A Summer of Maths (ASoM) 2016

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    A Summer of Maths 2016
    Purpose of this thread

    This thread should serve as an interactive resource to help people bridge the gap to undergraduate mathematics. I anticipate people will be discussing topics covered in a typical undergraduate course with emphasis on, though certainly not limited to, the first years of those respective courses. People should feel free to have tangential discussions including everything from the philosophy of mathematics to more practical things like choosing modules and subject areas. That said, we should aim to avoid discussing A-level material as well as topics from postgraduate courses as these will be of little interest to the majority of people participating in this thread. Also, most of the content here is taken from JKN's original post and all credit goes to him.

    Target audience

    I am assuming that the majority of people participating will be those who are currently preparing for the first year of their undergraduate course in mathematics at Cambridge. That said, this thread is in no way exclusive and so I hope that everybody feels welcome! This is also why I am choosing to use the topic headings for the Cambridge mathematics course as the basis for the resource library. I hope that many do not find this in any way elitist or discouraging as I believe that most undergraduate courses cover almost identical material in the first year.

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    Cambridge Part IA (schedule)
    Lecture notes (IA Course notes)

    Lecture notes from Cambridge
    -Official lecture notes.

    Lecture notes by 'Dexter Chua'.-At a glance seem very well-written and certainly up-to-date and relevant/useful.

    Lecture notes from 'The Archimedeans' (Cambridge Mathematical Society).
    -Whilst useful, many notes relate to the courses as taught many years ago.

    Lecture notes by 'William Chen'.
    -Not immediately relevant to the the specifics of the course, but seem a very accessible resource.

    Lecture notes by 'Paul Taylor'
    -Quite rather out-dated now, including this for completeness and because I think they're still interesting.

    Lecture notes from other sources.
    -This is essentially a catalogue of further reading recommendations and additional links to lecture notes.

    Lecture notes and course materials from Oxford University.
    -These look extremely good and also include problem sheets which would be particularly useful for those not going to Oxford.

    MIT OCW Lecture course
    -Whilst not directly synonymous to the Cambridge Tripos, it is nevertheless interesting and very useful.

    Problems:

    DPMMS Examples sheets from Cambridge
    - Click on course name to access sheets. Would recommend against recent ones.

    DAMTP Example sheets from Cambridge
    - Would recommend against doing recent example sheets.

    This thread
    - I expect many people on this thread to share interesting questions and problems from a variety of sources for others to try.

    Advanced integration techniques.
    -Undoubtedly this will prove to be a favourite amongst those on this thread.

    Selected problems, published by the Hong Kong mathematical society.
    -This is an excellent resource.

    Problems and solutions from miscellaneous competitions (Art of Problem Solving).
    -Contains some gaps, but is a reliable source of diverse and irregular problems.

    BMO and IMO.
    -For the confident problem-solver.

    '104 Number Theory Problems', from the training of the USA IMO team.
    -Co-authored by 'Zuming Feng' should be all you need to know about why this is worth a read. The other books in there series are also very useful. For example: '103 Trigonometry Problems'.

    STEP Mathematics.
    -Arguably the best preparation for the types of problem-solving skills required for university. The 'Problem-Solving Society' is also a great resource.

    'The Proof is Trivial'.
    -An on-going problem-solving marathon packed with interesting and exciting problems (not limited to those listed in the O.P).

    Calculus Challenge.
    -Not as high a level as the majority of the problems you will find above but the papers make up a good resource for non-standard questions.

    Introduction to Olympiad Inequalities, from the training of the Bangladeshi IMO team.
    -I've read through several booklets on Inequalities and feel that this provides the most accessible introduction. It also places emphasis appropriately, which is always helpful for the problem-solver.

    Courses

    Michaelmas Term:

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    Numbers and Sets:

    Syllabus:
    Syllabus

    Introduction to number systems and logic
    Overview of the natural numbers, integers, real numbers, rational and irrational numbers, algebraic and transcendental numbers. Brief discussion of complex numbers; statement of the Fundamental Theorem of Algebra. Ideas of axiomatic systems and proof within mathematics; the need for proof; the role of counter-examples in mathematics. Elementary logic; implication and negation; examples of negation of com-pound statements. Proof by contradiction. [2 lectures]
    Sets, relations and functions
    Union, intersection and equality of sets. Indicator (characteristic) functions; their use in establishing set identities. Functions; injections, surjections and bijections. Relations, and equivalence relations.Counting the combinations or permutations of a set. The Inclusion-Exclusion Principle. [4 lectures]
    The integers
    The natural numbers: mathematical induction and the well-ordering principle. Examples, including the Binomial Theorem. [2 lectures]
    Elementary number theory
    Prime numbers: existence and uniqueness of prime factorisation into primes; highest common factors and least common multiples. Euclid’s proof of the infinity of primes. Euclid’s algorithm. Solution in integers of ax+by = c.
    Modular arithmetic (congruences). Units modulo n. Chinese Remainder Theorem. Wilson’s Theorem;the Fermat-Euler Theorem. Public key cryptography and the RSA algorithm. [8 lectures]
    The real numbers
    Least upper bounds; simple examples. Least upper bound axiom. Sequences and series; convergence of bounded monotonic sequences. Irrationality of \sqrt{2} and e. Decimal expansions. Construction of a transcendental number. [4 lectures]
    Countability and uncountability
    Definitions of finite, infinite, countable and uncountable sets. A countable union of countable sets is countable. Uncountability of R. Non-existence of a bijection from a set to its power set. Indirect proof of existence of transcendental numbers. [4 lectures]


    Book Recommendations:
    Books

    Cambridge Recommendations:
    R.B.J.T. Allenby Numbers and Proofs. Butterworth-Heinemann 1997 (£19.50 paperback)
    R.P. Burn Numbers and Functions: steps into analysis. Cambridge University Press 2000 (£21.95 paperback)
    H. Davenport The Higher Arithmetic. Cambridge University Press 1999 (£19.95 paperback)
    A.G. Hamilton Numbers, sets and axioms: the apparatus of mathematics. Cambridge University Press 1983 (£20.95 paperback)
    C. Schumacher Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley 2001 (£42.95 hardback)
    I. Stewart and D. Tall The Foundations of Mathematics. Oxford University Press 1977 (£22.50 paper-back)
    Additional recommendations:
    G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers.
    -Encapsulates the 3rd year 'Number Theory' C course though not all of 'Numbers and Sets'. This book is excellent though it may be worth buying the updated edition as the edition linked-to above is extremely old.


    Problems and resources from this thread:
    Problems
    ...


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    Groups:

    Syllabus:
    Groups Syllabus

    Examples of groups
    Axioms for groups. Examples from geometry: symmetry groups of regular polygons, cube, tetrahedron.Permutations on a set; the symmetric group. Subgroups and homomorphisms. Symmetry groups as subgroups of general permutation groups. The Möbius group; cross-ratios, preservation of circles, thepoint at infinity. Conjugation. Fixed points of Möbius maps and iteration. [4 lectures]
    Lagrange’s theorem
    Cosets. Lagrange’s theorem. Groups of small order (up to order 8). Quaternions. Fermat-Euler theorem from the group-theoretic point of view. [5 lectures]
    Group actions
    Group actions; orbits and stabilizers. Orbit-stabilizer theorem. Cayley’s theorem (every group is isomorphic to a subgroup of a permutation group). Conjugacy classes. Cauchy’s theorem. [4 lectures]
    Quotient groups
    Normal subgroups, quotient groups and the isomorphism theorem. [4 lectures]
    Matrix groups
    The general and special linear groups; relation with the Möbius group. The orthogonal and specia lorthogonal groups. Proof (in ) that every element of the orthogonal group is the product of reflections and every rotation in has an axis. Basis change as an example of conjugation. [3 lectures]
    Permutations
    Permutations, cycles and transpositions. The sign of a permutation. Conjugacy in and in . Simple groups; simplicity of . [4 lectures]


    Book recommendations:
    Book Recommendations

    Cambridge Recommendations:
    M.A. Armstrong Groups and Symmetry. Springer–Verlag 1988 (£33.00 hardback)
    Alan F Beardon Algebra and Geometry. CUP 2005 (£21.99 paperback, £48 hardback).
    R.P. Burn Groups, a Path to Geometry. Cambridge University Press 1987 (£20.95 paperback)
    J.A. Green Sets and Groups: a first course in Algebra. Chapman and Hall/CRC 1988 (£38.99 paper-back)
    W. Lederman Introduction to Group Theory. Longman 1976 (out of print)
    Nathan Carter Visual Group Theory. Mathematical Association of America Textbooks (£45)
    Additional Recommendations:
    J.B. Fraleigh, A First Course in Abstract Algebra 2003 (free) Note that this book covers the second year "Groups, Rings and Modules" course.


    Problems and resources from this thread:
    Problems
    ...


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    Vectors and Matrices:

    Syllabus:
    Syllabus

    Complex numbers
    Review of complex numbers, including complex conjugate, inverse, modulus, argument and Argand diagram. Informal treatment of complex logarithm, ]n-th roots and complex powers. de Moivre’s theorem. [2 lectures]

    Vectors
    Review of elementary algebra of vectors in , including scalar product. Brief discussion of vectors in and ; scalar product and the Cauchy–Schwarz inequality. Concepts of linear span, linear independence, subspaces, basis and dimension.
    Suffix notation: including summation convention, δij and εijk. Vector product and triple product:definition and geometrical interpretation. Solution of linear vector equations. Applications of vectors to geometry, including equations of lines, planes and spheres. [5 lectures]

    Matrices
    Elementary algebra of 3 × 3 matrices, including determinants. Extension to n × n complex matrices.Trace, determinant, non-singular matrices and inverses. Matrices as linear transformations; examples of geometrical actions including rotations, reflections, dilations, shears; kernel and image. [4 lectures]
    Simultaneous linear equations: matrix formulation; existence and uniqueness of solutions, geometric interpretation; Gaussian elimination. [3 lectures]
    Symmetric, anti-symmetric, orthogonal, hermitian and unitary matrices. Decomposition of a general matrix into isotropic, symmetric trace-free and antisymmetric parts. [1 lecture]

    Eigenvalues and Eigenvectors
    Eigenvalues and eigenvectors; geometric significance. [2 lectures]
    Proof that eigenvalues of hermitian matrix are real, and that distinct eigenvalues give an orthogonal basis of eigenvectors. The effect of a general change of basis (similarity transformations). Diagonalization of general matrices: sufficient conditions; examples of matrices that cannot be diagonalized. Canonical forms for 2 × 2 matrices. [5 lectures]
    Discussion of quadratic forms, including change of basis. Classification of conics, cartesian and polar forms. [1 lecture]
    Rotation matrices and Lorentz transformations as transformation groups. [1 lecture]


    Book recommendations:
    Book Recommendations

    Cambridge Recommendations:
    Alan F Beardon Algebra and Geometry. CUP 2005 (£21.99 paperback, £48 hardback)
    Gilbert Strang Linear Algebra and Its Applications. Thomson Brooks/Cole, 2006 (£42.81 paperback)
    Richard Kaye and Robert Wilson Linear Algebra. Oxford science publications, 1998 (£23)
    D.E. Bourne and P.C. Kendall Vector Analysis and Cartesian Tensors. Nelson Thornes 1992 (£30.75 paperback)
    E. Sernesi Linear Algebra: A Geometric Approach. CRC Press 1993 (£38.99 paperback)
    James J. Callahan The Geometry of Spacetime: An Introduction to Special and General Relativity. Springer 2000 (£51)


    Problems and resources from this thread:

    Problems
    ...


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    [To add]

    Miscellaneous Maths:
    Spoiler:
    Show
    Proof of Goldbach's Weak Conjecture by H.A. Helfgott (major arcs, minor arcs).
    -For the advanced reader. This is one of the most exciting papers in mathematics at the moment.

    Selected theorems (and proofs) concerning Prime Numbers and Infinity.
    -For the lover of mathematical beauty.

    Resource for learning how to write computer code by 'Code Academy'.
    -Remarkably easy to use and will likely prepare you for a 'Computational Projects' course.

    The sinc function.
    -An interesting one to learn about, though perhaps a little obscure.

    The Feynman Lectures on Physics Volumes 1, 2 and 3.
    -An essential companion to the budding Physicist.

    An introduction to Topology.
    -Not at all tedious and will certainly help you to come to terms with the key ideas.

    Topics in generating functions.
    -Extremely useful and applies to many areas of maths. Another favourite is 'Generating Functionology'.

    The Problems of Philosophy by Bertrand Russell.
    -This is on every reading list for undergraduate Philosophy. It is extremely good.

    A Mathematician's Apology by G.H. Hardy.
    -Should speak for itself. This will change your life. An up-to-date satire on the education system containing similar ideas can be seen in 'A Mathematician's Lament'. This may also change your life.

    Good luck and have fun!
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    Reserved.
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    Memes
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    Nice one, Z.
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    (Original post by Zacken)
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    Dropbox error on Feynman`s lectures on Physics .
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    dad
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    (Original post by Kadak)
    Dropbox error on Feynman`s lectures on Physics .
    they're also hostedo n caltech website as they 're public domain if you do a cheeky googleroo
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    (Original post by Number Nine)
    they're also hostedo n caltech website as they 're public domain if you do a cheeky googleroo
    Ty .
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    (Original post by Kadak)
    Dropbox error on Feynman`s lectures on Physics .
    Fixed, I think.
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    I live here now
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    Might as well sign in now...
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    :woo::woo::woo::woo:
    (might as well decorate the place)
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    Hey lads. It's gonna be one great summer.
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    Zacken, damn you put some real work into that OP.Mathemagicien, krishdesai7 here it is m8.
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    Thanks for this
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    (Original post by Zacken)
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    Why do you need to reserve 2nd post if you have OP? Is there a limit to the length of posts or something?
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    (Original post by Mathemagicien)
    Is there a limit to the length of posts?
    Yes.
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    (Original post by Insight314)
    Zacken, damn you put some real work into that OP.Mathemagicien, krishdesai7 here it is m8.
    Thanks.

    Hallo everyone 😄
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    I read A Mathematician`s Apology a while ago,and while I did enjoy it,I still don`t get why it held in such reverence by pure mathematicians.
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    First non mathmo, yesssssss
 
 
 
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