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Algebraic topology --- Hodge theory --- Jordan matrix --- Singularities (Mathematics) --- Hodge, Théorie de --- Singularités (mathématiques) --- Geometry, Algebraic --- Form, Jordan --- Form, Jordan normal --- Jordan form --- Jordan normal form --- Matrix, Jordan --- Matrices --- Complex manifolds --- Differentiable manifolds --- Homology theory --- Hodge, Théorie de.

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This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process. Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high co-dimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right half-plane. Epstein and Mazzeo also demonstrate precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

Elliptic operators. --- Markov processes. --- Population biology --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Stochastic processes --- Differential operators, Elliptic --- Operators, Elliptic --- Partial differential operators --- Mathematical models. --- 1-dimensional integral. --- Euclidean model problem. --- Euclidean space. --- Hlder space. --- Hopf boundary point. --- Kimura diffusion equation. --- Kimura diffusion operator. --- Laplace transform. --- Schauder estimate. --- WrightІisher geometry. --- adjoint operator. --- backward Kolmogorov equation. --- boundary behavior. --- degenerate elliptic operator. --- doubling. --- elliptic Kimura operator. --- elliptic equation. --- forward Kolmogorov equation. --- function space. --- general model problem. --- generalized Kimura diffusion. --- heat equation. --- heat kernel. --- higher dimensional corner. --- higher regularity. --- holomorphic semi-group. --- homogeneous Cauchy problem. --- hybrid space. --- hypersurface boundary. --- induction hypothesis. --- induction. --- inhomogeneous problem. --- irregular solution. --- long time asymptotics. --- long-time behavior. --- manifold with corners. --- martingale problem. --- mathematical finance. --- model problem. --- normal form. --- normal vector. --- null-space. --- off-diagonal behavior. --- open orthant. --- parabolic equation. --- perturbation theory. --- polyhedron. --- population genetics. --- probability theory. --- regularity. --- resolvent operator. --- semi-group. --- solution operator. --- uniqueness.

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Riemann introduced the concept of a "local system" on P1-{a finite set of points} nearly 140 years ago. His idea was to study nth order linear differential equations by studying the rank n local systems (of local holomorphic solutions) to which they gave rise. His first application was to study the classical Gauss hypergeometric function, which he did by studying rank-two local systems on P1- {0,1,infinity}. His investigation was successful, largely because any such (irreducible) local system is rigid in the sense that it is globally determined as soon as one knows separately each of its local monodromies. It became clear that luck played a role in Riemann's success: most local systems are not rigid. Yet many classical functions are solutions of differential equations whose local systems are rigid, including both of the standard nth order generalizations of the hypergeometric function, n F n-1's, and the Pochhammer hypergeometric functions. This book is devoted to constructing all (irreducible) rigid local systems on P1-{a finite set of points} and recognizing which collections of independently given local monodromies arise as the local monodromies of irreducible rigid local systems. Although the problems addressed here go back to Riemann, and seem to be problems in complex analysis, their solutions depend essentially on a great deal of very recent arithmetic algebraic geometry, including Grothendieck's etale cohomology theory, Deligne's proof of his far-reaching generalization of the original Weil Conjectures, the theory of perverse sheaves, and Laumon's work on the l-adic Fourier Transform.

Differential equations --- Hypergeometric functions. --- Sheaf theory. --- Numerical solutions. --- Additive group. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic differential equation. --- Algebraically closed field. --- Algorithm. --- Analytic continuation. --- Automorphism. --- Axiom of choice. --- Bernhard Riemann. --- Big O notation. --- Calculation. --- Carlos Simpson. --- Coefficient. --- Cohomology. --- Commutator. --- Compactification (mathematics). --- Comparison theorem. --- Complex analytic space. --- Complex conjugate. --- Complex manifold. --- Conjecture. --- Conjugacy class. --- Convolution. --- Corollary. --- Cube root. --- Cusp form. --- De Rham cohomology. --- Differential equation. --- Dimension. --- Dimensional analysis. --- Discrete valuation ring. --- Disjoint union. --- Divisor. --- Duality (mathematics). --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Equation. --- Equivalence of categories. --- Exact sequence. --- Existential quantification. --- Finite field. --- Finite set. --- Fourier transform. --- Functor. --- Fundamental group. --- Generic point. --- Ground field. --- Hodge structure. --- Hypergeometric function. --- Integer. --- Invertible matrix. --- Isomorphism class. --- Jordan normal form. --- Level of measurement. --- Linear differential equation. --- Local system. --- Mathematical induction. --- Mathematics. --- Matrix (mathematics). --- Monodromy. --- Monomial. --- Morphism. --- Natural filtration. --- Parameter. --- Parity (mathematics). --- Perfect field. --- Perverse sheaf. --- Polynomial. --- Prime number. --- Projective representation. --- Projective space. --- Pullback (category theory). --- Pullback. --- Rational function. --- Regular singular point. --- Relative dimension. --- Residue field. --- Ring of integers. --- Root of unity. --- Sequence. --- Sesquilinear form. --- Set (mathematics). --- Sheaf (mathematics). --- Six operations. --- Special case. --- Subgroup. --- Subobject. --- Subring. --- Suggestion. --- Summation. --- Tensor product. --- Theorem. --- Theory. --- Topology. --- Triangular matrix. --- Trivial representation. --- Vector space. --- Zariski topology.