stable cohomology of spaces of non singular hypersurfaces abstract we prove that the rational cohomology of the space x d n of non singular complex homogeneous polynomials of degree d in n variables stabilizes to the cohomology of gl n c for d sufficiently large

existence and regularity of invariant measures for the three dimensional stochastic primitive equations we establish the continuity of the markovian semigroup associated with strong solutions of the stochastic 3d primitive equations and prove the existence of an invariant measure the proof is based on new moment bounds for strong solutions the invariant measure is supported on strong solutions and is furthermore shown to have higher regularity properties

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

homological stability for topological chiral homology of completions abstract by proving that several new complexes of embedded disks are highly connected we obtain several new homological stability results our main result is homological stability for topological chiral homology on an open manifold with coefficients in certain partial framed e n algebras using this we prove a special case of a conjecture of vakil and wood on homological stability for complements of closures of particular strata in the symmetric powers of an open manifold and we prove that the bounded symmetric powers of closed manifolds satisfy homological stability rationally

the indian rope trick for a parametrically excited flexible rod nonlinear and subharmonic analysis recently mullin has demonstrated experimentally that an upright column that is longer than its critical length for self weight buckling can be stabilized by subjecting it to vertical harmonic excitation this paper extends an earlier linearized analysis of a rod mechanics model of this set up to include three dimensionality and geometric nonlinearity the stability of the upright state is then analysed using weakly nonlinear asymptotic expansions in the limit of small amplitude excitation first the unforced problem is treated extending the classical result by greenhill to show that all bifurcations are supercritical the main results are for the forced problem near the simplest among the potential infinity of dynamic instabilities these correspond to pure bending modes and to resonances between a vibration mode of the column and the first harmonic or subharmonic of the drive the result is an asymptotic description of these instabilities including information on the stability of dynamically bifurcating states in terms of the three dimensionless parameters of the problem bending stiffness and the amplitude and frequency of excitation a qualitative explanation is offered of why the earlier linearized analysis fails to quantitatively match the experiments

random periodic solutions of spdes via integral equations and wiener sobolev compact embedding abstract in this paper we study the existence of random periodic solutions for semilinear spdes on a bounded domain with a smooth boundary we identify them as the solutions of coupled forward backward infinite horizon stochastic integral equations on l 2 d in general cases for this we use mercer s theorem and eigenvalues and eigenfunctions of the second order differential operators in the infinite horizon integral equations we then use the argument of the relative compactness of wiener sobolev spaces in c 0 0 t l 2 d and generalized schauder s fixed point theorem to prove the existence of a solution of the integral equations this is the first paper in literature to study random periodic solutions of spdes our result is also new in finding semi stable stationary solution for non dissipative spdes while in literature the classical method is to use the pull back technique so researchers were only able to find stable stationary solutions for dissipative systems

conics in the grothendieck ring abstract the subring of the grothendieck ring of k varieties generated by smooth conics is described giving many zero divisors the proof uses only elementary projective geometry

invariant measure of scalar first order conservation laws with stochastic forcing under an hypothesis of non degeneracy of the flux we study the long time behaviour of periodic scalar first order conservation laws with stochastic forcing in any space dimension for sub cubic fluxes we show the existence of an invariant measure moreover for sub quadratic fluxes we show uniqueness and ergodicity of the invariant measure also since this invariant measure is supported by l p for some p small we are led to generalize to the stochastic case the theory of l 1 solutions developed by chen and perthame ann inst h poincare anal non lineaire 20 4 645 668 2003

ergodic and mixing properties of the boussinesq equations with a degenerate random forcing we establish the existence uniqueness and attraction properties of an ergodic invariant measure for the boussinesq equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales the central challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system towards this aim we encounter a lie bracket structure in the associated vector fields with a complicated dependence on solutions this leads us to develop a novel hormander type condition for infinite dimensional systems demonstrating the sufficiency of this condition requires new techniques for the spectral analysis of the malliavin covariance matrix

averaging principle for one dimensional stochastic burgers equation abstract in this paper we consider the averaging principle for one dimensional stochastic burgers equation with slow and fast time scales under some suitable conditions we show that the slow component strongly converges to the solution of the corresponding averaged equation meanwhile when there is no noise in the slow component equation we also prove that the slow component weakly converges to the solution of the corresponding averaged equation with the order of convergence 1 r for any 0 r 1

ergodicity of the 2d navier stokes equations with random forcing we consider the navier stokes equation on a two dimensional torus with a random force acting at discrete times and analytic in space for arbitrarily small viscosity coefficient we prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time

gibbsian dynamics and invariant measures for stochastic dissipative pdes we present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative pdes it consists of two main steps the first step is the reduction to a finite dimensional gibbsian dynamics of the low modes the second step is to prove the equivalence between measures induced by different past histories using girsanov theorem as applications we prove ergodicity for ginzburg landau kuramoto sivashinsky and cahn hilliard equations with stochastic forcing

stochastic dissipative pde s and gibbs measures we study a class of dissipative nonlinear pde s forced by a random force omega t x with the space variable x varying in a bounded domain the class contains the 2d navier stokes equations under periodic or dirichlet boundary conditions and the forces we consider are those common in statistical hydrodynamics they are random fields smooth in t and stationary short correlated in time t in this paper we confine ourselves to kick forces of the form where the k s are smooth bounded identically distributed random fields the equation in question defines a markov chain in an appropriately chosen phase space a subset of a function space that contains the zero function and is invariant for the random flow of the equation concerning this markov chain we prove the following main result see theorem 2 2 the markov chain has a unique invariant measure to prove this theorem we present a construction assigning to any invariant measure a gibbs measure for a 1d system with compact phase space and apply a version of ruelle perron frobenius uniqueness theorem to the corresponding gibbs system we also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced pde

the grothendieck ring of varieties and piecewise isomorphisms let k0 var k be the grothendieck ring of algebraic varieties over a field k l et x y be two algebraic varieties over k which are piecewise isomorphic i e x and y admit finite partitions x1 xn y1 yn into locally closed subvarieties such that xi is iso morphic to yi for all i n then x y in k0 var k larsen and lunts ask whether the converse is true for characteristic zero and algebraically closedfield k we answer positively this question when dim x 1o rx is a smooth connected projective surface or if x contains only finitely many rational curves

coincidences between homological densities predicted by arithmetic abstract motivated by analogies with basic density theorems in analytic number theory we introduce a notion and variations of the homological density of one space in another we use weil s number field function field analogy to predict coincidences for limiting homological densities of various sequences z n d 1 d m x of spaces of 0 cycles on manifolds x the main theorem in this paper is that these topological predictions which seem strange from a purely topological viewpoint are indeed true one obstacle to proving such a theorem is the combinatorial complexity of all possible collisions of points this problem does not arise in the simplest and classical case m n 1 2 of configuration spaces to overcome this obstacle we apply the bjorner wachs theory of lexicographic shellability from algebraic combinatorics

elliptic genera of singular varieties the notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced and the relation between them is studied the elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of calabi yau hypersurfaces in fano gorenstein toric varieties introduced earlier the orbifold elliptic genus is given in terms of the fixed point sets of the action we show that the generating function for the orbifold elliptic genus ellorb x 6n p for symmetric groups 6n acting on n fold products coincides with the one proposed by r dijkgraaf g moore e verlinde and h verlinde the two notions of elliptic genera are conjectured to coincide

stability analysis of columns with imperfection in this paper geometrically exact fully intrinsic equations and variational asymptotic beam sections are used to study the buckling and postbuckling of a column under self weight initial curvature effects both generalized one dimensional beam equations and the cross sectional stiffness properties the effects of initial curvature have been studied on buckling and postbuckling in this paper this problem is a valuable test case for the utility of the fully intrinsic equations and variational asymptotic beam sections

probabilistic estimates for the two dimensional stochastic navier stokes equations we consider the navier stokes equation on a two dimensional torus with a random force white noise in time and analytic in space for arbitrary reynolds number r we prove probabilistic estimates for the long time behavior of the solutions that imply bounds for the dissipation scale and energy spectrum as r

spectral gaps in wasserstein distances and the 2d stochastic navier stokes equations we develop a general method to prove the existence of spectral gaps for markov semigroups on banach spaces unlike most previous work the type of norm we consider for this analysis is neither a weighted supremum norm nor an lp type norm but involves the derivative of the observable as well and hence can be seen as a type of 1 wasserstein distance this turns out to be a suitable approach for infinite dimensional spaces where the usual harris or doeblin conditions which are geared toward total variation convergence often fail to hold in the first part of this paper we consider semigroups that have uniform behavior which one can view its the analog of doeblin s condition we then proceed to study situations where the behavior is not so uniform but the system has a suitable lyapunov structure leading to a type of harris condition we finally show that the latter condition is satisfied by the two dimensional stochastic navier stokes equations even in situations where the forcing is extremely de generate using the convergence result we show that the stochastic navier stokes equations invariant measures depend continuously on the viscosity and the structure of the forcing

ergodicity of 2d navier stokes equations with random forcing and large viscosity the stochastically forced two dimensional incompressable navier stokes equations are shown to possess an unique invariant measure if the viscosity is taken large enough this result follows from a stronger result showing that at high viscosity there is a unique stationary solution which attracts solutions started from arbitrary initial conditions that is to say the system has a trivial random attractor along the way results controling the expectation and averaging time of the energy and enstrophy are given

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

e_n cell attachments and a local to global principle for homological stability we define bounded generation for e n algebras in chain complexes and prove that this property is equivalent to homological stability for n ge 2 using this we prove a local to global principle for homological stability which says that if an e n algebra a has homological stability or equivalently the topological chiral homology int mathbb r n a has homology stability then so has the topological chiral homology int m a of any connected non compact manifold m using scanning we reformulate the local to global homological stability principle so that it applies to compact manifolds we also give several applications of our results

anticipating random periodic solutions i sdes with multiplicative linear noise in this paper we study the existence of random periodic solutions for semilinear stochastic differential equations we identify them as solutions of coupled forward backward infinite horizon stochastic integral equations ihsies using the substitution theorem of stochastic differential equations with anticipating initial conditions in general random periodic solutions and the solutions of ihsies are anticipating for the linear noise case with the help of the exponential dichotomy given in the multiplicative ergodic theorem we can identify them as the solutions of infinite horizon random integral equations ihsies we then solve a localised forward backward ihrie in c r lloc2 using an argument of truncations the malliavin calculus the relative compactness of wiener sobolev spaces in c 0 t l2 c 0 t l2 and schauder s fixed point theorem we finally measurably glue the local solutions together to obtain a global solution in c r l2 c r l2 thus we obtain the existence of a random periodic solution and a periodic measure

the emergence of large scale coherent structure under small scale random bombardments we provide mathematical justification of the emergence of large scale coherent structure in a two dimensional fluid system under small scale random bombardments with small forcing and appropriate scaling assumptions the analysis shows that the large scale structure emerging out of the small scale random forcing is not the one predicted by equilibrium statistical mechanics but the error is very small which explains earlier successful prediction of the large scale structure based on equilibrium statistical mechanics 2005 wiley periodicals inc

the kolmogorov equation for a 2d navier stokes stochastic flow in a channel abstract we consider a 2d navier stokes equation in a channel with periodic conditions along the axis navier type conditions on the wall and perturbed by a stochastic driving force q w where q is a nonnegative self adjoint operator of trace class and w is a space time white noise this work is concerned with the construction of the kolmogorov operator associated with the corresponding stochastic process expressed in terms of vorticity the main result is that the kolmogorov operator defined on a space of smooth c 2 functions is essentially m dissipative in l 2 h 0 where h 0 is a state space and an invariant measure

postbuckling and vibration of linearly elastic and softening columns under self weight the critical height for buckling of a linearly elastic cantilevered column due to its self weight was determined by greenhill in 1881 postbuckling behavior also has been studied often assuming that the column is an elastica inextensible with its bending moment proportional to its curvature the bifurcation point at the critical height is supercritical so that the postbuckling path is stable as the height increases past its critical value subcritical bifurcation may occur if the column is nonlinearly elastic with a softening behavior this results in a sudden jump from the straight vertical configuration to a severely drooped shape the governing equation is solved numerically with the use of a shooting method to obtain the equilibrium paths also small vibrations about the straight and postbuckled equilibrium states are examined and vibration frequencies and hence stability properties are obtained an initial curvature of the column is included in the analysis experiments are conducted to verify the results qualitatively for linearly elastic and softening materials

gibbsian dynamics and ergodicity for the stochastically forced navier stokes equation we study stationary measures for the two dimensional navier stokes equation with periodic boundary condition and random forcing we prove uniqueness of the stationary measure under the condition that all determining modes are forced the main idea behind the proof is to study the gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time history of the low modes

algebraic independence in the grothendieck ring of varieties we give sufficient cohomological criteria for the classes of given varieties over a field k to be algebraically independent in the grothendieck ring of varieties over k and construct some examples

pathwise random periodic solutions of stochastic differential equations abstract in this paper we study the existence of random periodic solutions for semilinear stochastic differential equations we identify these as the solutions of coupled forward backward infinite horizon stochastic integral equations in general cases we then use the argument of the relative compactness of wiener sobolev spaces in c 0 0 t l 2 and generalized schauder s fixed point theorem to prove the existence of a solution of the coupled stochastic forward backward infinite horizon integral equations the condition on f is then further weakened by applying the coupling method of forward and backward gronwall inequalities the results are also valid for stationary solutions as a special case when the period can be an arbitrary number

bertini theorems over finite fields let x be a smooth quasiprojective subscheme of pn of dimension m i 0 over fq then there exist homogeneous polynomials f over fq for which the intersection of x and the hypersurface f 0 is smooth in fact the set of such f has a positive density equal to aex m 1 1 where aex s zx q s is the zeta function of x an analogue for regular quasiprojective schemes over z is proved assuming the abc conjecture and another conjecture

generalized couplings and convergence of transition probabilities we provide sufficient conditions for the uniqueness of an invariant measure of a markov process as well as for the weak convergence of transition probabilities to the invariant measure our conditions are formulated in terms of generalized couplings we apply our results to several spdes for which unique ergodicity has been proven in a recent paper by glatt holtz mattingly and richards and show that under essentially the same assumptions the weak convergence of transition probabilities actually holds true

law of large numbers and central limit theorem for randomly forced pde s we consider a class of dissipative pde s perturbed by an external random force under the condition that the distribution of perturbation is sufficiently non degenerate a strong law of large numbers slln and a central limit theorem clt for solutions are established and the corresponding rates of convergence are estimated it is also shown that the estimates obtained are close to being optimal the proofs are based on the property of exponential mixing for the problem in question and some abstract slln and clt for mixing type markov processes

motivic random variables and representation stability ii hypersurface sections abstract we prove geometric and cohomological stabilization results for the universal smooth degree d hypersurface section of a fixed smooth projective variety as d goes to infinity we show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed grothendieck ring of varieties and deduce from this the stabilization of the hodge euler characteristic of natural families of local systems constructed from the vanishing cohomology we prove explicit formulas for the stable values using a probabilistic interpretation along with the natural analogs in point counting over finite fields we explain how these results provide new geometric examples of a weak version of representation stability for symmetric symplectic and orthogonal groups this interpretation of representation stability was studied in the prequel 20 for configuration spaces

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

on inviscid limits for the stochastic navier stokes equations and related models we study inviscid limits of invariant measures for the 2d stochastic navier stokes equations as shown by kuksin j stat phys 115 1 2 469 492 2004 the noise scaling sqrt nu is the only one which leads to non trivial limiting measures which are invariant for the 2d euler equations using a moser type iteration for stochastic drift diffusion equations we show that any limiting measure mu 0 is in fact supported on bounded vorticities relationships of mu 0 to the long term dynamics of 2d euler in l infty with the weak topology are discussed we also obtain a drift independent modulus of continuity for a stationary deterministic model problem which leads us to conjecture that in fact mu 0 is supported on c 0 moreover in view of the batchelor krainchnan 2d turbulence theory we consider inviscid limits for a weakly damped stochastic navier stokes equation in this setting we show that only an order zero noise scaling with respect to leads to a nontrivial limiting measure in the inviscid limit

the topology of rational functions and divisors of surfaces segal studied the homotopy types of the spaces of holomorphic functions of the 2 sphere s 2 of closed surfaces of higher genus and of the spaces of divisors of these surfaces we continue segal s program by describing the entire stable homotopy types of these spaces in terms of the homotopy types of more familiar spaces

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

postbuckling and vibration of end supported elastica pipes conveying fluid and columns under follower loads abstract fluid conveying pipes with supported ends buckle when the fluid velocity reaches a critical value for higher velocities the postbuckled equilibrium shape can be directly related to that for a column under a follower end load however the corresponding vibration frequencies are different due to the coriolis force associated with the fluid flow clamped clamped pinned pinned and clamped pinned pipes are considered first axial sliding is permitted at the downstream end the pipe is modeled as an inextensible elastica the equilibrium shape may have large displacements and small motions about that shape are analyzed the behavior is conservative in the prebuckling range and nonconservative in the postbuckling range during which the coriolis force does work and the motions decay next related columns are studied first with a concentrated follower load at the axially sliding end and then with a distributed follower load in all cases a shooting method is used to solve the nonlinear boundary value problem for the equilibrium configuration and to solve the linear boundary value problem for the first four vibration frequencies the results for the three different types of loading are compared

ergodic theory of infinite dimensional systems with applications to dissipative parabolic pdes we consider a class of randomly perturbed dynamical systems satisfying conditions which reflect the properties of general nonlinear dissipative parabolic pdes results on invariant measures and their exponential mixing properties are proved and applications to 2d navier stokes systems are included

homological stability for configuration spaces of manifolds let cn m be the configuration space of n distinct ordered points in m we prove that if m is any connected orientable manifold closed or open the homology groups hi cn m are representation stable in the sense of church and farb arxiv 1008 1368 applying this to the trivial representation we obtain as a corollary that the unordered configuration space bn m satisfies classical homological stability for each i hi bn m hi bn 1 m for n i this improves on results of mcduff segal and others for open manifolds applied to closed manifolds this provides natural examples where rational homological stability holds even though integral homological stability fails

fi modules and stability for representations of symmetric groups in this paper we introduce and develop the theory of fi modules we apply this theory to obtain new theorems about the cohomology of the configuration space of n distinct ordered points on an arbitrary connected oriented manifold the diagonal coinvariant algebra on r sets of n variables the cohomology and tautological ring of the moduli space of n pointed curves the space of polynomials on rank varieties of n n matrices the subalgebra of the cohomology of the genus n torelli group generated by h1 and more the symmetric group sn acts on each of these vector spaces in most cases almost nothing is known about the characters of these representations or even their dimensions we prove that in each fixed degree the character is given for n large enough by a polynomial in the cycle counting functions that is independent of n in particular the dimension is eventually a polynomial in n in this framework representation stability in the sense of church farb for a sequence of sn representations is converted to a finite generation property for a single fi module

irreducibility of the three and two and a half dimensional hall magnetohydrodynamics system abstract the hall magnetohydrodynamics system has been considered by many to play an important role in astrophysical plasmas star formation and magnetic re connection the hall term mathematically elevates the magnetohydrodynamics system from a semi linear to a quasi linear type and standard results which are well known for the navier stokes equations and the magnetohydrodynamics system such as the local well posedness in the non diffusive case as well as the uniqueness of the weak solution in case the solution depends only on two spatial variables are completely open for the hall magnetohydrodynamics system because the hall magnetohydrodynamics system is not known to be globally well posed even if the solution depends only on two spatial variables a typical method to prove ergodicity for the two dimensional navier stokes equations such as hairer and mattingly 2006 seems to have no chance in this manuscript we take advantage of the structure of the hall term follow the work of flandoli 1997 and prove its irreducibility property in order to elaborate on the effect of the hall term we also prove analogous results for related systems such as the magnetohydrodynamics system benard problem magnetic benard problem micropolar fluid system as well as the magneto micropolar fluid system at the time of writing this manuscript due to a technical difficulty the strong feller property of the hall magnetohydrodynamics system is an open problem

on the stochastic p laplace equation the p laplace equation with random perturbation is studied for the singular case 1 p 2 in this paper some properties of the invariant measure and transition semigroups are obtained the main tool is the dimension free harnack inequality which is established by using the coupling argument as consequences some ergodicity compactness and contractive properties are derived for the associated transition semigroups the main results are applied to stochastic reaction diffusion equations and the stochastic p laplace equation in hilbert space

the dissipative scale of the stochastics navier stokes equation regularization and analyticity we prove spatial analyticity for solutions of the stochastically forced navier stokes equation provided that the forcing is sufficiently smooth spatially we also give estimates which extend to the stationary regime providing strong control of both of the expected rate of dissipation and fluctuations about this mean surprisingly we could not obtain non random estimates of the exponential decay rate of the spatial fourier spectra