2016–17–B

• Cesaro means: Convolutions, positive summability kernels and Fejer’s theorem.
• Applications of Fejer’s theorem: the Weierstrass approximation theorem for polynomials, Weyl’s equidistribution theorem, construction of a nowhere differentiable function (time permitting).
• Pointwise and uniform convergence and divergence of partial sums: the Dirichlet kernel and its properties, construction of a continuous function with divergent Fourier series, the Dini test.
• $L^2$ approximations. Parseval’s formula. Absolute convergence of Fourier series of $C^1$ functions. Time permitting, the isoperimetric problem or other applications.
• Applications to partial differential equations. The heat and wave equation on the circle and on the interval. The Poisson kernel and the Laplace equation on the disk.
• Fourier series of linear functionals on $C^n(\mathbb{T})$. The notion of a distribution on the circle.
• Time permitting: positive definite sequences and Herglotz’s theorem.
• The Fourier transform: convolutions, the inversion formula, Plancherel’s theorem, Hermite functions. Time permitting: tempered distributions, further applications to differential equations.
• Fourier analysis on finite cyclic groups, and the Fast Fourier Transform algorithm.

• 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

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.

• 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. Geometry of Curves. Parametrizations, arc length, curvature, torsion, Frenet equations, global properties of curves in the plane.
2. Extrinsic Geometry of Surfaces. Parametrizations, tangent plane, differentials, first and second fundamental forms, curves in surfaces, normal and geodesic curvature of curves.
3. Differential equations without coordinates. Vector and line fields and flows, frame fields, Frobenius theorem. Geometry of fixed point and singular points in ODEs.
4. Intrinsic and Extrinsic Geometry of Surfaces. Frames and frame fields, covariant derivatives and connections, Riemannian metric, Gaussian curvature, Fundamental Forms and the equations of Gauss and Codazzi-Mainardi.
5. Geometry of geodesics. Exponential map, geodesic polar coordinates, properties of geodesics, Jacobi fields, convex neighborhoods.
6. Global results about surfaces. The Gauss-Bonnet Theorem, Hopf-Rinow theorem, Hopf-Poincaret theorem.

1. An introductory sketch and some motivating examples. Degenerate critical points of functions. Singular (nonsmooth) points of curves.
2. Holomorphic functions of several variables. Weierstrass preparation theorem. Local Rings and germs of functions/sets.
3. Isolated critical points of holomorphic functions. Unfolding and morsication. Finitely determined function germs.
4. Classification of simple singularities. Basic singularity invariants. Plane curve singularities. Decomposition into branches and Puiseux expansion.
5. Time permitting we will concentrate on some of the following topics: a. Blowups and resolution of plane curve singularities; b. Basic topological invariants of plane curve singularities (Milnor fibration); c. Versal deformation and the discriminant.

The course covers central ideas and central methods in classical set theory, without the axiomatic development that is required for proving independence results. The course is aimed as 2nd and 3rd year students and will equip its participants with a broad variety of set theoretic proof techniques that can be used in different branches of modern mathematics.

Sylabus

• The notion of cardinality. Computation of cardinalities of various known sets.
• Sets of real numbers. The Cantor-Bendixsohn derivative. The structure of closed subsets of Euclidean spaces.
• What is Cantor’s Continuum Hypothesis.
• Ordinals. Which ordinals are order-embeddable into the real line. Existence theorems ordinals. Hartogs’ theorem.
• Transfinite recursion. Applications.
• Various formulations of Zermelo’s axiom of Choice. Applications in algebra and geometry.
• Cardinals as initial ordinals. Hausdorff’s cofinality function. Regular and singular cardinals.
• Hausdorff’s formula. Konig’s lemma. Constraints of cardinal arithmetic.
• Ideal and filters. Ultrafilters and their applications.
• The filter of closed and unbounded subsets of a regular uncountable cardinal. Fodor’s pressing down lemma and applications in combinatorics.
• Partition calculus of infinite cardinals and ordinals. Ramsey’s theorem. The Erdos-Rado theorem. Dushnik-Miller theorem. Applications.
• Combinatorics of singular cardinals. Silver’s theorem.
• Negative partition theorems. Todorcevic’s theorem.
• Other topics

Bibliography.

1. Winfried Just and Martin Wese. Discovering modern set theory I, II. Graduate Studies in Mathematics, vol. 8, The AMS, 1996.
2. Azriel Levy. Basic Set Theory. Dover, 2002.
3. Ralf Schindler. Set Theory. Springer 2014.

• Rings and modules, Polynomial rings in several variables over a field
• Monomial orders and the division algorithm in several variables
• Grobner bases and the Buchberger algorithm, Elimination and equation solving
• Applications of Grobner bases:
  • integer programming
  • graph coloring
  • robotics
  • coding theory
  • combinatorics and more
• The Hilbert function and the Hilbert series, Speeding up the Buchberger algorithm, The f4 and f5 algorithms