Foundations of Classical Mechanics, Newtonian Mechanics, The One-Body and Two-Body problems, Rigid Body Motion, Lagrangian Mechanics.

Hamiltonian Mechanics, Hamilton-Jacobi Theory, Canonical Transformations, Symplectic Manifolds, Liouville Integrability.

Group theory Sylow theorems. Solvable and nilpotent groups, normal and central series, free groups, simple groups, Jordan-H¨older theorem. Direct and semi-direct products, extensions. Category theory Categories and functors, natural transformations, universal properties, products and coproducts. Rings and modules ◦ Polynomial rings, elementary symmetric polynomials. Euclidean, Principal ideal, and Unique factorization domains, Gauss lemma. ◦ Structure theorem for modules over PID: elementary divisors and invariant factor forms. Noetherian and Artinian rings and modules, Hilbert basis theorem, simple modules, composition series, and Jordan-H¨older theorem for modules. ◦ Vector spaces and linear operators, characteristic and minimal polynomials, Cayley-Hamilton theorem, canonical Jordan form, Rational Canonical Form. ◦ Semi-simple rings, Artin-Wedderburn theorem. Prerequisites (Undergraduate courses in Abstract Algebra)

Commutative algebra: rings Local rings and Nakayama lemma, Integral extensions, Krull dimension, Noether normalization lemma, Hilbert Nullstellensatz, localization. Prime ideal spectrum and Zariski topology, Algebraic sets and rings of regular functions. Discrete valuation rings and Dedekind domains. and modules Tensor product, flatness, local properties of modules, exterior and symmetric powers. Graded rings and modules, Hilbert functions and polynomials. Homological algebra Exact sequences, 5-lemma and snake lemma, projective and injective modules, resolutions, chain complexes, (left and right) exact functors, adjoint functors, adjointenss of Hom and Tensor functors, Tor and Ext. Prerequisite Math-660

This course is usually an introduction to commutative algebra. Noetherian rings and modules, localization, integral dependence, Going Up and Going Down Theorems, Hilbert's Basis Theorem and Nullstellensatz, Noether Normalization Theorem, and Primary Decomposition. (Prerequisite Undergraduate Abstract Algebra or Instructor’s permission)

(Measure Theory) Construction of a measure, Sigma algebras product measure, Measure image change of variables, $L_p$-space, Regularity and density theorem, Convolution product, Fourier transform. Prerequisite MATH-651

(Measure Theory), Measurable functions, The Borel case, Measurement, Integral positive functions, Applications, Sequences, sequences and series sets, Cardinals theory, Measurable functions, The Borel case, Measure, Integral of positive functions, Applications, Inequality and $L_p$-spaces. Prequisites (Undergraduate courses in Mathematical Analysis)

(General Topology) Topological Spaces, Metric and Normed Spaces, Separation Axioms, Compactness, Connectedness, Homotopy Theory

(Algebraic Topology) The fundamental group and covering spaces, Homology, Cohomology.