A review of field extensions​ with emphasis on the following topics: Algebraic extensions, algebraic closure, normal extensions, separable extensions, finite fields, inseparable extensions. [DF-13, J1-4, L-V]

Galois theory:​ Galois extensions, linear independence of characters, norm, trace and discriminants, Hilbert theorem 90, cyclic extensions, solvable and radical extensions, Kummer theory, algebraic independence of homomorphisms, the normal basis theorem, Krull topology, projective limits, profinite groups, fundamental theorem of Galois theory for infinite extensions. [DF-14, J1-4, J2-8, L-VI]

Ring extensions:​ Integral extensions, integral Galois extensions, prime ideals in integral ring extensions, decomposition and inertia groups, ramification index and residue class degree, Frobenius map, extensions of homomorphisms. [DF-16, J2-7, L-VII]

Transcendental extensions:​ Transcendental bases, Noether normalization lemma, linearly disjoint extensions, separable and regular extensions, derivations, Hilbert Nullstellensatz. [DF-15, J2-8, L-VIII, L-IX.1]

Dedekind domains:​ Dedekind domains, unique factorisation of ideals. [DF-16, J2-10, L-VII]

Valuations and completions:​ Basic definitions, finite dimensional extensions of complete fields, local fields, discrete valuations, Hensel's lemma, Krasner's lemma, zeros of polynomials in complete fields. [DF-16, J2-9, L-XII]

Text/References

[DF] Dummit, Foote: Abstract algebra, second edition, Wiley student editions, 2005.

[J1] Jacobson: Basic algebra, I, Dover publications, 2009.

[J2] Jacobson: Basic algebra, II, Dover publications, 2009.

[L] Lang: Algebra, third edition, Springer-Verlag, GTM 211, 2002

A review of modules over a PID​. [DF-12, J1-3, L-III.7]

Noetherian modules and rings:​ Primary decomposition, Nakayama's lemma, filtered and graded modules, the Hilbert polynomial, Artinian modules and rings. [DF-15, J2-3, L-X]

Semisimple and simple rings:​ Semisimple modules, Jacobson density theorem, semisimple and simple rings, Wedderburn-Artin structure theorems, Jacobson radical, the effect of a base change on semisimplicity. [DF-18, J2-3, J2-4, L-XVII]

Representations of finite groups:​ Basic definitions, characters, class functions, orthogonality relations, induced representations and induced characters, Frobenius reciprocity, decomposition of the regular representation, supersolvable groups, representations of symmetric groups. [DF-18, DF-19, J2-5, L-XVIII]

Categories and functors:​ Definitions and examples, functors and natural transformations, the equivalence of categories, products and coproducts, the Hom functor, representable functors, universals and adjoints, direct and inverse limits, free objects. [DF-Appendix II, J2-1, L-I.11]

Homological algebra:​ Additive and abelian categories, complexes and homology, long exact sequences, homotopy, resolutions, derived functors, Ext, Tor, cohomology of groups, extensions of groups. [DF-17, J2-6, L-XX]

Text/References:

[DF] Dummit, Foote: Abstract algebra, second edition, Wiley student editions, 2005.

[J1] Jacobson: Basic algebra, I, Dover publications, 2009.

[J2] Jacobson: Basic algebra, II, Dover publications, 2009.

[L] Lang: Algebra, third edition, Springer-Verlag, GTM 211, 2002

Review of measure theory: monotone convergence theorem, dominated convergence theorem, complete measures. Borel measures: Riesz representation theorem, Lebesgue measure on Rk, Lp spaces

Complex measures: total variation, absolute continuity, Radon-Nikodym theorem, polar and Hahn decompositions, bounded linear functionals on Lp, generalised Riesz representation theorem.

Differentiation: Maximal function, Lebesgue points, absolute continuity of functions, fundamental theorem of calculus, Jacobian of a differentiable transformation, change of variable formula.

Product measures: Fubini's theorem, completion of product measures, convolutions, Fourier transform, Riemann-Lebesgue lemma, inversion theorem, Plancherel theorem, L1 as a Banach algebra.

Content on a locally compact Hausdorff space, existence and uniqueness of the Haar measure on a locally compact group.

Text/References:

K. Chandrasekharan, A Course on Topological Groups, Hindustan Book Agency, 1996.

L. Nachbin, The Haar Integral, van Nostrand, 1965.

I. K. Rana, An Introduction to Measure and Integration, 2nd Ed., American Mathematical Society, 2002.

H. L. Royden, Real Analysis, 3rd Ed., Prentice Hall of India, 1988.

W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.

Review of basic complex analysis: Cauchy's theorem, Liouville's theorem, power series representation, open mapping theorem, calculus of residues.

Harmonic functions, Poisson integral, Harnack's theorem, Schwarz reflection principle.

Maximum modulus principle, Schwarz lemma, Phragmen-Lindelof method,.

Runge's theorem, Mittag-Leffler theorem, Weierstrass theorem, conformal equivalence, Riemann mapping theorem, characterisation of simply connected regions, Jensen's formula.

Analytic continuation, monodromy theorem, little Picard theorem.

Text/References:

L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1996.

S. Lang, Complex Analysis, 4th Ed., Springer, 1999.

D. H. Luecking and L. A. Rubel, Complex Analysis: A Functional Analysis Approach, Springer-Verlag, 1984.

R. Narasimhan and Y. Nievergelt, Complex Analysis in One Variable, Birkh�user, 2001.

R. Remmert, Theory of Complex Functions, Springer (India), 2005.

W. Rudin, Real and Complex Analysis, McGraw Hill, 1987.

Review of differentiable manifolds, tangent and cotangent bundles, tensors.

DeRham complex, Poincare's Lemma, Mayer-Vietoris sequences, cohomology with compact supports, degree of a map, Poincare duality.

Vector bundles, cohomology with vertical compact supports, Thom isomorphism, twisted DeRham complex, Poincare duality for non-orientable manifolds.

Text/References:

R. Bott and L. W. Tu , Differential Forms in Algebraic Topology, Springer-Verlag, New York, 1982.

L. Conlon, Differentiable manifolds, 2nd Ed., Birkh�user, Boston, 2001.

G. E Bredon, Topology and Geometry, Springer-Verlag, New York, 1997.

Paths and homotopy, homotopy equivalence, contractibility, deformation retracts.

Basic constructions: cones, mapping cones, mapping cylinders, suspension.

Cell complexes, subcomplexes, CW pairs.

Fundamental groups. Examples (including the fundamental group of the circle) and applications (including Fundamental Theorem of Algebra, Brouwer Fixed Point Theorem and Borsuk-Ulam Theorem, both in dimension two). Van Kampen's Theorem. Covering spaces, lifting properties, deck transformations, universal coverings.

Simplicial complexes, barycentric subdivision, stars and links, simplicial approximation. Simplicial Homology. Singular Homology. Mayer-Vietoris sequences. Long exact sequence of pairs and triples. Homotopy invariance and excision.

Degree. Cellular Homology.

Applications of homology: Jordan-Brouwer separation theorem, Invariance of dimension, Hopf's Theorem for commutative division algebras with identity, Borsuk-Ulam Theorem, Lefschetz Fixed Point Theorem.

Optional Topics: Outline of the theory of: cohomology groups, cup products, Kunneth formulas, Poincare duality.

Text/References:

M.J. Greenberg and J. R. Harper, Algebraic Topology, Benjamin, 1981.

W. Fulton, Algebraic topology: A First Course, Springer-Verlag, 1995.

A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002.

W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, Berlin, 1991.

J.R. Munkres, Elements of Algebraic Topology, Addison Wesley, 1984.

J.J. Rotman, An Introduction to Algebraic Topology, Springer (India), 2004.

H. Seifert and W. Threlfall, A Textbook of Topology, Academic Press, 1980.

Distribution Theory and Sobolev Spaces: Distributional derivatives, Definitions and elementary properties of Sobolev Spaces, Approximations by smooth functions, Traces, Imbedding Theorems (without proof), Rellich-Kondrachov Compactness Theorem.

Second Order Linear Elliptic Equations: Weak Solutions, Lax-Milgram Theorem, Existence and Regularity Results, Maximum Principles, Eigenvalue Problems.

Second Order Linear Parabolic Equations: Existence of weak solutions and Regularity Results, Maximum Principles.

Second Order Linear Hyperbolic Equations: Existence of weak solutions and Regularity Results, Maximum Principles, Propagation of Disturbance

Text/References:

S. Kesavan, Topics in Functional Analysis and Applications, New Age International Pvt. Ltd., 1989.

L C. Evans, Partial Differential Equation, American Mathematical Society, 1998.

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, 2004.

G. B. Folland, Introduction to Partial Differential Equations, 2nd Ed., Prentice-Hall of India, 1995.

R. C. McOwen, Partial Differential Equations: Methods and Applications, 2nd Ed., Pearson Education, Inc., 2003.

Nonlinear First-Order Scalar Equations: Method of Characteristics, Weak Solutions and Uniqueness for Hamilton-Jacobi Equations, Scalar Conservation Laws: shocks and entropy condition, weak solutions and uniqueness, and long time behavior.

Calculus of Variations: Euler-Lagrange Equation, Second Variations, Existence of Minimizers: Coercivity, Lower-Semicontinuity, Convexity, and Constrained Minimization Problems.

Hamilton-Jacobi Equations: Viscosity Solutions, Uniqueness, Applications to Control Theory and Dynamic Programming.

System of Conservation Laws: Theory of Shock Waves, Traveling Waves, Entropy Criteria, Riemann Problem, Glimm Existence Result for System of Two Conservation Laws.

Text/References:

L C. Evans, Partial Differential Equations, American Mathematical Society, 1998.

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer, 2004.

M. Defermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, 2000.

B. Dacorogna, Direct Methods in Calculus of Variation, Springer 1989.

P. Prasad and R. Ravindran, Partial Differential Equations, Wiley Eastern, 1985.

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, 1993.

Review of conditional expectation : Conditional expectation and conditional probability, regular conditional distributions, disintegration, conditional independence.

Martingales and Stopping times : Stopping times, random time change, martingale property, optional sampling theorem, maximum and upcrossing inequalities, martingale convergence theorem.

Gaussian processes and Brownian motion: Symmetries of Gaussian distribution, existence and path properties of Brownian motion, law of iterated logarithm.

Weak convergence in metric spaces with special reference to C[0, 1], Martingale central limit theorem.

Text/References:

P. Billingsley, Convergence of probability measures, Wiley, 1999.

K. R. Parthasarathy, Probability measures on metric spaces, Academic press, 1967.

V S. Borkar, Probability theory : an advanced course, Springer, NewYork, 1995.

A. W. Van-der-Vaart and J. A. Wellner, Weak convergence and empirical processes: With applications to Statistics, Springer-Verlag, 1996.

D. Williams, Probability with martingales, Cambridge Mathematical textbooks, 1991.

Discrete time Markov chains- definition and examples, stopping times and strong Markov property,Classification of states, Limit theorems. Markov chain mixing- Coupling and total variation distance, Convergence theorem, Mixing time, Mixing and time reversal, Ergodic theorem, upper bound and lower bound on mixing time.

Continuous time Markov chains- definition and examples, embedded Markov chain, Kolmogorov forward and backward equations, classification of states, limit theorems. Random walk – in dimension one, two and three, The Reflection Principle, hitting probabilities of a finite sets, Last visits and Long leads, Maxima and first passages, Duality, position of maxima.

Poisson Process - definition and properties, inter arrival and waiting time distributions, conditional distribution of arrival times.

Text/References:

Hoel, Port and Stone, Introduction to Stochastic Processes.

David A. Levin, Yuval Peres and Elizabeth L. Wilmer, Markov chains and mixing times.

William Feller, An introduction to probability and its applications.

Frank Spitzer, Principles of Random Walk.

Review of probability space. Random variables in R and Rn, distribution of random variables, Expectation of a R-valued random variable, Change of variable formula, Fatou’s lemma, monotone convegence theorem, dominated convergence theorem, Jensen’s inequality, notion of independence of sigma-fields and random variables, product of distributions, Fubini’s theorem.

Convergence almost surely, in probability, in law, convergence in moments, Borel -Cantelli lemma, Uniform integrability of sequence of random variables. Characteristic functions, convolution of distributions, Uniqueness theoerm, Fourier inversion theorem.

Weak law of large numbers, strong law of large numbers, Lindberg-Feller central limit theorem,Law of iterated logarithms.

Radon Nikodym theorem (reading exercise), Condition expectation definition, existence andits properties, regular conditional law.

Text/References:

KL Chung, A course in probability theory.

P. Billingsley, Probability and measure.

Robert B. Ash, Probability and measure theory.

Review of normed linear spaces, Hahn-Banach theorems, uniform boundedness principle, open mapping theorem, closed graph theorem, Riesz representation theorem on Hilbert spaces.

Weak and weak* convergence, reflexivity in the setting of normed linear spaces.

Compact operators, Sturm-Liouville problems.

Spectral projections, spectral decomposition theorem, spectral theorem for a bounded normal operator, unbounded operators, spectral theorem for an unbounded normal operator.

Text/References:

M. Ahues, A. Largillier and B. V. Limaye, Spectral Computations for Bounded Operators, Chapman & Hall/CRC, 2001.

J. B. Conway, Functional Analysis, 2nd Ed., Springer-Verlag, 1990.

S. Lang, Complex Analysis, 4th Ed., Springer, 1999.

B. V. Limaye, Functional Analysis, 2nd Ed., New Age International Publishers, 1996.

F. Riesz and B. SzNagy, Functional Analysis, Dover Publications, 1990.

W. Rudin, Functional Analysis, Tata McGraw Hill, 1974.

K. Yosida, Functional Analysis, 5th Ed., Narosa, 1979.

Face rings of simplical complexes, rings of invariants of finite groups, local cohomology of modules and its applications to Cohen-Macaulay Gorenstein rings and face rings of simplicial complexes

Text/References:

W.Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press, 1992. J. Herzog and T. Hibi, Monomial Ideals, Springer 2011.

A selection of topics from the following:

Regular sequences, grade and depth. Projective dimension, Auslander-Buchsbaum formula. Koszul complex. Rank of modules. Buchsbaum-Eisenbud acyclicity criterion. Graded rings and modules. Basic properties of graded modules: associated primes, dimension etc.

Hensel's Lemma, Newton' Theorem and Weierstrass Preparation Theorem.

Chevalley's Theorem on invariants of a finite pseudo-reflection group acting on the polynomial ring.

The Jacobian criterion for regularity. Divisor class group of a noetherian normal domain and its properties under ring extensions etc. Applications to unique factorization.

Cohen-Macaulay rings. Homological characterization of regular local rings.

Injective hulls, Matlis Duality. Local cohomology. Basic properties. Invariance under flat and finite base changes. Canonical module: Existence and basic properties. Local duality and applications. Canonical module of graded rings.

Text/References:

S. S. Abhyankar, Lectures on Algebra, Vol. I, World Scientific, Hackensack, NJ, 2006.

W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised second edition, Cambridge University Press, 1998

H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1989.

A selection of topics from the following:

Cohen-Macaulay rings and modules, Canonical Module, Gorenstein rings.

Hilbert functions and multiplicities, Macaulay's Theorem

Stanley-Reisner rings, shellability.

Semigroup rings and rings of invariants

Determinantal rings, Straightening law.

Big Cohen-Macaulay modules, Hochster's finiteness theorem.

Text/References:

S. S. Abhyankar, Lectures on Algebra, Vol. I, World Scientific, Hackensack, NJ, 2006.

W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised second edition, Cambridge University Press, 1998

H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1989.

A selection of topics from the following:

Singular Integrals (Calderon-Zygmund theory), the Kakeya problem, the Uncertainty Principle, the almost everywhere convergence of Fourier series, multilinear operators between Lp spaces.

Pseudodifferential operators, Index theorems.

Advanced complex analysis in one variable: Nevanlina theory, the existence of quasi-conformal maps, iterated polynomial maps, complex dynamics, compact Riemann surfaces, the Corona theorem.

Holomorphic functions in several complex variables: elementary properties of functions of several complex variables, analytic continuation, subharmonic functions, Hartog's theorem, automorphisms of bounded domains.

Text/References:

R.C. Gunning, Introduction to holomorphic functions of several variables. Vol. I. Function theory, Wadsworth & Brooks/Cole, 1990.

A.W. Knapp, Advanced real analysis, Birkhauser, 2005.

S. Lang and W. Cherry, Topics in Nevanlinna theory, Springer-Verlag, 1990.

R. Narasimhan, Several complex variables, University of Chicago Press, 1995.

E.M. Stein, Harmonic Analysis: Real Variable Methods,Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993.

S. Thangavelu, An Introduction to the Uncertainty Principle: Hardy's Theorem on Lie Groups, Birkhauser, 2004.

A selection of topics from the following:

Fourier Series and Fourier Transforms: Orthonormal Sequences in Inner Product Spaces, Fourier Series, Riemann-Lebesgue Lemma, Convergence/Divergence of Fourier Series, Fejer Theory, Fourier Transform, Inversion Theorem, Approximate Identities, Plancherel Theorem

Hp spaces: Harmonic and Subharmonic Functions, Hp spaces, Nevanlinna Class of Functions, Boundary Values, Non-tangential Limits, F. and M. Riesz Theorem, Inner Functions, Outer Functions, Factorization Theorems, Beurling's Theorem

Banach Algebras: Examples of Banach Algebras, Spectrum, Gelfand Representation, C*-Algebras, Positive Linear Functionals, Gelfand-Naimark Representation

Elements of Operator Theory: Hilbert Space Operators, Parts of Spectrum, Orthogonal Projections, Invariant Subspaces, Reducing Subspaces, Shifts, Decompositions of Operators

Perturbation Theory for Linear Operators: Analyticity of the resolvent operator, spectral projection and the weighted mean of the eigenvalues, The method of majorizing series, Spectral Decomposition Theorem.

Spectral Approximation: Norm and nu- convergence, Iterative refinement methods such as the Rayleigh-Schrodinger series and methods based on the fixed point techniques, error estimates.

Approximate solutions of Operator Equations: Galerkin, Iterated Galerkin and Nystrom methods, Condition Numbers, Two Grid Methods.

Text/References:

M. Ahues, A. Largillier, B.V. Limaye, Spectral Computation for bounded operators, Chapman & Hall/CRC, 2001.

K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997.

G. Bachman, L. Narici and E. Beckenstein, Fourier and Wavelet Analysis, Springer-Verlag, 2000.

S. K. Berberian, Lectures in Functional Analysis and Operator Theory, Narosa Publishing House, 1979.

F. Chatelin, Spectral Approximation of Linear Operators, Academic Press, 1983.

J.B. Conway, A Course in Functional Analysis, Springer-Verlag, 1985.

P.L. Duren, Theory of Hp spaces, Dover Publications, 2000.

W. Hackbusch, Integral Equations: Theory and Numerical Treatment, Birkhauser, 1995.

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1995.

R. Kress, Linear Integral Equations, Second Edition, Springer-Verlag, 1999.

P. Koosis, Introduction to Hp spaces, 2nd Edition, Cambridge University Press, 1999.

C.S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhauser, 1997.

G.J. Murphy, C*-Algebras and Operator Theory, Academic Press Inc., 1990.

W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.

W. Rudin, Functional Analysis, McGraw Hill, 1991.

A. Vretblad, Fourier Analysis and its Applications, Springer-Verlag, 2005.

A selection of topics from the following:

Basic Combinatorial Objects : Sets, multisets, partitions of sets, partitions of numbers, finite vector spaces, permutations, graphs etc.

Basic Counting Coefficients: The twelve fold way, binomial, q-binomial and the Stirling coefficients, permutation statistics, etc.

Sieve Methods : Principle of inclusion-exclusion, permutations with restricted positions, Sign-reversing involutions, determinants etc.

Combinatorial reciprocity.

Theory of Symmetric functions.

Text/References:

C. Berge, Principles of Combinatorics, Academic Press, 1972.

I.G. Macdonald, Symmetric functions and Hall polynomials. Second edition, Oxford University Press, 1995.

R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole, 1986.

A selection of topics from the following:

Partially ordered sets, Mobius inversion.

Rational generating functions: P-partitions and linear Diophantine equations.

Polya theory and representation theory of the symmetric group.

Combinatorial algorithms, and symmetric functions.

Generating functions: Single and multivariable Lagrange inversion.

Young tableaux and plane partitions

Text/References:

M. Aigner, Combinatorial Theory, Springer-Verlag, New York, 1979.

I. G. Macdonald, Symmetric functions and Hall polynomials. Second edition, Oxford University Press, New York, 1995.

B.E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions, Wadsworth & Brooks/Cole, 1991.

R. P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole, Monterey, CA, 1986.

R. P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge University Press, Cambridge, 1999.

A selection of topics from the following:

Review of the theory of curves and surfaces in the Euclidean 3-space.

Differentiable manifolds, and Riemannian structures. Connections, and curvature tensor.

The theorems of Bonnet-Meyers and Hadamard. Manifolds of constant curvature.

Text/References:

J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer-Verlag, New York, 1997.

W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edition, Academic Press, 2002.

M. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.

S. Kumaresan, A Course in Differential Geometry and Lie Groups, Hindustan Book Agency, 2002.

J. Milnor, Morse Theory, Princeton University Press, 1963.

A selection of topics from the following:

Affine and projective varieties, rational maps, nonsingularity.

Algebraic Curves, Riemann Roch Theorem.

Sheaves and Schemes. Basic properties. Divisors and Differentials.

Cohomology of sheaves, Serre Duality Theorem.

Text/References:

S. S. Abhyankar, Algebraic Geometry for Scientists and Engineers, American Mathematical Society, Providence, RI, 1990.

D. Eisenbud and J. Harris, The Geometry of Schemes, Springer-Verlag, 2000.

R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.

I. R. Shafarevich, Basic Algebraic Geometry, Vol. 1 and 2, Second edition, Springer-Verlag, 1994.

A selection of topics from the following:

CW complexes, Homotopy groups, Cellular Approximation.

Whitehead's theorem, Hurewicz theorem.

Excision, Fibre bundles, Long exact sequences.

Postnikov Towers, Obstruction Theory.

Stable homotopy groups. Spectral Sequences, Serre Class of abelian groups.

Text/References:

B. Gray, Homotopy Theory, Academic Press, 1975.

A. Hatcher, Algebraic Topology, Cambridge University Press 2002.

G. W. Whitehead, Elements of Homotopy Theory, Springer Verlag, 1978.

P. Hilton, Homotopy Theory and Duality, Gordon and Beach Sc. Publishers, 1965.

N. Steenrod, The Topology of Fibre Bundles, 7th reprint, Princeton University Press, 1999.

R. M. Switzer, Algebraic topology: Homotopy and Homology, Springer Verlag, 2002.

A selection of topics from the following:

Basics of Topological groups, Lie group.

Group actions, homogeneous spaces examples.

G-spaces, existence of slice and tubes

Covering homotopy theorem, Classification of G-Spaces.

Finite group actions, homology spheres

G-coverings, Cech theory

Locally smooth actions, orbit types, principal orbits

Actions of tori.

Cohomology structure of fixed point sets, Z_p-actions projective spaces and product of spheres.

Text/References:

G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press 1972.

T. Br�cker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, New York, 1985.

W. Y. Hsiang, Cohomology Theory of Topological Transformation Groups, Springer-Verlag, 1975.

A selection of topics from the following:

Algebraic number theory, abelian and non-abelian reciprocity laws, the Langlands programme, automorphic forms and representations.

The arithmetic of algebraic groups.

Arithmetic algebraic geometry: counting rational points of varieties over finite fields

Galois representations and galois cohomology.

Additive number theory: partitions, compositions, Goldbach problem.

Text/References:

S. Lang, Algebraic number theory., Second edition, Springer-Verlag, New York, 1994.

D. Bump, Automorphic forms and representations, Cambridge University Press, Cambridge, 1997.

H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society, Providence, RI, 2004.

H. Hida, Modular forms and Galois cohomology, Cambridge University Press, Cambridge, 2000.

A selection of topics from the following:

Harmonic analysis on Lie groups, L-functions, l-adic representations and motives.

Diophantine equations and the applications of K-theory to number theory.

Analytic number theory and transcendental methods.

Applications of ergodic theory to number theory.

Text/References:

S. Lang, Algebraic number theory., Second edition, Springer-Verlag, New York, 1994.

D. Bump, Automorphic forms and representations, Cambridge University Press, Cambridge, 1997.

H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society, Providence, RI, 2004.

H. Hida, Modular forms and Galois cohomology, Cambridge University Press, Cambridge, 2000.

A selection of topics from the following:

Schauder theory, regularity for second order elliptic equations. Nonlinear analysis and its applications to nonlinear PDEs: Fixed point methods, variational methods, monotone iteration, degree theory.

Evolution equations: Existence via semigroup theory

Nonlinear Hyperbolic systems: Theory of well posedness, compensated compactness,

Young measures; propagation of oscillations, weakly nonlinear geometric optics.

Text/References:

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1984.

D. Serre, Systems of Conservation Laws, Vols. 1, 2, Cambridge University Press, 2000.

L. Evans, Weak Convergence Methods for Nonlinear PDEs, CBMS Regional Conference series in Math., American Mathematical Society, Providence RI, 1990

A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North Holland, 1978.

M. Struwe, Variational Methods: Applications to nonlinear PDEs and Hamiltonian systems, Springer-Verlag, 1990.

A selection of topics from the following:

Diffeomorphisms and flows: Elementary dynamics of diffeomorphisms, flows and differential equations, conjugacy, equivalence of flows, Sternberg's theorem on smooth conjugacy (statement only), Hamiltonian flows and Poincare maps.

Local properties of flows and diffeomorphisms: Hyperbolic fixed points, Hartman-Grobman theorems for maps and flows, Normal forms for vector fields, Centre manifolds.

Structural stability and hyperbolicity: Structural stability for linear systems, Flows on 2-dimensional manifolds, Peixoto's characterisation of structural stability on unit disc, Anosov and Horseshoe diffeomorphisms, Homoclinic points, Melnikov function.

Bifurcations and Perturbations: Saddle-node and Hopf bifurcations, Andronov-Hopf bifurcation, The logistic map, Arnold's circle map; Perturbation theory: Melnikov's method for the study of perturbation of completely integrable systems.

Floquet theory and Hill's equation and some of its applications.

Two dimensional systems: Poincare-Bendixon theorem, Index of planar vector fields and the Poincare Hopf index theorem for two dimensional manifolds.

Van der Pol's equation, Duffing's equation, Lorenz's equation.

First integrals and functional independence of first integrals, notion of complete integrability, Jacobi multipliers, Liouville's theorem on preservation of phase volume, Jacobi's last multiplier theorem and its applications.

Text/References:

D. K. Arrowsmith, C. M. Place: An Introduction to Dynamical Systems, Cambridge University Press, 1990.

C. Chicone, Ordinary Differential Equations. Springer-Verlag, 1999.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, 2002.

P.Glendinning, Stability, instability and chaos: An Introduction to the Theory of Nonlinear Differential Equations, Cambridge University Press, 1994.

J. Palis and W. C. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, 1982.

R. Grimshaw, Nonlinear Ordinary Differential Equations. CRC press, 1991.

N.A. Magnitskii and S.V. Sidorov, New Methods for Chaotic Dynamics, World Scientific, 2006.

L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 2001.

A selection of topics from the following:

Review of finite difference methods for elliptic, parabolic and hyperbolic problems. Stability, consistency and convergence theory.

Finite difference schemes for scalar conservation laws (Lax-Friedrichs, Upwind, Lax-Wendroff, etc.), Conservative schemes and their numerical flux functions, Consistency, Lax-Wendroff Theorem, CFL Condition, Nonlinear Stability and TVD property, Monotone Difference schemes, Numerical entropy condition, Convergence result.

Finite difference Schemes for one-dimensional system of conservation laws, approximate Riemann solvers, Godunov's method, High resolution methods, Multidimensional approaches.

Large Scale Scientific Computing: Classical Iterative Methods for solving Linear systems, Large Sparse Linear Systems, Storage Schemes, GMRES algorithm, Preconditioned Conjugate Gradient method and Multi-grid method, Newton's Method and some of its variations for solving nonlinear systems.

Text/References:

Axelsson, O. Iterative Solution Methods, Cambridge University Press, 1994.

Briggs, W. L., Henson, V. E. and McCormick, S. F. A Multigrid tutorial, SIAM, 2000.

Godlewski, E. and Raviart, P. –A. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1995.

Kroner, D. Numerical Schemes for Conservation Laws. John Wiley, 1997.

LeVeque, R. J. Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.

LeVeque, R. J. Numerical Methods for Conservation Laws. Birkhauser, 1992.

Quarteroni, A. and Valli, A. Numerical Approximation of Partial Differential Equations, Springer, 1997.

Ueberrhuber, C. W. Numerical Computation: Methods, Software and Analysis, Springer-Verlag, 1997.

A selection of topics from the following:

Mixed Finite Element Methods: Examples of mixed variational formulations- primal, dual formulations; abstract mixed formulations, discrete mixed formulations, existence-uniqueness of solutions, convergence analysis, implementation procedures.

Adaptive FEM: A study of -Explicit A posteriori error estimators, Implicit A posteriori estimators, Recovery based error estimators, Goal Oriented adaptive mesh refinement for second order elliptic boundary value problems.

Discontinuous Galerkin Methods for second order elliptic boundary value problems: Global element methods, Symmetric Interior Penalty Method, Discontinuous hp- Galerkin Method, Non-symmetric interior penalty method: Consistency, approximation properties, existence and uniqueness of solutions, error estimates, implementation procedures.

FEM for parabolic problems: The standard Galerkin method, semi-discretization in space. discretization in space and time, the discontinuous Galerkin Method, a mixed method, implementation procedures.

Elements of Multigrid Methods: Multigrid Components - Interpolation, restriction Coarse-grid correction, V, W, and FMG cycles, Implementation, Convergence analysis, Performance diagnostics.

Text/References:

Z. Chen, Finite Element Methods And Their Applications, Springer-Verlag, New York, 2005.

S. C. Brenner and R. L. Scott, The Mathematical Theory of Finite Element Methods, 2nd Edition, Springer-Verlag, New York, 2002.

M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, John Wiley and Sons, 2000.

V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, 2nd Edition, Springer-Verlag, Berlin, 2006.

A selection of topics from the following:

Stochastic optimal control: compactness of laws, dynamic programming principle.

Malliavin calculus and applications to finance: Wiener-Ito chaos expansion, Shorohod integral, Integration by parts formula, Clark- Ocone formula and application to finance.

Text/References:

V.S. Borkar, Optimal control of diffusion processes, Longman Scientific and Technical, Harlow (copublished by John Wiley), 1989.

D. Nualart, The Malliavin calculus and related topics, Springer-Verlag, 1995.

A selection of topics from the following:

Univariate Stochastic Orders-hazard rate order, likelihood ratio order, mean residual rate order. Univariate variability orders- convex order, dispersive order, peakedness order. Univariate monotone convex and related orders. Multivariate stochastic orders. Multivariate variability and related orders. Statistical Inference for stochastic ordering. Applications in reliability theory, biology, economics and scheduling.

Text/References:

J. George Shanthikumar and Moshe Shaked (1994) Stochastic Orders and their Applications, Academic press.

C.D. Lai and M. Xie (2006) Stochastic Ageing and Dependence for Reliability, Springer Verlag.

A selection of topics from the following:

Inference in Semi-parametric models: Models with infinite imensional parameters, Efficient estimation and the delta method, Score and information operators, Estimating equations, Maximum Likelihood estimation, Testing.

Generalized linear models: Components of a GLM, estimation techniques, diagnostics, continuous response models, Binomial response models, Poisson response models, overdispersion, multivariate GLMs, quasi likelihoods, generalized estimating equations, generalized linear mixed models, programming in R and SAS.

Text/References:

A. W. Van der Vaart, Asymptotic Statistics, Cambridge University Press, 2000.

U. Grenander, Abstract Inference, John Wiley, 1981.

P. McCullagh and J. A. Nelder, Generalized Linear Models, 2nd Edition, Chapman and Hall/CRC, 1994.

L. Fahrmeir and G. Tutz, Multivariate Statistical Modeling based on Generalized Linear Models, 2nd Edition, Springer-Verlag, 1994.

R. H. Myers, D. C. Montgomery and G. Geoffrey Vining, Generalized Linear Models with applications in Engineering and Sciences, Wiley-Interscience, 2001.

Basic Extremal Combintorics: Chains and antichains, Sunflower Lemmas , Intersecting Families , Density theorems

Basic Enumerative Combinatorics : Ordinary Generating functions , Exponential Generating Functions , Quasi-polynomials and applications to Ehrhart theory, Transfer Matrix Method, Stanley's Reciprocity Theorem, Exponential Structures, Trees , Lagrange inversion Theorem, Composition of generating functions and Exponential Formula

Extremal Set Theory : Sperner`s Theorem, Theorems of Erdos-Ko-Rado, KruskalKatona, Dilworth`s theorem, Kleitman`s lemma for ideals and correlation inequalities.

Graph theory : Matching theory, Hamiltonicity, Extremal graph theory, Graph colorings, Ramsey theory

Text/References:

Enumerative Combinatorics - Stanley, Vol.1 (2nd Edition) and 2, Cambridge University Press.

Extremal Combinatorics With Applications in Computer Science - Stasys Jukna, Springer, 2nd Edition.

Computing the Continuous Discretely : Integer-point Enumeration in Polyhedra - Beck and Robbins, Springer, 2nd edition.

Combinatorics of Finite Sets - Anderson, Dover Books on Mathematics.

Modern Graph theory - Ballobas, Graduate Texts in Mathematics, Springer.

Matchings and SDRs , Linear Algebra method , Polynomial method , combinatorial Nullstellensatz and applications.

Mobius Inversion on Posets

Advanced Enumeration : Permutation Statistics and generalizations to Coxeter groups, Enumeration with Symmetric Functions, RSK Algorithm, Frobenius characteristic, The Jacobi-Trudi identity, Murnaghan-Nakayama Lemma, LittlewoodRichardson rule.

Text/References:

Enumerative Combinatorics - Stanley, Vol.1 (2nd Edition) and 2, Cambridge University Press.

Extremal Combinatorics With Applications in Computer Science - Stasys Jukna, Springer, 2nd Edition

The Symmetric Group : Representations, Combinatorial Algorithms, and symmetric functions - Bruce Sagan, Graduate texts in Mathematics, Springer, 2nd ed.

Representation Theory : A combinatorial Viewpoint - A. Prasad, Cambridge University Press

Combinatorics of Coxeter Groups : Bjorner and Brenti, Graduate Texts in Mathematics, Springer.

Symmetric Functions and Hall Polynomials - Macdonald, Oxford Mathematical monographs.

Linear Algebra methods in Coombinatorics - Babai/Frankl, lecture notes.

The Polynomial method in Combinatorics - survey paper by T. Tao

Incidence Theorems and Their Applications - Z. Dvir, Foundations and Trends in Theoretical Computer Science, Now Publishers Inc.

1) Parametric models, exponential and location-scale family, Sufficiency, MinimalSufficiency, Complete Statistic, Decision Rule, Loss Function and Risk, Pointestimators, consistency, asymptotic bias, variance and MSE, asymptoticinference.[Chapter 2]

2) UMVUE, U-statistics, Asymptotic Unbiased estimator, V-statistics [Chapter 3]

3) Bayes Decision and Bayes estimators, Invariance, Minimaxity and admissibility,MLE and efficient estimation method. [Chapter 4]

4) The NP Lemma, monotone likelihood ratio, UMP test for one sided and two sidedhypothesis, UMP Unbiased test, UMP invariant test, likelihood ratio test, chi-squaredtest, Sign, permutation and rank test, Kolmogorov- Smirnov and Cramer-von Misestest and asymptotic test [Chapter 6.]

Text/References:

Main text: Jun Shao, Mathematical Statistics, 2nd Ed., Springer, 2003.Additional Texts: Theoretical Statistics D.R. Cox, D.V. Hinkley CRC PressE. L. Lehmann, Theory of Statistical Inference, Wiley, 1983.E. L. Lehmann, Testing Statistical Hypotheses, Wiley, 1986.

Categories, functors, natural transformations.

Limits, colimits, complete and cocomplete categories.

Adjoint functors, universal constructions, free and cofree objects.

Functor categories, comma categories, quotient categories, derived categories.

Representable functors, Yoneda lemma.

Cauchy completeness, Karoubi envelopes.

Cartesian categories, group objects. The above concepts can be motivated and discussed by connecting them to other areas of mathematics depending on the interests of the instructor and students.

Text/References:

Aguiar and Mahajan, Monoidal functors, species and Hopf algebras, American Mathematical Society, 2010.

Awodey, Category theory, Oxford University Press, 2010.

Borceau, Handbook of categorical algebra, Volumes 1, 2 and 3, Cambridge University Press, 1994.

Goerss and Jardine, Simplicial homotopy theory, Birkhauser, 1997.

Hirschhorn, Model categories and their localizations, American Mathematical Society, 2003.

Leinster, Higher categories, Higher operads, Cambridge University Press, 2004.

Leinster, Basic category theory, Cambridge University Press, 2014.

Mac Lane, Categories for the working mathematician, Springer, 1998

Monoidal categories, monoids, comonoids.

Symmetric monoidal categories, braidings, Hopf monoids.

Higher monoidal categories.

Enriched categories, $2$-categories, bicategories, higher categories.

Monads, distributive laws, higher monads. The above concepts can be motivated and discussed by connecting them to other areas of mathematics depending on the interests of the instructor and students.

Text/References:

Aguiar and Mahajan, Monoidal functors, species and Hopf algebras, American Mathematical Society, 2010.

Awodey, Category theory, Oxford University Press, 2010.

Borceau, Handbook of categorical algebra, Volumes 1, 2 and 3, Cambridge University Press, 1994.

Goerss and Jardine, Simplicial homotopy theory, Birkhauser, 1997.

Hirschhorn, Model categories and their localizations, American Mathematical Society, 2003.

Leinster, Higher categories, Higher operads, Cambridge University Press, 2004.

Leinster, Basic category theory, Cambridge University Press, 2014.

Mac Lane, Categories for the working mathematician, Springer, 1998

Full rank model (Chapters 3 and 4)Models with rank deficiency (Chapter 5: Sections 5.1,5.2,5.3,5.4,5.5)One-way classification model (Chapter 6: Sections 6.1,6.2,6.3,6.4)Two-way Crossed Classification model (Chapter 7: Sections 7.1,7.2)Fixed, Random and Mixed models for Balanced Data (Chapter 9.1-9.5, 9.8, 9.9)

Text/References:

Main Text: Linear Models by S.R. Searle (1971) Wiley & SonsOther References: Linear Model Methodology by A. I. Khuri (2009) CRC Press