Topics in Local Algebra

Mathematical Sciences Research Institute
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Commutative Algebra

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Pluripotential theory and holomorphic dynamical systems. Iyer , Apoorva Khare , R.

Department of Mathematics

Juni 1 of Diagonalization: Eigenvalues and eigenvectors, diagonalizability, Invariant subspaces , adjoint of an operator, normal, unitary and self adjoint operators, Schur's Lemma, diagonalization of normal matrices, spectral decompositions and spectral theorem, applications of spectral theorem, Cayley-Hamilton theorem, primary decomposition theorem, Jordon canonical form, minimal polynomials, Introduction to bilinear and Quadratic forms: Bilinear and quadratic forms, Sylvester's law of inertia. Preliminaries to Complex analysis: Basic properties: convergence, compactness, connectedness; continuous functions, Holomorphic functions, power series, integration along curves and properties. Douglas C. MacDonald: Introduction to commutative algebra. Reference materials: H. Complete reducibility and Weyls theorem.

Venkatesh Algebraic and enumerative combinatorics, random geometric graphs, graph limits, symmetric functions, Schur polynomials, determinantal identities; Coxeter groups, root systems, structure theory of Borcherds—Kac—Moody algebras and connections to algebraic graph theory, lattice polytopes and polyhedra; Combinatorial aspects of simplicial complexes, Tessellation and tiling problems.

Algebraic and Combinatorial Topology Basudeb Datta Combinatorial manifolds, PL-manifolds, minimal triangulation of manifolds, triangulation of spheres and projective planes with few vertices, pseudomanifolds with small excess, equivelar polyhedral maps.

Topology of three-manifolds and smooth four-manifolds, hyperbolic geometry, geometric group theory, Heegaard Floer theory and its relations to geometric topology. Automated Theorem Proving Siddhartha Gadgil Homotopy type theory and its applications to automated theorem proving.

Research Areas

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as Classification of finite simple groups · Sylow theorems · Local analysis. Constructions. Free group · Presentation of a group · Word problem for. Commutative algebra is the branch of abstract algebra that studies commutative rings, their Regular local ring · Localization of a module · Valuation ( mathematics) · Discrete valuation · Discrete valuation ring · I-adic topology · Weierstrass.

Theoretical Physics Arvind Ayyer , Vamsi Pritham Pingali Statistical mechanics, exactly solvable models, partial differential equations arising from string theory and general relativity. Couded dynamical systems, Synchronization, Turing patterns, applications of Lie algebraic methods to nonlinear Hamiltonian systems, fractal dimensional analysis, generalized replicator dynamics.

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