# Elementary Differential Geometry Oneill Homework Market

## Course Catalogue

#### MA 200: Multivariable Calculus (3:1)

Functions on $R^n$ , directional derivatives, total derivative, higher order derivatives and Taylor series.The inverse and implicit function theorem, Integration on $R^n$ , differential forms on $R^n$ , closed and exact forms. Green’s theorem, Stokes’ theorem and the Divergence theorem.

#### Suggested books :

- Rudin,
*Principles of Mathematical Analysis*, McGraw-Hill, 1986. - B. V. Limaye and S. Ghorpade,
*A course in Calculus and Real Analysis*, Springer. - Spivak, M.,
*Calculus on Manifolds*, W.A. Benjamin, co., 1965.

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#### MA 210: Logic, Types and Spaces (3:0)

#### Pre-requisites :

- No prior knowledge of logic is assumed.
- Some background in algebra and topology will be assumed.
- It will be useful to have some familiarity with programming.

This course is an introduction to logic and foundations from both a modern point of view (based on type theory and its relations to topology) as well as in the traditional formulation based on first-order logic.

Topics:

- Basic type theory: terms and types, function types, dependent types, inductive types.
- First order logic: First order languages, deduction and truth, Models, Godel’s completeness and compactness theorems.
- Godel’s incompleteness theorem
- Homotopy Type Theory: propositions as types, the identity type family, topological view of the identity type, foundations of homotopy type theory.
- Most of the material will be developed using the dependently typed language/proof assistant Agda. Connections with programming in functional languages will be explored.

#### Suggested books :

*Homotopy Type Theory: Univalent Foundations of Mathematics*, Institute for Advanced Studies, Princeton 2013; available at http://homotopytypetheory.org/book/.- Manin, Yu. I.,
*A Course in Mathematical Logic for Mathematicians, Second Edition*, Graduate Texts in Mathematics, Springer-Verlag, 2010. - Srivastava, S. M.,
*A Course on Mathematical Logic, Universitext, Springer-Verlag, 2008*.

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#### MA 211: Matrix theory (3:0)

Vector spaces, Bases and dimension, Direct ums, linear transformations, Matrix algebra, Eigenvalues and eigenvectors, Cayley Hamilton Theorem, Jordan canonical form., Orthogonal matrices and rotations, Polar decomposition., Bilinear forms.

#### Suggested books :

- Artin, M.,
*Algebra*, Prentice-Hall of India, 1994. - Hoffman, K and Kunze R.,
*Linear Algebra*, Prentice-Hall of India, 1972. - Halmos, P.R.,
*Finite dimensional vector spaces*, van Nostrand, 1974 . - Greub, W.H.,
*Linear algebra*, Springer-Verlag, 1967.

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#### MA 212: Algebra I (3:0)

Part A

- Groups: definitions & basic examples;
- Normal subgroups, quotients;
- Three isomorphism theorems;
- Centralizer and normalizer of a subset, centre of a group;
- Permutations, symmetrc groups and Cayley’s theorem;
- Group actions and their applications, Sylow’s theorems.

Part B

- Rings and ideals: basic definitions, quotient rings;
- The Chinese Remainder Theorem;
- Maximal and prime ideals;
- Unique factorization, unique factorization domains, principal ideal domains, Euclidean domains, polynomial rings;
- Modules: basic definitions and examples, Hom and tensor products, the Structure Theorem for finitely generated modules over PIDs;
- Fields: basic definitions and examples, algebraic & trancendental numbers;
- Finite fields, characteristic, the order of a finite field.

#### Suggested books :

- Artin,
*Algebra*, M. Prentice-Hall of India, 1994. - Dummit, D. S. and Foote, R. M.,
*Abstract Algebra*, McGraw-Hill, 1986. - Herstein, I. N.,
*Topics in Algebra*, John Wiley & Sons, 1995. - Lang, S.,
*Algebra (3rd Ed.)*, Springer, 2002.

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#### MA 213: Algebra II (3:1)

Part A

- Introduction to categories and functors, direct and inverse limits;
- Field of fractions of an integral domain, localization of rings;
- I-adic completion of rings;
- Tensor products, short exact sequences of modules;
- Noetherian rings and modules, Hilbert Basis Theorem, Jordan-Holder Theorem;
- Artinian rings, Artinian implies Noetherian, Krull-Schmidt Theorem.

Part B

- Splitting fields, normal and separable extensions;
- Application to finite fields;
- The Fundamental Theorem of Galois Theory;
- The Primitive Element Theorem.

#### Suggested books :

- Artin, M.,
*Algebra*, Prentice_Hall of India, 1994. - Dummit, D. S. and Foote, R. M.,
*Abstract Algebra*, McGraw-Hill, 1986. - Lang, S.,
*Algebra (3rd Ed.)*, Springer, 2002. - Atiyah, M. and MacDonald, R.,
*Introduction to Commutative Algebra*, Addison-Wesley(or any reprint).

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#### MA 214: Topics in Commutative Algebra (3:0)

Abstract relations and Dickson’s Lemma; Hilbert Basis theorem, Buchberger Criterion for Grobner Bases and Elimination Theorem; Field Extensions and the Hilbert Nullstellensatz; Decomposition, Radical, and Zeroes of Ideals; Syzygies, Grobner Bases for Modules, Computation of Hom, Free Resolutions; Universal Grobner Bases and Toric Ideals.

#### Suggested books :

- T. Becker and V. Weispfenning,
*Grobner Bases–a Computational Approach to Commutative Algebra*, Springer 1993. - W.W. Adams and P. Loustaunau,
*An Introduction to Grobner Bases*, Graduate Studies in Mathematics, Vol. 3, American Mathematical Society, 1994. - B. Sturmfels,
*Grobner bases and convex polytopes*, American Mathematical Society 1996.

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#### MA 215: Introduction to Modular Forms (3:0)

#### Pre-requisites :

- MA 224 (Complex Analysis) or equivalent

The modular group and its subgroups, the fundamental domain. Modular forms, examples, Eisenstein series, cusp forms. Valence (dimension) formula, Petersson inner product. Hecke operators. L-functios: definition, analytic continution and functional equation.

#### Suggested books :

- Serre, J.P.,
*A Course in Arithmetic, Graduate Texts in Mathematics no. 7*, Springer-Verlag, 1996. - Koblitz, N.,
*Introdution to Modular Forms*, Graduate Texts in Mathematics no. 97, Springer-Verlag, 1984. - Iwaniec, H.,
*Topics in Classical Automorphic Forms*, Graduate Texts in Mathematics 17, AMS, 1997. - Diamond, F. and Schurman, J.,
*A First Course in Modular Forms*, Graduate Texts in Mathematics no. 228, Springer-Verlag, 2005.

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#### MA 217: Discrete Mathematics (3:0)

**Combinatorics:** Basic counting techniques. Principle of inclusion and exclusion. Recurrence relations and generating functions. Pigeon-hole principle, Ramsey theory. Standard counting numbers, Polya enumeration theorem.

**Graph Theory:** Elementary notions, Shortest path problems. Eulerian and Hamiltonian graphs, The Chinese postman problem. Matchings, the personal assignment prolem. Colouring or Graphs.

**Number Theory:** Divisibility Arithmetic functions. Congruences. Diophantine equations. Fermat’s big theorem, Quadratic reciprocity laws. Primitive roots.

#### Suggested books :

- Bondy, J. A. and Muirty, U. S. R.,
*Graph theory with applications*, Elsevier-North Holland, 1976. - Burton, D.,
*Elementary Number Theory*, McGraw Hill, 1997. - Clark, J. and Holton, D. A.,
*A first book at Graph Theory*, World Scientific Cp., 1991. - Polya G. D., Tarjan, R. E. and Woods, D. R.,
*Notes on Introductory Combinations*, Springer-Verlag, 1990.

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#### MA 218: Number Theory (3:0)

**Algebraic Number Theory:** Algebraic numbers and algebraic integers, Class groups, Groups of units, Quadratic fields, Quadratic reciprocity law, Class number formula.

**Analytic Number Theory:** Fundamental theorem of arithmetic, Arithmetical functions, Some elementary theorems on the distribution of prime numbers, Congruences, Finite Abelian groups and their characters, Dirichlet theorem on primes in arithmetic progression.

#### Suggested books :

- Narasimham, R., Raghavan, S., Rangachari, S. S. and Sunder Lal.,
*Algebraic Number Theory*, Lecture Notes in Mathematics, TIFR, 1966. - Niven, I. and Zuckerman, H. S.,
*An Introduction to the Theory of numbers*, Wiley Eastern Limited, 1989. - Apostol, T. M.,
*Introduction to Analytic Number Theory*, Springer International Student Edition, 1989. - Ireland, K. and Rosen, M.,
*Classical Introduction to Modern Number Theory*, Springer-Verlag (GTM), 1990.

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#### MA 219: Linear Algebra (3:1)

Vector spaces: Definition, Basis and dimension, Direct sums. Linear transformations: Definition, Rank-nullity theorem, Algebra of linear transformations, Dual spaces, Matrices.

Systems of linear equations:elementary theory of determinants, Cramer’s rule. Eigenvalues and eigenvectors, the characteristic polynomial, the Cayley- Hamilton Theorem, the minimal polynomial, algebraic and geometric multiplicities, Diagonalization, The Jordan canonical form. Symmetry: Group of motions of the plane, Discrete groups of motion, Finite groups of S0(3). Bilinear forms: Symmetric, skew symmetric and Hermitian forms, Sylvester’s law of inertia, Spectral theorem for the Hermitian and normal operators on finite dimensional vector spaces.

#### Suggested books :

- Artin, M.,
*Algebra*, Prentice_Hall of India, 1994. - Halmos, P.,
*Finite dimensional vector spaces*, Springer-Verlag (UTM), 1987. - Hoffman, K. and Kunze, R.,
*Linear Algebra (2nd Ed.)*, Prentice-Hall of India, 1992.

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#### MA 220: Representation theory of Finite groups (3:0)

Representation of finite groups, irreducible representations, complete reducibility, Schur’s lemma, characters, orthogonality, class functions, regular representations and induced representation, the group algebra.

Linear groups: Representation of the group $SU(2)$

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#### MA 221: Analysis I - Real Analysis (3:0)

Construction of the field of real numbers and the least upper-bound property. Review of sets, countable & uncountable sets. Metric Spaces: topological properties, the topology of Euclidean space. Sequences and series. Continuity: definition and basic theorems, uniform continuity, the Intermediate Value Theorem. Differentiability on the real line: definition, the Mean Value Theorem. The Riemann-Stieltjes integral: definition and examples, the Fundamental Theorem of Calculus. Sequences and series of functions, uniform convergence, the Weierstrass Approximation Theorem. Differentiability in higher dimensions: motivations, the total derivative, and basic theorems. Partial derivatives, characterization of continuously-differentiable functions. The Inverse and Implicit Function Theorems. Higher-order derivatives.

#### Suggested books :

- Rudin, W.,
*Principles of Mathematical Analysis*, McGraw-Hill, 1986. - Apostol, T. M.,
*Mathematical Analysis*, Narosa, 1987.

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#### MA 222: Analysis II - Measure and Integration (3:1)

Construction of Lebesgue measure, Measurable functions, Lebesgue integration, Abstract measure and abstract integration, Monotone convergence theorem, Dominated convergence theorem, Fatou’s lemma, Comparison of Riemann integration and Lebesgue integration, Product sigma algebras, Product measures, Sections of measurable functions, Fubini’s theorem, Signed measures and Randon-Nikodym theorem, Lp-spaces, Characterization of continuous linear functionals on Lp - spaces, Change of variables, Complex measures, Riesz representation theorem.

#### Suggested books :

- Royden, H. L.,
*Real Analysis*, Macmillan, 1988. - Folland, G.B.,
*Real Analysis: Modern Techniques and their Applications (2nd Ed.)*, Wiley. - Hewitt, E. and Stromberg, K.,
*Real and Abstract Analysis*, Springer, 1969.

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#### MA 223: Functional Analysis (3:0)

Basic topological concepts, Metric spaces, Normed linear spaces, Banach spaces, Bounded linear functionals and dual spaces,Hahn-Banach theorem. Bounded linear operators, open-mapping theorem, closed graph theorem. The Banach- Steinhaus theorem. Hilbert spaces, Riesz representation theorem, orthogonal complements, bounded operators on a Hilbert space up to (and including) the spectral theorem for compact, self-adjoint operators.

#### Suggested books :

- Rudin,
*Functional Anaysis (2nd Ed.)*, McGraw-Hill, 2006. - Yosida, K.,
*Functional Anaysis (4th Edition)*, Narosa, 1974. - Goffman, C. and Pedrick, G.,
*First Course in Functional Analysis*, Prentice-Hall of India, 1995.

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#### MA 224: Complex Analysis (3:1)

Complex numbers, complex-analytic functions, Cauchy’s integral formula, power series, Liouville’s theorem. The maximum-modulus theorem. Isolated singularities, residue theorem, the Argument Principle, real integrals via contour integration. Mobius transformations, conformal mappings. The Schwarz lemma, automorphisms of the dis. Normal families and Montel’s theorem. The Riemann mapping theorem.

#### Suggested books :

- Ahlfors, L. V.,
*Complex Analysis*, McGraw-Hill, 1979. - Conway, J. B.,
*Functions of One Complex Variable*, Springer-veriag, 1978.

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#### MA 226: Complex Analysis II (3:0)

Harmonic and subharmonic functions, Green’s function, and the Dirichlet problem for the Laplacian; the Riemann mapping theorem (revisited) and characterizing simple connectedness in the plane; Picard’s theorem; the inhomogeneous Cauchy–Riemann equations and applications; covering spaces and the monodromy theorem.

#### Suggested books :

- Narasimhan, R.,
*Complex Analysis in One Variable, 1st ed. or 2nd ed. (with Y. Nievergelt)*, Birkhauser (2nd ed. is available in Indian reprint, 2004). - Greene, R.E. and Krantz, S.G.,
*Functions Theory of One Complex Variable, 2nd ed.*, AMS 2002 (available in Indian reprint, 2009, 2011).

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#### MA 229: Calculus on Manifolds (3:0)

Functions of several variables, Directional derivatives and continuity, total derivative, mean value theorem for differentiable functions, Taylor’s formula. The inverse function and implicit function theorems, extreme of functions of several variables and Lagrange multipliers. Sard’s theorem. Manifolds: Definitions and examples, vector fields and differential forms on manifolds, Stokes theorem.

#### Suggested books :

- Spivak, M.,
*Calculus on Manifolds*, W.A. Benjamin, co., 1965. - Hirsh, M.W.,
*Differential Topology*, Springer-Verlag, 1997.

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#### MA 231: Topology (3:1)

Point-set topology: Open and closed sets, continuous functions, Metric topology, Product topology, Connectedness and path-connectedness, Compactness, Countability axioms, Separation axioms, Complete metric spaces, Quotient topology, Topological groups, Orbit spaces.

The fundamental group: Homotopic maps, Construction of the fundamental group, Fundamental group of the circle, Homotopy type, Brouwer’s fixed-point theorem, Separation of the plane.

#### Suggested books :

- Armstrong, M. A., Basic Topology, Springer (India), 2004.,
*Functional Anaysis (2nd Ed.)*, McGraw-Hill, 2006. - Munkres, K. R., Topology,Pearson Education, 2005,
*Functional Anaysis (4th Edition)*, Narosa, 1974. - Viro, O.Ya., Ivanov, O.A., Netsvetaev, N., and Kharlamov, V.M.,
*Elementary Topology: Problem Textbook*, AMS, 2008.

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#### MA 232: Introduction to algebraic topology (3:0)

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. Simplicial and singular

Homology: Simplicial complexes, chain complexes, definitions of the simplicial and singular homology groups, properties of homology groups, applications

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#### MA 233: Differential Geometry (3:0)

Curves in Euclidean space: Curves in R3, Tangent vectors, Differential derivations, Principal normal and binomial vectors, Curvature and torsion, Formulae of Frenet.

Surfaces in R3: Surfaces, Charts, Smooth functions, Tangent space, Vector fields, Differential forms, Regular Surfaces, The second fundamental form, Geodesies, Parellel transport, Weingarten map, Curvatures of surfaces, Rules surfaces, Minimal surfaces, Orientation of surfaces.

#### Suggested books :

- do Carmo, M. P.,
*Differential Geometry of curves and surfaces*, Prentice-Hall, 1976. - Thorpe, J. A.,
*Elementary topics in Differential Geometry*, Springer-Verlag (UTM), 1979. - O'Neill, B.,
*Elementary Differential Geometry*, Academic, 1996. - Gray, A.,
*Modern Differential Geometry of Curves and Surfaces*, CRC Press, 1993.

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#### MA 234: Metric Geometry of Spaces and Groups (3:0)

#### 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. This course will introduce concepts, examples and basic results of Metric Geometry and Geometric Group theory.

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#### MA 241: Ordinary Differential Equations (3:1)

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. Existence and Uniqueness Theorems :Peano’s and Picard’s theorems, Grownwall’s inequality, Dependence on initial conditions and associated flows. Linear system:The fundamental matrix, stability of equilibrium points, Phase- plane analysis, Sturm-Liouvile theory . Nonlinear system and their stability:Lyapunov’s method, Non-linear Perturbation of linear systems, Periodic solutions and Poincare- Bendixson theorem.

#### Suggested books :

- Hartman,
*Ordinary Differential Equations*, P. Birkhaeuser, 1982. - Coddington, E. A. and Levinson, N.,
*Theory of Ordinary Differential Equations*, Tata McGraw-Hill, 1972. - Perko, L.,
*Differential Equations and Dynamical Systems*, Springer-Verlag, 1991.

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#### MA 242: Partial Differential Equations (3:0)

First order partial differential equation and Hamilton-Jacobi equations; Cauchy problem and classification of second order equations, Holmgren’s uniqueness theorem; Laplace equation; Diffusion equation; Wave equation; Some methods of solutions, Variable separable method.

#### Suggested books :

- Garabedian, P. R.,
*Partial Differential Equations*, John Wiley and Sons, 1964. - Prasad. P. and Ravindran, R.,
*Partial Differential Equations*, Wiley Eastern, 1985. - Renardy, M. and Rogers, R. C.,
*An Introduction to Partial Differential Equations*, Springer-Verlag, 1992. - Fritz John,
*Partial Differential Equations*, Springer (International Students Edition), 1971.

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#### MA 246: Mathematical Methods (3:0)

**Matrix Algebra:** Systems of linear equations, Nullspace, Range, Nullity, Rank, Similarity, Eigenvalues, Eigenvectors, Diagonalization, Jordan Canonical form. Ordinary Differential Equations: Singular points, Series solution Sturm Liouville problem, Linear Systems, Critical points, Fundamental matrix, Classification of critical points, Stability.

**Complex Variables:** Analytic functions, Cauchy’s integral theorem and integral formula, Taylor and Laurent series, isolated singularities, Residue and Cauchy’s residue theorem chwarz lemma.

#### Suggested books :

- Hoffman, K. and Kunze, R.,
*Linear Algebra (2nd Ed.)*. - Herstein, I. N. and Winter, D. J.,
*Matrix Theory and Linear Algebra*, Macmillan, 1989. - Simmons G. F.,
*Differential Equations*, Tata McGraw-Hill, 1985. - Churchill, R. V.,
*Complex Variables and Applications*, McGraw-Hill, 1960.

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#### MA 251: Numerical Methods (3:0)

Numerical solution of algebraic and transcendental equations, Iterative algorithms, Convergence, Newton Raphson procedure, Solutions of polynomial and simultaneous linear equations, Gauss method, Relaxation procedure, Error estimates, Numerical integration, Euler-Maclaurin formula. Newton-Cotes formulae, Error estimates, Gaussian quadratures, Extensions to multiple integrals.

Numerical integration of ordinary differential equations: Methods of Euler, Adams, Runge-Kutta and predictor - corrector procedures, Stability of solution. Solution of stiff equations.

Solution of boundary value problems: Shooting method with least square convergence criterion, Quasilinearization method, Parametric differentiation technique and invariant imbedding technique.

Solution of partial differential equations: Finite-difference techniques, Stability and convergence of the solution, Method of characteristics. Finite element and boundary element methods.

#### Suggested books :

- Gupta, A. and Bose, S. C.,
*Introduction to Numerical analysis*, Academic Publishers, 1989. - Conte, S. D. and Carl de Boor.,
*Elementary Numerical Analysis*, McGraw-Hill, 1980. - Hildebrand, F. B.,
*Introduction to Numerical Analysis*, Tata McGraw-Hill, 1988. - Froberg, C. E.,
*Introduction to Numerical Analysis*, Wiley, 1965.

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#### MA 253: Numerical Methods for Partial Differential Equations (3:0)

Finite difference methods for two point boundary value problems, Laplace equation on the square, heat equation and symmetric hyperbolic systems in 1 D. Lax equivalence theorem for abstract initial value problems. Introduction to variational formulation and the Lax-Milgram lemma. Finite element methods for elliptic and parabolic equations.

#### Suggested books :

- Smith, G. D.,
*Numerical solution of partial differential equations: Finite Difference Methods*, Calarendon Press, 1985. - Evans, G. Blackledge, J. and Yardley, P.,
*Numerical methods of partial differential equations*, Springer-Verlag, 1999.

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#### MA 254: Numerical Analysis (3:0)

**Introduction:** Floating point representation of numbers and roundoff errors, Interpolation Numerical integration.

**Linear systems and matrix theory:** Various factorizations of inversion of matrices, Condition number and error analysis.

**Non-linear systems:** Fixed point iteration, Newton-Rapson and other methods, Convergence acceleration.

**Numerical methods for ODE:** Introduction and analysis of Taylor, Runge-kutta and other methods.

**Numerical methods for PDE:** Finite difference method for Laplace, Heat and wave equations.

#### Suggested books :

- Faires, J. D. and Burden, R.,
*Numerical Methods*, Brooks/Cole Publishing Co., 1998. - Conte, S. D. and Carl de Boor.,
*Elementary Numerical Analysis*. - Stoer, J. and Bilrisch, R.,
*Introduction to Numerical Analysis*, Springer- Verlag, 1993. - Iserlas, A.,
*First course in the numerical analysis of differential equations*, Cambridge, 1996.

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#### MA 261: Probability Models (3:0)

Sample spaces, events, probability, discrete and continuous random variables, Conditioning and independence, Bayes’ formula, moments and moment generating function, characteristic function, laws of large numbers, central limit theorem, Markov chains, Poisson processes.

#### Suggested books :

- Ross, S.M. ,
*Introduction to Probability Models*, Academic Press 1993. - Taylor, H.M., and Karlin, S.,
*An Introduction to Stochastic Modelling*, Academic Press, 1994.

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#### MA 277: Nonlinear Dynamical Systems and Chaos (3:0)

**Conservative Systems:** Hamiltonians, canonical transformations, nonlinear pendulum, perturbative methods, standard map, Lyapunov exponents, chaos, KAM theorem, Chirikov criterion.

**Dissipative Systems:** logistics map, period doubling, chaos, strange attractors, fractal dimensions, Smale horseshoe, coupled maps, synchronization, control of chaos.

#### Suggested books :

- Lichtenberg, A. J. and Lieberman, M. A.,
*Regular and Stochastic motion*, Springer-Verlag, 1983. - Guckenheimer, J. and Holmes, P.,
*Nonlinear Oscillations, Dynamical Systems and bifurcations of vector fields*, Springer-Verlag, 1983.

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#### MA 278: Introduction to Dynamical Systems Theory (3:0)

The course introduces basic mathematical techniques to understand qualitatively the long-term behaviour of systems evolving in time. Most of the phenomena occurring in nature, and around us, are nonlinear in nature and often these exhibit interesting behaviour which could be unpredictable and counterintuitive. Tools and techniques of dynamical systems theory help in understanding the behaviour of systems and in gaining control over their behaviour, to a certain extent. Dynamical systems theory has wide applications in the study of complex systems, including physical & biological systems, engineering, aerodynamics, economics, etc.

#### Suggested books :

- S. Strogatz,
*Nonlinear Dynamics and Chaos: with Applications to physics, Biology, Chemistry, and Engineering*, Westview, 1994. - S. Wiggins,
*Introduction to applied nonlinear dynamics & chaos*, Springer-Verlag, 2003. - K. Alligood, T. Sauer, & James A.Yorke,
*Chaos: An Introduction to Dynamical Systems*, Springer-Verlag, 1996. - M.Tabor,
*Chaos and Integrability in Non-linear Dynamics*, 1989.

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#### MA 301: Real Analysis (3:0)

REAL ANALYSIS

The Lebesgue Integral:Riemann-Stieltjes integral, Measures and measurable sets, measurable functions, the abstract Lebesgue integral. Product measures and Fubini’s theorem. Complex measures and the lebesgue - Radon - Nikodym theorem and its applications. Function Spaces and Banach Spaces: Lpspaces, Abstract Banach Spaces. The conjugate spaces. Abstract Hilbert spaces.

#### Suggested books :

- Royden, H. L.,
*Real Analysis*, The Macmillan Company, New York, 1963. - Rudin, W.,
*Principles of Mathematical Analysis*, McGraw-Hill, 1986. - Rudin,
*Functional Anaysis (2nd Ed.)*, McGraw-Hill, 2006. - Hewitt, E. and Stromberg, K.,
*Real and Abstract Analysis*, Springer International Student Edition, Springer-Verlag/Narosa Pub. House, New Delhi, 1978.

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#### MA 302: Advanced Calculus (3:0)

Differentiation:Basic definitions and theorems, Partial derivatives, Derivatives (as linear maps), Inverse and Implicit function theorems. Integration:Basic definitions and theorems, Integrable functions, Partitions of unity, Change of variables. Manifolds:Basic definitions, forms on manifolds, Stokes theorem on manifolds, Volume element, classical theorems (Green’s and divergence).

#### Suggested books :

- Rudin, W.,
*Principles of Mathematical Analysis*, McGraw-Hill, 1986. - Apostol, T. M.,
*Mathematical Analysis*, Narosa Publishing House, New Delhi, 1992. - Spivak, M.,
*Calculus on Manifolds*, The Benjamin Publishing Company, New York, 1965. - Munkres, J., R.,
*Analysis on Manifolds*, Addison-Wesley Publishing Company, 1991.

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#### MA 303: Topics in Operator Theory (3:0)

$C^*$-algebras, Calkin algebra, Compact and Fredholm operators, Index spectral theorem, the Weyl-von Neumann-Berg Theorem and the Brown-Douglas-Fillmore Theorem.

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#### MA 312: Commutative Algebra (3:0)

Noetherian rings and Modules, Localisations, Exact Sequences, Hom, Tensor Products, Hilbert’s Null-stellensatz, Integral dependence, Going-up and Going down theorems, Noether’s normalization lemma , Discrete valuation rings and Dedekind domains.

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#### MA 313: Algebraic Number Theory (3:0)

#### Pre-requisites :

- Linear algebra (MA 219 or equivalent)
- Basic algebra : Groups, rings, modules (MA 212 or equivalent), and algebraic field extensions

Algebraic preliminaries: Algebraic field extensions: Normal, separable and Galois extensions. Euclidean rings, principal ideal domains and factorial rings. Quadratic number fields. Cyclotomic number fields. Algebraic integers: Integral extensions: Algebraic number fields and algebraic integers. Norms and traces. Resultants and discriminants. Integral bases. Class numbers:Lattices and Minkowski theory. Finiteness of class number. Dirichlet’s unit theorem. Ramification Theory: Discriminants. Applications to cryptography.

#### Suggested books :

- Artin, E.,
*Galois Theory*, University of Notre Dame Press, 1944. - Borevich, Z. and Shafarevich, I.,
*Number Theory*, Academic Press, New York, 1966. - Cassels, J.W. and Frohlich, A.,
*Algebraic Number Theory*, Academic Press, New York, 1948. - Hasse, H.,
*Zahlentheorie*, Akademie Verlag, Berlin, 1949. - Hecke, E.,
*Vorlesungen uber die Theorie der algebraischen Zahlen*, Chelsea, New York, 1948. - Samuel, P.,
*Algebraic Theory of Numbers*, Hermann, 1970.

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#### MA 314: Introduction to Algebraic Geometry (3:0)

Affine algebraic sets, Hilbert basis theorem, Hilbert Nullstellensatz, function field, plane curves, Bezout’s theorem, product, normality, morohisms, Noether normalisation, Graded rings, projective varieties, rational functions, tangent spaces, non- singularity, blowing up points, Riemann-Roch for curves. Schemes examples.

#### Suggested books :

- Shafarevich, I.R.,
*Basic Algebraic Geometry 1, 2nd edition*, Springer-Verlag, 1994. - Smith, K., Kahanpaa, L., Kekalainen, P., Traves, W.,
*An Invitation to Algebraic Geometry*, Springer-Verlag, 2000. - Reid, M.,
*Undergraduate Algebraic Geometry*, London Mathematical Society Student Texts 12, Cambridge University Press, 1988. - Fulton, W.,
*Algebraic Curves (http://math.lsa.umich.edu/CurveBook.pdf)*. - Holme, A.,
*A Royal Road to Algebraic Geometry*, Springer,2012.

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#### MA 315: Lie Algebras and their Representations (3:0)

LIE ALGEBRAS AND THEIR REPRESENTATIONS

Finite dimensional Lie algebras, Ideals, Homomorphisms, Solvable and Nilpotent Lie algebras, Semisimple Lie algebras, Jordan decomposition, Kiling form, root space decomposition, root systems, classification of complex semisimple Lie algebras Representations Complete reducibility, weight spaces, Weyl character formula, Kostant, steinberg and Freudenthal formulas

#### Suggested books :

- J E Humphreys,
*Introduction to Lie algebras and Representation theory*, Springer-Verlag, 1972. - J P Serre,
*Complex Semisimple Lie Algebras*, Springer, 2001. - Fulton. W., and Harris J.,
*Representation theory*, Springer-Verlag. 1991.

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#### MA 316: Introduction to Homological Algebra (3:0)

Polynomial ring, Projective modules, injective modules, flat modules, additive category, abelian category, exact functor, adjoint functors, (co)limits, category of complexes, snake lemma, derived functor, resolutions, Tor and Ext, dimension, local cohomology,group (co)homology, sheaf cohomology, Cech cohomology, Grothendieck spectral sequence, Leray spectral sequence.

#### Suggested books :

- Cartan and Eilenberg,
*Homological Algebra*. - Weibel,
*Introduction to Homological Algebra*. - Rotman,
*Introduction to Homological Algebra*.

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#### MA 317: Introduction to Analytic Number Theory (3:0)

#### Pre-requisites :

- MA 224 (complex Analysis) or equivalent
- An introductory course in Number Theory, or Consent of instructor

Review of arithmetical functions, averages of arithmetical functions, elementary results on the distribution of prime numbers, Dirichlet characters, Dirichlet’s theorem on primes in arithmetic functions, Dirichlet series and Euler products, the Riemann zeta function and related objects, the prime number theorem. (Time permitting: advanced topics like sieves, bounds on exponential sums, zeros of functions. the circle method.)

#### Suggested books :

- Apostol, T.M.,
*Introduction to Analytic Number Theory*, Springer-Verlag, 1976. - Davenport, H.,
*Multiplicative Number theory, 3rd edition*, Springer, 2000.

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#### MA 318: Combinatorics (3:0)

#### Pre-requisites :

- Calculus, Linear algebra and some exposure to proofs and abstract mathematics.
- Programming in Sage will be a part of every lecture. Students will need to bring a laptop with access to the IISc WLAN.

Counting problems in sets, multisets, permutations, partitions, trees, tableaux; ordinary and exponential generating functions; posets and principle of inclusion-exclusion, the transfer matrix method; the exponential formula, Polya theory; bijections, combinatorial identities and the WZ method.

#### Suggested books :

- Herbert Wilf,
*Generatingfunctionology*, ISBN-13 - 978-1568812793; Freely downloadable from http://www.math.upenn.edu/~wilf/DownldGF.html. - Richard P. Stanley,
*Enumerative Combinatorics: Volume 1 (Second Edition)*, ISBN-13 - 978-1107602625 Older version freely downloadable from http://www-math.mit.edu/~rstan/ec/ec1/.

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#### MA 319: Algebraic Combinatorics (3:0)

The algebra of symmetric functions, Schur functions, RSK algorithm, Murnaghan- Nakayama Rule, Hillman-Grassl correspondence, Knuth equivalence, jeu de taquim, promotion and evacuation, Littlewood-Richardson rules.

No prior knowledge of combinatorics is expected, but a familiarity with linear algebra and finite groubs will be assumed.

#### Suggested books :

- Stanley, R.,
*Enumerative Combinatorics, volume 2*, Cambridge University Press, 2001. - Sagan, B.,
*The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions*, Graduate Texts in Mathematics vol. 203, Springer-Verlag, 2001. - Prasad, A.,
*Representation Theory : A Combinational Viewpoint*, Cambridge Studies in Advanced Mathematics vol. 147, 2014. - Stanley, R.,
*Lecture notes on Topics in Algebraic Combinatorics*.

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#### MA 320: Representation theory of compact Lie groups (3:0)

Lie groups, Lie algebras, matrix groups , representations, Schur’s orthogonality relations, Peter-Weyl theorem, structure of compact semisimple Lie groups, maximal tori, roots and rootspaces, classification of fundamental systems Weyl group, Highest weight theorem, Weyl integration formula, Weyl’s character formula.

#### Suggested books :

- V. S. Varadarajan,
*Lie groups, Lie algebras and their representations*, Sringer 1984. - A. C. Hall,
*Lie groups, Lie algebras and representations*, Springer 2003. - Barry Simon,
*Representations of finite and compact groups*, AMS 1996. - A. W. Knapp,
*Representation theory of semismiple lie groups. An overview based on examples*, Princeton university press 2002.

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#### MA 321: Analysis III (3:0)

Theory of Distributions: Introduction, Topology of test functions, Convolutions, Schwartz Space, Tempered distributions.

Fourier transform and Sobolev-spaces:Definitions, Extension operators, Continuum and Compact imbeddings, Trace results.

Elliptic boundary value problems: Variational formulation, Weak solutions, Maximum Principle, Regularity results.

#### Suggested books :

- Barros-Nato,
*An Introduction to the Theory of Distributions*, Marcel Dekker Inc., New York, 1873. - Kesavan, S.,
*Topics in Functional Analysis and Applications*, Wiley Eastern Ltd., 1989. - Evans, L. C.,
*Partial Differential Equations*, Univ. of California, Berkeley, 1998. - Schwartz, L. Hermann,
*Theorie des Distributions*, 1966.

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#### MA 322: Harmonic Analysis (3:0)

Harmonic Analysis on the Poincare disc-Fourier transform, Spherical functions, Jacobi transform, Paley-Wiener theorem, Heat kernels, Hardy’s theorem etc.,

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#### MA 323: Operator Theory (3:0)

Review of basic notions from Banach and Hilbert space theory.

**Bounded linear operators:** Spectral theory of compact, Self adjoint and normal operators, Sturm-Liouville problems, Green’s function, Fredholm integral operators.

**Unbound linear operators on Hilbert spaces:** Symmetric and self adjoint operators, Spectral theory. Banach algebras Gelfand representation theorem. $C^*$-algebras, Gelfand-Naimark-Segal construction.

#### Suggested books :

- Conway, J. B.,
*A course in Functional Analysis*, Springer-Verlag, 1990. - Rudin, W.,
*Functional Analysis*, Tata Mcgraw-Hill, 1974. - Berberian, S. K.,
*Lectures in Functional Analysis*, Frederic Ungar, 1955.

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#### MA 324: Topics in Complex Analysis (3:0)

The general theory of holomorphic mappings between bounded domains, automorphisms of bounded domains, discussions on the non-existence of a classical Riemann Mapping Theorem in several variables, discussion of the various forms of the one-variable Riemann Mapping Theorem, the Rosay-Wong Theorem, other Riemann-Rosay-Wong-type results (e.g., the work of Pinchuk) to the extent that time permits.

#### Suggested books :

- Krantz, S. G.,
*Geometric analysis and function spaces*, CBMS Regional Conference Series in Mathematics, 81 (A M S, Providence, USA). - Rudin, W.,
*Function theory in the unit ball of $\mathbb{C^n}$*, Grundlehren der Mathematischen Wissenschaften (Springer-Verlag, New York-Berlin, 1980). - Krantz., S. G.,
*Function theory of several complex variables*, AMS Chelsea Publishing, Providence, RI, 2001.

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#### MA 325: Operator Theory II (3:0)

**Sz.-Nagy Foias theory:** Dilation of contractions on a Hilbert space, minimal isometric dilation, unitary dilation. Von Neumann’s inequality.

**Ando’s theorem:** simultaneous dilation of a pair of commuting contractions. Parrott’s example of a triple of contractions which cannot be dilated simultaneously. Creation operators on the full Fock space and the symmetric Fock space.

Operators spaces. Completely positive and completely bounded maps. Endomorphisms. Towards dilation of completely positive maps. Unbounded operators: Basic theory of unbounded self-adjoint operators.

#### Suggested books :

- John B. Conway,
*A course in Functional Analysis*, Springer, 1985. - Vern Paulson,
*Completely Bounded Maps and Dilations*, Pitman Research Notes, 1986.

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#### MA 326: Fourier Analysis (3:0)

Introduction to Fourier Series; Plancherel theorem, basis approximation theorems, Dini’s Condition etc. Introduction to Fourier transform; Plancherel theorem, Wiener-Tauberian theorems, Interpolation of operators, Maximal functions, Lebesgue differentiation theorem, Poisson representation of harmonic functions, introduction to singular integral operators.

#### Suggested books :

- Dym, H. and Mckean,
*H.P., Fourier Series and Integrals*, 1972. - Stein, E.M.,
*Singular Integrals and Differentiability Properties of Functions*, 1970. - Stein, E.M., and Weiss, G.,
*Introduction to Fourier Analysis on Euclidean Spaces*, 1975. - Sadosky, C.,
*Interpolation of Operators and Singular integrals*, 1979.

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#### MA 327: Topics in Analysis (3:0)

#### Pre-requisites :

- Real analysis
- Complex analysis
- Basic probability
- Linear algebra
- Groups
- It would help to know or to concurrently take a course in measure theory and /or functional analysis.

In this course we begin by stating many wonderful theorems in analysis and proceed to prove them one by one. In contrast to usual courses (where we learn techniques and see results as “applications of those techniques). We take a somewhat experimental approach in stating the results and then exploring the techniques to prove them. The theorems themselves have the common feature that the statements are easy to understand but the proofs are non-trivial and instructive. And the techniques involve analysis.

We intend to cover a subset of the following theoremes: Isoperimetric inequality, infinitude of primes in arithmetic progressions, Weyl’s equidistribution theorem on the circle, Shannon’s source coding theorem, uncertainty, principles including Heisenberg’s Wigner’s law for eigenvalue of a random matrix, Picard’s theorem on the range of an entire function, principal component analysis to reduce dimensionality of data.

#### Suggested books :

- Korner, I. T. W.,
*Fourier Analysis (1st Ed.)*, Cambridge Univ., Press, 1988. - Robert Ash.,
*Information Theory*, Dover Special Priced, 2008. - Serre, J. P.,
*A course in Arithmetic*, Springer-Verlag, 1973. - Thangavelu, S.,
*An Introduction to the Uncertainity Principle*, Birkhauser, 2003. - Rudin W.,
*Real and Complex Analysis (3rd Edition)*, Tata McGraw Hill Education, 2007.

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#### MA 328: Introduction to Several Complex Variables (3:0)

**Preliminaries:** Holomorphic functions in $C^n$ : definition , the generalized Cauchy integral formula, holomorphic functions: power series development(s), circular and Reinhardt domains, analytic continuation : basic theory and comparisons with the one- variable theory.

**Convexity theory:** Analytic continuation: the role of convexity, holomorphic convexity, plurisub-harmonic functions, the Levi problem and the role of the d-bar equation.

**The d- bar equation:** Review of distribution theory, Hormander’s solution and estimates for the d-bar operator.

#### Suggested books :

- Lars Hormander,
*An Introduction to Complex Analysis in Several Variables, 3rd edition*, North-Holland Mathematical Library, North-Holland, 1989. *Function Theory of Several Complex Variables, 2nd edition*, Wadsworth & Brooks/Cole, 1992.- Raghavan Narasimhan,
*Several Complex Variables*, Chicago Lectures in Mathematics Series, The University of Chicago Press, 1971.

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#### MA 329: Topics in Several Complex Variables (3:0)

#### Pre-requisites :

- MA 224 (i.e., the first course in Complex Analysis)
- preferably, some exposure to complex dynamics in one variable (the latest iteration of the topics course MA 324, Topics in Complex Analysis in One Variable, for instance).
- Students who have not seen any one-dimensional complex dynamics earlier but are highly interested in this course are encouraged to speak to the instructor.

This topics course is being run as an experiment in approaching the basic concepts in several complex variables with the eventual aim of studying some topics in multi-variable complex dynamics. By “complex dynamics”, we mean the the study of the dynamical system that arises in iterating a holomorphic map.

The course will begin with a complete and rigorous introduction to holomorphic functions in several variables and their basic properties. This will pave the way to motivating and studying a concept that is, perhaps, entirely indigenous to several complex variables: the notion of plurisubharmonicity.

Next, we shall look at some of the motivations behind the study of complex dynamics in several variables. Using the tools developed, we shall undertake a crash-course in currents, which are objects central to the study of some aspects of complex dynamics. We shall then cover as much of the following topics as time permits:

- Properties of fixed points
- The existence of proper subdomains of $C^n$, $n \geq 2$, that are holomorphically equivalent to $C^n$
- The Fatou and the Julia set for a dominant holomorphic self-map of $CP^n$, $n \geq 2$
- The Green current associated to a dominant holomorphic self-map of $CP^n$, and the dynamical information that it provides.

#### Suggested books :

- L. Hormander,
*Complex Analysis in Several Variables, 3rd edition*, North-Holland Publishing Co. Amsterdam, 1990. - J.E. Fornaess,
*Dynamics in Several Complex Variables*, CBMS Series, No. 87, American Mathematical Society, Providence, Rhode Island, 1996.

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#### MA 330: Topology - II (3:0)

TOPOLOGY - II

**Point Set Topology:** Continuous functions, metric topology, connectedness, path connectedness, compactness, countability axioms, separation axioms, complete metric spaces, function spaces, quotient topology, topological groups, orbit

**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.

#### Suggested books :

- Armstrong, M. A.,
*Basic Topology*, Springer (India), 2004. - Hatcher, A.,
*Algebraic Topology*, Cambridge Univ. Press, 2002. - Janich, K.,
*Topology*, Springer-Verlag (UTM), 1984. - Kosniowski, C.,
*A First Course in Algebraic Topology*, Cambridge Univ. Press, 1980. - Munkres, K. R.,
*Topology*, Pearson Education, 2005.

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#### MA 331: Topology and Geometry (3:0)

**Manifolds:** Differentiable manifolds, differentiable maps and tangent spaces, regular values and Sard’s theorem, vector fields, submersions and immersions, Lie groups, the Lie algebra of a Lie group.

**Fundamental Groups:** 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.

#### Suggested books :

- Brickell, F. and Clark, R. S.,
*Differentiable Manifolds*, Van Nostrand Reinhold Co., London, 1970. - Guillemin, V. and Pollack, A.,
*Differential Topology*, Prentice Hall, 1974. - Kosniowski, C., A,
*First Course in Algebraic Topology*, Cambridge Univ. Press, 1980. - Milnor, John W.,
*Topology from the Differentiable Viewpoint*, Princeton Landmarks in Mathematics, Princeton Univ. Press, 1997. - Munkres, J. R.,
*Elements of Algebraic Topology*, Addison-Wesley, 1984.

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#### MA 332: Algebraic Topology (3:0)

Homology : Singular homology, excision, Mayer-Vietoris theorem, acyclic models, CW-complexes, simplicial and cellular homology, homology with coefficients./p> Cohomology : Comology groups, relative cohomology,cup products, Kunneth formula, cap product, orientation on manifolds, Poincare duality.

#### Suggested books :

- Hatcher, A.,
*Algebraic Topology*, Cambridge Univ. Press, 2002 (Indian edition is available). - Rotman, J,
*An Introduction to Algebraic Topology*, Graduate Texts in Mathematics, 119, Springer-Verlag, 1988. - Munkres, I. R.,
*Elements of Algebraic Topology*, Addison-Wesiley, 1984. - Shastri, A. R.,
*Basic Algebraic Topology*, CRC Press, 2014.

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#### MA 333: Riemannian Geometry (3:0)

Riemannian metric, Levi-Civita connection, geodesics, exponential map, Hopf-Rinow theorem, curvature tensior, first and second variational formula, jacobi fields, Myers Bonnet theorem, Bishop-Gromov volume comparison theorem, Cartan-Hadamard theorem, Synge’s theorem, de Rham cohomology and the Bochner techniques. Topological implications of positive or negative curvature.

#### Suggested books :

- Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine,
*Riemannian geometry, Third edition.*, Universitext. Springer-Verlag, Berlin, 2004. - Peter Petersen,
*Riemannian geometry*, Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998.

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#### MA 334: Introduction to Homotopy Type Theory (3:0)

#### Pre-requisites :

- Basic topology and mathematical maturity will be assumed.
- Some familiarity with algebraic topology will be helpful, as will some familiarity with programming in a functional language.

This course introduces homotopy type theory, which provides alternative foundations for mathematics based on deep connections between type theory, from logic and computer science, and homotopy theory, from topology. This connection is based on interpreting types as spaces, terms as points and equalities as paths. Many homotopical notions have type-theoretic counterparts which are very useful for foundations.

Such foundations are far closer to actual mathematics than the traditional ones based on set theory and logic, and are very well suited for use in computer-based proof systems, especially formal verification systems. We will use the Lean Theorem Prover in this course. Note that the latest version (Lean 3) doen not support Homotopy Type Theory (yet), so you must use Lean 2.

#### Suggested books :

*Homotopy Type Theory: Univalent Foundations of Mathematics*, Institute for Advanced Studies, Princeton 2013.- Hatcher, A.,
*Algebraic topology*, Cambridge University Press, Cambridge, 2002. *Tutorial for the Lean Theorem Prover*.

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#### MA 336: Topics in Riemannian Geometry (3:0)

#### Pre-requisites :

- MA 333 - Riemannian Geometry
- MA 332 - Algebraic Topology

The first half of the course will focus on convergence theory of Riemannian manifolds. Gromov-Hausdorff convergence, Lipschitz convergence and collapsing theory will be discussed.

The second half will be about the Ricci flow. Existence and uniqueness, maximum principles and Hamilton’s theorem for 3-manifolds with positive Ricci curvature will be covered.

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#### MA 337: Computer Assisted Topology and Geometry (1:3)

#### Pre-requisites :

- familiarity with constructing proofs (e.g., having taken an Algebra/Linear Algebra/Analysis course in the mathematics department)
- a basic familiarity with programming.

The goal of this course is to use computers to address various questions in Topology and Geometry, with an emphasis on arriving at rigourous proofs. The course will consist primarily of assignments and projects. The computing tools used will include wrting own programs, existing Automated Theorem Proving Programs and packages like Maple, Mathematica and Matlab.

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#### MA 338: Differentiable manifolds and Lie groups (3:0)

#### Pre-requisites :

- Point set topology. A first course in algebraic topology is helpful but not necessary.
- Real analysis in more than one variable.
- Linear algebra.

Differentiable manifolds, differentiable maps, regular values and Sard’s theorem, submersions and immersions, tangent and cotangent bundles as examples of vector bundles, vector fields and flows, exponential map, Frobenius theorem, Lie groups and Lie algebras, exponential map , tensors and differential forms, exterior algebra, Lie derivative, Orientable manifolds, integration on manifolds and Stokes Theorem . Covariant differentiation, Riemannian metrics, Levi-Civita connection, Curvature and parallel transport, spaces of constant curvature.

#### Suggested books :

- Spivak M.,
*A comprehensive introduction to differential geometry (Vol. 1) (3rd Ed.)*, Publish or Perish, Inc., Houston, Texas, 2005. - Kumaresan S.,
*A course in differential geometry and Lie groups*, Texts and Readings in Mathematics, 22. Hindustan Book Agency, New Delhi, 2002. - Warner F.,
*Foundations of differentiable manifolds and Lie groups*, Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983. - Lee J.,
*Introduction to smooth manifolds*, Graduate Texts in Mathematics, 218., Springer, New York, 2013.

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#### MA 339: Geometric Analysis (3:0)

#### Pre-requisites :

- A first course on manifolds (MA 338 should do).
- Analysis (multivariable calculus, some measure theory, function spaces).
- Functional analysis (The Hahn-Banach theorem, Riesz representation theorem, Open mapping theorem. Ideally, the spectral theory of compact self-adjoint operators too, but we will recall the statement if not the proof)

Basics of Riemannian geometry (Metrics, Levi-Civita connection, curvature, Geodesics, Normal coordinates, Riemannian Volume form), The Laplace equation on compact manifolds (Existence, Uniqueness, Sobolev spaces, Schauder estimates), Hodge theory, more general elliptic equations (Fredholmness etc), Uniformization theorem.

#### Suggested books :

- Do Carmo,
*Riemannian Geometry*. - Griffiths and Harris,
*Principles of Algebraic Geometry*. - S. Donaldson,
*Lecture Notes for TCC Course “Geometric Analysis”*. - J. Kazdan,
*Applications of Partial Differential Equations To Problems in Geometry*. - L. Nicolaescu,
*Lectures on the Geometry of Manifolds*. - T. Aubin,
*Some nonlinear problems in geometry*. - C. Evans,
*Partial differential equations*. - Gilbarg and Trudinger,
*Elliptic partial differential equations of the second order*. - G. Szekelyhidi,
*Extremal Kahler metrics*.

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#### MA 340: Advanced Functional Analysis (3:0)

Banach algebras, Gelfand theory, $C^{*}$-algebras the GNS construction, spectral theorem for normal operators, Fredholm operators. The L-infinity functional calculus for normal operators.

#### Suggested books :

- Conway, J.B.,
*A Course in Functional Analysis*, Springer, 1985. - Douglas, R. G.,
*Banach Algebra Techniques in Operator Theory*, Academic Press, 1972.

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#### MA 341: Matrix Analysis and Positivity (3:0)

#### Pre-requisites :

- A course in linear algebra, and a course in calculus/real analysis.

This course explores matrix positivity and operations that preserve it. These involve fundamental questions that have been extensively studied in the mathematics literature over the past century, and are still being studied owing to modern applications to high-dimensional covariance estimation. The course will bring together techniques from different areas: analysis, linear algebra, combinatorics, and symmetric functions.

**List of topics** (time permitting):

**1.***The cone of positive semidefinite matrices. Totally positive/non-negative matrices.* Examples of PSD and TP/TN matrices (Gram, Hankel, Toeplitz, Vandermonde, $\mathbb{P}_G$). Matrix identities (Cauchy-Binet, Andreief). Generalized Rayleigh quotients and spectral radius. Schur complements.

**2.***Positivity preservers.* Schur product theorem. Polya-Szego observation. Schoenberg’s theorem. Positive definite functions to correlation matrices. Rudin’s (stronger) theorem. Herz, Christensen-Ressel.

**3.***Fixed-dimension problem.* Introduction and modern motivations. H.L. Vasudeva’s theorem and simplifications. Roger Horn’s theorem and simplifications.

**4.***Proof of Schoenberg’s theorem.* Characterization of (Hankel total) positivity preservers in the dimension-free setting.

**5.***Analytic/polynomial preservers – I.* Which coefficients can be negative? Bounded and unbounded domains: Horn-type necessary conditions.

**6.***Schur polynomials.* Two definitions and properties. Specialization over fields and for real powers. First-order approximation.

**7.***Analytic/polynomial preservers – II.* Sign patterns: The Horn-type necessary conditions are best possible. Sharp quantitative bound. Extension principle I: dimension increase.

**8.***Entrywise maps preserving total positivity.* Extension principle II: Hankel TN matrices. Variants for all TP matrices and for symmetric TP matrices. Matrix completion problems.

**9.***Entrywise powers preserving positivity.* Application of Extension principle I. Low-rank counterexamples. Tanvi Jain’s result.

**10.***Characterizations for functions preserving $\mathbb{P}_G$.* Extension principle III: pendant edges. The case of trees. Chordal graphs and their properties. Functions and powers preserving $\mathbb{P}_G$ for $G$ chordal. Non-chordal graphs.

#### Suggested books :

- Rajendra Bhatia,
*Matrix Analysis*, vol. 169 of Graduate Texts in Mathematics, Springer, 1997. - Rajendra Bhatia,
*Positive definite matrices*, Princeton Series in Applied Mathematics, 2007. - Roger A. Horn and Charles R. Johnson,
*Matrix analysis*, Cambridge University Press, 1990. - Roger A. Horn and Charles R. Johnson,
*Topics in matrix analysis*, Cambridge University Press, 1991. - Samuel Karlin,
*Total positivity*, Stanford University Press, 1968.

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#### MA 342: Partial Differential Equations II (3:0)

Introduction to distribution theory and Sobolev spaces, Fundamental solutions for Laplace, heat and wave operations.

Second order elliptic equations: Boundary value problems, Regularity of weak solutions, Maximum principle, Eigenvalues.

Semi group theory:Hille-Yosida theorem, Applications to heat, Schroedinger and wave equations.

System of first order hyperbolic equations: Bicharacteristics, Shocks, Ray theory, symmetric hyperbolic systems.

#### Suggested books :

- Evans, L. C.,
*Partial Differential Equations*, AMS, 1998. - Kesavan, S.,
*Topics in Functional Analysis and Applications*, Wiley Eastern, 1988. - Pazy, A.,
*Semigroups of Linear Operators and Applications to Partial Differential Equations*, Springer-Verlag, 1983. - Prasad, P. and Ravindran, R.,
*Partial Differential Equations*, Willey Eastern, 1985. - Treves, J. E.,
*Basic Linear Partial Differential Equations*, Academic Press, 1975.

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#### MA 344: Homogenization of Partial Differential Equations (3:0)

Review of Distributions, Sobolev spaces and Variational formulation. Introduction to Homogenization. Homogenization of elliptic PDEs. Specific Cases: Periodic structures and layered materials. Convergence Results: Energy method, Two-scale multi-scale methods, H-Convergence, Bloch wave method. General Variational convergence: G -convergence and G- convergence, Compensated compactness. Study of specific examples and applications

#### Suggested books :

- A. Bensoussan, J. L., Lions and G., Papanicolaon.,
*Asymptotic Analysis for Periodic Structures*, North Holland (1978). - G. Dal Maso, An introduction to $\\Gamma$ convergence, Birkauser (1993)., .
- V. V. Jikov, S. M. Kozlov, and O. A. Oleinik,
*Homogenization of Differential Operators and Integral Functionals*, Springer (1991). - E. Sanchez Palencia,
*Non homogeneous Media and Vibration Theory*, Springer lecture Notes in Physics, 127 (1980).

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#### MA 345: Nonlinear Functional Analysis and Applications to Differential Equations (3:0)

Introduction to Calculus of Variations and Morse Theory. Critical Point Theory for Gradient Mappings: Mountain Pass theorem, linking Theorems, Saddle Point theorem, Dual formulation etc.

#### Suggested books :

- Rabinowitz,
*Minimax methods in critical point theory with applications to differential equations*, C. B. M. S. No. 65, 1986. - Ghoussoub, N.,
*Duality and perturbation methods in critical point theory*, Cambridge Univ. Press, 1993. - Struwe, M.,
*Variational methods and their applications to nonlinear partial differential equations and Hamiltonian Systems*, Springer-Verlag, 1990.

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#### MA 346: Integral Equations with Applications (3:0)

Classification of integral equations and occurrence in boundary value problems for Ordinary and Partial Differential Equations, Abel’s integral equations, Integral equations of the second kind, Degenerate Kernels, The Neumann series solutions, Fredholm theorems, The eigenvalue problems, Rayleigh-Ritz method, Galerkin method, The Hilbert-Schmidt theory, Singular integral equations, Riemann-Hilbert problems, The Wiener-Hopf equations, The Wiener-Hopf technique, Numerical methods, Applications of integral equations of problems of Elasticity, Fluid Mechanics and Electromagnetic theory.

#### Suggested books :

- Porter, D. and Stirling, S. G.,
*Integral Equations, A Practical Treatment*, Cambridge Univ. Press, 1990. - Gakhov, F. D.,
*Boundary Value Problems*, Addision Wesley, 1966. - Muskhelishvilli, N. I.,
*Singular Integral Equations*, Noordhoff, 1963. - Nobe, B.,
*The Wiener-Hopf Technique*, Pergamon, 1958. - Jones. D. S.,
*The Theory of Electromagnetism*, Pergamon, 1964.

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#### MA 347: Advanced Partial Differential Equations and Finite Element Method (3:0)

**Distribution Theory** - Introduction, Topology of Test functions, Convolutions, Schwartz Space, Tempered Distributions, Fourier Transform;

**Sobolev Spaces** - Definitions, Extension Operators, Continuous and Compact Imbeddings, Trace results; Weak Solutions - Variational formulation of Elliptic Boundary Value Problems, Weak solutions, Maximum Principle, Regularity results;

**Finite Element Method (FEM)** - Introduction to FEM, Finite element solution of Elliptic boundary value problems.

#### Suggested books :

- L. Schwartz,
*Theorie des Distributions*, Hermann, (1966). - S. Kesavan,
*Topics in Functional Analysis and applications*, John Wiley & Sons (1989). - P. G. Ciarlet,
*Lectures on Finite Element Method*, TIFR Lecture Notes Series, Bombay (1975). - J. T. Marti,
*Introduction to Finite Element Method and Finite Element Solution of Elliptic Boundary Value Problems*, Academic Press (1986).

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#### MA 350: Topics in Analytic Number Theory (3:0)

#### Pre-requisites :

- Basics of number theory
- Complex analysis
- Preferably some familiarity with MA 352 (=Introduction to Analytic number theory)

Arithmetical functions, Primes in Arithmetic Progressions, Prime number theorem for arithmetic progressions and zeros of Dirichlet L-functions, Bombieri-Vinogradov theorem, Equidistribution, circle method and applications (ternary Goldbach in mind), the Large Sieve and applications, Brun’s theorem on twin primes.

(Further topics if time permits: more on sieves, automorphic forms and L-functions, Hecke’s L-functions for number fields, bounds on exponential sums etc.)

#### Suggested books :

- H. Davenport,
*Multiplicative Number Theory*, Springer GTM 74. - M. Ram Murty,
*Problems in Analytic Number Theory*, Springer GTM 206. - H. Iwaniec and E. Kowalski.,
*Analytic Number Theory*, AMS Colloquium Publ. 53.

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#### MA 351: Topics in Modular Forms (3:0)

#### Pre-requisites :

- Basics of number theory
- Complex analysis
- Preferably some familiarity with MA 215 (Introduction to Modular Forms) but not necessary.

Holomorphic Modular forms: motivation and introduction, Eisentein series, cusp forms, Fourier expansion of Poincare series and Petersson trace formula, Hecke operators and overview of newform theory, Kloosterman sums and bounds for Fourier coefficients, Automorphic L-functions, Dirichlet-twists and Weil’s converse theorm, Theta functions and representation by quadratic forms, Convolution: the Rankin-Selberg method. (Further topics if time permits: Non-holomorphic modular forms (overview), Siegel modular forms (introduction), Elliptic curves and cusp forms, spectral theory, analytic questions related to modular forms.)

#### Suggested books :

- J.P. Serre.,
*A Course in Arithmetic*, Springer GTM, 2007. - N.Koblitz.,
*Introduction to Elliptic Curves and Modular Forms*, Springer GTM, 1997. - H. Iwaniec,
*Topics in Classical Automorphic Forms*, GTM 17, AMS,1997. - F. Diamond and J.Schurman,
*A First Course in Modular forms*, Springer GTM 228.

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#### MA 352: Introduction to Analytic Number Theory (3:0)

#### Pre-requisites :

- Introductory courses in basic number theory
- Complex analysis

Review of arithmetical functions, Averages of arithmetical functions, Elementary results on the distribution of prime numbers, Dirichlet characters, Dirichlet’s theorem on primes in arithmetic functions, Dirichlet series and Euler products, Riemann zeta function and related objects, The prime number theorem.

(Time permitting: More advanced topics like Sieves, bounds on exponential sums, zeros of zeta functions, circle method etc.)

#### Suggested books :

- H. Davenport.,
*Multiplicative Number Theory*, Springer GTM 74 (third ed.) 2000. - Tom. M. Apostol.,
*Introduction to Analytic Number Theory*, Springer-Verlag, 1976.

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#### MA 360: Random Matrix Theory (3:0)

Wigner’s semicircle law: (a) combinatorial method, (b) Stieltjes’ transform method, (c) Chatterjee’s invariance principle method.

Gaussian unitary and orthogonal ensembles: (a) Exact density of eigenvalues. (b) Orthogonal polynomials and determinantal formulas leading to another proof of Wigner’s semicircle law.

Tridiagonal reduction for GUE and GOE: (a) Another derivation of eigenvalue density. (b) Another proof of Wigner’s semicircle law. (c) Matrix models for Beta ensembles. (d) Selberg’s integral.

Other models of random matrices - Wishart and Jacobi ensembles.

Free probability: (a) Noncommutative probability space and free independence. (b) Combinatorial approach to freeness. (c) Limiting spectra of sums of random matrices.

Non-hemitian random matrices: (a) Ginibre ensemble. (b) Circular law for matrices with i.i.d entries.

Fluctuation behaviour of eigenvalues (if time permits).

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#### MA 361: Probability Theory (3:0)

Probability measures and randown variables, pi and lambda systems, expectation, the moment generating function, the characteristic function, laws of large numbers, limit theorems, conditional contribution and expectation, martingales, infinitely divisible laws and stable laws.

#### Suggested books :

- Durrett, R.,
*Probability: Theory and Examples (4th Ed.)*, Cambridge University Press, 2010. - Billingsley, P.,
*Probability and Measure (3rd Ed.)*, Wiley India, 2008. - Kallenberg, O.,
*Foundations of Modern Probability (2nd Ed.)*, Springer-Verlag, 2002. - Walsh, J.,
*Knowing the Odds: An Introduction to Probability*, AMS, 2012.

Top

#### MA 362: Stochastic Processes (3:0)

First Construction of Brownian Motion, convergence in $C[0,\infty)$, $D[0,\infty)$, Donsker’s invariance principle, Properties of the Brownian motion, continuous-time martingales, optional sampling theorem, Doob-Meyer decomposition, stochastic integration, Ito’s formula, martingale representation theorem, Girsanov’s theorem, Brownian motion and the heat equation, Feynman- Kac formula, diffusion processes and stochastic differential equations, strong and weak solutions, martingale problem.

#### Suggested books :

- P. Billingsley,
*Convergence of probability measures*. - Karatzas and Shreve,
*Brownian motion and stochastic calculus*. - Revuz and Yor,
*Continuous martingales and Brownian motion*. - A. Oksendal,
*Introduction to stochastic differential equations*.

Top

#### MA 363: Stochastic Finance I (3:0)

Financial market. Financial instruments: bonds, stocks, derivatives. Binomial no-arbitrage pricing model: single period and multi-period models. Martingale methods for pricing. American options: the Snell envelope. Interest rate dependent assets: binomial models for interest rates, fixed income derivatives, forward measure and future. Investment portfolio: Markovitz’s diversification. Capital asset pricing model (CAPM). Utility theory.

#### Suggested books :

- Luenberger, D.V.,
*Investment Science*, Oxford University Press, 1998. - Shiryaev, A.N.,
*Essentials of Stochastic Finance*, World Scientific, 1999. - Shreve, S.E.,
*Stochastic Calculus for Finance I: The Binomial Asset pricing Model*, Springer, 2005.

Top

#### MA 364: Linear and Non-linear Time Series Analysis (3:0)

**Linear time series analysis** - modelling time series using stochastic processes, stationarity, autocovariance, auto correlation, multivariate analysis - AR, MA, ARMA, AIC criterion for order selection;

**Spectral analysis** - deterministic processes, concentration problem, stochastic spectral analysis, nonparametric spectral estimation (periodogram, tapering, windowing), multitaper spectral estimation; parametric spectral estimation (Yule-Walker equations, Levinson Durbin)(recursions);

**Multivariate analysis** - coherence, causality relations; bootstrap techniques for estimation of parameters;

**Nonlinear time series analysis** - Lyapunov exponents, correlation dimension, embedding methods, surrogate data analysis.

#### Suggested books :

- Box, G. E. P. and jenkins, G. M.,
*Time series analysis*, Holden-Day, 1976. - Jenkins, G. M. and Watts, D. G.,
*Spectral analysis and its applications*, Holden-Day, 1986. - Efron, B.,
*The Jackknife, the bootstrap and other resampling plans*, SIAM, 1982. - Parker, T. S. and Chua, L. O.,
*Practical numerical algorithms for chaotic systems*, Springer-Verlag, 1989.

Top

#### MA 365: Topics in Gaussian Processes (3:0)

A course in Gaussian processes. At first we shall study basic facts about Gaussian processes - isoperimetric inequality and concentration, comparison inequalities, boundedness and continuity of Gaussian processes, Gaussian series of functions, etc. Later we specialize to smooth Gaussian processes and their nodal sets , in particular expected length and number of nodal sets, persistence probability and other such results from recent papers of many authors.

#### Suggested books :

- Robert Adler and Jonathan Taylor,
*Gaussian Random Fields*, Springer, New York, 2007. - Svante Janson,
*Gaussian Hilbert Spaces*, Cambridge University Press, Cambridge, 1997. - A. I. Bogachev,
*Gaussian Measures*, American Mathematical Society, Providence, RI, 1998. - Michel Ledoux and Michel Talagrand,
*Probability in Banach spaces. Isoperimetry and processes*, Springer-Verlag, Berlin, 2011. - Michel Ledoux,
*Isoperimetry and Gaussian analysis*, St. Flour lecture notes-1994.

Top

#### MA 366: Stochastic Finance II (3:0)

Trading in continuous time : geometric Brownian motion model. Option pricing : Black-Scholes-Merton theory. Hedging in continuous time : the Greeks. American options. Exotic options. Market imperfections. Term-structure models. Vasicek, Hull-White and CIR models. HJM model. LIBOR model. Introduction to credit Rsik Models: structural and intensity models. Credit derivatives.

#### Suggested books :

**James Cook's Elementary Differential Geometry Homepage**

__The initial big picture:__

Elementary differential geometry is centered around problems of curves and surfaces in three dimensional euclidean space. We're using Barret Oneil's excellent text this semester.

Oneil uses linear algebra and differential forms throughout his text. I am excited about learning the method of moving frames for surfaces in 3-space. Ideally, I'd like to start reading some papers which generalize this method to surfaces in more exotic three dimensional manifolds. If our course goes very well, perhaps I'll be able to show you something about that other world of ideas at the end.

This course is the natural bridge to abstract manifold theory. I will avoid the temptation to generalize our work as we go. In some sense, advanced calculus is a more abstract course as we insist on treating many questions in n-dimensions. Here, in this course, usually n=3.

__Guide to Self-Paced Differential Geometry Course:__

In Summer 2015 I wrote these notes:

from which I gave the Lectures based on O'neill, Kuhnel for Test 1. Then for Test 2 I simply recycled my old course notes plus a few new hand-written pages for Chapter 4. Then I talked through my notes from Tapp to help build-up to the final exam project. I have links to videos which are not absolutely essential to watch, I think you could get enough from the texts and the notes alone. But, perhaps the videos help. I also included scans of what I wrote in landscape for the Lectures.

**Lecture 1**: [ notes from video ] video (5:23)

overview, goals, prerequisites:**Lecture 2**: [ notes from video ] part 1 (23:30) part 2 (39:18)

(Section 1.1-1.3 of Oneill and Chapter 1 of Kuhnel): basic notations of points and vectors, tangent spaces, directional derviatives, derivation notation for vectors.**Lecture 3**: [ notes from video ] part 1 (39:42) part 2 (12:21) part 3 (2:56)

(Section 1.5-1.6 of Oneill): differential forms in Euclidean three dimensional space. Wedge product and exterior differentiation as you might see at the end of an ambitious multivariate calculus course. Kuhnel introduces this topic much later at a higher level of sophistication.**Lecture 4**: [ notes from video ] part 1 (32:59) part 2 (36:28)

(Section 1.4 and 1.7 of Oneill and Chapter 1 of Kuhnel): curves and velocity and an operator, major theorems and terms from advanced calculus. Component functions, derivative of vector-valued functions of several variables. Here I will be a bit more explicit about derivations than Oneill.**Lecture 5**: [ notes from video ] part 1 (53:32) part 2 (46:27)

(Sections 2.1-2.3 of Oneill): tangent space as an inner product space, curves in three dimensional space, velocity and acceleration (or first and second derivatives of parametrized curve if you prefer)**Lecture 6**: [ notes from video ] part 1 (57:01) part 2 (20:08)

(Section 2.3- 2.4 of Oneill): Frenet frame and the Frenet formulas, non-unit speed curves, further examples of curves in three dimensions.**Lecture 7**: [ notes from video ] part 1 (31:07) part 2 (17:49) part 3 (32:25)

(Section 2.5-2.7 of Oneill): covariant derivative, frame fields and connection form introduced. This is the first showcase of the frame field technique. We also prove some essential calculus for matrices of forms in Part 3.**Lecture 8**: [ notes from video ] video (59:51)

(Section 2.8): the structural equations. Here we use matrix-valued differential forms to understand the geometry of frame fields. This section is important as it is later specialized to surfaces.**Lecture 9**: [ notes from video ] part 1 (25:00) part 2 (28:30)

(Sections 3.1-3.3 of Oneill): isometries and orientation in Euclidean space.**Lecture 10**: [ notes from video ] video (34:36)

(Chapter 2 of Kuhnel): theory of Frenet curves generalized to curves in n-dimensional Euclidean space. Naturally, in n=3 case this is redundant, but, I think we�ll gain much deeper understanding of Oneill from this perspective. On the other hand, Kuhnel does not discuss covariant derivatives here as the approach to surfaces in Kuhnel does not use Cartan�s equations as primary.**Lecture 11**: [ notes from video ] part 1 (27:28) part 2 (51:12)

(Sections 3.4-3.5 of Oneill): Euclidean geometry and congruence of curves.- [
**MISSION 1**:] in preparation for Test 1 (also should do the recommended problems for which solutions are posted towards base of this page). **TEST 1**: covers Chapters 1-3 of Oneil and Chapters 1 and 2 of Kuhnel with focus on curves and frames.**Lecture 12**: [ notes from video ] part 1 (10:59) part 2 (19:02) part 3 (8:37) part 4 (16:17) part 5 (49:12)

(Sections 4.1-4.3 of Oneill): surfaces in three dimensions, patches, examples, differentiable functions and tangent vectors to a surface.**Lecture 13**: [ notes from video ] part 1 (40:22) part 2 (5:18) part 3 (41:32) part 4 (10:48) part 5 (19:07) part 6 (11:15)

(Sections 4.4-4.6 of Oneill): differential forms, mappings of surfaces, integration of forms on surfaces.**Lecture 14**: [ notes from video ] part 1 (43:02) part 2 (15:24)

(Sections 4.7-4.8 of Oneill): topological concepts on surfaces, manifolds.**Lecture 15**: [ notes from video ] part 1 (41:04) part 2 (35:04) part 3 (38:49)

(Sections 5.1-5.3 of Oneill): shape operator, normal and Gaussian curvature.**Lecture 16**: [ notes from video ] video (35:01)

(Sections 5.4-5.5 of Oneill): alphabet soup of classical formulas E,F,G, L,M,N and also the formulas for level surfaces, computational techniques.**Lecture 17**: [ notes from video ] video (56:21)

(Sections 5.6 - 5.7 of Oneill): special curves and surfaces of revolution.**Lecture 18**: [ notes from video ] video (41:46)

(Sections 6.1-6.2 of Oneill): frame fields adapted to surfaces, fundamental structural equations, form calculations.**Lecture 19**: [ notes from video ] video (13:31)

(Section 6.3 of Oneill): some global theorems about vanishing curvature, planes, spheres, compact and umbilic surfaces.**Lecture 20**: [ notes from video ] part 1 (26:30) part 2 (23:34)

(Sections 6.4-6.5 of Oneill): local isometries and the intrinsic geometry of surfaces. Gauss� theorema egregium as argued in the language of differential forms.**Lecture 21**: [ notes from video ] part 1 (43:02) part 2 (13:01)

(Sections 6.6-6.8 of Oneill): orthogonal coordinates give nice formulas, orientation and integration, total curvature. (the total curvature on its own is perhaps not so exciting, but, when you see how it appears in the Gauss-Bonnet Theorem in 7.6.4 you�ll appreciate studying it here)**Lecture 22**: [ notes from video ] video (6:32)

(Section 6.9 of Oneill): congruence of surfaces, here we see that having the same shape operator amounts to having the same shape in the sense that one surface is related to the other by a rigid motion. This generalizes Lecture 10.**Lecture 23**: [ notes from video ] video (26:57)

(Section 7.1 of Oneill): concept of an abstract geometric surface.**Lecture 24**: [ notes from video ] video (42:03)

(Section 7.2 of Oneill): Gaussian curvature developed in the abstract case.**Lecture 25**: [ notes from video ] video (41:16)

(Section 7.3 of Oneill): covariant derivatives motivated and discussed for an abstract geometric surface.**Lecture 26**: [ notes from video ] video (29:56)

(Sections 7.4-7.5 of Oneill): geodesics and Clairaut parametrizations.**Lecture 27**: [ notes from video ] part 1 (55:54) part 2 (3:42)

(Section 7.6 of Oneill): Gauss Bonnet Theorem**Lecture 28**: [ notes from video ] video (29:56)

(Section 7.7 of Oneill): applications of Gauss Bonnet.**Lecture 29**: [ notes from video ] video (29:56)

(Section 3A-3B of Kuhnel): the I, II and III forms on a surface.**Lecture 30**: (Section 3C of Kuhnel): surfaces of revolution and ruled surfaces (not given)**Lecture 31**: (Section 3D of Kuhnel): minimal surfaces (not given)- [
**MISSION 2**:] in preparation for Test 2 (also should do the recommended problems for which solutions are posted towards base of this page). **TEST 2**on Chapters 4-7 of Oneill, that is, on surfaces.**Lecture 32**: (Sections 5A-5B of Kuhnel): n-dimensional manifold and tangent space (not yet given)**Lecture 33**: (Section 5C of Kuhnel): Riemannian metrics (not yet given)**Lecture 34**: (Section 5D of Kuhnel): Riemannian connections (not yet given)**Lecture 35**: (Section 6A of Kuhnel): tensors (not yet given)**Lecture 36**: (Section 6B of Kuhnel): sectional curvature (not yet given)**Lecture 37**: (Section 6C of Kuhnel): Einstein and Ricci tensors (not yet given)**Lecture 38-46 relabled under their own title of "crash course in matrix algebras"**: on my You Tube Channel where I posted 14 little talks which cover the main results from Tapp's text:**Part 1:**(13:41)

In this talk we cover pages 1 to 4 of my notes where the K-notation for K=R,C,H is explained and the general linear group of nxn matrices over K is defined. We also begin to explain mapping which embeds nxn complex matrices in 2nx2n real matrices.**Part 2:**(20:16)

Pages 4 to 6 of my notes. Here we study the complex matrices represented as real matrices and quaternionic matrices represented as complex matrices. These natural homomorphisms allow us to reduce everything to a problem of sufficiently large real matrices. Conversely, for a set of real matrices which satisfies the right conditions with respect to the complex or quaternionic structure we can trade the real problem for a corresponding complex or quaternionic problem.**Part 3:**(15:25)

Based on pages 7 to 9 of my notes. Inner product for n-tuples over K=R,C, H are described using appropriate conjugations. Also, the isometries of these standard inner products naturally give rise to On(K) which produces at once the seemingly distinct matrix groups O(n) over R, U(n) over C and Sp(n) over H; that is orthogonal, unitary and symplectic matrices**Part 4:**(13:47)

Covers pages 10 to 12 of my notes. Calculus on matrix groups exposed. In particular, we examine how curves of matrices through the identity matrix are used to define the tangent space to the identity which we find to have a Lie Algebra structure.**Part 5:**(16:57)

Covers page 13 to 15 of my notes. We characterize the Lie algebras of several standard examples by studying the velocity of matrix-curves through the identity. Then, we spend a few minutes deriving a beautiful formula through the manifest multilinearity of the levi-civita formula for the determinant. One of my favorite calculations.**Part 6:**(13:33)

Covers pages 16 to 19 of my notes. Here we sketch how transformations on Kn can be viewed as vector fields on Kn as points correspond naturally to vectors in the vector space Kn. Probably could use a lot more pictures here. Lie algebra of SO(n) proves a bit more challenging to derive than our previous examples.**Part 7:**(13:47)

Covers pages 20 to 21 of my notes. We study the matrix exponential which gives us a map between the Lie algebra and the Lie group.**Part 8:**(5:18)

Covers page 22 to 23 of my notes. We use the matrix exponential to generate the one-parameter groups for the matrix Lie group.**Part 9:**(21:27)

Covers pages 23 to 27 of my notes. The big adjoint is introduced as the derivative of the conjugate (in the group-theoretic sense of the term). Then we derive a few basic calculational tools which allow us to see the fascinating theorem about morphisms of Lie Groups transferring naturally to the corresponding Lie Algebras. Once more, the matrix exponential serves a deep and meaningful role in all this.**Part 10:**(7:08)

Covers pages 28 to 29 of my notes.We give an example of Lie algebras which are seemingly distinct, yet, share the same bracket-structure (hence are isomorphic). However, we caution the isomorphism of groups brings global topological questions to bear of which we only touch on here... Finally the homomorphism Ad is introduced to provide a homomorphism of the Lie group G on sets of invertible matrices of size dxd.**Part 11:**(9:11)

Covers pages 30 to 32 of my notes. Here we study yet more about the adjoint map, an interesting identity connecting the Lie algebra with the exponential of the adjoint. There is more to read in Tapp, I didn't touch on the Baker-Cambell-Hausdorff relation etc.**Part 12:**(7:54)

Covers page 33 of my notes. Lie correspondence theorem given and the concept of Spin(n) is briefly described.**Part 13:**(25:31)

Covers pages 34 to 38 of my notes. Higher dimensional torus is described. The maximal torus of a matrix group is defined and a standard presentation of the torus is given for each of the standard examples. A theorem about the center of the group is studied with the help of the maximal tori. We also describe how conjugation (in the group theoretic sense) moves us from the standard torus to other points in the group. We mention certain parts of the proof which are interesting to the study of e-vectors and and diagonalization of symmetric matrices in linear algebra.**Part 14:**(11:00)

Covers pages 39 to 40 of my notes. Lie algebra of the tori are detailed. Further detail on the conjugates of tori are also given as to place regular elements on just one such torus. Finally, the classification of compact matrix groups is given. We explain that all there is to find is direct products of SO(n), SU(n), Sp(n) and the 5 exceptional Lie groups. Of course, these results are due to an entirely different course of study. See Erdmann and Wildon's "Introduction to Lie Algebras" for a treatment which is essentially at the same level as Tapp.

- Final Exam Project.

__Rough Notes from my Reading in Oneil:__

- Chapter 1: preliminary notes on Chapter 1 (the next set has more of exterior derivatives and wedges to help those who are rusty and/or have never seen them).
- Chapter 1: summary notes on Chapter 1.
- Chapter 2 (2.1-2.4): notes on vector fields in R3 and the Frenet-Serret equations.
- Chapter 2 (2.5-2.7): notes on the covariant derivative in R3, frame fields and the reformulation of the covariant derivative in terms of the connection form of a frame field.
- Chapter 2 (2.8): notes on the structure equations of Cartan. In this section we begin to see how matrix-valued differential forms are used to calculate geometric data.
- Chapter 3 (3.1-3.3): notes on isometries of three dimensional space and their tangent maps.
- Chapter 4: calculus on surfaces in R3 ( unfinished, beware some typos on last couple pages).
- Chapter 5: shape operators and some classic formulas of Gauss
- Chapter 6: on the intrinsic geometry of surfaces via form calculus.
- Chapter 6: notes on integration and total curvature again for surfaces in euclidean 3-space
- Chapter 7: on geometric surfaces: how to create non-euclidean metrics on plane, pullback metrics, flat spheres and curved planes. Gauss's awesome theorem made a definition.
- Chapter 7: covariant derivatives, geodesic curves, intrinsic angles and geodesic curvatures.

__Homework:__

As I mentioned, my personal goal is to solve most the problems. However, that seems like a bit much for the class. Missions are given below:

I'll probably just let you look at my global solutions for this class. At a minimum, the solutions below should include most of the problems assigned in the Missions.

- My Solutions from Chapter 1 of Oneil: I solved most of the problems here, enjoy.
- My Solutions from Chapter 2.1-2.3 of Oneil: I solved most of the problems here, enjoy.
- My Solutions from Chapter 2.4-2.5 of Oneil: I solved some of the problems here, enjoy.
- My Solutions from Chapter 2.6-2.8 of Oneil: I solved some of the problems here, enjoy.
- My Solutions from Chapter 3 of Oneil: I solved some of the problems here, enjoy.
- My Solutions from Chapter 4 of Oneil: I solved some of the problems here, enjoy. (so much more to solve here, so many nice problems left undone...)
- My Solutions from Chapter 5 of Oneil: I solved some of the problems here, enjoy. (so much more to solve here, so many nice problems left undone...)

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Last Modified: 6-26-15

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