Graduate Algebra I
Galois theory: separable and normal extensions, purely inseparable extensions, fundamental theorem of Galois theory. Module theory, structure theorem for modules over PIDs. multilinear algebra: tensor, symmetric and exterior products, tensor product of algebras. Categories and functors, some notions of homological algebra.
Graduate Algebra II
Non-commutative rings, semisimplicity, Jacobson theory, Artin-Wedderburn theorem, group-rings, matrix groups, introduction to representations.
Graduate Analysis I
Measure spaces, monotone classes, outer measures, Lebesgue measure, Regularity, measurable functions, integration, MCT, LDCT, product measures, Fubini, normed linear spaces, boundedness, completeness of B(X,Y), C(X), Hahn-Banach (statement only proof later) , Heine-Borel, all norms are equivalent in finite dimensions, Hilbert spaces, Cauchy-Schwartz inequality, existence of orthonormal basis, Riesz lemma, Radon-Nikodym derivative. Basic Fourier analysis on the circle and on R (up to Plancheral theorem). If time permits: Cantor set, complex measures, Riesz representation theorem.
Graduate Analysis II
Hahn-Banach, Dual space, Lp-Lq duality, Stone-Weierstrass theorem, Arzela-Ascoli theorem, Baire's theorem, nowhere differentiable functions are second category, open mapping, Closed graph, uniform boundedness principle and applications to Fourier series. Weak, weak* topology, Banach-Alaoglu theorem, projections, unitaries, isometries, spectral theorem for compact normal operators. If time permits: trace class, Hilbert-Schmidt.
Graduate Topology I
Point-set topology: Connectedness and compactness, Tychonoff's theorem, separation axioms, normal and regular spaces, Urysohn-Tietze theorems. Fundamental groups and covering spaces, Van Kampen's theorem.
Graduate Topology II
Singular homology theory. Cohomology theory.
Introduction to Manifolds
Inverse and implicit function theorem. derivatives and tangents, submersions, transversality, homotopy and stability, Sard's theorem, Morse functions. embedding manifolds in Euclidean spaces. Differential forms, integration on manifolds, exterior derivative, Stoke's theorem. Poincare Lemma. De Rham cohomology.
Complex numbers and geometric representation — analytic functions — power series — exponential and logarithmic functions — conformality — Mobius transformations — complex integration — Cauchy's theorem — Cauchy's integral formula — singularities — Taylor's theorem — The maximum principle — The residue theorem and applications — Invariance of integrals under homotopy — Topology on space of holomorphic and meromorphic functions — Hadamard theory of entire functions — Order of entire functions, Picard's theorem — Automorphisms of Complex and Upper half plane — Analytic continuation — meromorphic continuation along a path — monodromy theorm — Riemann surfaces — branch points — analytic, meromorphic and holomorphic functions on Riemann surfaces.
A student is expected to read, under the guidance of a faculty advisor, preparatory material for the MSc thesis. As a part of this course the student is expected give a seminar talk to an audience of other graduate students and faculty. The content of the talk is to be decided in consultation with the advisor.
Each student is expected to write a dissertation on a research problem of current interest, under the guidance of a faculty advisor. This may be substituted by two courses upon the recommendation by the advisor(s)/monitoring committee.