# Undergraduate
Linear Algebra: Finite dimensional vector spaces (over R) and linear maps between them – subspaces, quotient spaces, dimension, bases, matrix representations. Positive definite inner products, orthonormal bases, extensions of orthonormal subsets. Eigenvalues and eigenvectors for automorphisms. Characteristic polynomial.
References: M. Artin, ‘Algebra‘ (Prentice Hall, 1991), Chapters 1,3,4
K. Hoffman and R. Kunze, ‘Linear Algebra‘, Chapters 1-6, (Prentice-Hall, 1971).
Abstract Algebra: Definitions of groups, rings, fields, and modules over a ring. Homomorphisms of these objects. Subgroups, normal subgroups, quotient groups. Cyclic groups, finite abelian groups (structure theorem). Ideals, prime and maximal and their quotients — basic examples Z, k[X], rings of algebraic integers. Field extensions, splitting fields of polynomials, normal extensions.
References: M. Artin, ‘Algebra‘, Chapters 2, 10, 11, 12, 13, 14
I. Herstein, ‘Topics in Algebra‘ (Blaisdell Publishers, 1964).
Point-set Topology: Open and closed sets, continuous functions. Connectedness, compactness, Hausdorff, normality. Metric spaces, Rn. Heine-Borel theorem.
Reference: J. Munkres, ‘Topology, A First Course‘, Part I (Prentice-Hall).
Calculus: Differential of a smoothing mapping between open subsets in Euclidean spaces. Matrix of partial derivatives. Inverse and implicit functions. Multivariable Riemann integration.
References: W. Rudin, ‘Principles of Mathematical Analysis‘ (McGraw-Hill, 1964)
A. Browder, ‘Mathematical Analysis: An Introduction‘ (Springer, 1996).
Complex Analysis: Definition of holomorphic functions, Cauchy integral formula, power series representations of holomorphic functions, radius of convergence, meromorphic functions, residues.
Reference: L. Ahlfors, ‘Complex Analysis‘, (McGraw-Hill, 1973), Chapters 1- 5.
Real Analysis: A thorough working knowledge of advanced calculus, at the level of the books of W. Rudin or A. Browder as listed under Calculus. Additional topics: Pointwise uniform convergence of functions, equi-continuity, l2, L2(S1), Hilbert spaces, orthonormal bases.
Other Introductory Material
For those entering students whose backgrounds are strong and who wish to do reading before entering graduate school to enrich their knowledge of mathematics the Department suggests the following books. The Department considers each of these to be a well-written introduction which conveys the flavor of the subject. Looking at these books may help give a feeling for various subspecialities within mathematics.
L. Ahlfors, ‘Complex Analysis‘ (McGraw-Hill, 1973).
V. Arnold, ‘Ordinary Differential Equations‘ (MIT Press)
M. Artin, ‘Algebra‘ (Prentice Hall, 1991)
F. Kirwan, ‘Complex Algebraic Curves‘, London Mathematical Series Student Texts 23 (Cambridge, 1992).
W. Massey, ‘A Basic Course in Algebraic Topology‘ Graduate Texts in Mathematics Vol. 127 (Springer-Verlag, 1991).
J. Milnor, ‘Topology from the Differentiable Viewpoint‘
S. Ross, ‘A First Course in Probability‘ (5th edition, Prentice-Hall, 1997).
W. Rudin, ‘Real and Complex Analysis‘
J-P. Serre, ‘A course in Arithmetic‘
J-P. Serre, ‘Linear Representations of Finite Groups‘
# First Year Graduate
1) Algebra
Group theory: Sylow theorems, p-groups, solvable groups, free groups.
Rings and modules: tensor products, determinants, Jordan canonical form, PID's, UFD's, polynomials rings.
Field theory: splitting fields, separable and inseparable extensions.
Galois theory: Fundamental theorems of Galois theory, finite fields, cyclotomic fields.
Representations of Finite Groups: character theory, induced representations, structure of the group ring.
Basics of Lie groups and Lie algebras: exponential map, nilpotent and semi-simple Lie algebras and Lie groups.
References: Dummit and Foote: Abstract Algebra, 2nd edition, except chapters 15, 16 and 17, Serre: Representations of Finite Groups (Sections 1-6). Fulton-Harris: Representation Theory: A First Course (Graduate Texts in Mathematics/Readings in Mathematics) Lie groups and algebras, Chapters 7-10.
2) Algebraic Geometry
Affine and projective varieties; regular functions and maps; cones and projections
Projective space and Grassmannian
Ideals of varieties; the Nullstellensatz
Rational functions, rational maps and blowing up
Dimension and degree of a variety; the Hilbert function and Hilbert polynomial
Smooth and singular points of varieties; the Zariski tangent space; tangent cones; dual varieties
Families of varieties (Chow varieties and Hilbert schemes)
algebraic curves: genus; the genus formula for plane curves,
the Riemann-Hurwitz formula. Riemann-Roch theorem.
References: Shafarevich: Basic Algebraic Geometry 1, 2nd edition, Harris: Algebraic Geometry: A First Course
3) Complex Analysis
Holomorphic and meromorphic functions
Conformal maps, linear fractional transformations, Schwarz's lemma
Complex integrals: Cauchy's theorem, Cauchy integral formula, residues
Harmonic functions: the mean value property; the reflection principle; Dirichlet's problem
Series and product developments: Laurent series, partial fractions expansions, and canonical products
Special functions: the Gamma function, the zeta functions and elliptic functions
basics of Riemann surfaces
Riemann mapping theorem. Picard theorems.
References: Ahlfors: Complex Analysis (3rd edition)
4) Algebraic Topology
Fundamental groups
Covering spaces
Higher homotopy groups.
Fibrations and the long exact sequence of a fibration
Singular homology and cohomology
Relative homology
CW complexes and the homology of CW complexes.
Mayer-Vietoris
Universal coefficient theorem
Kunneth formula
Poincare duality
Lefschetz fixed point formula
Hopf index theorem
Cech cohomology and de Rham cohomology.
Equivalence between singular, Cech and de Rham cohomology
References: A. Hatcher: Algebraic Topology, W. Fulton: Algebraic Topology, E. Spanier: Algebraic Topology, Greenberg and Harper: Algebraic Topology: A First Course
5) Differential Geometry
Basics of smooth manifolds: Inverse function theorem, implicit function theorem, submanifolds, integration on manifolds
Basics of matrix Lie groups over R and C: The definitions of Gl(n), SU(n), SO(n), U(n), their manifold structures, Lie algebras, right and left invariant vector fields and differential forms, the exponential map.
Definition of real and complex vector bundles, tangent and cotangent bundles, basic operations on bundles such as dual bundle, tensor products, exterior products, direct sums, pull-back bundles.
Definition of differential forms, exterior product, exterior derivative, de Rham cohomology, behavior under pull-back.
Metrics on vector bundles.
Riemannian metrics, definition of a geodesic, existence and uniqueness of geodesics.
Definition of a principal Lie group bundle for matrix groups.
Associated vector bundles: Relation between principal bundles and vector bundles
Definition of covariant derivative for a vector bundle and connection on a principal bundle. Relations between the two.
Definition of curvature, flat connections, parallel transport.
Definition of Levi-Civita connection and properties of the Riemann curvature tensor.
References: Taubes: Differential geometry: Bundles, Connections, Metrics and Curvature Lee: Manifolds and Differential Geometry (Graduate Studies in Math 107, AMS), S. Kobayashi and K. Nomizu: Foundations of Differential Geometry
6) Real Analysis
Convergence theorems for integrals, Borel measure, Riesz representation theorem
Lp space, Duality of Lp space, Jensen inequality
Lebesgue differentiation theorem, Fubini theorem, Hilbert space
Complex measures of bounded variation, Radon-Nikodym theorem.
Fourier series, Fourier transform, convolution.
Heat equation, Dirichlet problem, fundamental solutions
Central limit theorem, law of large numbers, conditional probability and conditional expectation.
Distributions, Sobolev embedding theorem.
Maximum principle.
References: Rudin: Real and complex analysis is a general reference but the following books have more useful techniques Stein and Shakarchi: Real analysis. Stein's book does not have Lp spaces. A good source of Lp spaces and convexity is Lieb-Loss: Analysis, Chapter 2. Fourier series: Stein and Shakarchi: Fourier Analysis. This book is very elementary but more than sufficient chapters 2 and 3 are Fourier series, chapter 5 is Fourier transform. Sobolev spaces: Evans: Partial Differential Equations. Chapter 5. Probability: Shiryayev: Probability. Feller: An Introduction To Probability Theory And Its Applications Durrett: Probability: Theory And Examples