## Mathematics Department - Graduate Faculty

Generalized Functions, Volume 5: Integral Geometry and Representation Theory is devoted to the theory of representations, focusing on the group of two-dimensional complex matrices of determinant one. This book emphasizes that the theory of representations is a good example of the use of algebraic and geometric methods in functional analysis, in which transformations are performed not on the points of a space, but on the functions defined on it.

The topics discussed include Radon transform on a real affine space, integral transforms in the complex domain, and representations of the group of complex unimodular matrices in two dimensions. The properties of the Fourier transform on G, integral geometry in a space of constant curvature, harmonic analysis on spaces homogeneous with respect to the Lorentz Group, and invariance under translation and dilation are also described.

This volume is suitable for mathematicians, specialists, and students learning integral geometry and representation theory. Kortext is an ebook and etextbook provider for universities, university students and for anyone who loves to read. We have a wide range of , ebooks in our portfolio and the number of titles are increasing daily. We offer a free ebook reader to download with our books where users can freely make notes, highlight texts and do citations and save them in their accounts. Best selling etextooks. Principles and Practice of Marketing.

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## Integral Geometry and Representation Theory

Email, fax, or send via postal mail to:. Graev and N. Translated by Eugene Saletan. The first systematic theory of generalized functions also known as distributions was created in the early s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics.

Some basics of set theory: Relations, Partitions, Functions and sets of functions.

Families of sets, Cartesian products of families. Principles of weak and strong mathematical induction and their equivalence. Schroder-Bernstein theorem. Countable and uncountable sets. Cantor's theorem. Classical propositional calculus PC : Syntax. Valuations and truth tables, Truth functions, Logical equivalence relation. Semantic consequence and satisfiability.

Compactness theorem with application. Adequacy of connectives. Normal forms.

Applications to Circuit design. Axiomatic approach to PC: soundness, consistency, completeness. Other proof techniques: Sequent calculus, Computer assisted formal proofs: Tableaux. Decidability of PC, Boolean algebras: Order relations.

### Bibliographic Information

Boolean algebras as partially ordered sets. Atoms, Homomorphism, sub-algebra.

Stone's representation sketch. Completeness of PC with respect to the class of all Boolean algebras. Classical first order logic FOL and first order theories, Syntax. Satisfaction, truth, validity in FOL. Axiomatic approach, soundness. Computer assisted formal proofs: Tableaux. Consistency of FOL and completeness sketch. Examples of first order theories with equality. Peano's arithmetic.

## Glossary of areas of mathematics

Zermelo-Fraenkel axioms of Set theory. Axiom of choice, Well-ordering theorem, Zorn's lemma and their equivalence; illustrations of their use. Well-ordering principle and its equivalence with principles of weak and strong induction. Completeness of first order theories, Isomorphism of models, Categoricity-illustrations through theories such as those of finite Abelian groups, dense linear orders without end points and Peano's arithmetic. Oxford Logic Guides, Chiswell and W. Hodges: Mathematical Logic. Oxford, Cori and D. Lascar: Mathematical Logic, Oxford, Goubalt-Larrecq and J.

Topological spaces; open sets, closed sets, basis, sub-basis, closure, interior and boundary. Subspace topology. Continuous maps, open maps, closed maps, Homeomorphisms. Product Topology. Hausdorff spaces, Countability and separation axioms. Compact spaces and its properties, Locally compact spaces, one point compactification, Tychonoff's Theorem, Statement and Applications of Urysohn Lemma, Tietz extension theorem and Urysohn metrization theorem. Connectedness, path connectedness, components, its properties.

Quotient Topology, various type of examples, cone, suspension, surfaces as quotient spaces. Group actions, orbit spaces. Time permitting : Homotopy, Fundamental group, deformation retract, contractible spaces, simply connected spaces, computation of ,, Brouwer fixed point theorem. Differentiation: Differentiable functions, Directional derivatives, Composition of differentiable functions and chain rule, Mean value inequalities. Inverse mapping theorem and Implicit mapping theorem.

Regular value of differentiable maps, Lagrange multipliers. Higher order derivatives, Taylor's Theorem. Definition of parametrised curves. Arc length of a regular curve and arc length parametrisation of regular curves. Curvature of plane curves and Frenet-Serret formula for regular space curves. Statement of Green's Theorem.

Isoperimetric inequality for plane curves.

Tangent planes. Differentiable functions on surfaces and differentiable maps between surfaces. Tangent Plane. First fundamental form. Local isometries. Gauss map, Weingartent map and second fundamental form.

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Principal curvatures, Gauss curvature and mean curvature of surfaces. Surfaces of revolution and classification of surfaces of revolution of constant curvature. Umbilic points on surface. Classification of totally umbilical surfaces. Hilbert's theorem on compact surfaces. Geodesics and examples. Tom M. A Pressley: Elementary differential geometry, Springer India.