Microdifferential Systems in the Complex Domain

ISBN-13: 9783642649042 / Angielski / Miękka / 2011 / 216 str.

P. Schapira

Opis

The words "microdifferential systems in the complex domain" refer to seve ral branches of mathematics: micro local analysis, linear partial differential equations, algebra, and complex analysis. The microlocal point of view first appeared in the study of propagation of singularities of differential equations, and is spreading now to other fields of mathematics such as algebraic geometry or algebraic topology. How ever it seems that many analysts neglect very elementary tools of algebra, which forces them to confine themselves to the study of a single equation or particular square matrices, or to carryon heavy and non-intrinsic formula tions when studying more general systems. On the other hand, many alge braists ignore everything about partial differential equations, such as for example the "Cauchy problem," although it is a very natural and geometri cal setting of "inverse image." Our aim will be to present to the analyst the algebraic methods which naturally appear in such problems, and to make available to the algebraist some topics from the theory of partial differential equations stressing its geometrical aspects. Keeping this goal in mind, one can only remain at an elementary level."

Spis treści

I. Microdifferential Operators.- Summary.- § 1. Construction of the Ring ?x.- 1.1. Differential Operators.- 1.2. Formal Microdifferential Operators.- 1.3. Microdifferential Operators.- Exercises.- § 2. Division Theorems.- 2.1. A Banach Algebra of Operators.- 2.2. The Spaäth and Weierstrass Theorems.- Exercises.- § 3. Refined Microdifferential Cauchy-Kowalewski Theorem.- 3.1. Statement of the Theorem.- 3.2. The Abstract Cauchy-Kowalewski Theorem in Scales of Banach Spaces.- 3.3. Proof of Theorem 3.1.1.- Exercises.- § 4. Microdifferential Modules Associated to a Submanifold.- 4.1. The Sheaf CZ|X.- The Case when Z is a Hypersurface.- 4.3. The Sheaf ?Y?X.- Exercises.- § 5. Quantized Contact Transformations.- 5.1. Division by an Ideal.- 5.2. Adjoint.- 5.3. Quantized Contact Transformations.- 5.4. Examples.- Exercises.- § 6. Systems with Simple Characteristics.- 6.1. Equivalence of Operators.- 6.2. The Regular Involutive Case.- 6.3. Holonomic Systems with Simple Characteristics.- Exercises.- Notes.- II. ?X-modules.- Summary.- § 1. Filtered Rings and Modules.- 1.1. Noetherian and Zariskian Filtrations.- 1.2. Homological Properties.- 1.3. Characteristic Ideal.- 1.4. Sheaves of Filtered Modules.- 1.5. Examples.- Exercises.- § 2. Structure of the Ring ?X.- 2.1. The Ring ?X(0).- 2.2. Main Properties of ?X.- 2.3. Characteristic Cycle.- 2.4. Holonomic Modules.- 2.5. Adjunction of a Dummy Variable.- 2.6. DX-Modules.- Exercises.- § 3. Operations on ?X-modules.- 3.1. Definitions.- 3.2. Operations on BS|X.- 3.3. CS|X.- 3.4. Operations on ?X-modules.- 3.5. Complement on Inverse Images.- Exercises.- Notes.- III. Cauchy Problem and Propagation.- Summary.- § 1. Microcharacteristic Varieties.- 1.1. Normal Cones.- 1.2. 1-microcharacteristic Variety.- 1.3. Characteristic Varieties Associated to a Submersion.- 1.4 Characteristic Varieties Associated to an Embedding.- 1.5. Characteristic Variety of the Systems Induced on a Submanifold.- 1.6. Coherency of the Systems Induced on a Submanifold.- Exercises.- § 2. The Cauchy Problem.- 2.1. The Cauchy Problem for a System.- 2.2. Application: The Cauchy Problem with Data Ramified along Hypersurfaces.- Exercises.- § 3. Propagation.- 3.1. Propagation at the Boundary for one Operator.- 3.2. Propagation for Systems.- Exercises.- § 4. Constructibility.- 4.1. Real and Complex Stratifications.- 4.2. Micro-support of Sheaves.- 4.3. Micro-supports and Microcharacteristic Varieties.- Exercises.- Notes.- Appendices.- A. Symplectic Geometry.- A.1. Symplectic Vector Spaces.- A.2. Symplectic Manifolds.- A.3. Homogeneous Symplectic Structures.- A.4. Contact Transformations.- B. Homological Algebra.- B.1. Categories and Derived Functors.- B.2. Rings and Modules.- B.3. Graded Rings and Modules.- B.4 Koszul Complexes.- B.5. The Mittag-Leffler Condition.- C. Sheaves.- C.1. Presheaves and Sheaves.- C.2. Cohomology of Sheaves.- C.3. ?ech Cohomology.- C.4. An Extension Theorem.- C.5. Coherent Sheaves.- D.1. Support and Multiplicities.- D.2. Homological Dimension.- List of Notations and Conversions.