Table of contents

CONTENTS § 1. Introduction § 2. Definition of generalized functions § 3. Properties of generalized functions § 4. Weak equality § 5. Generalized functions and boundary-value problems for differential equations § 6. The Colombeau theory § 7. On the Maslov theory § 8. Shock waves References

CONTENTS Introduction Part I. Generalized Jacobi conditions 1. The regular representation. Elementary examples 2. Quasi-commutation relations 3. Quadratic relations. Reduction to triangular form 4. The spectrum and cospectrum Part II. Quantization of the cospectrum 5. The spectrum and cospectrum of quadratic-linear relations 6. The transformation of the scale and structure constants by a graded semigroup 7. Algebras equivalent to Lie algebras. L-operator equations References

CONTENTS Introduction Chapter I. The Riemann zeta-function and its connection with primes § 1. Definition of and Euler's identity § 2. Continuation of to the half-plane § 3. Continuation of to the whole plane § 4. Functional properties of and § 5. Zeros of and primes § 6. Elementary theorems on the complex zeros of § 7. Theorems of de la Vallée Poussin § 8. Elementary consequences of the Riemann hypothesis Notes Chapter II. Approximate functional equations for § 1. An approximation of an exponential sum by a sum of integrals § 2. An asymptotic calculation for a class of exponential integrals § 3. An approximation of an exponential sum by a shorter one § 4. Approximate functional equations for Notes Chapter III. Vinogradov's method in the theory of the Riemann zeta-function § 1. The mean-value of a power of the modulus of an exponential sum § 2. Simple lemmas § 3. The main recurrence inequality § 4. Vinogradov's mean-value theorem § 5. An estimate for a zeta-sum and its consequences § 6. The current boundary of the zero-free region and some consequences Notes Chapter IV. A zero-density theorem and primes in short intervals § 1. Auxiliary lemmas § 2. A zero-density theorem § 3. Primes in short intervals Notes Appendix References

CONTENTS Introduction Part I. Supersymmetry 1. Supersymmetric quantum systems 2. Quantum mechanics. Discrete spectrum 3. Index theory 4. Wess-Zumino quantum mechanics with two bosons () 5. Continuous spectrum and scattering theory 6. Quantum algebras 7. Super KMS-functionals 8. Infinite-dimensional quantum algebras 9. Quantum field theory 10. Pfaffians 11. Quantum differential geometry Part II. Quantum solitons 1. The two-dimensional Ising model with continuous time 2. The connection with the one-dimensional quantum Ising model 3. Hamiltonian construction of soliton sectors 4. Euclidean construction of soliton sectors Part III. The boson-fermion correspondence 1. Continuous analogues of infinite-dimensional Lie algebras 2. Central extensions 3. Representations in Dirac space. Bosonization 4. Euclidean bosonization References