a class of solutions of the time dependent reaction diffusion equation for the processes a a to a and a a to oe consideration is given to the treatment of the nonlinear partial differential equation describing the diffusion of an assembly of particles which simultaneously coagulate or annihilate it is shown how a class of solutions may be obtained in terms of the solutions of an ordinary differential equation and detailed application is made to the case of an initially localized particle distribution

minimal rational curves on moduli spaces of stable bundles let c be a smooth projective curve of genus g 2 and l be a line bundle on c of degree d let m uc r l be the moduli space of stable vector bundles on c of rank r and with the fixed determinant l assume that r d 1 then m is a smooth projective fano variety with picard number 1 for any projective curve on m we can define its degree with respect to the ample anti canonical line bundle km the first result of this paper determines all rational curves of minimal degree passing through a generic point of m which answers a question of jun muk hwang see question 1 in hw

modica type gradient estimates for an inhomogeneous variant of the normalized p laplacian evolution abstract in this paper we study an inhomogeneous variant of the normalized p laplacian evolution which has been recently treated in banerjee and garofalo 2013 does 2011 manfredi et al 2010 and juutinen 2014 we show that if the initial datum satisfies the pointwise gradient estimate 1 6 a e then the unique solution to the cauchy problem 1 2 satisfies the same gradient estimate a e for all later times see 1 7 a general pointwise gradient bound for the entire bounded solutions of the elliptic counterpart of 1 2 was first obtained in caffarelli et al 1994 such estimate generalizes one obtained by l modica for the laplacian and it has connections to a famous conjecture of de giorgi

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

on de giorgi s conjecture in dimensions 4 and 5 in this paper we develop an approach for establishing in some important cases a conjecture made by de giorgi more than 20 years ago the problem originates in the theory of phase transition and is so closely connected to the theory of minimal hypersurfaces that it is sometimes referred to as the eversion of bernstein s problem for minimal graphs the conjecture has been completely settled in dimension 2 by the authors 15 and in dimension 3 in 2 yet the approach in this paper seems to be the first to use in an essential way the solution of the bernstein problem stating that minimal graphs in euclidean space are necessarily hyperplanes provided the dimension of the ambient space is not greater than 8 we note that the solution of bernstein s problem was also used in 18 to simplify an argument in 9 here is the conjecture as stated by de giorgi 12 conjecture 1 1 suppose that u is an entire solution of the equation

field theoretic formulation of an epidemic process with immunisation the author describes a model of the spread of an epidemic on a lattice in which sites may be infected by their neighbours after which they recover and are subsequently immune it corresponds to a modification of the directed percolation problem and also to a growth model of clusters the model is analysed using a continuum field theory both within a self consistent approximation and using an epsilon expansion below six transverse dimensions

nonlinear evolution equations that are non local in space and time abstract we deal with a nonlocal nonlinear evolution problem of the form r n r j x y t s v y s v x t p 2 v y s v x t d y d s 0 for x t r n 0 here p 2 j r n 1 r is a nonnegative kernel compactly supported inside the set x t r n 1 t 0 with r n r j x t d x d t 1 and v stands for an extension of a given initial value f that is v x t v x t t 0 f x t t 0 for this problem we prove existence and uniqueness of a solution in addition we show that the solutions approximate viscosity solutions to the local nonlinear pde u p 2 u t p u when the kernel is rescaled in a suitable way

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

flat connections and geometric quantization using the space of holomorphic symmetric tensors on the moduli space of stable bundles over a riemann surface we construct a projectively flat connection on a vector bundle over teichmuller space the fibre of the vector bundle consists of the global sections of a power of the determinant bundle on the moduli space both dolbeault and cech techniques are used

regularity of flat level sets in phase transitions we consider local minimizers of the ginzburg landau energy functional 1 2 u 2 1 4 1 u 2 2 dx and prove that if the 0 level set is included in a flat cylinder then in the interior it is included in a flatter cylinder as a consequence we prove a conjecture of de giorgi which states that level sets of global solutions of u u 3 u such that u 1 n u 0 lim u x x n 1 x n are hyperplanes in dimension n 8

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

entire solutions of semilinear elliptic equations in r 3 and a conjecture of de giorgi this paper is concerned with the study of bounded solutions of semilinear elliptic equations u f u in the whole space r under the assumption that u is monotone in one direction say nu in r n the goal is to establish the one dimensional character or symmetry of u namely that u only depends on one variable or equivalently that the level sets of u are hyperplanes this type of symmetry question was raised by de giorgi in who made the following conjecture we quote literally page of dg

fluctuation effects in coagulation processes abstract we study diffusion controlled coagulation processes in the system with a finite density of droplets using the fock space formalism we find that the effective reaction rate has a correction to smoluchowski s classical rate constant for three dimensional systems we also predict the scaling form of the scattering function and find that our results qualitatively agree with experimental results

a pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature abstract we prove a pointwise gradient bound for bounded solutions of u f u 0 in possibly unbounded proper domains whose boundary has nonnegative mean curvature we also obtain some rigidity results when equality in the bound holds at some point

spatial fluctuations in reaction diffusion systems a model for exponential growth the spatial fluctuations in an exactly soluble model for the irreversible aggregation of clusters are treated the model is characterized byrate constants k ij i j for the clustering of ani and aj mer anddiffusion constants d j d it is assumed thatd 1 reaction limited aggregation explicit expressions for the correlation functions at equal and at different times are calculated the equal time correlation functions show scaling behavior in the scaling limit the correlation length remains finite ast and the fluctuations becomelarge at large times t t d inany dimension the crossover timet d at which the mean field theory smoluchowski s equation breaks down is given byt d ind these exact results imply that the upper critical dimension of this model isd c and hence that there isno unique upper critical dimension in models for the irreversible aggregation of clusters

scaling of conformal blocks and generalized theta functions over overline mathcal m _ g n by way of intersection theory on overline mathcal m g n we show that geometric interpretations for conformal blocks as sections of ample line bundles over projective varieties do not have to hold at points on the boundary we show such a translation would imply certain recursion relations for first chern classes of these bundles while recursions can fail geometric interpretations are shown to hold under certain conditions

on the evolution governed by the infinity laplacian we investigate the basic properties of the degenerate and singular evolution equation ut d 2 u du du du du which is a parabolic version of the increasingly popular infinity laplace equa tion we prove existence and uniqueness results for both dirichlet and cauchy problems establish interior and boundary lipschitz estimates and a harnack inequality and also provide numerous explicit solutions

tug of war games and pdes in this talk we review some recent results concerning tug of war games and their relation to some well known pdes see 1 2 4 3 in particular we will show that solutions to certain pdes like the p laplacian and the parabolic p laplacian can be obtained as limits of values of tug of war games when the parameter that controls the length of the possible movements goes to zero since the equations under study are nonlinear and not in divergence form we will make extensive use of the concept of viscosity solutions seccion en el cedya 2011 edp

coagulation diffusion systems derivation and existence of solutions for the diffuse interface structure equations abstract this paper considers an infinite system of partial differential equations the coagulation diffusion equations which add spatial diffusion to the classical coagulation equations the main emphasis is placed on deriving an infinite system of ordinary differential equations which described the structured interface between reacting coagulation and dilute concentration existence of solutions to interfacial equaitons is proven under spatial boundary conditions

renormalisation of fluctuation effects in the a a to a reaction the field theoretical description of the a a to a reaction is exactly renormalised to all orders in perturbation theory it is confirmed that it belong to the same universality class as the a a to oe reaction and that the concentration decay like t d 2 for space dimensionality d smaller than two and like ln t t for d 2 are exact results

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

field theoretic approach to a source enhanced aggregation process a diffusion controlled cluster cluster aggregation system with a source is analysed using the fock space formalism and diagrammatic perturbation technique the effective coagulation kernel and equation for low dimensional cases d or 2 are derived the asymptotic size distribution obeying a power law is obtained for any dimension the application to the aggregation process of particles undergoing levy flight is discussed through the analysis in this paper the author clarifies a relation between critical phenomena and a source enhanced aggregation

the dirichlet problem for the fractional p laplacian evolution equation abstract we consider a model of fractional diffusion involving a natural nonlocal version of the p laplacian operator we study the dirichlet problem posed in a bounded domain of r n with zero data outside of for which the existence and uniqueness of strong nonnegative solutions is proved and a number of quantitative properties are established a main objective is proving the existence of a special separate variable solution u x t t 1 p 2 f x called the friendly giant which produces a universal upper bound and explains the large time behaviour of all nontrivial nonnegative solutions in a sharp way moreover the spatial profile f of this solution solves an interesting nonlocal elliptic problem we also prove everywhere positivity of nonnegative solutions with any nontrivial data a property that separates this equation from the standard p laplacian equation

smoluchowski and beyond a field theoretical study of coagulation fragmentation processes adopting simple field theoretical techniques the author demonstrates how the smoluchowski approach to reaction diffusion models can be systematically corrected focusing on coagulation fragmentation processes he identifies the upper critical dimension dc 2 below which finite mobility corrections are important and full universality lattice independence etc can be achieved trends seem by direct simulation of simple lattice models for d 2 are confirmed whilst problems of interpretation exist for d 2

fluctuations in reaction diffusion systems a new exactly soluble growth model the method of compounding moments devised by van kampen is used to study the spatial fluctuations in a model describing the irreversible formation of clusters the reaction and diffusion constants in this model are chosenindependent of the cluster sizes for a monodisperse initial distribution explicit expressions are calculated for the equal time and two time correlation functions of the concentrations ofm andn mers for general initial conditions the fluctuations in the mass density are considered and a scaling theory is presented for the fluctuations at large times extensions to more general models are discussed

a gradient bound and a liouville theorem for nonlinear poisson equations soit f c 2 r une fonction non negative et u c 3 r n une solution dans tout r n de l equation u f u ou f f est la derivee 1ere de f si u est borne sur r n et s il existe x 0 r n tel que f u x 0 0 alors u est constante

on de giorgi s conjecture in dimension n 9 a celebrated conjecture due to de giorgi states that any bounded solution of the equation u 1 u 2 u 0 in r n with ynu 0 must be such that its level setsfu g are all hyperplanes at least for dimension n 8 a counterexample for n 9 has long been believed to exist starting from a minimal graph which is not a hyperplane found by bombieri de giorgi and giusti in r n n 9 we prove that for any small 0 there is a bounded solution u y with yn u 0 which resembles tanh a t p 2 a where t t y denotes a choice of signed distance to the blown up minimal graph 1 this solution is a counterexample to de giorgi s conjecture for n 9

monodromy of the sl n and gl n hitchin fibrations we compute the monodromy of the hitchin fibration for the moduli space of l twisted sl n mathbb c and gl n mathbb c higgs bundles for any n on a compact riemann surface of genus g 1 we require the line bundle l to either be the canonical bundle or satisfy deg l 2g 2 the monodromy group is generated by picard lefschetz transformations associated to vanishing cycles of singular spectral curves we construct such vanishing cycles explicitly and use this to show that the sl n mathbb c monodromy group is a skew symmetric vanishing lattice in the sense of janssen using the classification of vanishing lattices over mathbb z we completely determine the structure of the monodromy groups of the sl n mathbb c and gl n mathbb c hitchin fibrations as an application we determine the image of the restriction map from the cohomology of the moduli space of higgs bundles to the cohomology of a non singular fibre of the hitchin fibration