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diff --git a/md/research/oxford-math.md b/md/research/oxford-math.md index c2306c7..7388d06 100644 --- a/md/research/oxford-math.md +++ b/md/research/oxford-math.md @@ -6,7 +6,7 @@ keywords:math,oxford https://www.ox.ac.uk/admissions/undergraduate/courses-listing/mathematics http://mmathphys.physics.ox.ac.uk/ -https://courses.maths.ox.ac.uk/course_planner +https://courses-archive.maths.ox.ac.uk/course_planner https://www.ox.ac.uk/admissions/undergraduate/courses-listing/mathematics-and-statistics http://www.stats.ox.ac.uk/student-resources/bammath/course-materials/ @@ -32,7 +32,7 @@ Statistics - Applied and computational statistics,Statistical inference,Statisti ## Topics -https://courses.maths.ox.ac.uk/overview/undergraduate/#50879 +https://courses-archive.maths.ox.ac.uk/overview/undergraduate/#50879 ### A0: Linear Algebra @@ -91,7 +91,7 @@ Further Reading: ### A1: Differential Equations 1 -https://courses.maths.ox.ac.uk/node/50806 +https://courses-archive.maths.ox.ac.uk/node/50806 Picard's Existence Theorem: Picard's Theorem for first-order scalar ODEs with proof. @@ -124,7 +124,7 @@ J. Ockendon, S. Howison, A. Lacey & A. Movchan, Applied Partial Differential Equ ### A2: Metric Spaces and Complex Analysis -https://courses.maths.ox.ac.uk/node/50681 +https://courses-archive.maths.ox.ac.uk/node/50681 Basic definitions: metric spaces, isometries, continuous functions (epsilon gamma definition), homeomorphisms, open sets, closed sets. @@ -196,7 +196,7 @@ Reinhold Remmert, Theory of Complex Functions (Springer, 1989) (Graduate Texts i ### A3: Rings and Modules -https://courses.maths.ox.ac.uk/node/50723 +https://courses-archive.maths.ox.ac.uk/node/50723 Recap on rings (not necessarily commutative) and examples: Z, fields, polynomial rings (in more than one variable), matrix rings. Zero-divisors, integral domains. @@ -243,7 +243,7 @@ Covers almost all material of the course. Out of print but many libraries should ### A4: Integration -https://courses.maths.ox.ac.uk/node/50730 +https://courses-archive.maths.ox.ac.uk/node/50730 Measure spaces. Outer measure, null set, measurable set. @@ -278,7 +278,7 @@ R. L. Schilling, Measures, Integrals and Martingales (CUP first ed. 2005, or sec ### A5: Topology -https://courses.maths.ox.ac.uk/node/50742 +https://courses-archive.maths.ox.ac.uk/node/50742 Axiomatic definition of an abstract topological space in terms of open sets. Basic definitions: closed sets, continuity, homeomorphism, convergent sequences, connectedness and comparison with the corresponding definitions for metric spaces. Examples to include metric spaces (definition of topological equivalence of metric spaces), discrete and indiscrete topologies, cofinite topology. The Hausdorff condition. Subspace topology. [2 lectures] @@ -307,7 +307,7 @@ J. Dugundji, Topology (Allyn and Bacon, 1966), chapters 3, 4, 5, 6, 7, 9, 11. [A ### A6: Differential Equations 2 -https://courses.maths.ox.ac.uk/node/50812 +https://courses-archive.maths.ox.ac.uk/node/50812 Models leading to two point boundary value problems for second order ODEs @@ -333,7 +333,7 @@ J. D. Logan, Applied Mathematics, (3rd Ed. Wiley Interscience, 2006). ### A7: Numerical Analysis -https://courses.maths.ox.ac.uk/node/50826 +https://courses-archive.maths.ox.ac.uk/node/50826 Lagrange interpolation [1 lecture] @@ -358,7 +358,7 @@ Linear multi-step methods and Dahlquist s theorem [2 lectures] ### A8 Probability -https://courses.maths.ox.ac.uk/node/50703 +https://courses-archive.maths.ox.ac.uk/node/50703 Continuous random variables. Jointly continuous random variables, independence, conditioning, functions of one or more random variables, change of variables. @@ -383,7 +383,7 @@ D. R. Stirzaker, Elementary Probability (Second edition, CUP, 2003). Chapters 7- ### A9 Statistics -https://courses.maths.ox.ac.uk/node/50850 +https://courses-archive.maths.ox.ac.uk/node/50850 Order statistics, probability plots. @@ -414,7 +414,7 @@ A. C. Davison, Statistical Models (Cambridge) ### A10: Fluids and Waves -https://courses.maths.ox.ac.uk/node/50862 +https://courses-archive.maths.ox.ac.uk/node/50862 Incompressible flow. Convective derivative, streamlines and particle paths. Euler's equations of motion for an inviscid fluid. Bernoulli's Theorem. Vorticity, circulation and Kelvin's Theorem. The vorticity equation and vortex motion. @@ -431,7 +431,7 @@ M. van Dyke, An Album of Fluid Motion (Parabolic Press, 1982). ### A11: Quantum Theory -https://courses.maths.ox.ac.uk/node/50872 +https://courses-archive.maths.ox.ac.uk/node/50872 Wave-particle duality; Schrodinger's equation; stationary states; quantum states of a particle in a box (infinite squarewell potential). @@ -464,11 +464,11 @@ A.I.M. Rae, Quantum Mechanics (4th Edition, Taylor & Francis Ltd, 2002). Chapter ### ASO: Integral Transforms -https://courses.maths.ox.ac.uk/node/50748 +https://courses-archive.maths.ox.ac.uk/node/50748 ### ASO: Number Theory -https://courses.maths.ox.ac.uk/node/50752 +https://courses-archive.maths.ox.ac.uk/node/50752 #### Course Overview: Number theory is one of the oldest parts of mathematics. For well over two thousand years it has attracted professional and amateur mathematicians alike. Although notoriously `pure' it has turned out to have more and more applications as new subjects and new technologies have developed. Our aim in this course is to introduce students to some classical and important basic ideas of the subject. @@ -499,7 +499,7 @@ basic version of the RSA encryption method. ### ASO: Group Theory -https://courses.maths.ox.ac.uk/node/50756 +https://courses-archive.maths.ox.ac.uk/node/50756 #### Course Overview: This group theory course develops the theory begun in prelims, and this course will build on that. After recalling basic concepts, the focus will be on two circles of problems. @@ -530,7 +530,7 @@ Applications including classification of groups of small order. [2] ### ASO: Projective Geometry -https://courses.maths.ox.ac.uk/node/50765 +https://courses-archive.maths.ox.ac.uk/node/50765 #### Course Overview: Projective Geometry might be viewed as the geometry of perspective. Two observers of a painting - one looking obliquely, one straight on - will not agree on angles and distances but will both sees lines as lines and will agree on whether they meet. So projective transformations (such as relate the two observers views) are less rigid than Euclidean, or even affine, transformations. Projective geometry also introduces the idea of points at infinity - points where parallel lines meet. These points fill in the missing gaps/address some special cases of geometry in a similar way to which complex numbers resolve such problems in algebra. From this point of view ellipses, parabolae and hyperbolae are all projectively equivalent and just happen to include no, one or two points at infinity. The study of such conics also has applications to the study of quadratic Diophantine equations. @@ -562,7 +562,7 @@ Simple applications to Diophantine Equations. ### ASO: Introduction to Manifolds -https://courses.maths.ox.ac.uk/node/50777 +https://courses-archive.maths.ox.ac.uk/node/50777 Course Overview: In this course, the notion of the total derivative for a function f:Rm->Rn is introduced. Roughly speaking, this is an approximation of the function near each point in Rn by a linear transformation. This is a key concept which pervades much of mathematics, both pure and applied. It allows us to transfer results from linear theory locally to nonlinear functions. For example, the Inverse Function Theorem tells us that if the derivative is an invertible linear mapping at a point then the function is invertible in a neighbourhood of this point. Another example is the tangent space at a point of a surface in R3, which is the plane that locally approximates the surface best. @@ -603,7 +603,7 @@ Stephen G. Krantz and Harold R. Parks, The Implicit Function Theorem: History, T ### ASO: Graph Theory -https://courses.maths.ox.ac.uk/node/50793 +https://courses-archive.maths.ox.ac.uk/node/50793 #### Course Overview: This course introduces some central topics in graph theory. @@ -631,7 +631,7 @@ D.B. West, Introduction to Graph Theory, 2nd edition, Prentice Hall, 2001. ### B1.1 Logic -https://courses.maths.ox.ac.uk/node/48789 +https://courses-archive.maths.ox.ac.uk/node/48789 The notation, meaning and use of propositional and predicate calculus. The formal language of propositional calculus: truth functions; conjunctive and disjunctive normal form; tautologies and logical consequence. The formal language of predicate calculus: satisfaction, truth, validity, logical consequence. @@ -651,7 +651,7 @@ R. Cori and D. Lascar, Mathematical Logic: A Course with Exercises (Part II) (Ox ### B1.2 Set Theory -https://courses.maths.ox.ac.uk/node/48811 +https://courses-archive.maths.ox.ac.uk/node/48811 What is a set? Introduction to the basic axioms of set theory. Ordered pairs, cartesian products, relations and functions. Axiom of Infinity and the construction of the natural numbers; induction and the Recursion Theorem. @@ -673,7 +673,7 @@ J. Stillwell, Roads to Infinity (CRC Press, 2010). ### B2.1 Introduction to Representation Theory -https://courses.maths.ox.ac.uk/node/48817 +https://courses-archive.maths.ox.ac.uk/node/48817 Noncommutative rings, one- and two-sided ideals. Associative algebras (over fields). Main examples: matrix algebras, polynomial rings and quotients of polynomial rings. Group algebras, representations of groups. @@ -692,7 +692,7 @@ P. Etingof, Introduction to representation theory (Online course notes, MIT 2011 ### B2.2 Commutative Algebra -https://courses.maths.ox.ac.uk/node/48825 +https://courses-archive.maths.ox.ac.uk/node/48825 Modules, ideals, prime ideals, maximal ideals. @@ -717,7 +717,7 @@ D. Eisenbud: Commutative Algebra with a view towards Algebraic Geometry, (Spring ### B3.1 Galois Theory -https://courses.maths.ox.ac.uk/node/48832 +https://courses-archive.maths.ox.ac.uk/node/48832 Review of polynomial rings, factorisation, integral domains. Reminder that any nonzero homomorphism of fields is injective. @@ -744,7 +744,7 @@ Herstein, Topics in Algebra (Wiley, 1975). ### B3.2 Geometry of Surfaces -https://courses.maths.ox.ac.uk/node/48839 +https://courses-archive.maths.ox.ac.uk/node/48839 #### Course Overview: Different ways of thinking about surfaces (also called two-dimensional manifolds) are introduced in this course: first topological surfaces and then surfaces with extra structures which allow us to make sense of differentiable functions (`smooth surfaces'), holomorphic functions (`Riemann surfaces') and the measurement of lengths and areas ('Riemannian 2-manifolds'). @@ -784,7 +784,7 @@ M. P. do Carmo, Differential Geometry of Curves and Surfaces (Dover, 2016) ### B3.3 Algebraic Curves -https://courses.maths.ox.ac.uk/node/48854 +https://courses-archive.maths.ox.ac.uk/node/48854 #### Course Overview: A real algebraic curve is a subset of the plane defined by a polynomial equation p(x,y)=0. The intersection properties of a pair of curves are much better behaved if we extend this picture in two ways: the first is to use polynomials with complex coefficients, the second to extend the curve into the projective plane. In this course projective algebraic curves are studied, using ideas from algebra, from the geometry of surfaces and from complex analysis. @@ -811,7 +811,7 @@ W. Fulton, Algebraic Curves, 3rd ed., downloadable at http://www.math.lsa.umich. ### B3.4 Algebraic Number Theory -https://courses.maths.ox.ac.uk/node/48862 +https://courses-archive.maths.ox.ac.uk/node/48862 Field extensions, minimum polynomial, algebraic numbers, conjugates, discriminants, Gaussian integers, algebraic integers, integral basis Examples: quadratic fields @@ -834,59 +834,59 @@ D. Marcus, Number Fields (Springer-Verlag, New York-Heidelberg, 1977). ISBN 0-38 ### B3.5 Topology and Groups -https://courses.maths.ox.ac.uk/node/48869 +https://courses-archive.maths.ox.ac.uk/node/48869 ### B4.1 Functional Analysis I -https://courses.maths.ox.ac.uk/node/48881 +https://courses-archive.maths.ox.ac.uk/node/48881 ### B4.2 Functional Analysis II -https://courses.maths.ox.ac.uk/node/48890 +https://courses-archive.maths.ox.ac.uk/node/48890 ### B4.3 Distribution Theory -https://courses.maths.ox.ac.uk/node/48897 +https://courses-archive.maths.ox.ac.uk/node/48897 ### B4.4 Fourier Analysis -https://courses.maths.ox.ac.uk/node/48912 +https://courses-archive.maths.ox.ac.uk/node/48912 ### B5.1 Stochastic Modelling of Biological Processes -https://courses.maths.ox.ac.uk/node/48922 +https://courses-archive.maths.ox.ac.uk/node/48922 ### B5.2 Applied Partial Differential Equations -https://courses.maths.ox.ac.uk/node/48944 +https://courses-archive.maths.ox.ac.uk/node/48944 ### B5.3 Viscous Flow -https://courses.maths.ox.ac.uk/node/48954 +https://courses-archive.maths.ox.ac.uk/node/48954 ### B5.4 Waves and Compressible Flow -https://courses.maths.ox.ac.uk/node/48960 +https://courses-archive.maths.ox.ac.uk/node/48960 ### B5.5 Further Mathematical Biology -https://courses.maths.ox.ac.uk/node/48968 +https://courses-archive.maths.ox.ac.uk/node/48968 ### B5.6 Nonlinear Systems -https://courses.maths.ox.ac.uk/node/48977 +https://courses-archive.maths.ox.ac.uk/node/48977 ### B6.1 Numerical Solution of Differential Equations I -https://courses.maths.ox.ac.uk/node/48986 +https://courses-archive.maths.ox.ac.uk/node/48986 ### B6.2 Numerical Solution of Differential Equations II -https://courses.maths.ox.ac.uk/node/49027 +https://courses-archive.maths.ox.ac.uk/node/49027 ### B6.3 Integer Programming -https://courses.maths.ox.ac.uk/node/49036 +https://courses-archive.maths.ox.ac.uk/node/49036 Week 1: Classical examples of Integer Programming problems (IP), modelling and basic terminology. @@ -926,17 +926,17 @@ L. A. Wolsey, Integer Programming (John Wiley & Sons, 1998), parts of chapters 1 ### B8.1 Probability, Measure and Martingales -https://courses.maths.ox.ac.uk/node/49105 +https://courses-archive.maths.ox.ac.uk/node/49105 ### B8.2 Continuous Martingales and Stochastic Calculus ### B8.3 Mathematical Models of Financial Derivatives -https://courses.maths.ox.ac.uk/node/49119 +https://courses-archive.maths.ox.ac.uk/node/49119 ### B8.4 Information Theory -https://courses.maths.ox.ac.uk/node/49135 +https://courses-archive.maths.ox.ac.uk/node/49135 ### B8.5 Graph Theory @@ -974,7 +974,7 @@ D. West, Introduction to Graph Theory, Second edition, (Prentice-Hall, 2001). ### BO1.1 History of Mathematics -https://courses.maths.ox.ac.uk/node/49179 +https://courses-archive.maths.ox.ac.uk/node/49179 Introduction: ancient mathematical knowledge and its transmission to early modern Europe; the development of symbolic notation up to the end of the sixteenth century. Seventeenth century: analytic geometry; the development of calculus; Newton's Principia. @@ -1014,7 +1014,7 @@ June Barrow-Green, Jeremy Gray and Robin J. Wilson, The history of mathematics : ### SB3.1 Applied Probability -https://courses.maths.ox.ac.uk/node/49150 +https://courses-archive.maths.ox.ac.uk/node/49150 Poisson processes and birth processes. Continuous-time Markov chains. Transition rates, jump chains and holding times. Forward and backward equations. Class structure, hitting times and absorption probabilities. Recurrence and transience. Invariant distributions and limiting behaviour. Time reversal. Renewal theory. Limit theorems: strong law of large numbers, strong law and central limit theorem of renewal theory, elementary renewal theorem, renewal theorem, key renewal theorem. Excess life, inspection paradox. @@ -1031,7 +1031,7 @@ D. R. Stirzaker, Elementary Probability (2nd edition, Cambridge University Press ### C1.1 Model Theory -https://courses.maths.ox.ac.uk/node/49212 +https://courses-archive.maths.ox.ac.uk/node/49212 @@ -1043,15 +1043,15 @@ https://courses.maths.ox.ac.uk/node/49212 ### C2.4 Infinite Groups -https://courses.maths.ox.ac.uk/node/49282 +https://courses-archive.maths.ox.ac.uk/node/49282 ### C2.7 Category Theory -https://courses.maths.ox.ac.uk/node/49315 +https://courses-archive.maths.ox.ac.uk/node/49315 ### C3.1 Algebraic Topology -https://courses.maths.ox.ac.uk/node/49323 +https://courses-archive.maths.ox.ac.uk/node/49323 ### C3.3 Differentiable Manifolds @@ -1059,7 +1059,7 @@ https://courses.maths.ox.ac.uk/node/49323 ### C3.8 Analytic Number Theory -https://courses.maths.ox.ac.uk/node/49389 +https://courses-archive.maths.ox.ac.uk/node/49389 ### C3.10 Additive and Combinatorial Number Theory @@ -1079,19 +1079,19 @@ https://courses.maths.ox.ac.uk/node/49389 ### C5.12 Mathematical Physiology -https://courses.maths.ox.ac.uk/node/49542 +https://courses-archive.maths.ox.ac.uk/node/49542 ### C6.1 Numerical Linear Algebra -https://courses.maths.ox.ac.uk/node/49550 +https://courses-archive.maths.ox.ac.uk/node/49550 ### C6.3 Approximation of Functions -https://courses.maths.ox.ac.uk/node/49601 +https://courses-archive.maths.ox.ac.uk/node/49601 ### C6.5 Theories of Deep Learning -https://courses.maths.ox.ac.uk/node/49612 +https://courses-archive.maths.ox.ac.uk/node/49612 ### C7.1 Theoretical Physics (C6) @@ -1101,7 +1101,7 @@ https://courses.maths.ox.ac.uk/node/49612 ### C8.3 Combinatorics -https://courses.maths.ox.ac.uk/node/49710 +https://courses-archive.maths.ox.ac.uk/node/49710 ### CCD Dissertations on a Mathematical Topic @@ -1111,7 +1111,7 @@ https://courses.maths.ox.ac.uk/node/49710 ### C1.4 Axiomatic Set Theory -https://courses.maths.ox.ac.uk/node/49234 +https://courses-archive.maths.ox.ac.uk/node/49234 ### C2.3 Representation Theory of Semisimple Lie Algebras @@ -1123,11 +1123,11 @@ https://courses.maths.ox.ac.uk/node/49234 ### C3.5 Lie Groups -https://courses.maths.ox.ac.uk/node/49375 +https://courses-archive.maths.ox.ac.uk/node/49375 ### C3.7 Elliptic Curves -https://courses.maths.ox.ac.uk/node/49382 +https://courses-archive.maths.ox.ac.uk/node/49382 #### Course Overview: Elliptic curves give the simplest examples of many of the most interesting phenomena which can occur in algebraic curves; they have an incredibly rich structure and have been the testing ground for many developments in algebraic geometry whilst the theory is still full of deep unsolved conjectures, some of which are amongst the oldest unsolved problems in mathematics. The course will concentrate on arithmetic aspects of elliptic curves defined over the rationals, with the study of the group of rational points, and explicit determination of the rank, being the primary focus. Using elliptic curves over the rationals as an example, we will be able to introduce many of the basic tools for studying arithmetic properties of algebraic varieties. @@ -1174,17 +1174,17 @@ J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate T ### C5.4 Networks -https://courses.maths.ox.ac.uk/node/49460 +https://courses-archive.maths.ox.ac.uk/node/49460 ### C5.6 Applied Complex Variables ### C5.9 Mathematical Mechanical Biology -https://courses.maths.ox.ac.uk/node/49511 +https://courses-archive.maths.ox.ac.uk/node/49511 ### C6.2 Continuous Optimisation -https://courses.maths.ox.ac.uk/node/49577 +https://courses-archive.maths.ox.ac.uk/node/49577 ### C6.4 Finite Element Method for PDEs @@ -1196,21 +1196,21 @@ https://courses.maths.ox.ac.uk/node/49577 ### C7.7 Random Matrix Theory -https://courses.maths.ox.ac.uk/node/50988 +https://courses-archive.maths.ox.ac.uk/node/50988 ### C8.2 Stochastic Analysis and PDEs ### C8.4 Probabilistic Combinatorics -https://courses.maths.ox.ac.uk/node/49719 +https://courses-archive.maths.ox.ac.uk/node/49719 ### C8.5 Introduction to Schramm-Loewner Evolution -https://courses.maths.ox.ac.uk/node/49728 +https://courses-archive.maths.ox.ac.uk/node/49728 ### C8.6 Limit Theorems and Large Deviations in Probability -https://courses.maths.ox.ac.uk/node/49733 +https://courses-archive.maths.ox.ac.uk/node/49733 ### CCD Dissertations on a Mathematical Topic |