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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. Programming in Sage will be a part of every lecture. Richard P. Suggested books : Stanley, R. Sagan, B. Prasad, A. Stanley, R.

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Model theory

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. Linear algebra. Suggested books : Spivak M.

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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.

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