Suggested books : Shafarevich, I. 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. 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. 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. Sadosky, C.

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

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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.

## Professional homepage of Raf Cluckers

Kumaresan S. Hindustan Book Agency, New Delhi, Warner F.

- Valuation Theory and Its Applications, Volume II.
- The Mathematics of Boolean Algebra!
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- Staphylococcus Aureus : Infection and Disease (Infectious Agents and Pathogenesis) (Infectious Agents and Pathogenesis).
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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.

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. Suggested books : Rajendra Bhatia, Matrix Analysis , vol.

Roger A.

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Horn and Charles R. Johnson, Matrix analysis , Cambridge University Press, Johnson, Topics in matrix analysis , Cambridge University Press, Semi group theory:Hille-Yosida theorem, Applications to heat, Schroedinger and wave equations. Suggested books : Evans, L. Pazy, A. Prasad, P. Treves, J. Ando, On a pair of commutative contractions , Acta Sci. Szeged 24 88— Das, B. Krishna and Sarkar, Jaydeb, Ando dilations, von Neumann inequality, and distinguished varieties. Bensoussan, J. Jikov, S. Kozlov, and O. Suggested books : Rabinowitz, Minimax methods in critical point theory with applications to differential equations , C.

Ghoussoub, N. Struwe, M. Suggested books : Porter, D. Gakhov, F. Muskhelishvilli, N. Nobe, B. Schwartz, Theorie des Distributions , Hermann, MA Topics around the Grothendieck inequality Top. Suggested books : H.

Iwaniec and E. Suggested books : J. Diamond and J. Other models of random matrices - Wishart and Jacobi ensembles.