Syllabi for the Core Courses

Topology/Geometry I (MAT 530)

(Fall Semester)


  1. Basic point set topology
  2. Introduction to algebraic topology

Typical references:

James R. Munkres, Topology: a first course,
Prentice Hall, Englewood Cliffs NJ, 1975;
William S. Massey, Algebraic topology an introduction,
4th corrected printing, Springer-Verlag, 1977.
Topology/Geometry II (MAT 531) (Spring Semester)
1. Differentiable manifolds and maps Inverse and implicit function theorems Submanifolds, immersions and submersions 2. The tangent bundle Vector bundles, transition functions Reconstruction of a vector bundle from transition functions Equivalence classes of curves and derivations; tangent vectors The tangent bundle of a manifold as a vector bundle, examples Vector fields, differential equations and flows Lie derivatives and bracket 3. Differential forms Exterior differential, closed and exact forms Distributions, foliations and Frobenius integrability theorem Poincar=E9 Lemma 4. Integration Stokes' Theorem Integration and volume on manifolds De Rham cohomology Chain and cochain complexes Homotopy theorem The degree of a map The Mayer-Vietoris theorem Typical references: Michael Spivak, A Comprehensive introduction to differential Geometry, 2nd ed., Publish or Perish, Berkeley1979; Glen Bredon, Topology and Geometry, Springer-Verlag, 1993.

Algebra I (MAT 534)

(Fall Semester)

1. Vector spaces over R and C Subspaces, quotient spaces, linear maps, kernel, image Linear dependence and independence, bases, dimension Matrices of linear maps, base change, solving systems of linear equations Homomorphism theorem, sum formula for dimensions Dual spaces, duality theorem for finite-dimensional vector spaces Direct sums and products Determinants 2. Groups and group actions Division and Euclidean algorithms, unique factorization Groups and subgroups Examples: permutation groups, general linear groups, orthogonal groups Cayley's theorem Orbits and cosets, Lagrange's theorem Homomorphisms, normal subgroups, isomorphism theorem Group actions, examples Simple groups 3. Theory of a single linear transformation Eigenvectors and characteristic polynomial Triangular form Cramer's Rule Cayley-Hamilton Theorem, determinant-trace formula Minimal polynomial Projection operators Primary decomposition Jordan form 4. Inner product spaces Inner products and orthonormal sets Diagonalization of symmetric forms and normalization of alternating forms Hermitian forms, spectral theorem Orthogonal and unitary groups 5. Multilinear algebra Multilinear maps and tensor products Tensor, symmetric and exterior algebras as vector spaces Induced linear maps on tensor, symmetric and exterior powers, eigenvalues Typical references: Michael Artin, Algebra, Prentice Hall, Englewood Cliffs, NJ 1991; Paul Halmos, Finite-dimensional vector spaces, Springer-Verlag, 1986; Kenneth Hoffman and Ray Kunze, Linear Algebra, 2nd ed., Prentice Hall, Englewood Cliffs, NJ 1971; Anthony W. Knapp, Lie groups, Lie algebras and cohomology, Princeton Univ. Press, 1988.
Algebra II (MAT 535)
(Spring Semester)
1. Advanced group theory Structure of finitely generated abelian groups Sylow theorems Free groups Groups given by generators and relations 2. Rings and modules Rings and fields, examples Homomorphisms, subrings, ideals, quotients, isomorphism theorems= Vector spaces and modules, examples Polynomial rings, polynomial functions, factor theorem Integral domains, prime and maximal ideals, field of fractions Principal ideal domains (=93PID=94), unique factorization theorem Structure theorem for finitely generated modules over PID, examples Tensor, symmetric and exterior algebras as algebras 3. Fields and Galois theory Algebraic elements Construction of field extensions Characteristics, finite fields Geometric constructions by ruler and compass Algebraic closure Splitting field of a polynomial, examples Normal and separable extensions, automorphisms of field extensions Fundamental theorem of Galois theory Applications: construction of regular polygons, fundamental theorem of algebra, nonsolvable equations Typical references: Michael Artin, Algebra, Prentice Hall, Englewood Cliffs, NJ 1991; Serge Lang, Algebra, 2nd ed., Addison-Wesley, Menlo Park, CA 1984, Nathan Jacobson, Basic algebra, vol. 1, 2nd ed., W.H. Freeman & Co, San Francisco1985; Van der Waerden, Algebra 1, 9th ed., Springer-Verlag, 1994.
Complex Analysis I (MAT 542)
(Spring Semester)
1. The field of complex numbers, geometric representation of complex numbers 2. Analytic functions Definition, Cauchy-Riemann equations Elementary theory of power series, uniform convergence Elementary functions: rational, exponential and trigonometric functions The logarithm 3. Analytic functions as mappings Conformality Linear fractional transformations Elementary conformal mappings 4. Complex integration Line integrals and Cauchy's theorem for disk and rectangle Cauchy's integral formula Cauchy's inequalities Morera's theorem, Liouville's theorem and fundamental theore= m of algebra The general form of Cauchy's theorem 5. Local properties of analytic functions Removable singularities, Taylor's theorem Zeros and poles, classification of isolated singularities The local mapping theorem The maximum modulus principle, Schwarz's lemma 6. The calculus of residues The residue theorem The argument principle Rouche's theorem Evaluation of definite integrals 7. Power series Weierstrass theorem The Taylor and Laurent series Partial fractions and infinite products Normal families 8. The Riemann mapping theorem 9. Harmonic functions The mean-value property Harnack's inequality The Dirichlet problem Typical references: Lars V. Ahlfors, Complex analysis: an introduction = to the theory of analytic functions of one complex variable, 3rd ed., McGraw-Hill, 1979; John B. Conway, Functions of one complex variable, = 2nd ed., Springer-Verlag, 1978. Analysis (MAT 544) (Fall Semester) 1. Advanced Calculus/Ordinary Differential Equations (=93ODE=94) Review of the real number system Metric spaces, continuity, uniform convergence Contraction mapping principle Existence and uniqueness theorems for ODE Global existence theorem for linear ODE Linear transformations, orthogonal projections and matrix expone= ntial Linear systems of ODE with constant coefficients Derivatives in Rn and in Banach spaces Newton's method and the inverse function theorem The implicit function theorem 2. Measure theory Riemann integral in Rn Cantor-type sets, dyadic decompositions in Rn Measures arising from volume functions on open sets Basic properties of the Lebesgue measure Measurable and integrable functions Convergence theorems for Lebesgue integrals: monotone and dominated convergence theorems and Fatou's lemma Criterion for Riemann integrability Typical references: Daryl Geller, A first graduate course in real analysis. Part I, Solutions Custom Publishing (can be ordered from t= he campus bookstore); Walter Rudin, Principles of mathematical analysis= , 3rd ed., McGraw-Hill, New York 1976; Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, New York 1987. Real Analysis I (MAT 550) (Spring Semester) 1. Brief discussion of the measure theory Riesz Representation Theorem Tonelli's and Fubini's Theorems The dual of L=B9 Radon-Nykodim Theorem Lebesgue's Theorem Hahn Decomposition Theorem 2. L^p spaces, convergence in measure, the dual of L^p 3. Fourier series Riemann-Lebesgue lemma Convergence of Fourier series for differentiable functions Parseval's formula 4. Functional analysis Open mapping and closed graph theorems Uniform boundedness principle Hahn-Banach theorem Existence of orthonormal bases for Hilbert spaces Maximal operator controlling sequences of operators between Bana= ch spaces 5. More measure theory Maximal operators controlling almost everywhere convergence The fundamental theorems of calculus for the Lebesgue integral Change of variables of integration Polar coordinates 6. Partial Differential Equations Separation of variables The heat equation Laplace's equation, the fundamental solution The strong maximum principle and the Liouville theorem The mean-value theorem The Poisson kernel Approximate identities and the Weierstrass theorem on approximation by polynomials The wave equation, d'Alembert's solution Typical references: Daryl Geller, A first graduate course in real analysis. Part II, Solutions Custom Publishing (can be ordered from the campus bookstore); Walter Rudin, Principles of mathematical analysis, 3rd ed.,= McGraw-Hill, New-York 1976; Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, New-York 1987; Michael Taylor, Partial differential equations, Springer-Verlag, 1996.