Manual Elements of Advanced Mathematical Analysis

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Basic notions from set theory, countable and uncountable sets. Metric spaces: definition and examples, basic topological notions. Sequences and series: essential definitions, absolute versus conditional convergence of series, some tests of convergence of series. Continuous functions: properties, the sequential and the open- set characterizations of continuity, uniform continuity. Differentiation in one variable. The Riemann integral: formal definitions and properties, continuous functions and integration, the Fundamental Theorem of Calculus.

Uniform convergence: definition, motivations and examples, uniform convergence and integration, the Weierstrass Approximation Theorem. Integrated Ph. Interdisciplinary Ph.

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Programme Integrated Ph. Limaye and S. Spivak, M.

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Benjamin, co. Some background in algebra and topology will be assumed. It will be useful to have some familiarity with programming. Topics: Basic type theory: terms and types, function types, dependent types, inductive types. Most of the material will be developed using the dependently typed language Idris. Connections with programming in functional languages will be explored. Manin, Yu. Srivastava, S.

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Becker and V. Adams and P. Sturmfels, Grobner bases and convex polytopes , American Mathematical Society Suggested books : Serre, J. Koblitz, N. Iwaniec, H. Diamond, F. Suggested books : Adrian Bondy and U. Springer-Verlag, Berlin, ISBN: Douglas B. West, Introduction to graph theory , Prentice Hall, Inc. Suggested books : Bondy, J. Burton, D.

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Course Descriptions: Math

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Hoffman, K. American Mathematical Society. Serre, Linear representations of finite groups , Graduate Texts in Mathematics. New York-Heidelberg.

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Hirsh, M. Suggested books : Armstrong, M. Munkres, K. Viro, O. MA Introduction to algebraic topology Prerequisite courses: MA and MA The fundamental group: Homotopy of maps, multiplication of paths, the fundamental group, induced homomorphisms, the fundamental group of the circle, covering spaces, lifting theorems, the universal covering space, Seifert-van Kampen theorem, applications. Hatcher, A. Kosniowski, C. Press, Croom, F. Suggested books : do Carmo, M.

Thorpe, J.

4 Degree Plans – Mathematical Analysis

O'Neill, B. Gray, A. MA Metric Geometry of Spaces and Groups Prerequisite courses: MA Pre-requisites : A first course in Topology can be taken concurrently Metric geometry is the study of geometric properties such as curvature and dimensions in terms of distances, especially in contexts where the methods of calculus are unavailable, An important instance of this is the study of groups viewed as geometric objects, which constitutes the field of geometric group theory.

MA Ordinary Differential Equations Prerequisite courses: MA Basics concepts:Introduction and examples through physical models, First and second order equations, general and particular solutions, linear and nonlinear systems, linear independence, solution techniques. Birkhaeuser, Coddington, E.