2019–20–B

Embedded differentiable manifolds with boundary in Euclidean space. The tangent space, normal, vector fields. Orientable manifolds, the outer normal orientation. Smooth partitions of unity. Differential forms on embedded manifolds, the exterior derivative. Integration of differential forms and the generalized Stokes theorem. Classical formulations (gradient, curl and divergence and the theorems of Green, Stokes and Gauss). Closed and exact forms. Conservative vector fields and existence of potentials. Application to exact ordinary differential equations. Introduction to differential geometry: curvature of curves and surfaces in 3 dimensional space, the Gauss map, the Gauss-Bonnet theorem (time permitting).

Ordinary differential equations of first order, existence and uniqueness theorems, linear equations of order $n$ and the Wronskian, vector fields and autonomous equations, systems of linear differential equations, nonlinear systems of differential equations and stability near equilibrium

Topological spaces and continuous functions (product topology, quotient topology, metric topology). Connectedness and Compactness. Countabilty Axioms and Separation Axioms (the Urysohn lemma, the Urysohn metrization theorem, Partition of unity). The Tychonoff theorem and the Stone-Cech compactification. Metrization theorems and paracompactness.

1. Preliminaries: floating point arithmetic, round-off errors and stability. Matrix norms and the condition number of a matrix.
2. Introduction to numerical solutions for ODE’s:initial value problems, Euler’s method, introduction to multistep methods. Boundary value problems.
3. Numerical solution of linear equations: Gauss elimination with pivoting, LU decomposition. Iterative techniques: Jacobi, Gauss-Seidel, conjugate gradient. Least squares approximation.
4. Numerical methods for finding eigenvalues: Gershgorin circles. The power method. Stability considerations in Gram-Schmidt: Hausholder reflections and Givens rotations. Hessenberg and tridiagonal forms. QR decomposition and the QR algorithm.

• Complex numbers. Analytic functions, Cauchy–Riemann equations.
• Conformal mappings, Mobius transformations.
• Integration. Cauchy Theorem. Cauchy integral formula. Zeroes, poles, Taylor series, Laurent series. Residue calculus.
• The theorems of Weierstrass and of Mittag-Leffler. Entire functions. Normal families.
• Riemann Mapping Theorem. Harmonic functions, Dirichlet problem.

1. The Fourier transform: convolutions, the inversion formula, Plancherel’s theorem, Hermite functions, tempered distributions. The Poisson summation formula. The Fourier transform in R^n.
2. The Laplace transform. Connections with convolutions and the Fourier transform. Laguerre polynomials. Applications to ODE’s. Uniqueness, Lerch’s theorem.
3. Classification of the second order PDE: elliptic, hyperbolic and parabolic equations, examples of Laplace, Wave and Heat equations.
4. Elliptic equations: Laplace and Poisson equations, Dirichlet and Neumann boundary value problems, Poisson kernel, Green’s functions, properties of harmonic functions, Maximum principle
5. Analytical methods for resolving partial differential equations: Sturm-Liouville problem and the method of separation of variables for bounded domains, applications for Laplace, Wave and Heat equations including non-homogenous problems. Applications of Fourier and Laplace transforms for resolving problems in unbounded domains.

Bibliography

1. Stein E. and Shakarchi R., Fourier analysis, Princeton University Press, 2003.
2. Korner T.W., Fourier analysis, Cambridge University Press, 1988.
3. Katznelson Y., An Introduction to Harmonic Analysis, Dover publications. 4. John, Partial differential equations, Reprint of the fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1991.
4. Evans Lawrence C. Partial Differential Equations, Second Edition.
5. Gilbarg D.; Trudinger N. S. Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Ver lag, Berlin, 2001.
6. Zauderer E. Partial differential equations of applied mathematics, Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1989. xvi+891 pp. ISBN: 0-471-61298-7.

• Fields: basic properties and examples, the characteristic, prime fields
• Polynomials: irreducibility, the Eisenstein criterion, Gauss’s lemma
• Extensions of fields: the tower property, algebraic and transcendental extensions, adjoining an element to a field
• Ruler and compass constructions
• Algebraic closures: existence and uniqueness
• Splitting fields
• Galois extensions: automorphisms, normality, separability, fixed fields, Galois groups, the fundamental theorem of Galois theory.
• Cyclic extensions
• Solving polynomial equations by radicals: the Galois group of a polynomial, the discriminant, the Cardano-Tartaglia method, solvable groups, Galois theorem
• Roots of unity: cyclotomic fields, the cyclotomic polynomials and their irreducibility
• Finite fields: existence and uniqueness, Galois groups over finite fields, primitive elements

1. Rings and ideals (revisited and expanded).
2. Modules, exact sequences, tensor products.
3. Noetherian rings and modules over them.
4. Hilbert’s basis theorem.
5. Finitely generated modules over PID.
6. Hilbert’s Nullstellensatz.
7. Affine varieties.
8. Prime ideals and localization. Primary decomposition.
9. Discrete valuation rings.

An introduction to the basic notions of probability theory:

sample spaces limits of events conditional probability independent events sigma algebras, continuous spaces, Lebesgue measure random variables and distributions independence expectation variance and covariance convergence of random variables: almost-sure, in Lp, in probability law of large numbers convergence in law central limit theorem

Coding Theory investigates error-detection and error-correction. Such errors can occur in various communication channels: satellite communication, cellular telephones, CDs and DVDs, barcode reading at the supermarket, etc. A mathematical analysis of the notions of error detection and correction leads to deep combinatorial problems, which can be sometimes solved using techniques ranging from linear algebra and ring theory to algebraic geometry and number theory. These techniques are in fact used in the above-mentioned communication technologies.

Topics

1. The main problem of Coding Theory
2. Bounds on codes
3. Finite fields
4. Linear codes
5. Perfect codes
6. Cyclic codes
7. Sphere packing
8. Asymptotic bounds

Bibliography:

R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford 1986

• The reflection theorem
• Mostowski collapse theorem
• Absoluteness of formulas
• The constructible universe
• Forcing
• Consistency of the negation of the continuum hypothesis
• Consistency of the negation of the Axiom of choice

In this course we will discuss Cantor minimal systems. These are dynamical systems that admit a precise “digital representation” in the sense that the state space can be viewed as an infinite sequence of digits.There has been considerable recent interest in such systems. In this course we will consider formal versions of the question “when are two systems actually the same” . We will introduce and study the notions of isomorphism and orbit equivalence for dynamical systems. The objective of this course is to introduce students to basic notions in dynamical systems through contemporary results. Introduction: Dynamical systems in topological dynamics and ergodic theory, ergodicity and minimality, isomorphism of dynamical systems, the Cantor set, toplogical equivalence relations, orbit equivalence. Invariant measures. Cantor mimimal systems, Bratteli diagrams and the Bratteli-Vershik Model, topological equivalence relations, Orbit cocyles, strong orbit equivalence, invariants for orbit equivalence and topological orbit equivalence.

1. Review of differentiable manifolds, definition of a Lie group. Quotients in the category of Lie groups, homogeneous manifolds, haar measure, connected components.
2. Algebraic groups, matrix groups, the classical groups.
3. Lie algebras and connection to Lie groups.
4. Nilpotent, solvable and semisimple Lie algebras and Lie groups, Lie theorem, Engel theorem, Levi decomposition.
5. Cartan-Killing form.
6. Representation of a Lie algebra over the complex numbers.
7. Root and weights, root systems, Dynkin diagrams, classication of complex semisimple Lie algebras.

Description: This is a graduate level course. Undergraduate students can register with my permission.

The prerequisite course is “Commutative Algebra” that I gave in the fall semester (or an equivalent course). We will need the following topics from that course: categories and functors; additive and exact functors; free modules; products and coproducts; tensor products of modules and rings.

The pace of the course, and the amount of material covered, will be determined by the background and capabilty of the audience. There will be many examples and exercises. I will upload typed notes after every lecture.

Course Grade: pass/fail grade. Passing the course requires attending all lectures and submitting most of the homework.

Homework: To be assigned every week. Checking will be sporadic.

See the first day handout for more administrative information.

Course Topics:

1. Adjoint functors.
2. Morita Theory.
3. Projective and Injective modules.
4. Complexes of modules.
5. Homotopies and homotopy equivalences.
6. The long exact cohomology sequence.
7. Projective, flat and injective resolutions.
8. Left and right derived functors.
9. Applications of derived functors to commutative algebra.
10. Further applications of derived functors and cohomology.