhdan lusztig polynomials ehrhart polynomials and a q analogue of eule rian posets many applications and examples are given throughout department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address rstan math mit edu this content downloaded from 157 55 39 210 on sat 30 jul 2016 05 59 16 utc all use subject to http about jstor org terms

the rem zeros in the complex temperature and magnetic field planes abstract the distribution of zeros of the partition function of the random energy model is described both for the complex temperature and magnetic field planes there are regions where these zeros become dense it is shown that for t t c a dense region of zeros reaches the real magnetic field axis as the thermodynamic limit is taken for a fixed sample this manifests itself as jumps in the equilibrium magnetization m whose position is correlated with the closest zeros

a degeneration formula of gw invariants this is the sequel to the paper 23 in this paper we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in 23 we also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety based on these we prove a degeneration formula of the gromov witten invariants

the zeroes of the partition function of the random energy model abstract the random energy model is one of the simplest disordered models containing some of the physics of spin glasses the zeroes of the partition function in the complex temperature plane are determined for this model they either lie on simple lines or are dense in some regions of the complex plane

shellable nonpure complexes and posets ii this is a direct continuation of shellable nonpure complexes and posets i which appeared in transactions of the american mathematical society 348 1996 1299 1327 8 interval generated lattices and dominance order in this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets let f i1 i2 in be a family of intervals of integers by which is meant sets of the form a b a a 1 b a b we assume that there are no containments among these intervals and that they are ordered so that their left and right endpoints are increasing let l f be the lattice of all sets that are unions of subfamilies of f ordered by inclusion such interval generated lattices l f were introduced and studied by greene g define an edge labeling of l f as follows if a b is a covering and a max b a then a b a if a 1 a and a is the left endpoint of some i f

euler flag enumeration of whitney stratified spaces abstract the flag vector contains all the face incidence data of a polytope and in the poset setting the chain enumerative data it is a classical result due to bayer and klapper that for face lattices of polytopes and more generally eulerian graded posets the flag vector can be written as a c d index a non commutative polynomial which removes all the linear redundancies among the flag vector entries this result holds for regular cw complexes we relax the regularity condition to show the c d index exists for whitney stratified manifolds by extending the notion of a graded poset to that of a quasi graded poset this is a poset endowed with an order preserving rank function and a weighted zeta function this allows us to generalize the classical notion of eulerianness and obtain a c d index in the quasi graded poset arena we also extend the semi suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the simplicial shelling components

phase transition in the jarzynski estimator of free energy differences the transition between a regime in which thermodynamic relations apply only to ensembles of small systems coupled to a large environment and a regime in which they can be used to characterize individual macroscopic systems is analyzed in terms of the change in behavior of the jarzynski estimator of equilibrium free energy differences from nonequilibrium work measurements given a fixed number of measurements the jarzynski estimator is unbiased for sufficiently small systems in these systems the directionality of time is poorly defined and the configurations that dominate the empirical average but which are in fact typical of the reverse process are sufficiently well sampled as the system size increases the arrow of time becomes better defined the dominant atypical fluctuations become rare and eventually cannot be sampled with the limited resources that are available asymptotically only typical work values are measured the jarzynski estimator becomes maximally biased and approaches the exponential of minus the average work which is the result that is expected from standard macroscopic thermodynamics in the proper scaling limit this regime change has been recently described in terms of a phase transition in variants of the random energy model in this paper this correspondence is further demonstrated in two examples of physical interest the sudden compression of an ideal gas and adiabatic quasistatic volume changes in a dilute real gas

complex random energy model zeros and fluctuations the partition function of the random energy model at inverse temperature beta is a sum of random exponentials mathcal z n beta sum k 1 n exp beta sqrt n x k where x 1 x 2 ldots are independent real standard normal random variables random energies and n log n we study the large n limit of the partition function viewed as an analytic function of the complex variable beta we identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature we prove limit theorems for the random partition function at complex beta both on the logarithmic scale and on the level of limiting distributions our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent

analyticity of the partition function of the random energy model we consider the random energy model with complex temperature for beta lying in a suitable set the function zn beta ezn beta is at the thermodynamic limit an analytic function of beta analytic properties of the high temperature free energy follow directly

mean field theory of directed polymers with random complex weights we show that for the problem of directed polymers on a tree with i i d random complex weights on each bond three possible phases can exist the phase of a particular system is determined by the distribution of the random weights for each of these three phases we give the expression of the free energy per unit length in the limit of infinitely long polymers our proofs require several hypotheses on the distribution most importantly that the amplitude and the phase of each complex weight be statistically independent the main steps of our proofs use bounds on noninteger moments of the partition function and self averaging properties of the free energy we illustrate our results by some examples and discuss possible generalizations to a larger class of distributions to random energy models and to the finite dimensional case we note that our results are not in agreement with the predictions of a recent replica approach to a similar problem

relative virtual localization and vanishing of tautological classes on moduli spaces of curves we prove a localization formula for the moduli space of stable relative maps as an application we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i g 1 genus 0 components as consequences we prove and generalize various conjectures and theorems about various moduli spaces of curves due to getzler ionel faber looijenga pandharipande diaz and others this theorem appears to be the geometric content behind these results the rest is straightforward graph combinatorics the theorem also suggests the importance of the stratification of the moduli space by number of rational components

peak algebras paths in the bruhat graph and kazhdan lusztig polynomials we obtain a nonrecursive combinatorial formula for the kazhdan lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one and which cannot be linearly simplified our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions

the symplectic sum formula for gromov witten invariants in the symplectic category there is a connect sum operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds this paper establishes a formula for the gromov witten invariants of a symplectic sum z x y in terms of the relative gw invariants of x and y several applications to enumerative geometry are given gromov witten invariants are counts of holomorphic maps into symplectic manifolds to define them on a symplectic manifold x w one introduces an almost complex structure j compatible with the symplectic form w and

lifting inequalities for polytopes we present a method of lifting linear inequalities for the flag f vector of polytopes to higher dimensions known inequalities that can be lifted using this technique are the non negativity of the toric g vector and that the simplex minimizes the cd index we obtain new inequalities for six dimensional polytopes in the last section we present the currently best known inequalities for dimensions 5 through 8

numerical complex zeros of the random energy model abstract the distribution of zeros of the partition function of the random energy model rem is numerically estimated in order to do this we devise a discrete version of the model drem for which the partition function is a polynomial and whose roots can therefore be numerically obtained a continuous limit transforms drem into rem so the zeros of the latter can be obtained from those of the former

monotonicity of the cd index for polytopes we prove that the cd index of a convex polytope satisfies a strong monotonicity property with respect to the cd indices of any face and its link as a consequence we prove for d dimensional polytopes a conjecture of stanley that the cd index is minimized on the d dimensional simplex moreover we prove the upper bound theorem for the cd index namely that the cd index of any d dimensional polytope with n vertices is at most that of c n d the d dimensional cyclic polytope with n vertices

the donaldson thomas invariants under blowups and flops using the degeneration formula for doanldson thomas invariants we proved formulae for blowing up a point and simple flops given a smooth projective calabi yau 3 fold x the moduli space of stable sheaves on x has virtual dimension zero donaldson and thomas d t defined the holomorphic casson invariant of x which essentially counts the number of sta ble bundles on x however the moduli space has positive dimension and is singular in general making use of virtual cycle technique see b f and l t thomas showed in thomas that one can define a virtual moduli cycle for some x including calabi yau and fano 3 folds as a consequence one can define donaldson type in variants of x which are deformation invariant donaldson thomas invariants pro vide a new vehicle to study the geometry and other aspects of higher dimensional varieties it is important to understand these invariants much studied gromov witten invariants of x are the counting of stable maps from curves to x in mnop1 mnop2 maulik nekrasov okounkov and pandharipande discovered relations between gromov witten invarints of x and donaldson thomas invariants constructed from moduli spaces of ideal sheaves of curves on x they conjectured that these two invariants can be identified via the equations of partition functions of both theory this suggests that many phe nomena on gromov witten theory have the counterparts in donaldson thomas theory donaldson thomas invariants are deformation independent in the birational geometry of 3 folds we have blowups and flops donaldson thomas invariants couldn t be effective in studying birational geometry unless we understand how invariants change under birational operations li and ruan in l r studied how gromov witten invariants change under a flop for calabi yau 3 fold they proved that one can identify the 3 point functions of x and the flop x f of x up to some transformation of the q variables the same question was aslo studied by liu and yau in l y recently using the j li s degeneration formula from algebraic geometry in hu1 hu2 the first author studied the change of gromov witten

statistical mechanics of conducting phase transitions the critical behavior of the effective conductivity of the random resistor network in zd near its percolation threshold is considered the network has bonds assigned the conductivities 1 and e 0 in the volume fractions p and 1 p motivated by the statistical mechanics of an ising ferromagnet at temperature t in a field h we introduce a partition function and free energy for the resistor network which establishes a direct correspondence between the two problems in particular we show that the free energies for the resistor network and the ising model both have the same type of integral representation which has the interpretation of the complex potential due to a charge distribution on 0 1 in the s 1 1 e plane for the resistor network and on the unit circle in the z exp 2 h plane for the ferromagnet based on this correspondence we develop a yang lee picture of the onset of nonanalytic behavior of the effective conductivity so that the percolation threshold p pc is characterized as a