P.hd in Mathematics

P.hd in Mathematics

Algebra

* Groups: Jordan Holder theorem; solvable groups; symmetric and alternating groups; nilpotent groups; groups acting on sets; Sylow theorems; free groups.
* Rings and Modules: Noetherian and Artinian rings and modules; semisimple rings; Hilbert basis theorem; Principal ideal domains and unique factorisation domains; modules over PID; linear algebra and Jordan canonical form; structure theorems for semisimple rings.
* Representation theory of finite groups.
* Field theory: Steinitz theorem; algebraic extensions; Galois theory and applications; purely transcendental extensions; Luroth's theorem* .
* Homological Algebra: Categories and functors; adjoint functors Homand Tensor; their exactness properties and derived functors; Tensor, symmetric and exterior algebras.
* Commutative Rings: Integral extensions; Noether normalisation theorem; Hilbert's Nullstellensatz; discrete valuation rings and Dedekind domains and some applications to arithmetic; primary decomposition* .

Topology

* General and metric topology: Proper maps; quotient space construction; examples of spheres, real and complex projective spaces, Grassmannians; normal and Hausdorff spaces; paracompact spaces; topological groups and continuous actions; classical groups*.
* Homotopy theory: Covering spaces; homotopy of maps, homotopy equivalence of spaces, contractible spaces, deformation retractions; fundamental group: universal cover and lifting problem for covering maps; Van Kampen's theorem, Galois coverings.
* Homology theory: Simplicial complexes; barycentric subdivision; simple approximations; singular homology - basic properties - excision, Mayer-Vietoris; cellular homology, and basic examples using cellular homology; Kunneth formula; universal coefficient theorem; singular cohomology; cup product, Poincaré duality; CW complexes; basic facts about topology of CW complexes; CW structures for standard examples.
* Smooth manifolds: Smooth manifolds, tangent and cotangent spaces; Vector fields, integral curves, Frobenius theorem, flows; Immersions and submersions; Implicit and inverse functions theorems; Sard's theorem.

Analysis

* Measure and integration: Abstract theory; convergence theorems; product measure and Fubini's theorem; Borel measures on locally compact Hausdorff space, and Riesz representation theorem; Lebesgue measure; regularity properties of Borel measures; Haar measures - concept and examples; complex measures, differentiation and decomposition of measures; Radon Nikodym theorem; maximal function; Lebesgue differentiation theorem; functions of bounded variation.
* Elementary functional analysis: Topological vector spaces; Banach spaces; Hilbert spaces; Hahn Banach theorem; open mapping theorem; uniform boundedness principle; bounded linear transformation;linear functionals and dual spaces.
* $ L^p$ spaces: Basic theory, Hölder's inequality, Minkowsky inequality.
* Elementary Harmonic analysis: Analysis on ${I\!\!R}^n$; convolutions; approximate identity; approximation theorems; Fourier transform; Fourier inversion formula; Plancherel theorem, Hausdorff-Young inequality*.
* Operator theory: Spectral theorems for bounded normal operators, compact normal operators; Hilbert-Schmidt operators; Peter-Weyl theorem*. Banach algebras,Gelfand-Naimark theorem*.
* Distribution Theory: The spaces $ D(\Omega),E(\Omega)$, for $ \Omega$ open in ${I\!\!R}^n$;
$S({I\!\!R}^n)$ and their duals; convolution; Fourier transform; Paley-Wiener theorems; fundamental solutions of constant coefficient partial differential operators; Sobolev spaces.
* Cauchy-Riemann equation and holomorphic functions: Basic properties of holomorphic functions; open mapping theorem; maximum modulus theorem; zeros of holomorphic functions, Weierstrass factorisation theorem Riemann mapping theorem; meromorphic functions; essential singularities; Picard's theorem.

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