Göllnitz-Gordon Partitions with Weights and Parity Conditions -- Partition Identities for the Multiple Zeta Function -- A Perturbative Theory of the Evolution of the Center of Typhoons -- Algebraic Aspects of Multiple Zeta Values -- On the Local Factor of the Zeta Function of Quadratic Orders -- Sums Involving the Hurwitz Zeta-Function Values -- Crystal Symmetry Viewed as Zeta Symmetry -- Sum …
Absolute Values -- Valuations -- Extension of Valuations -- Henselian Fields -- Structure Theory -- Applications of Valuation Theory.Absolute values and their completions -like the p-adic number fields- play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valu…
Problems -- Elementary Number Theory -- Euclidean Rings -- Algebraic Numbers and Integers -- Integral Bases -- Dedekind Domains -- The Ideal Class Group -- Quadratic Reciprocity -- The Structure of Units -- Higher Reciprocity Laws -- Analytic Methods -- Density Theorems -- Solutions -- Elementary Number Theory -- Euclidean Rings -- Algebraic Numbers and Integers -- Integral Bases -- Dedekind Do…
Primes! -- Number-Theoretical Tools -- Recognizing Primes and Composites -- Primality Proving -- Exponential Factoring Algorithms -- Subexponential Factoring Algorithms -- Elliptic Curve Arithmetic -- The Ubiquity of Prime Numbers -- Fast Algorithms for Large-Integer Arithmetic.Prime numbers beckon to the beginner, as the basic notion of primality is accessible even to children. Yet, some of th…
Arithmetic over Function Fields: A Cohomological Approach -- Algebraic Stacks Whose Number of Points over Finite Fields is a Polynomial -- On a Problem of Miyaoka -- Monodromy Groups Associated to Non-Isotrivial Drinfeld Modules in Generic Characteristic -- Irreducible Values of Polynomials: A Non-Analogy -- Schemes over -- Line Bundles and p-Adic Characters -- Arithmetic Eisenstein Classes on…
Squares -- Numbers, Numbers Everywhere -- Fibonacci Numbers -- Prime Numbers -- Choice and Chance -- Geometrical Constructions -- The Algebra of Group.Mathematics Is Not a Spectator Sport challenges the reader to become an active mathematician. Beginning at a gentle pace, the author encourages the reader to get involved, with discussions of an exciting variety of topics, each placed in its hist…
Problems and Tricks -- Number Theory -- Some Applications of Elementary Number Theory -- Ideas and Theories -- Induction and Recursion -- Arithmetic of algebraic numbers -- Arithmetic of algebraic varieties -- Zeta Functions and Modular Forms -- Fermat’s Last Theorem and Families of Modular Forms -- Analogies and Visions -- Introductory survey to part III: motivations and description -- Arake…
Numerical Invariants of a Rees Algebra -- Hilbert Functions and Multiplicities -- Depth and Cohomology of Rees Algebras -- Divisors of a Rees Algebra -- Koszul Homology -- Integral Closure of Algebras -- Integral Closure and Normalization of Ideals -- Integral Closure of Modules -- HowTo.Integral Closure gives an account of theoretical and algorithmic developments on the integral closure of alg…
Part I: Foundations and Computations: Deloopings in Algebraic K- theory -- The Motivic Spectral Sequence -- K-theory of truncated polynomial algebras -- Bott Periodicity in Topological, Algebraic and Hermitian K-theory -- Algebraic K-theory of Rings and Integers in Local and Global Fields. Part II: K-theory and Algebraic Geometry: Motivic Cohomology, K-theory and topological cyclic Homology -- …