Technical Books

This book explains the following topics, Introduction, Fundamental group, Classification of compact surfaces, Covering spaces, Homology, Basics of Cohomology, Cup Product in Cohomology, Poincar� Duality, Basics of Homotopy Theory, Spectral Sequences. Applications, Fiber bundles, Classifying spaces, Applications, Vector Bundles, Characteristic classes, Cobordism, Applications.

This PDF Lectures covers the following topics related to Algebraic Topology, Singular homology, Introduction, singular simplices and chains, Homology, Categories, functors, and natural transformations, Basic homotopy theory, The homotopy theory of CW complexes, Vector bundles and principal bundles, Spectral sequences and Serre classes, Characteristic classes, Steenrod operations, and cobordism.

This note explains the following topics, Chain Complexes, Homology, and Cohomology, Homological Algebra, Products, Fiber Bundles, Homology with Local Coefficients, Fibrations, Cofibrations and Homotopy Groups, Obstruction Theory and EilenbergMacLane Spaces, Bordism, Spectra, and Generalized Homology, Spectral Sequences.

This book contains below topics, Point Set Topology Background, Abstract Algebra Background, Traditional Homology Theory, Persistent Homology, Related Techniques, Some Unfinished Ideas, Cohomology, Homotopy Theory, Obstruction Theory, Steenrod Squares and Reduced Powers, Homotopy Groups of Spheres.

This note contains below topics, Gelfand category and Gelfand functor, Gelfand space and Ktheory, Algebraic topology of the noncommutative torus, Hausdorff blowingup, Groupoids foliations and their coverings, Gelfand spaces of continuous trace C algebras, Appendices, Category theory, Topology, Lattices and filters, Sheaves and cohomology, C algebras, Continuous trace C algebras, C algebras of groupoids, Noncommutative torus Tn.

This book contains below topics, Configuration spaces and braids, Cohomology of ConfkRn, Covering theorems, Deferred proofs, Mapping space models, Homology calculations from stable splittings, Mod p cohomology, Postscript Lie algebra methods, Split simplicial spaces, Homotopy colimits, The Spanier Whitehead category, Tate cohomology and periodicity.

The contents of this book include, Course Introduction, Zariski topology, Affine Varieties, Projective Varieties, Noether Normalization, Grassmannians, Finite and Affine Morphisms, More on Finite Morphisms and Irreducible Varieties, Function Field, Dominant Maps, Product of Varieties, Separateness, Sheaf Functors and Quasicoherent Sheaves, Quasicoherent and Coherent Sheaves, Invertible Sheaves, Quasi coherent sheaves on Projective Spaces, Divisors and the Picard Group, Bezouts Theorem, AbelJacobi Map, Elliptic Curves, KSmoothness, Canonical Bundles, the Adjunction Formulaahler Differentials, Cotangent Bundles of Grassmannians, Bertinis Theorem, Coherent Sheves on Curves, Derived Functors, Existence of Sheaf Cohomology, BirkhoffGrothendieck, RiemannRoch, Serre Duality, Proof of Serre Duality.

This PDF book covers the following topics related to Algebraic Geometry, General Remarks on Computer Algebra Systems, The GeometryAlgebra Dictionary, Affine Algebraic Geometry, Ideals in Polynomial Rings, Affine Algebraic Sets, Hilberts Nullstellensatz, Irreducible Algebraic Sets, Removing Algebraic Sets, Polynomial Maps, The Geometry of Elimination, Noether Normalization and Dimension, Local Studies, Projective Algebraic Geometry, The Projective Space, Projective Algebraic Sets, Affine Charts and the Projective Closure, The Hilbert Polynomial, Computing, Standard Bases and Singular, Applications, Ideal Membership, Elimination, Radical Membership, Ideal Intersections, Ideal Quotients, Kernel of a Ring Map, Integrality Criterion, Noether Normalization, Subalgebra Membership, Homogenization, Dimension and the Hilbert Function, Primary Decomposition and Radicals, Buchbergers Algorithm and Field Extensions, Sudoku, A Problem in Group Theory Solved by Computer Algebra, Finite Groups and Thompsons Theorem, Characterization of Finite Solvable Groups.

This note covers classical result, Affine varieties, Local properties of affine varieties, Varieties and Schemes, Projective varieties and Curves.

This note covers notation, What is algebraic geometry, Affine algebraic varieties, Projective algebraic varieties, Sheaves, ringed spaces and affine algebraic varieties, Algebraic varieties, Projective algebraic varieties, revisited, Morphisms, Products, Dimension, The fibres of a morphism, Sheaves of modules, Hilbert polynomials and Bezouts theorem, Products of preschemes, Proj and projective schemes, More properties of schemes, More properties of schemes, Relative differentials, Locally free sheaves and vector bundles, Cartier divisors, Rational equivalence and the chow group, Proper push forward and flat pull back, Chern classes of line bundles.

This note contains below topics, Preliminary Material, Lelong Numbers and Intersection Theory, Holomorphic Vector Bundles, Connections and Curvature, Bochner Technique and Vanishing Theorems, Numerically Eective Line Bundles, Seshadri Constants and the Fujita Conjecture, Algebraic Approach of the Fujita Conjecture, Regularization of Currents and Selfintersection Inequ alities, Use of MongeAmp ere Equations, Numerical Criteria for Very Ample Line Bundles, Holomorphic Morse Inequalities.

This note contains below topics, Examples of homogeneous varieties, Auxilliary varieties, Topology and consequences, Projective dierential invariants, Varieties with degenerate Gauss images, When can a uniruled variety be smooth, Dual varieties, Linear systems of bounded and constant rank, Secant and tangential varieties, Systems of quadrics with tangential defects, Recognizing uniruled varieties.

This note contains below topics, Overview Higher algebra, Derived commutative algebra, Derived algebraic geometry, The stack of perfect modules.

This book contains below topics, Introduction and dgalgebras, Infinity categories and model categories, Consequences for dgalgebras, Simplicial structures, Geometric nstacks, Derived geometric nstacks.

This note contains below topics, What is the Universal Algebraic Geometry Lecture, Algebraic sets and algebraic varieties Lecture, Geometric properties and geometric relations of algebras Lecture, Algebraic geometry in groupbased algebras Lecture, Isomorphisms of categories of algebraic sets Lecture, Algebraic geometry in the First Order Logic FOL Lecture, Generic view, Some information from universal algebra, Algebras with a fixed algebra of constants, Systems of equations and algebraic sets, Lattices of algebraic sets, Categories of algebraic sets.