fractional laplacian phase transitions and boundary reactions a geometric inequality and a symmetry result we deal with symmetry properties for solutions of nonlocal equations of the type sv f v in rn where s 0 1 and the operator s is the so called fractional laplacian the study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation div x u 0on rn 0 x ux f u on rn 0 where 1 1 y rn x 0 and u u y x this equation is related to the fractional laplacian since the dirichlet to neumann operator u r n 1 x ux r n 1 is 1 2 more generally we study the so called boundary reaction equations given by div x u g x u 0on rn 0 x ux f u on rn 0 under some natural assumptions on the diffusion coefficient and on the nonlinearities f and g we prove a geometric formula of poincare type for stable solutions from which we derive a symmetry result in the spirit of a conjecture of de giorgi

symmetrized models of last passage percolation and non intersecting lattice paths it has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups u l sp 2l and o l we present a theory of such results based on non intersecting lattice paths and integration techniques familiar from the theory of random matrices detailed derivations of probabilities relating to two further symmetrizations are also given

tropical combinatorics and whittaker functions we establish a fundamental connection between the geometric robinson schensted knuth rsk correspondence and gl n r whittaker functions analogous to the well known relationship between the rsk correspondence and schur functions this gives rise to a natural family of measures associated with gl n r whittaker functions which are the analogues in this setting of the schur measures on integer partitions the corresponding analogue of the cauchy littlewood identity can be seen as a generalization of an integral identity for gl n r whittaker functions due to bump and stade as an application we obtain an explicit integral formula for the laplace transform of the law of the partition function associated with a 1 dimensional directed polymer model with log gamma weights recently introduced by one of the authors

geometric rsk correspondence whittaker functions and symmetrized random polymers we show that the geometric lifting of the rsk correspondence introduced by a n kirillov physics and combinatorics proc nagoya 2000 2nd internat workshop pp 82 150 2001 is volume preserving with respect to a natural product measure on its domain and that the integrand in givental s integral formula for mathit gl n mathbb r whittaker functions arises naturally in this context apart from providing further evidence that whittaker functions are the natural analogue of schur polynomials in this setting our results also provide a new combinatorial framework for the study of random polymers when the input matrix consists of random inverse gamma distributed weights the probability distribution of a polymer partition function constructed from these weights can be written down explicitly in terms of whittaker functions next we restrict the geometric rsk mapping to symmetric matrices and show that the volume preserving property continues to hold we determine the probability law of the polymer partition function with inverse gamma weights that are constrained to be symmetric about the main diagonal with an additional factor on the main diagonal the third combinatorial mapping studied is a variant of the geometric rsk mapping for triangular arrays which is again showed to be volume preserving this leads to a formula for the probability distribution of a polymer model whose paths are constrained to stay below the diagonal we also show that the analogues of the cauchy littlewood identity in the setting of this paper are equivalent to a collection of whittaker integral identities conjectured by bump number theory trace formulas and discrete groups pp 49 109 1989 and bump and friedberg festschrift in honor of piatetski shapiro part ii pp 47 65 1990 and proved by stade am j math 123 121 161 2001 israel j math 127 201 219 2002 our approach leads to new combinatorial proofs and generalizations of these identities with some restrictions on the parameters

schur dynamics of the schur processes abstract we construct discrete time markov chains that preserve the class of schur processes on partitions and signatures one application is a simple exact sampling algorithm for q volume distributed skew plane partitions with an arbitrary back wall another application is a construction of markov chains on infinite gelfand tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite dimensional unitary group

shuffling algorithm for boxed plane partitions we introduce discrete time markov chains that preserve uniform measures on boxed plane partitions elementary markov steps change the size of the box from a b c to a 1 b 1 c or a 1 b 1 c algorithmic realization of each step involves o a b c operations one application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions trajectories of our markov chains can be viewed as random point configurations in the three dimensional lattice we compute the bulk limits of the correlation functions of the resulting random point process on suitable two dimensional sections the limiting correlation functions define a two dimensional determinantal point processes with certain gibbs properties

plancherel representations of u infty and correlated gaussian free fields we study asymptotics of traces of noncommutative monomials formed by images of certain elements of the universal enveloping algebra of the infinitedimensional unitary group in its plancherel representations we prove that they converge to commutative moments of a gaussian process that can be viewed as a collection of simply yet nontrivially correlated two dimensional gaussian free fields the limiting process has previously arisen via the global scaling limit of spectra for submatrices of wigner hermitian random matrices the results of the present work were announced in 4

stochastic higher spin six vertex model and q taseps abstract we present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the macdonald processes hierarchy first we show how macdonald q difference operators with t 0 an algebraic tool crucial for studying the corresponding macdonald processes can be utilized to get q moments of the height function h in the higher spin six vertex model first computed in 21 using bethe ansatz this result in particular implies that for the vertex model with the step bernoulli boundary condition the value of h at an arbitrary point n 1 t z 2 z 1 has the same distribution as the last component n of a random partition under a specific t 0 macdonald measure on the other hand it is known that x n n n can be identified with the location of the nth particle in a certain discrete time q tasep started from the step initial configuration the second construction we present is a coupling of this q tasep and the higher spin six vertex model with the step bernoulli boundary condition along time like paths providing an independent probabilistic explanation of the equality of h n 1 t and x n n in distribution as an illustration of our main results we obtain gue tracy widom asymptotics of a certain discrete time q tasep with the step initial configuration and special jump parameters by means of schur measures which are t q macdonald measures this analysis combines our results with the identification of averages of observables between the stochastic higher spin six vertex model and schur measures obtained recently in 8

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

markov processes on the path space of the gelfand tsetlin graph and on its boundary we construct a four parameter family of markov processes on infinite gelfand tsetlin schemes that preserve the class of central gibbs measures any process in the family induces a feller markov process on the infinite dimensional boundary of the gelfand tsetlin graph or equivalently the space of extreme characters of the infinite dimensional unitary group u the process has a unique invariant distribution which arises as the decomposing measure in a natural problem of harmonic analysis on u posed in olshanski 2003 44 as was shown in borodin and olshanski 2005 11 this measure can also be described as a determinantal point process with a correlation kernel expressed through the gauss hypergeometric function

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

harnack inequality for fractional sub laplacians in carnot groups in this paper we prove an invariant harnack inequality on carnot caratheodory balls for fractional powers of sub laplacians in carnot groups the proof relies on an abstract formulation of a technique recently introduced by caffarelli and silvestre in addition we write explicitly the poisson kernel for a class of degenerate subelliptic equations in product type carnot groups

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

a path transformation for random walks and the robinson schensted correspondence the author and marc yor recently introduced a path transformation g k with the property that for x belonging to a certain class of random walks on z k the transformed walk g k x has the same law as the original walk conditioned never to exit the weyl chamber x x 1 less than or equal to less than or equal to x k in this paper we show that g k is closely related to the robinson schensted algorithm and use this connection to give a new proof of the above representation theorem the new proof is valid for a larger class of random walks and yields additional information about the joint law of x and g k x the corresponding results for the brownian model are recovered by donsker s theorem these are connected with hermitian brownian motion and the gaussian unitary ensemble of random matrix theory the connection we make between the path transformation g k and the robinson schensted algorithm also provides a new formula and interpretation for the latter this can be used to study properties of the robinson schensted algorithm and moreover extends easily to a continuous setting

conditioned random walks and the rsk correspondence we consider the stochastic evolution of three variants of the rsk algorithm giving both analytic descriptions and probabilistic interpretations symmetric functions play a key role and the probabilistic interpretations are obtained by elementary doob hunt theory in each case the evolution of the shape of the tableau obtained via the rsk algorithm can be interpreted as a conditioned random walk this is intuitively appealing and can be used for example to obtain certain relationships between orthogonal polynomial ensembles in a certain scaling limit there is a continous version of the rsk algorithm which inherits much of the structure exhibited in the discrete settings intertwining relationships between conditioned and unconditioned random walks are also given in the continuous limit these are related to the harish chandra itzyksen zuber integral

the q pushasep a new integrable model for traffic in 1 1 dimension we introduce a new interacting stochastic particle system q pushasep which interpolates between the q tasep of borodin and corwin probab theory relat fields 158 1 2 225 400 2014 see also borodin et al ann probab 42 6 2314 2382 2014 borodin and corwin int math res not 2 499 537 2015 o connell and pei electron j probab 18 95 1 25 2013 borodin et al comput math 2013 and the q pushtasep introduced recently borodin and petrov adv math 2013 in the q pushasep particles can jump to the left or to the right and there is a certain partially asymmetric pushing mechanism present this particle system has a nice interpretation as a model of traffic on a one lane highway using the quantum many body system approach we explicitly compute the expectations of a large family of observables for this system in terms of nested contour integrals we also discuss relevant fredholm determinantal formulas for the distribution of the location of each particle and connections of the model with a certain two sided version of macdonald processes and with the semi discrete stochastic heat equation

fractional laplacian in conformal geometry abstract in this note we study the connection between the fractional laplacian operator that appeared in the recent work of caffarelli and silvestre and a class of conformally covariant operators in conformal geometry

pseudoholomorphic curves in symplectizations with applications to the weinstein conjecture in dimension three l in t roduct ion 515 1 1 the weinstein conjecture 515 1 2 the main results 517 1 3 sketch of the proof 522 2 local f redho lm theory 525 2 1 similar i ty principle and consequences 525 2 2 n m as a lmost complex manifold and local fillings 529 2 3 local propert ies of ho lomorphic curves 531 3 bubbl ing off analysis and global fredholm theory 533 3 l grad ien t bounds imply ca bounds 533 3 2 bubbing off analysis of ho lomorphic planes 534 3 3 grad ien t bounds near the boundary 537 4 proof of the main results 544 4 1 set up for a global fredholm theory 544 4 2 bishop s theorem 545 4 3 proof of theorem 5 546 4 4 proof of theorems 8 and 11 547 4 5 uniqueness results for foliations 547 5 appendix 550 5 1 per turbing surfaces 550 5 2 ki l l ing crit ical points of hypersurfaces 552 5 3 adding a thin tube to a sphere 560

an extension problem for the cr fractional laplacian we show that the conformally invariant fractional powers of the sub laplacian on the heisenberg group are given in terms of the scattering operator for an extension problem to the siegel upper halfspace remarkably this extension problem is different from the one studied among others by caffarelli and silvestre we also prove an energy identity that yields a sharp trace sobolev embedding

spectral theory for interacting particle systems solvable by coordinate bethe ansatz we develop spectral theory for the q hahn stochastic particle system introduced recently by povolotsky that is we establish a plancherel type isomorphism result that implies completeness and biorthogonality statements for the bethe ansatz eigenfunctions of the system owing to a markov duality with the q hahn tasep a discrete time generalization of tasep with particles jump distribution being the orthogonality weight for the classical q hahn orthogonal polynomials we write down moment formulas that characterize the fixed time distribution of the q hahn tasep with general initial data the bethe ansatz eigenfunctions of the q hahn system degenerate into eigenfunctions of other not necessarily stochastic interacting particle systems solvable by the coordinate bethe ansatz this includes the asep the asymmetric six vertex model and the heisenberg xxz spin chain all models are on the infinite lattice in this way each of the latter systems possesses a spectral theory too in particular biorthogonality of the asep eigenfunctions which follows from the corresponding q hahn statement implies symmetrization identities of tracy and widom for asep with either step or step bernoulli initial configuration as corollaries another degeneration takes the q hahn system to the q boson particle system dual to q tasep studied in detail in our previous paper 2013 thus at the spectral theory level we unify two discrete space regularizations of the kardar parisi zhang equation stochastic heat equation namely q tasep and asep

fractional adams moser trudinger type inequality on heisenberg group abstract the aim of this paper is to establish singular fractional adams moser trudinger inequality for both bounded and unbounded domains in the heisenberg group h n we first establish fractional adams moser trudinger type inequality on domain r n with finite measure theorem 1 12 and then using this inequality and hardy littlewood sobolev inequality adapted to the result of r o neil 1963 we establish singular fractional adams moser trudinger type inequality on domain r n with finite measure theorem 1 13 we also establish singular fractional adams moser trudinger type inequality in h n using the approach of n lam and g lu 2012 2013 the main idea of our approach is that any function in higher order fractional sobolev space in heisenberg group can be represented in terms of riesz potential and then using techniques from harmonic analysis and kernel properties of the associated operator we establish fractional adams moser trudinger type inequality in h n in this paper our approach is quite simple and free from symmetrization arguments as an applications of our theorems we establish the existence of solution to the following class of problems t u f u a b u 1 u in u 0 in h n where is a bounded subset of h n of class c 0 1 with bounded boundary and 0 a q f satisfies either the subcritical exponential growth or critical exponential growth condition and b is a small l 2 perturbation that is there exists a small 0 with 0 b l 2 0 1 and q 2

regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian we use a characterization of the fractional laplacian as a dirichlet to neumann operator for an appropriate differential equation to study its obstacle problem we write an equivalent characterization as a thin obstacle problem in this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

directed polymers and the quantum toda lattice we characterize the law of the partition function of a brownian directed polymer model in terms of a diffusion process associated with the quantum toda lattice the proof is via a multidimensional generalization of a theorem of matsumoto and yor concerning exponential functionals of brownian motion it is based on a mapping which can be regarded as a geometric variant of the rsk correspondence

from duality to determinants for q tasep and asep we prove duality relations for two interacting particle systems the q deformed totally asymmetric simple exclusion process q tasep and the asymmetric simple exclusion process asep expectations of the duality functionals correspond to certain joint moments of particle locations or integrated currents respectively duality implies that they solve systems of odes these systems are integrable and for particular step and half stationary initial data we use a nested contour integral ansatz to provide explicit formulas for the systems solutions and hence also the moments we form laplace transform like generating functions of these moments and via residue calculus we compute two different types of fredholm determinant formulas for such generating functions for asep the first type of formula is new and readily lends itself to asymptotic analysis as necessary to reprove gue tracy widom distribution fluctuations for asep while the second type of formula is recognizable as closely related to tracy and widom s asep formula comm math phys 279 2008 815 844 j stat phys 132 2008 291 300 comm math phys 290 2009 129 154 j stat phys 140 2010 619 634 for q tasep both formulas coincide with those computed via borodin and corwin s macdonald processes probab theory related fields 2014 158 225 400 both q tasep and asep have limit transitions to the free energy of the continuum directed polymer the logarithm of the solution of the stochastic heat equation or the hopf cole solution to the kardar parisi zhang equation thus q tasep and asep are integrable discretizations of these continuum objects the systems of odes associated to their dualities are deformed discrete quantum delta bose gases and the procedure through which we pass from expectations of their duality functionals to characterizing generating functions is a rigorous version of the replica trick in physics

a c 2 smooth counterexample to the hamiltonian seifert conjecture in r 4 we construct a proper c 2 smooth function on r 4 such that its hamiltonian flow has no periodic orbits on at least one regular level set this result can be viewed as a c 2 smooth counterexample to the hamiltonian seifert conjecture in dimension four

anisotropic 2 1 d growth and gaussian limits of q whittaker processes we consider a discrete model for anisotropic 2 1 dimensional growth of an interface height function owing to a connection with q whittaker functions this system enjoys many explicit integral formulas by considering certain gaussian stochastic differential equation limits of the model we are able to prove a space time limit of covariances to those of the 2 1 dimensional additive stochastic heat equation or edwards wilkinson equation along characteristic directions in particular the bulk height function converges to the gaussian free field which evolves according to this stochastic pde

stochastic higher spin vertex models on the line we introduce a four parameter family of interacting particle systems on the line which can be diagonalized explicitly via a complete set of bethe ansatz eigenfunctions and which enjoy certain markov dualities using this for the systems started in step initial data we write down nested contour integral formulas for moments and fredholm determinant formulas for laplace type transforms taking various choices or limits of parameters this family degenerates to many of the known exactly solvable models in the kardar parisi zhang universality class as well as leads to many new examples of such models in particular asymmetric simple exclusion process the stochastic six vertex model q totally asymmetric simple exclusion process and various directed polymer models all arise in this manner our systems are constructed from stochastic versions of the r matrix related to the six vertex model one of the key tools used here is the fusion of r matrices and we provide a probabilistic proof of this procedure

generalized counterexamples to the seifert conjecture using the theory of plugs and the self insertion construction due to the second author we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise linear fashion so that all minimal sets have codimension 1 in particular the 3 sphere s 3 has a real analytic dynamical system such that all limit sets are 2 dimensional we also prove that a 1 dimensional foliation of a manifold of dimension at least 3 can be modified in a piecewise linear fashion so that there are no closed leaves but all minimal sets are 1 dimensional these theorems provide new counterexamples to the seifert conjecture which asserts that every dynamical system on s 3 with no singular points has a periodic trajectory

nearest neighbor markov dynamics on macdonald processes abstract macdonald processes are certain probability measures on two dimensional arrays of interlacing particles introduced by borodin and corwin in 7 they are defined in terms of nonnegative specializations of the macdonald symmetric functions and depend on two parameters q t 0 1 our main result is a classification of continuous time nearest neighbor markov dynamics on the space of interlacing arrays that act nicely on macdonald processes the classification unites known examples of such dynamics and also yields many new ones when t 0 one dynamics leads to a new integrable interacting particle system on the one dimensional lattice which is a q deformation of the pushtasep long range tasep when q t the macdonald processes become the schur processes of okounkov and reshetikhin 41 in this degeneration we discover new robinson schensted type correspondences between words and pairs of young tableaux that govern some of our dynamics

quasiregular mappings of the heisenberg group the condition 1 1 means that for all d g the coordinate functions o f f belong to the sobolev space w d of functions in l d whose distributional first order partial derivatives are also l integrable in d in the above definition f x is the formal derivative of f a t x defined by means of the partial derivatives i f x l is the operator norm of f x and ji x is the jacobian determinant of f a t x these are uniquely defined almost everywhere by 1 1 it turns out that quasiregular mappings are differentiable almost everywhere and a nonconstant quasiregular map is discrete open and sense preserving for these and other basic properties of qr maps we refer to books re by reshetnyak vu by vuorinen and also to a forthcoming book r6 by the second author the definition extends immediately to the case f m n where m and n are connected oriented riemannian n manifolds we say that a continuous map f m n is quasiregular if it has a local representation at each point of m which is quasiregular in the above sense and if there exists k 1 k oe such that

operateurs uniformement sous elliptiques sur les groupes de lie given a connected unimodular lie group g having polynomial volume growth and left invariant vector fields x1 xk which generate the lie algebra of g consider the operator la i j 1kxiaijxj where a g rk rk is a smooth symmetric matrix valued function which satisfies a 2 i j 1k aij x i j 1 2 x g rk for some 0 1 what we show is that the corresponding heat flow semigroup etla admits a kernel which satisfies two sided gaussian estimates in terms of the control distance determined by the xi s moreover the estimates can be made to depend on a only through we also prove harnack inequalities and holder regularity of solutions we end with a discussion of some other sub elliptic situations

taubes s proof of the weinstein conjecture in dimension three does every smooth vector field on a closed three manifold for example the three sphere have a closed orbit no according to counterexamples by k kuperberg and others on the other hand there is a special class of vector fields called reeb vector fields which are associated to contact forms the three dimensional case of the weinstein conjecture asserts that every reeb vector field on a closed oriented three manifold has a closed orbit this conjecture was recently proved by taubes using seiberg witten theory we give an introduction to the weinstein conjecture the main ideas in taubes s proof and the bigger picture into which it fits taubes s proof of the weinstein conjecture is the culmination of a large body of work both by taubes and by others in an attempt to make this story accessible to nonspecialists much of the present article is devoted to background and context and taubes s proof itself is only partially explained hopefully this article will help prepare the reader to learn the full story from taubes s paper 62 more exposition of this subject which was invaluable in the preparation of this article can be found in the online video archive from the june 2008 msri hot topics workshop 44 and in the article by auroux 5 below in 1 3 we introduce the statement of the weinstein conjecture and discuss some examples in 4 6 we discuss a natural strategy for approaching the weinstein conjecture which proves it in many but not all cases and provides background for taubes s work in 7 we give an overview of the big picture surrounding taubes s proof of the weinstein conjecture readers who already have some familiarity with the weinstein conjecture may wish to start here in 8 9 we recall necessary material from seiberg witten theory in 10 we give an outline of taubes s proof and in 11 we explain some more details of it to conclude in 12 we discuss some further results and open problems related to the weinstein conjecture 1 statement of the weinstein conjecture the weinstein conjecture asserts that certain vector fields must have closed orbits before stating the conjecture at the end of this section we first outline its origins this is discussion is only semi historical because only a sample of the relevant works will be cited and not always in chronological order 1 1 closed orbits of vector fields let y be a closed manifold in this article all manifolds and all objects defined on them are smooth unless otherwise stated 2000 mathematics subject classification 57r17 57r57 53d40 partially supported by nsf grant dms 0806037

an index for closed orbits in beltrami fields abstract we consider the class of beltrami fields eigenfields of the curl operator on three dimensional riemannian solid tori such vector fields arise as steady incompressible inviscid fluids and plasmas using techniques from contact geometry we construct an integer valued index for detecting closed orbits in the flow which are topologically inessential they have winding number zero with respect to the solid torus the most important feature of the index is its computability it can be rigorously determined from a c 1 approximation to the vector field on any meridional disc of the solid torus as a result we obtain a test for detecting the non existence of a global cross section to the 3d flow via purely 2d information