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metric vs norm
Translation Invariance
Scaling Property
math.CA  QA  analysis 
27 days ago by coltongrainger
Analysis Prelim Workshop, Fall 2013


Hilbert Spaces

Metric Spaces, Banach Spaces and Fourier Transforms

Distributions and Sobolev Spaces
Practice Problems


Hilbert Spaces

Metric Spaces, Banach Spaces and Fourier Transforms

Distributions and Sobolev Spaces
prelims  analysis  math.CA 
5 weeks ago by coltongrainger
Introduction to Analysis Notes
Lecture Notes: By Chapter

Chapter 1: Sets and Functions
Chapter 2: Numbers
Chapter 3: Sequences
Chapter 4: Series
Chapter 5: Topology of the Real Numbers
Chapter 6: Limits of Functions
Chapter 7: Continuous Functions
Chapter 8: Differentiable Functions
Chapter 9: Sequences and Series of Functions
Chapter 10: Power Series
Chapter 11: The Riemann Integral
Chapter 12: Applications of the Riemann Integral
Chapter 13: Metric, Normed, and Topological Spaces
prelims  math.CA 
5 weeks ago by coltongrainger
Measure Theory Notes
Lecture Notes: By Chapter

Chapter 1: Measures
Chapter 2: Lebesgue Measure
Chapter 3: Measurable Functions
Chapter 4: Integration
Chapter 5: Product Measures
Chapter 6: Differentiation
Chapter 7: Lp Spaces
prelims  math.CA 
5 weeks ago by coltongrainger
Curriculum of the First Year Courses | Department of Mathematics
There are introductory graduate courses in Algebra, Analysis, and Geometry-Topology. Each is offered in two versions: a masters level version and a Ph.D. level version. The masters level algebra course is Math 502/503, and the Ph.D. level course is Math 602/603. In analysis, the corresponding numbers are 508/509 and 608/(609 or 610). In geometry-topology, the corresponding numbers are 500/501 and 600/601.

All Ph.D. students are required to take (or place out of) these 600-level courses, although it is possible to take the 500-level courses first if this seems appropriate.

In addition, all Ph.D. students are required to take a year of algebraic topology, Math 618/619. Below are descriptions of Math 602/603, 608/(609 or 610), and 600/601. See the full list of course descriptions for information about other courses.
The First Year Curriculum in Algebra

Groups, normal subgroups and conjugacy classes, finite groups of order 12.

Readings: Lang, Algebra, Chapter 1; Jacobson, Basic Algebra, Part I, chapter 1.

Rings, polynomial rings in one variable, unique factorization, non-commutative rings - matrix ring.

Readings: Lang, Chapters 5,6,9,10 (para. 1-4), 17. Jacobson, Part I, Chap. 2, Part II, Chap. 4 (para. 1-6), Chap. 7. Symmetric and Hermitian matrices, spectral theorem, quadratic forms and signature.

Readings: Lang, Chap. 3,13,14,15,16. Jacobson, Part I, Chap. 3,6 (para. 1-3), Part II, Chap. 3,5 (para. 1-3).

Definition of a field, field of fractions of an integral domain.
Course content, Algebra I,II (Math 602, 603)

Groups: Sylow's theorem and its applications, finite abelian groups, nilpotent and solvable groups.

Rings: commutative noetherian rings, Hilbert basis theorem, prime and maximal ideals and localizations, primary decomposition, integral extensions and normal rings, Dedekind domains, Eisenstein irreducibility criteria, group ring, semisimple rings and Wedderburn's theorem.

Modules: tensor product, symmetric and exterior algebras and induced maps, exact functors, projective and injective modules, finitely generated modules over a Principal Ideal Domain with application to canonical forms of a matrix over a field, elementary theory of group representations.

Field extensions and Galois theory: separable and inseparable extensions, norm and trace, algebraic and transcendental extensions, transcendence basis, algebraic closure, fundamental theorem of Galois theory, solvability of equations, cyclotomic extensions and explicit computations of Galois groups.

Readings: Lang, Chap. 7,8,10 (para. 1-4); Jacobson, Part I, Chap. 4, Part II, Chap. 8.

General references: Lang - Algebra; Jacobson - Basic Algebra I, II; Atiyah-Macdonald - Introduction to commutative algebra; Kaplansky - Fields and Rings; Artin - Galois Theory; van der Waerden - Modern Algebra; Kaplansky - Commutative rings; Serre - Linear representations of finite groups; Zariski-Samuel, Commutative algebra (Vol. 1).
The First Year Curriculum in Analysis

Axiomatic development of the real number system, especially the completeness axiom; Abstract metric spaces, open and closed sets, completeness, compactness; Continuous functions from one metric space to another, uniform continuity; Continuous functions on a compact metric space have compact image and are uniformly continuous; Pointwise and uniform convergence of sequences and series of functions; continuity of a uniform limit of continuous functions. Differentiation: mean value theorem, Taylor's theorem and Taylor's series, partial differentiation and total differentiability of functions of several variables.
Riemann integration: definition and elementary properties, fundamental theorem of calculus. Interchange of limit operations, of order of partial differentiation, integration of spaces term-by-term. Implicit function theorem. Fourier analysis up to pointwise convergence for piecewise smooth functions. Use of Fourier analysis to solve heat and vibration equations. Differential equations, solution of common forms. Complex numbers, power series and Fourier series (an undergraduate course in complex analysis would be helpful).

Readings: Except for the material on Fourier analysis, the above is all in Rosenlicht's "Introduction to Analysis", Rudin's "Principles of Mathematical Analysis", Boyce and de Prima's "Elementary Differential Equations" and many other books.
Course Content, Analysis I (Math 608)

The first two-thirds of the semester concerns conplex analysis: analyticity, Cauchy theory, meromorphic functions, isolated singularities, analytic continuation, Runge's theorem, d-bar equation, Mittlag-Leffler theorem, harmonic and sub-harmonic functions, Riemann mapping theorem, Fourier transform from the analytic perspective. The last third of the semester provides an introduction to real analysis: Weierstrass approximation, Lebesgue measure in Euclidean spaces, Borel measures and convergence theorems, C^0 and the Riesz-Markov theorem, L^p spaces, Fubini theorem.
Course Content, Analysis II (Math 609)

The first third of the semester continues the study of real analysis begun in Math 608. Topics will include: general measure theory, outer measures and Cartheodory construction, Hausdorff measures, Radon-Nikodym theorem, Fubini's theorem, Hilbert space and L^2-theory of the Fourier transform. The last two-thirds of the semester concerns functional analysis: normed linear spaces, convexity, the Hahn-Banach Theorem, duality for Banach spaces, weak convergence, bounded linear operators, Baire category theorem, uniform boundedness principle, open mapping theorem, closed graph theorem, compact operators, Fredholm theory, interpolation theorems, L^p theory for the Fourier transform.
The First Year Curriculum in Geometry-Topology

Basic familiarity with point-set (general) topology: metric spaces, topological spaces, separation axioms, compactness, completeness.
Course Content, Geometry-Topology I (Math 600)

Differentiable functions, inverse and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Introduction to Lie groups and Lie group actions.

Readings: M. Spivak, "A Comprehensive Introduction to Differential Geometry", vol.I, 2nd edition. Publish or Perish, 1979. Supplementary: V. Guillemin & A. Pollack, "Differential topology", Prentice-Hall, 1974.
Course content, Geometry-Topology II (Math 601)

Covering spaces and fundamental groups, van Kampen's theorem and classification of surfaces. Basics of homology and cohomology, singular and cellular; isomorphism with de Rham cohomology. Brouwer fixed point theorem, CW complexes, cup and cap products, Poincare duality, Kunneth and universal coefficient theorems, Alexander duality, Lefschetz fixed point theorem.

Readings: M.J. Greenberg & J. Harper, "algebraic Topology, a first course". Math Lecture Note Series, vol.58. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. Supplementary: A. Hatcher, "Algebraic topology", Cambridge University Press, 2002.

General references:
1. I. Singer - J. Thorpe, Lecture notes on elementary topology and geometry.
2. J. Kelley, General topology (look at the exercises).
3. J. Djugundji, Topology.
4. J. Munkres, Topology: a first course.
5. F. Warner, Foundations of differentiable manifolds and Lie groups (first half).
6. M. Spivak, A comprehensive introduction to differential geometry, Vol. 1.
7. H. Flanders, Differential forms, with applications to the physical sciences.
8. W. Fleming, Functions of several variables.
9. G. Simmons, Introduction to topology and modern analysis.
10. F. Croom, Basic concepts of algebraic topology (good for fundamental groups and homology).
11. W. Massey, Algebraic topology, an introduction (good for fundamental groups).
12. C.T. Wall, A geometric introduction to topology.
13. M.A. Armstrong, Basic topology.
14. J. Milnor, Topology from the differentiable viewpoint.
15. Greenberg, Lectures on algebraic topology (first two chapters are good for fundamental groups and covering spaces).
16. Bourbaki, General topology.
17. Guillemin and Pollack, Differential topology (good exercises).
18. Y. Matsushima, Differentiable manifolds.
prelims  math.AT  math.RA  math.CA 
7 weeks ago by coltongrainger
A.G. Kovalev: teaching materials
Part IB: Geometry (Lent Term 2018)

examples sheet 1, sheet 2, sheet 3

Part III: Riemannian geometry (Lent Term 2017)

examples sheet 1 (updated 22 February, minor changes in q. 2, 7 and 8), sheet 2, sheet 3

the final course schedule

Last example class: Monday 8 May, 2.00-3.30pm in MR14

Past exam papers: 2012

Part III: Differential Geometry (Michaelmas Term 2014)

examples sheet 1, sheet 2, sheet 3, sheet 4

the final course schedule

skeleton notes (updated Nov-2014: minor clarification on page 2, details added on the Levi–Civita):
smooth manifolds, vector bundles, Riemannian geometry
reference card on multilinear algebra
Note. In any given year only one of the topics `geodesics' or `Riemannian submanifolds' (found in the Riemannian geometry chapter) was lectured. There were minor variations in the smooth manifolds chapter.

Past exam papers: 2014

Part II(C): Topics in Analysis (Lent Term 2014)

examples sheet 1, sheet 2, sheet 3, sheet 4

Part II(D): Differential geometry (Michaelmas Term 2010)

examples sheet 1, sheet 2, sheet 3, sheet 4

supporting materials:
Triangulations and the Euler characteristic (a picture is missing as it was drawn by hand)

A set of notes (here is a direct link to the pdf file) by Prof. Gabriel Paternain (updated 28/11/12).
a link (taken from Gabriel Paternain's notes) to a great site about minimal surfaces

An animated gif showing an isometric deformation between catenoid and helicoid (taken from Wikipedia).

Imaginary is yet another great site featuring visualizations (including curves and surfaces), with free software and the mathematics behind it.

Part IB: Analysis II (Michaelmas Term 2009)

examples sheet 1, sheet 2, sheet 3, sheet 4

supporting materials:
Term by term integration and differentiation

Part IB: Complex Analysis (Lent Term 2009)

examples sheet 1, sheet 2, sheet 3

supporting materials:
Topology in the plane, Evaluation of definite integrals: case study,

Part III: Complex Manifolds (Lent Term 2009)

examples sheet 1, sheet 2, sheet 3, sheet 4

Part II(D): Riemann Surfaces (Michaelmas Term 2007)

examples sheet 1, sheet 2, sheet 3, sheet 4

course notes (24-lecture version, pictures are missing as these were drawn by hand), updated 25-Feb-2013
math.DG  math.CA  undergrad  prelims 
7 weeks ago by coltongrainger
analysis - Baby/Papa/Mama/Big Rudin - Mathematics Stack Exchange
Baby = Principles of Mathematical Analysis;
Papa/Big = Real and Complex Analysis
Grandpa = Functional Analysis;
math.HO  math.CA  QA 
7 weeks ago by coltongrainger
JIBLM Linear Topology
This is a traditional treatment of the topology of the line, addressing limit points, least upper bound, greatest lower bound, sequence convergence, Cauchy sequences, compactness, connectedness, and elementary measure theory.
math.CA  math.GT  undergrad 
10 weeks ago by coltongrainger
real analysis - Every power series is the Taylor series of some $C^{infty}$ function - Mathematics Stack Exchange
There is a huge caveat however: you can't go from the series to the function f .
Firstly, the series might not be convergent at any x≠0∈R ! An example is ΣanXn=ΣnnXn whose radius of convergence is zero.
Secondly, even if it does converge it might converge to the wrong function! For example if you start with Cauchy's function you get the zero Taylor series. It converges to zero, of course, but that is definitely not the Cauchy function you started with. So we should not read too much in Borel's theorem: it cannot force a non-analytic function to become analytic!
math.CA  undergrad  QA 
july 2018 by coltongrainger

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