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Core Courses
PS 641M: Algebra (3 credits)
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Groups: Nilpotent and solvable groups, Sylow’s theorems, free groups.
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Representation theory of finite groups, Peter-Weyl theorem.
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Rings and Modules: Commutative rings, Noetherian and Artinian rings and modules, principal ideal domains (PID), unique factorization domain, modules over PID, tensor products.
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Field Theory: Algebraic and transcendental extensions, introduction to Galois theory.
Suggested Texts:
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M. Artin. Algebra.
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I.N. Herstein. Topics in Algebra.
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S. Lang. Algebra.
PS 642M: Analysis (3 credits)
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Prerequsites: Real Analysis, Basic Measure Theory, Topology and basics of Hilbert and Banach Spaces.
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Review of Basic Measure Theory: Outer measures, Lebesgue measure, non-measurable sets, general measure spaces, Egorov's Theorem, sigma finite and complete measures, completion of measures, integration on measure spaces, approximation theorems for measurable and integrable functions, product measures, Fubini and Tonelli Theorem.
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Signed and Complex measures: Signed and complex measures, different variations of signed measures, Hahn decomposition Theorem, Jordan decomposition Theorem, total variation of complex measures, complex regular measures, discrete and continuous measures, absolute continuity, mutual singularity, continuous and discrete decomposition, Lebesgue decomposition theorem, Radon-Nikodym Theorem.
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Lp spaces: Holder and Minkowski inequalities, completeness, approximation by simple and continuous functions, duality, Vitali-Luzin theorem and denseness of C c(X) in L p(X).
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Riesz Representation Theorem (RRT): Original RRT, RRT for positive linear functionals, RRT for complex measures.
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Fourier Analysis: Fourier transforms on L 1(R), boundedness and continuity of Fourier transforms, Riemann-Lebesgue Lemma, convolution on L 1(R), approximate identities, inversion formula, Fourier transforms on L 2(R), Plancheral Theorem, Parseval Formula, Existence and uniqueness of Haar measure on compact abelian group using Kakutani-Markov fixed point Theorem.
Main text:
General references:
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Real and Complex Analysis, W. Rudin, McGraw-Hill, 2006.
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A course in Abstract Analysis, J. B. Conway, AMS
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Analysis Now, G. K. Pederson, Springer, GTM
PS 643M: Topology (3 credits)
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General Topology: Introduction, metric topology, separation axioms, compactness, Connectedness, product topology, introduction to manifolds, submanifolds.
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Homotopy Theory. Covering spaces, homotopy maps, homotopy equivalence,Contractible spaces, deformation retraction.
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Fundamental Groups: Universal cover and lifting problem for covering maps, Fundamental groups of S1 and Sn.
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Introduction to Homology Theory.
Suggested texts:
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C.O. Christenson and W.L. Voxman. Aspects of Topology.
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J.R. Munkres. General Topology.
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I.M. Singer and J.A. Thorpe. Lecture Notes in Elementary Topology and Geometry.
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Optional Courses
PS 644M: Topological Groups and Lie Groups (2 credits)
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Topological Groups: Introduction, integration on locally compact spaces, Haar Measure, Character groups, group action.
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Lie groups and lie algebras: Basic theory, linear groups.
Suggested texts:
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K. Chandrasekharan. A Course on Topological Groups.
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W. Fulton and J. Harris. Representation Theory.
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F.W. Warner. Foundations of Differentiable Manifolds and Lie Groups.
PS 645M: Functional Analysis and Operator Theory (2 credits)
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Functional Analysis
Topological vector spaces: Separation properties, Linear Mappings, Finite dimensional spaces, Metrizability, Seminorms and local convexity.
Completeness: Baire Category Theorem, Banach-Steinhauss Theorem, Open Mapping Theorem, Closed Graph Theorem.
Convexity: Hahn-Banach Theorems, Weak topologies, Banach-Alaouglu Theorem, Exteme points, Krein-Milman Theorem.
Some Applications: Stone-Weirstrass Theorem, Kakutani-Markov fixed point Theorem and Haar measure for compact groups
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Operator Theory
Compact operators on Banach spaces.
Bounded operators on a Hilbert space: Riesz Representation Theorem, Bounded operators, Adjoints, Normal Operators, Unitary Operators.
Spectral Theorem: Spectrum of an operator, Resolution of Identity, Spectral Theorem, Functional Calculus of normal operators, Spectral theorem for compact normal operators.
Suggested texts:
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W. Rudin, Functional Analysis, ISPAM, McGraw-Hill, 2006.
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J. B. Conway, A Course in Functional Analysis, GTM, Springer, 1990.
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R. J. Zimmer, Essential Results of Functional Analysis, Chicago Lecture in Mathematics, 1990.
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G. F. Simmons, Introduction to Topology and Modern Analysis, Tata McGraw-Hill, 2004.
PS 712M: Algebraic Number Theory (2 credits)
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Number fields, number rings and their structure as Dedekind domains.
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Factorization of prime ideals, quadratic and cyclotomic extensions.
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Decomposition group, inertia group.
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Group of units, ideal class group, theorems of Dedekind and Minkowski.
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Introduction to zeta function, Dirichlet character.
Suggested texts:
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D. Marcus. Number Fields
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Borevich and Shafarevich. Number Theory
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Esmonde and Murty. Problems in Algebraic Number theory
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Frohlich and Taylor. Algebraic Number Theory
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Hasse. Number Theory
PS 646M, 647M: Research Courses I & II (3 credits each)
The research courses are advanced courses to prepare students to work in a specific area. The details of these courses are usually decided by the instructors.
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