Press, Princeton, Zariski spectrum as a frame Refresher on categories : Categories, functors, Yoneda Lemma, equivalence of categories, adjoints. Presheaves and Sheaves Locally ringed spaces and schemes Separated schemes, proper schemes, irreducible schemes, reduced schemes, integral schemes, noetherian schemes. Morphisms : separated, proper, finite morphisms, finite type morphisms, affine morphisms Sheaves of algebras : affine morphisms as sheaves of algebras Sheaves of modules over a scheme, Quasi-coherent and coherent sheaves Divisors and Line Bundles, Weil divisors, Cartier divisors, Line bundles on Projective spaces, Serre sheaves.

Springer-Verlag, New York-Heidelberg, Robin Hartshorne, Residues and duality , Lecture notes of a seminar on the work of A. With an appendix by P. Lecture Notes in Mathematics, No.

Triangulated categories, Derived categories of abelian categories. Injective and flasque resolutions. Suggested books : Artin, E.

Borevich, Z. Cassels, J. Hasse, H. Hecke, E. Samuel, P. Suggested books : Shafarevich, I.

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Smith, K. Reid, M. Fulton, W. Holme, A. MA Introduction to Homological Algebra 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. Suggested books : Apostol, T. Davenport, H. MA Combinatorics Pre-requisites : Calculus, Linear algebra and some exposure to proofs and abstract mathematics.

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Programming in Sage will be a part of every lecture. Richard P. Suggested books : Stanley, R. Sagan, B. Prasad, A. Stanley, R. Suggested books : V. Varadarajan, Lie groups, Lie algebras and their representations , Sringer Hall, Lie groups, Lie algebras and representations , Springer Knapp, Representation theory of semismiple lie groups.

An overview based on examples , Princeton university press Kesavan, S. Evans, L.

## P Adic Numbers

Schwartz, L. Hermann, Theorie des Distributions , Suggested books : Conway, J. Berberian, S. MA Topics in Complex Analysis 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.

Suggested books : Krantz, S. Suggested books : John B. Conway, A course in Functional Analysis , Springer, Suggested books : Dym, H. Stein, E.

## P Adic Numbers Introduction by Gouvea Fernando

Sadosky, C. Suggested books : Korner, I. Robert Ash. Serre, J.

## P-Adic Numbers by Gouvea, Fernando Quadros

Thangavelu, S. Rudin W. 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. Suggested books : L. Amsterdam, Janich, K. Suggested books : Brickell, F. Guillemin, V. Milnor, John W. Suggested books : Hatcher, A. Press, Indian edition is available. Munkres, I. Shastri, A. MA Riemannian Geometry Review of differentiable manifolds and tensors, Riemannian metrics, Levi-Civita connection, geodesics, exponential map, curvature tensor, first and second variation formulas, Jacobi fields, conjugate points and cut locus, Cartan-Hadamard and Bonnet Myers theorems.

Springer-Verlag, New York, MA Introduction to Homotopy Type Theory Prerequisite courses: MA , MA , MA Pre-requisites : Algebraic Topology, Dependent Type Theory 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.

Tutorial for the Lean Theorem Prover. Benedetti-Petronio, Lectures on Hyperbolic Geometry. Martelli, Introduction to Geometric Topology. A first course in algebraic topology is helpful but not necessary. Real analysis in more than one variable.

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Linear algebra. Suggested books : Spivak M. Kumaresan S. Hindustan Book Agency, New Delhi, Warner F. Springer-Verlag, New York-Berlin, Lee J. Analysis multivariable calculus, some measure theory, function spaces. 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. Nicolaescu, Lectures on the Geometry of Manifolds. Aubin, Some nonlinear problems in geometry. Evans, Partial differential equations. Gilbarg and Trudinger, Elliptic partial differential equations of the second order.

Szekelyhidi, Extremal Kahler metrics. Douglas, R. List of topics time permitting : 1.