Supplement to the Course Handbook

for the

MSc in Mathematics and the

Foundations of Computer Science2016–2017

Course Synopses

2016-2017

Version 1.4

Contents

1 COURSES OFFERED IN 2016/2017 iv

2 SECTION A: MATHEMATICAL FOUNDATIONS vi

2.1 Schedule I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

2.2 Algebraic Number Theory — Prof. Minhyong Kim — 16 HT . . . . . . . . . . . . . vi

2.3 Algebraic Topology — Prof. Christopher Douglas — 16MT . . . . . . . . . . . . . . viii

2.4 Analytic Number Theory —Prof. Ben Green—16MT . . . . . . . . . . . . . . . . . . ix

2.5 Analytic Topology — Dr Rolf Suabedissen — 16MT . . . . . . . . . . . . . . . . . . x

2.6 Commutative Algebra — Prof. Nikolay Nikolov — 16HT . . . . . . . . . . . . . . . xi

2.7 Godel’s Incompleteness Theorems — Dr Dan Isaacson — 16HT . . . . . . . . . . . . xii

2.8 Introduction to Representation Theory — Prof. Nikolay Nikolov — 16 MT . . . . . xiii

2.9 Lambda Calculus and Types — Dr Steven Ramsay — 16 lectures HT . . . . . . . . xv

2.10 Lie Algebras — Prof. Dan Ciubotaru — 16MT . . . . . . . . . . . . . . . . . . . . . xvii

2.11 Model Theory — Prof Ehud Hrushovski — 16MT . . . . . . . . . . . . . . . . . . . xviii

2.12 Modular Forms — Prof Alan Lauder — 16HT . . . . . . . . . . . . . . . . . . . . . . xix

2.13 Topology and Groups — Prof Lackenby — 16 MT . . . . . . . . . . . . . . . . . . . xx

2.14 Schedule II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

2.15 Algebraic Geometry — Prof. Alexander Ritter — 16MT . . . . . . . . . . . . . . . . xxii

2.16 Axiomatic Set Theory — Dr Rolf Suabedissen — 16HT . . . . . . . . . . . . . . . . xxiii

2.17 Homological Algebra — Prof. Andre Henriques — 16MT . . . . . . . . . . . . . . . xxiv

2.18 Infinite Groups — Prof. Dan Segal — 16HT . . . . . . . . . . . . . . . . . . . . . . . xxv

2.19 Introduction to Schemes — Prof Damian Rossler — 16HT . . . . . . . . . . . . . . . xxvi

2.20 Non-Commutative Rings — Dr Thomas Bitaun — 16HT . . . . . . . . . . . . . . . . xxvii

2.21 Geometric Group Theory — Prof. Andre Henriques — 16HT . . . . . . . . . . . . . xxviii

2.22 Representation Theory of Semisimple Lie Algebra —Prof Dan Ciubotaru —HT . . . xxix

3 SECTION B: APPLICABLE THEORIES xxx

3.1 Schedule I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx

3.2 Advanced Machine Learning — Dr Varun Kanade — 16 HT . . . . . . . . . . . . . . xxx

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3.3 Applied Probability — Prof Chleboun — MT . . . . . . . . . . . . . . . . . . . . . . xxxi

3.4 Categories, Proofs and Processes — Dr Kishida — 20 lectures + extra reading MT . xxxii

3.5 Communication Theory — Prof Harold Oberhauser — 16 MT . . . . . . . . . . . . . xxxiv

3.6 Computer-Aided Formal Verification — Dr Alessandro Abate —16MT . . . . . . . . xxxv

3.7 Concurrency — Dr Thomas Gibson-Robson — 16 lectures + extra reading HT . . . xxxvii

3.8 Foundations of Computer Science — Prof Paul Goldberg — 16MT . . . . . . . . . . xxxix

3.9 Graph Theory — Prof. Oliver Riordan — 16 MT . . . . . . . . . . . . . . . . . . . . xli

3.10 Introduction to Cryptology — Dr Ali El Kaafarani— MT . . . . . . . . . . . . . . . xlii

3.11 Quantum Computer Science—Prof Bob Coecke —24 Lectures MT . . . . . . . . . . xliii

3.12 Schedule II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliv

3.13 Automata, Logic and Games — Prof Luke Ong —24 lectures + reading MT . . . . . xliv

3.14 Advanced Cryptology — Dr Christophe Petit — HT . . . . . . . . . . . . . . . . . . xlvi

3.15 Categorical Quantum Mechanics–Dr Dan Marsden & Jamie Vicary—16 Lectures HT xlvii

3.16 Combinatorics — Prof. Alex Scott — 16MT . . . . . . . . . . . . . . . . . . . . . . . xlviii

3.17 Computational Algebraic Topology — Prof Ulrike Tillmann & Prof Samson Abramsky16HT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlix

3.18 Computational Number Theory — Prof Richard Pinch — Reading course TT . . . . li

3.19 Computational Game Theory — Prof Edith Elkind & Prof Mike Aldridge — MT . . lii

3.20 Distributional Models of Meaning — Prof Bob Coecke— Reading course HT . . . . lv

3.21 Elliptic Curves — Prof Victor Flynn — 16HT . . . . . . . . . . . . . . . . . . . . . . lvii

3.22 Networks — Dr Heather Harrington — 16HT . . . . . . . . . . . . . . . . . . . . . . lix

3.23 Probabilistic Combinatorics — Prof. Oliver Riordan — 16MT . . . . . . . . . . . . . lx

3.24 Probability and Computing— Prof Stefan Kiefer& Prof Stanislav Zivny — 20MT . . lxi

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1 COURSES OFFERED IN 2016/2017

Section A

Schedule I

Algebraic Number Theory Prof Kim HTAlgebraic Topology Prof Douglas MTAnalytic Number Theory Prof Green MTAnalytic Topology Dr Suabedissen MTCommutative Algebra Prof Nikolov HTGodel’s Incompleteness Theorems Dr Isaacson HTIntroduction to Representation Theory Prof Nikolov MTLambda Calculus and Types Dr Ramsay HTLie Algebras Prof Ciubotaru MTModel Theory Prof Hrushouski MTModular Forms Prof Lauder HTTopology and Groups Prof Lackenby MT

Schedule II

Algebraic Geometry Prof Ritter MTAxiomatic Set Theory Dr Suabedissen HTHomological Algebra Dr Henriques MTInfinite Groups Prof Segal HTIntroduction to Schemes Prof Rossler HTNon-Commutative Rings Dr Bitaun HTGeometric Group Theory Prof Papazoglou HTRepresentation Theory of Semisimple Lie Algebra Prof Ciubotaru HT

Section B

Schedule I

Advanced Machine Learning Dr Kanade HTApplied Probability Prof Chleboun MTCategories, Proofs and Processes Dr Kishida MTCommunication Theory Prof Oberhauser MTComputer Aided Formal Verification Dr Abate MTConcurrency Dr Gibson-Robinson HTFoundations of Computer Science Prof Goldberg MTGraph Theory Prof Riordan MTIntroduction to Cryptology Dr El Kaafarani MTQuantum Computer Science Prof Coecke MT

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Schedule II

Automata, Logic and Games Prof Ong MTAdvanced Cryptology Dr Petit HTCategorical Quantum Mechanics Dr Marsden & Dr Vicary HTCombinatorics Prof Scott MTComputational Algebraic Topology Prof Tillmann & Prof Abramsky HTComputational Number Theory * Prof Pinch TTComputational Game Theory Prof Edith Elkind & Prof Mike

WooldridgeMT

Distributional Models of Meaning* Prof Coecke HTElliptic Curves Prof Flynn HTNetworks Dr Harrington HTProbabilistic Combinatorics Prof Riordan HTProbability and Computing Prof Zivny & Prof Kiefer HT

*These courses are offered as directed reading courses, with syllabuses provided as in the case oflecture courses. There may be one or two more reading courses to be added later.

WE REGRET THAT DUE TO TIMETABLING RESTRICTIONS THERE WILL BE A NUMBEROF CLASHES BETWEEN LECTURE COURSES. PLEASE CHECK THE LECTURE TIMETABLECAREFULLY.

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2 SECTION A: MATHEMATICAL FOUNDATIONS

2.1 Schedule I

2.2 Algebraic Number Theory — Prof. Minhyong Kim — 16 HT

Prerequisites: Rings and Modules and Number Theory. B3.1 Galois Theory is an essential pre-requisite.

Recommended Prerequisites: All second-year algebra and arithmetic. Students who have nottaken Part A Number Theory should read about quadratic residues in, for example, the appendixto Stewart and Tall. This will help with the examples.

Overview An introduction to algebraic number theory. The aim is to describe the properties ofnumber fields, but particular emphasis in examples will be placed on quadratic fields, where it iseasy to calculate explicitly the properties of some of the objects being considered. In such fields thefamiliar unique factorisation enjoyed by the integers may fail, and a key objective of the course isto introduce the class group which measures the failure of this property.

Learning Outcomes Students will learn about the arithmetic of algebraic number fields. Theywill learn to prove theorems about integral bases, and about unique factorisation into ideals. Theywill learn to calculate class numbers, and to use the theory to solve simple Diophantine equations.

Synopsis

1. field extensions, minimum polynomial, algebraic numbers, conjugates, discriminants, Gaussianintegers, algebraic integers, integral basis

2. examples: quadratic fields

3. norm of an algebraic number

4. existence of factorisation

5. factorisation in Q(√d)

6. ideals, Z-basis, maximal ideals, prime ideals

7. unique factorisation theorem of ideals

8. relationship between factorisation of number and of ideals

9. norm of an ideal

10. ideal classes

11. statement of Minkowski convex body theorem

12. finiteness of class number

13. computations of class number to go on example sheets

Reading

1. I. Stewart and D. Tall, Algebraic Number Theory and Fermat’s Last Theorem. (Third Edition,Peters, 2002).

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Further Reading

1. D. Marcus, Number Fields (Springer-Verlag, New York–Heidelberg, 1977). ISBN 0-387-90279-1.

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2.3 Algebraic Topology — Prof. Christopher Douglas — 16MT

Recommended Prerequisites Helpful but not essential: Part A Topology, B3.5 Topology andGroups.

Overview Homology theory is a subject that pervades much of modern mathematics. Its basicideas are used in nearly every branch, pure and applied. In this course, the homology groups oftopological spaces are studied. These powerful invariants have many attractive applications. Forexample we will prove that the dimension of a vector space is a topological invariant and the factthat ‘a hairy ball cannot be combed’.

Learning Outcomes At the end of the course, students are expected to understand the basicalgebraic and geometric ideas that underpin homology and cohomology theory. These include thecup product and Poincare Duality for manifolds. They should be able to choose between the differenthomology theories and to use calculational tools such as the Mayer-Vietoris sequence to computethe homology and cohomology of simple examples, including projective spaces, surfaces, certainsimplicial spaces and cell complexes. At the end of the course, students should also have developeda sense of how the ideas of homology and cohomology may be applied to problems from otherbranches of mathematics.

Synopsis Chain complexes of free Abelian groups and their homology. Short exact sequences.Delta complexes and their homology. Euler characteristic.

Singular homology of topological spaces. Relative homology and the Five Lemma. Homotopyinvariance and excision (details of proofs not examinable). Mayer-Vietoris Sequence. Equivalenceof simplicial and singular homology.

Degree of a self-map of a sphere. Cell complexes and cellular homology. Application: the hairy balltheorem.

Cohomology of spaces and the Universal Coefficient Theorem (proof not examinable). Cup products.Kunneth Theorem (without proof). Topological manifolds and orientability. The fundamental classof an orientable, closed manifold and the degree of a map between manifolds of the same dimension.Poincare Duality (without proof).

Reading

1. A. Hatcher, Algebraic Topology (Cambridge University Press, 2001). Chapters 2 and 3.

2. G. Bredon, Topology and Geometry (Springer, 1997). Chapters 4 and 5.

3. J. Vick, Homology Theory, Graduate Texts in Mathematics 145 (Springer, 1973).

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2.4 Analytic Number Theory —Prof. Ben Green—16MT

Recommended Prerequisites Basic ideas of complex analysis. Elementary number theory.Some familiarity with Fourier series will be helpful but not essential.

Overview The aim of this course is to study the prime numbers using the famous Riemann ζ-function. In particular, we will study the connection between the primes and the zeros of theζ-function. We will state the Riemann hypothesis, perhaps the most famous unsolved problem inmathematics, and examine its implication for the distribution of primes. We will prove the primenumber theorem, which states that the number of primes less than X is asymptotic to X/ logX.

Learning Outcomes In addition to the highlights mentioned above, students will gain experiencewith different types of Fourier transform and with the use of complex analysis.

Synopsis Introductory material on primes. Arithmetic functions: Mobius function, Euler’s φ-function, the divisor function, the σ-function. Multiplicativity. Dirichlet series and Euler products.The von Mangoldt function.

The Riemann ζ-function for <(s) > 1. Euler’s proof of the infinitude of primes. ζ and the vonMongoldt function.

Schwarz functions on R, Z, R/Z and their Fourier transforms. *Inversion formulas and uniqueness*.The Poisson summation formula. The θ-function and its functional equation. The Γ-function andthe meromorphic continuation and functional equation of the ζ-function. Poles and zeros of ζ andstatement of the Riemann hypothesis. Basic estimates for ζ.

*Product theorems for entire functions*. Hadamard’s product formula for ζ and the partial fractionexpansion.

The Mellin transform of a smooth compactly supported function. The Mellin inversion formula.Decay of the Mellin transform in vertical strips. Statement and proof of the explicit formula.

The classical zero-free region. Proof of the prime number theorem. Implications of the Riemannhypothesis for the distribution of primes.

Reading Full printed notes will be provided for the course, including the non-examinable topics(marked with asterisks above). The following books are relevant to the course.

1. G. H. Hardy and E. M. Wright, An introduction to the Theory of Numbers (Sixth edition,OUP 2008). Chapters 16, 17, 18.

2. H. Davenport, Multiplicative number theory (Third Edition, Springer Graduate texts 74),selected parts of the first half.

3. M. du Sautoy, Music of the primes (this is a popular book which could be useful backgroundreading for the course).

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2.5 Analytic Topology — Dr Rolf Suabedissen — 16MT

Recommended Prerequisites Part A Topology; a basic knowledge of Set Theory, includingcardinal arithmetic, ordinals and the Axiom of Choice, will also be useful.

Overview The aim of the course is to present a range of major theory and theorems, both im-portant and elegant in themselves and with important applications within topology and to math-ematics as a whole. Central to the course is the general theory of compactness, compactificationsand Tychonoff’s theorem, one of the most important in all mathematics (with applications acrossmathematics and in mathematical logic) and computer science.

Synopsis Bases and initial topologies (including pointwise convergence and the Tychonoff producttopology). Separation axioms, continuous functions, Urysohn’s lemma. Separable, Lindelof andsecond countable spaces. Urysohn’s metrization theorem. Filters and ultrafilters. Tychonoff’stheorem. Compactifications, in particular the Alexandroff One-Point Compactification and theStone–Cech Compactification. Completeness, connectedness and local connectedness. Componentsand quasi-components. Totally disconnected compact spaces. Paracompactness; Bing MetrizationTheorem.

Reading

1. S. Willard, General Topology (Addison–Wesley, 1970), Chs. 1–8.

2. R. Engelking, General Topology (Sigma Series in Pure Mathematics, Vol 6, 1989)

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2.6 Commutative Algebra — Prof. Nikolay Nikolov — 16HT

Recommended Prerequisites Rings and Modules is essential. Representation Theory and GaloisTheory are recommended.

Overview Amongst the most familiar objects in mathematics are the ring of integers and thepolynomial rings over fields. These play a fundamental role in number theory and in algebraicgeometry, respectively. The course explores the basic properties of such rings.

Synopsis Modules, ideals, prime ideals, maximal ideals.Noetherian rings; Hilbert basis theorem. Minimal primes.Localization.Polynomial rings and algebraic sets. Weak Nullstellensatz.Nilradical and Jacobson radical; strong Nullstellensatz.Integral extensions. Prime ideals in integral extensions.Noether Normalization Lemma.Krull dimension; dimension of an affine algebra.Noetherian rings of small dimension, Dedekind domains.

Reading

1. M. F. Atiyah and I. G. MacDonald: Introduction to Commutative Algebra, (Addison-Wesley,1969).

2. D. Eisenbud: Commutative Algebra with a view towards Algebraic Geometry, (Springer GTM,1995).

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2.7 Godel’s Incompleteness Theorems — Dr Dan Isaacson — 16HT

Recommended Prerequisites This course presupposes knowledge of first-order predicate logicup to and including soundness and completeness theorems for a formal system of first-order predicatelogic (B1 Logic).

Overview The starting point is Godel’s mathematical sharpening of Hilbert’s insight that manip-ulating symbols and expressions of a formal language has the same formal character as arithmeticaloperations on natural numbers. This allows the construction for any consistent formal system con-taining basic arithmetic of a ‘diagonal’ sentence in the language of that system which is true butnot provable in the system. By further study we are able to establish the intrinsic meaning of sucha sentence. These techniques lead to a mathematical theory of formal provability which generalizesthe earlier results. We end with results that further sharpen understanding of formal provability.

Learning Outcomes Understanding of arithmetization of formal syntax and its use to establishincompleteness of formal systems; the meaning of undecidable diagonal sentences; a mathemati-cal theory of formal provability; precise limits to formal provability and ways of knowing that anunprovable sentence is true.

Synopsis Godel numbering of a formal language; the diagonal lemma. Expressibility in a formallanguage. The arithmetical undefinability of truth in arithmetic. Formal systems of arithmetic;arithmetical proof predicates. Σ0-completeness and Σ1-completeness. The arithmetical hierarchy.ω-consistency and 1-consistency; the first Godel incompleteness theorem. Separability; the Rosserincompleteness theorem. Adequacy conditions for a provability predicate. The second Godel incom-pleteness theorem; Lob’s theorem. Provable Σ1-completeness. The ω-rule. The provability logicsGL; fixed point theorems for GL. The Bernays arithmetized completeness theorem; undecidable∆2-sentences of arithmetic.

Reading

1. Lecture notes for the course.

Further Reading

1. Raymond M. Smullyan, Godel’s Incompleteness Theorems (Oxford University Press, 1992).

2. George S. Boolos and Richard C. Jeffrey, Computability and Logic (3rd edition, CambridgeUniversity Press, 1989), Chs 15, 16, 27 (pp 170–190, 268-284).

3. George Boolos, The Logic of Provability (Cambridge University Press, 1993).

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2.8 Introduction to Representation Theory — Prof. Nikolay Nikolov —16 MT

Recommended Prerequisites: Rings and Modules is essential. Group Theory is recommended.

Overview This course gives an introduction to the representation theory of finite groups and finitedimensional algebras. Representation theory is a fundamental tool for studying symmetry by meansof linear algebra: it is studied in a way in which a given group or algebra may act on vector spaces,giving rise to the notion of a representation.

A large part of the course will deal with the structure theory of semisimple algebras and theirmodules (representations). We will prove the Jordan-Holder Theorem for modules. Moreover, wewill prove that any finite-dimensional semisimple algebra is isomorphic to a product of matrix rings(Wedderburn’s Theorem over C ).

In the later part of the course we apply the developed material to group algebras, and classify whengroup algebras are semisimple (Maschke’s Theorem). All of this material will be applied to the studyof characters and representations of finite groups.

Learning Outcomes They will know in particular simple modules and semisimple algebras andthey will be familiar with examples. They will appreciate important results in the course such asthe Jordan-Holder Theorem, Schur’s Lemma, and the Wedderburn Theorem. They will be familiarwith the classification of semisimple algebras over C and be able to apply this to representationsand characters of finite groups.

Synopsis Noncommutative rings, one- and two-sided ideals. Associative algebras (over fields).Main examples: matrix algebras, polynomial rings and quotients of polynomial rings. Group alge-bras, representations of groups.

Modules and their relationship with representations. Simple and semisimple modules, compositionseries of a module, Jordan-Holder Theorem. Semisimple algebras. Schur’s Lemma, the WedderburnTheorem, Maschke’s Theorem. Characters of complex representations. Orthogonality relations,finding character tables. Tensor product of modules. Induction and restriction of representations.Application: Burnside’s paqb Theorem.

Reading

1. K. Erdmann, B2 Algebras, Mathematical Institute Notes (2007).

2. G. D. James and M. Liebeck, Representations and Characters of Finite Groups (2nd edition,Cambridge University Press, 2001).

Further Reading

1. J. L. Alperin and R. B. Bell, Groups and Representations, Graduate Texts in Mathematics162 (Springer-Verlag, 1995).

2. P. M. Cohn, Classic Algebra (Wiley & Sons, 2000). (Several books by this author available.)

3. C. W. Curtis, and I. Reiner, Representation Theory of Finite Groups and Associative Algebras(Wiley & Sons, 1962).

4. L. Dornhoff, Group Representation Theory (Marcel Dekker Inc., New York, 1972).

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5. I. M. Isaacs, Character Theory of Finite Groups (AMS Chelsea Publishing, American Math-ematical Society, Providence, Rhode Island, 2006).

6. J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42(Springer-Verlag, 1977).

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2.9 Lambda Calculus and Types — Dr Steven Ramsay — 16 lectures HT

Recommended Prerequisites: There are no prerequisites, but the course will assume familiaritywith constructing mathematical proofs. Some basic knowledge of computability would be useful forone of the topics (the Models of Computation course is much more than enough), but is certainlynot necessary.

Overview As a language for describing functions, any literate computer scientist would expect tounderstand the vocabulary of the lambda calculus. It is folklore that various forms of the lambdacalculus are the prototypical functional programming languages, but the pure theory of the lambdacalculus is also extremely attractive in its own right. This course introduces the terminology andphilosophy of the lambda calculus, and then covers a range of self-contained topics studying thelanguage and some related structures. Topics covered include the equational theory, term rewrit-ing and reduction strategies, combinatory logic, Turing completeness and type systems. As such,the course will also function as a brief introduction to many facets of theoretical computer science,illustrating each (and showing the connections with practical computer science) by its relation tothe lambda calculus. There are no prerequisites, but the course will assume familiarity with con-struting mathematical proofs. Some basic knowledge of computability would be useful for one of thetopics (the Models of Computation course is much more than enough), but is certainly not necessary.

Learning Outcomes The course is an introductory overview of the foundations of computerscience with particular reference to the lambda-calculus. Students will

• understand the syntax and equational theory of the untyped lambda-calculus, and gain famil-iarity with manipulation of terms;

• be exposed to a variety of inductive proofs over recursive structures;

• learn techniques for analysing term rewriting systems, with particular reference to beta-reduction;

• see the connections between lambda-calculus and computabilty, and an example of how anundecidability proof can be constructed;

• see the connections and distinctions between lambda-calculus and combinatory logic;

• learn about simple type systems for the lambda-calculus, and how to prove a strong normal-ization result;

• understand how to deduce types for terms, and prove correctness of a principal type algorithm.

Synopsis Chapter 0 (1 lecture)

Introductory lecture. Preparation for use of inductive definitions and proofs.

Chapters 1–3 (5 lectures)

Terms, free and bound variables, alpha-conversion, substitution, variable convention, contexts, theformal theory lambda beta, the eta rule, fixed point combinators, lambda-theories. Reduction.Compatible closure, reflexive transitive closure, diamond and Church-Rosser properties for generalnotions of reduction. beta-reduction, proof of the Church-Rosser property (via parallel reduction),connection between beta-reduction and lambda beta, consistency of lambda beta. Inconsistency of

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equating all terms without beta-normal form. Reduction strategies, head and leftmost reduction.Standard reductions. Proof that leftmost reduction is normalising. Statement, without proof, ofGenericity Lemma, and simple applications.

Chapter 4 (2 lectures)

Church numerals, definability of total recursive functions. Second Recursion Theorem, Scott-CurryTheorem, undecideability of equality in lambda beta. Briefly, extension to partial functions.

Chapter 5 (2 lectures)

Untyped combinatory algebras. Abstraction algorithm, combinatory completeness, translations toand from untyped lambda-calculus, mismatches between combinary logic and lambda-calculus, ba-sis. Term algebras.

Chapters 6–8 (6 lectures)

Simple type assignment a la Curry using Hindley’s TA lambda system. Contexts and deductions.Subject Construction Lemma, Subject Reduction Theorem and failure of Subject Expansion. Briefly,a system with type invariance under equality. Informal and cursory treatment of Curry-Howardisomorphism. Tait’s proof of strong normalisation. Consequences: no fixed point combinators, poordefinability power. Pointer to literature on PCF as the obvious extension of simple types to coverall computable functions. Type substitutions and unification, Robinson’s algorithm. Principal Typealgorithm and correctness.

Syllabus Terms, formal theories lambda beta and lambda beta eta , fixed point combinators;reduction, Church-Rosser property of beta-reduction and consistency of lambda beta; reductionstrategies, standard reduction sequences, proof that leftmost reduction is normalising; Church nu-merals, definability of total recursive functions in the lambda-calculus, Second Recusion Theoremand undecidability results; combinatory algebras, combinatory completeness, basis; simple types a laCurry, type deductions, Subject Reduction Theorem, strong normalisation and consequences; typesubstitutions, unification, correctness of Principal Type Algorithm.

Reading List Essential

• Andrew Ker, lecture notes. Available online and handed out in the lectures. Comprehensivenotes on the entire course, including practice questions and class exercises.

Useful Background

• H. P. Barendregt, The Lambda Calculus, North-Holland, revised edition, 1984.

• J. R. Hindley, Basic Simple Type Theory, CUP Cambridge Tracts in Theoretical ComputerScience 42, 1997.

• J-Y. Girard, Y.Lafont and P. Taylor, Proofs and Types, CUP Cambridge Tracts in TheoreticalComputer Science 7, 1989.

• C. Hankin, Lambda Calculi, A Guide for Computer Scientists, OUP Graduate Texts in Com-puter Science, 1994.

• J. R. Hindley & J. P. Seldin, Introduction to Combinators and Lambda-Calculus (CambridgeUniversity Press, 1986).

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2.10 Lie Algebras — Prof. Dan Ciubotaru — 16MT

Recommended Prerequisites Part B course B2.1 Introduction to Representation Theory. Athorough knowledge of linear algebra and the second year algebra courses; in particular familiaritywith group actions, quotient rings and vector spaces, isomorphism theorems and inner productspaces will be assumed. Some familiarity with the Jordan–Holder theorem and the general ideas ofrepresentation theory will be an advantage.

Overview Lie Algebras are mathematical objects which, besides being of interest in their ownright, elucidate problems in several areas in mathematics. The classification of the finite-dimensionalcomplex Lie algebras is a beautiful piece of applied linear algebra. The aims of this course are tointroduce Lie algebras, develop some of the techniques for studying them, and describe parts of theclassification mentioned above, especially the parts concerning root systems and Dynkin diagrams.

Learning Outcomes Students will learn how to utilise various techniques for working with Liealgebras, and they will gain an understanding of parts of a major classification result.

Synopsis Definition of Lie algebras, small-dimensional examples, some classical groups and theirLie algebras (treated informally). Ideals, subalgebras, homomorphisms, modules.

Nilpotent algebras, Engel’s theorem; soluble algebras, Lie’s theorem. Semisimple algebras andKilling form, Cartan’s criteria for solubility and semisimplicity, Weyl’s theorem on complete re-ducibility of representations of semisimple Lie algebras.

The root space decomposition of a Lie algebra; root systems, Cartan matrices and Dynkin diagrams.Discussion of classification of irreducible root systems and semisimple Lie algebras.

Reading

1. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts inMathematics 9 (Springer-Verlag, 1972, reprinted 1997). Chapters 1–3 are relevant and part ofthe course will follow Chapter 3 closely.

2. B. Hall, Lie Groups, Lie Algebras, and Representations. An Elementary Introduction, Gradu-ate Texts in Mathematics 222 (Springer-Verlag, 2003).

3. K. Erdmann, M. J. Wildon, Introduction to Lie Algebras (Springer-Verlag, 2006), ISBN:1846280400.

Further Reading

1. J.-P. Serre, Complex Semisimple Lie Algebras (Springer, 1987). Rather condensed, assumesthe basic results. Very elegant proofs.

2. N. Bourbaki, Lie Algebras and Lie Groups (Masson, 1982). Chapters 1 and 4–6 are relevant;this text fills in some of the gaps in Serre’s text.

3. William Fulton, Joe Harris, Representation theory: a first course, GTM, Springer.

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2.11 Model Theory — Prof Ehud Hrushovski — 16MT

Recommended Prerequisites This course presupposes basic knowledge of First Order PredicateCalculus up to and including the Soundness and Completeness Theorems. A familiarity with (atleast the statement of) the Compactness Theorem would also be desirable.

Overview The course deepens a student’s understanding of the notion of a mathematical struc-ture and of the logical formalism that underlies every mathematical theory, taking B1 Logic as astarting point. Various examples emphasise the connection between logical notions and practicalmathematics.

The concepts of completeness and categoricity will be studied and some more advanced technicalnotions, up to elements of modern stability theory, will be introduced.

Learning Outcomes Students will have developed an in depth knowledge of the notion of analgebraic mathematical structure and of its logical theory, taking B1 Logic as a starting point. Theywill have an understanding of the concepts of completeness and categoricity and more advancedtechnical notions.

Synopsis Structures. The first-order language for structures. The Compactness Theorem forfirst-order logic. Elementary embeddings. Lowenheim–Skolem theorems. Preservation theorems forsubstructures. Model Completeness. Quantifier elimination.

Categoricity for first-order theories. Types and saturation. Omitting types. The Ryll Nardzewskitheorem characterizing aleph-zero categorical theories. Theories with few types. Ultraproducts.

Reading

1. C.C. Chang and H. Jerome Keisler, Model Theory (Third Edition (Dover Books on Mathe-matics)Paperback)

2. Tent and Ziegler, A Course in Model Theory, Cambridge University Press, April 2012

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2.12 Modular Forms — Prof Alan Lauder — 16HT

Recommended Prerequisites Part A Number Theory, Topology and Part B Geometry of Sur-faces, Algebraic Curves are useful but not essential.

Overview The course aims to introduce students to the beautiful theory of modular forms, one ofthe cornerstones of modern number theory. This theory is a rich and challenging blend of methodsfrom complex analysis and linear algebra, and an explicit application of group actions.

Learning Outcomes The student will learn about modular curves and spaces of modular forms,and understand in special cases how to compute their genus and dimension, respectively. They willsee that modular forms can be described explicitly via their q-expansions, and they will be familiarwith explicit examples of modular forms. They will learn about the rich algebraic structure onspaces of modular forms, given by Hecke operators and the Petersson inner product.

Synopsis

1. Overview and examples of modular forms. Definition and basic properties of modular forms.

2. Topology of modular curves: a fundamental domain for the full modular group; fundamentaldomains for subgroups Γ of finite index in the modular group; the compact surfaces XΓ; explicittriangulations of XΓ and the computation of the genus using the Euler characteristic formula;the congruence subgroups Γ(N),Γ1(N) and Γ0(N); examples of genus computations.

3. Dimensions of spaces of modular forms: general dimension formula (proof non-examinable);the valence formula (proof non-examinable).

4. Examples of modular forms: Eisenstein series in level 1; Ramanujan’s ∆ function; some arith-metic applications.

5. The Petersson inner product.

6. Modular forms as functions on lattices: modular forms of level 1 as functions on lattices;Eisenstein series revisited.

7. Hecke operators in level 1: Hecke operators on lattices; Hecke operators on modular forms andtheir q-expansions; Hecke operators are Hermitian; multiplicity one.

Reading

1. F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathe-matics 228, Springer-Verlag, 2005.

2. R.C. Gunning, Lectures on Modular Forms, Annals of mathematical studies 48, PrincetonUniversity Press, 1962.

3. J.S. Milne, Modular Functions and Modular Forms:www.jmilne.org/math/CourseNotes/mf.html

4. J.-P. Serre, Chapter VII, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer-Verlag, 1973.

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2.13 Topology and Groups — Prof Lackenby — 16 MT

Recommended Prerequisites Part A Topology is essential and Group Theory is recommended.

Overview This course introduces the important link between topology and group theory. On theone hand, associated to each space, there is a group, known as its fundamental group. This can beused to solve topological problems using algebraic methods. On the other hand, many results aboutgroups are best proved and understood using topology. For example, presentations of groups, wherethe group is defined using generators and relations, have a topological interpretation. The endpointof the course is the Nielsen–Shreier Theorem, an important, purely algebraic result, which is provedusing topological techniques.

Learning Outcomes Students will develop a sound understanding of simplicial complexes, cellcomplexes and their fundamental groups. They will be able to use algebraic methods to analysetopological spaces. They will also be able to address questions about groups using topologicaltechniques.

Synopsis Homotopic mappings, homotopy equivalence. Simplicial complexes. Simplicial approx-imation theorem.

The fundamental group of a space. The fundamental group of a circle. Application: the fundamentaltheorem of algebra. The fundamental groups of spheres.

Free groups. Existence and uniqueness of reduced representatives of group elements. The funda-mental group of a graph.

Groups defined by generators and relations (with examples). Tietze transformations.

The free product of two groups. Amalgamated free products.

The Seifert–van Kampen Theorem.

Cell complexes. The fundamental group of a cell complex (with examples). The realization of anyfinitely presented group as the fundamental group of a finite cell complex.

Covering spaces. Liftings of paths and homotopies. A covering map induces an injection betweenfundamental groups. The use of covering spaces to determine fundamental groups: the circle again,and real projective n-space. The correspondence between covering spaces and subgroups of thefundamental group. Regular covering spaces and normal subgroups.

Cayley graphs of a group. The relationship between the universal cover of a cell complex, and theCayley graph of its fundamental group. The Cayley 2-complex of a group.

The Nielsen–Schreier Theorem (every subgroup of a finitely generated free group is free) provedusing covering spaces.

Reading

1. John Stillwell, Classical Topology and Combinatorial Group Theory (Springer-Verlag, 1993).

Additional Reading

1. D. Cohen, Combinatorial Group Theory: A Topological Approach, Student Texts 14 (LondonMathematical Society, 1989), Chapters 1–7.

2. A. Hatcher, Algebraic Topology (CUP, 2001), Chapter. 1.

3. M. Hall, Jr, The Theory of Groups (Macmillan, 1959), Chapters. 1–7, 12, 17 .

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4. D. L. Johnson, Presentations of Groups, Student Texts 15 (Second Edition, London Mathe-matical Society, Cambridge University Press, 1997). Chapters. 1–5, 10,13.

5. W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory (Dover Publications,1976). Chapters. 1–4.

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2.14 Schedule II

2.15 Algebraic Geometry — Prof. Alexander Ritter — 16MT

Recommended Prerequisites Part A Rings and Modules. B3.3 Algebraic Curves useful butnot essential.

Overview Algebraic geometry is the study of algebraic varieties: an algebraic variety is roughlyspeaking, a locus defined by polynomial equations. One of the advantages of algebraic geometry isthat it is purely algebraically defined and applied to any field, including fields of finite characteristic.It is geometry based on algebra rather than calculus, but over the real or complex numbers it providesa rich source of examples and inspiration to other areas of geometry.

Synopsis Affine algebraic varieties, the Zariski topology, morphisms of affine varieties. Irreduciblevarieties.

Projective space. Projective varieties, affine cones over projective varieties. The Zariski topologyon projective varieties. The projective closure of affine variety. Morphisms of projective varieties.Projective equivalence.

Veronese morphism: definition, examples. Veronese morphisms are isomorphisms onto their image;statement, and proof in simple cases. Subvarieties of Veronese varieties. Segre maps and productsof varieties. Categorical products: the image of the Segre map gives the categorical product.

Coordinate rings. Hilbert’s Nullstellensatz. Correspondence between affine varieties (and morphismsbetween them) and finitely generated reduced k-algebras (and morphisms between them). Gradedrings and homogeneous ideals. Homogeneous coordinate rings.

Categorical quotients of affine varieties by certain group actions. The maximal spectrum.

Primary decomposition of ideals.

Discrete invariants of projective varieties: degree, dimension, Hilbert function. Statement of theoremdefining Hilbert polynomial.

Quasi-projective varieties, and morphisms between them. The Zariski topology has a basis of affineopen subsets. Rings of regular functions on open subsets and points of quasi-projective varieties.The ring of regular functions on an affine variety is the coordinate ring. Localisation and relationshipwith rings of regular functions.

Tangent space and smooth points. The singular locus is a closed subvariety. Algebraic re-formulationof the tangent space. Differentiable maps between tangent spaces.

Function fields of irreducible quasi-projective varieties. Rational maps between irreducible varieties,and composition of rational maps. Birational equivalence. Correspondence between dominant ratio-nal maps and homomorphisms of function fields. Blow-ups: of affine space at a point, of subvarietiesof affine space, and of general quasi-projective varieties along general subvarieties. Statement ofHironaka’s Desingularisation Theorem. Every irreducible variety is birational to a hypersurface.Re-formulation of dimension. Smooth points are a dense open subset.

Reading KE Smith et al, An Invitation to Algebraic Geometry, (Springer 2000), Chapters 1–8.

Further Reading

1. M Reid, Undergraduate Algebraic Geometry, LMS Student Texts 12, (Cambridge 1988).

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2. K Hulek, Elementary Algebraic Geometry, Student Mathematical Library 20. (AmericanMathematical Society, 2003).

3. A Gathmann, Algebraic Geometry lecture notes, online: www.mathematik.uni-kl.de/en/

agag/members/professors/gathmann/notes/alggeom

4. I Shafarevich, Basic Algebraic Geometry 1, (Springer, 1994).

5. D Mumford, The Red Book of Varieties and Schemes, (Springer, 2009).

2.16 Axiomatic Set Theory — Dr Rolf Suabedissen — 16HT

Recommended Prerequisites This course presupposes basic knowledge of First Order PredicateCalculus up to and including the Soundness and Completeness Theorems, together with a courseon basic set theory, including cardinals and ordinals, the Axiom of Choice and the Well OrderingPrinciple.

Overview Inner models and consistency proofs lie at the heart of modern Set Theory, historicallyas well as in terms of importance. In this course we shall introduce the first and most importantof inner models, Godel’s constructible universe, and use it to derive some fundamental consistencyresults.

Synopsis A review of the axioms of ZF set theory. Absoluteness, the recursion theorem. TheCumulative Hierarchy of sets and the consistency of the Axiom of Foundation as an example of themethod of inner models. Levy’s Reflection Principle. Godel’s inner model of constructible sets andthe consistency of the Axiom of Constructibility (V = L). V = L is absolute. The fact that V = Limplies the Axiom of Choice. Some advanced cardinal arithmetic. The fact that V = L implies theGeneralized Continuum Hypothesis.

Reading For the review of ZF set theory and the prerequisites from Logic:

1. D. Goldrei, Classic Set Theory (Chapman and Hall, 1996).

2. K. Kunen, The Foundations of Mathematics (College Publications, 2009).

For course topics (and much more):

1. K. Kunen, Set Theory (College Publications, 2011) Chapters (I and II).

Further Reading

1. K. Hrbacek and T. Jech, Introduction to Set Theory (3rd edition, M Dekker, 1999).

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2.17 Homological Algebra — Prof. Andre Henriques — 16MT

Recommended Prerequisites Part A Rings and Modules. Introduction to Representation The-ory B2.1 is recommended but not essential.

Overview Homological algebra is one of the most important tools in mathematics with applicationranging from number theory and geometry to quantum physics. This course will introduce the basicconcepts and tools of homological algebra with examples in module theory and group theory.

Learning Outcomes Students will learn about abelian categories and derived functors and willbe able to apply these notions in different contexts. They will learn to compute Tor, Ext, and groupcohomology and homology.

Synopsis Chain complexes: complexes of R-modules, operations on chain complexes, long exactsequences, chain homotopies, mapping cones and cylinders (4 hours) Derived functors: delta functors,projective and injective resolutions, left and right derived functors (5 hours) Tor and Ext: Tor andflatness, Ext and extensions, universal coefficients theorems, Koszul resolutions (4 hours) Grouphomology and cohomology: definition, interpretation of H1 and H2, universal central extensions,the Bar resolution (3 hours).

Reading Weibel, Charles An introduction to Homological algebra (see Google Books)

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2.18 Infinite Groups — Prof. Dan Segal — 16HT

Recommended Prerequisites A thorough knowledge of the second-year algebra courses; inparticular, familiarity with group actions, quotient rings and quotient groups, and isomorphismtheorems will be assumed. Familiarity with the Commutative Algebra course will be helpful but notessential.

Overview The concept of a group is so general that anything which is true of all groups tendsto be rather trivial. In contrast, groups that arise in some specific context often have a rich andbeautiful theory. The course introduces some natural families of groups, various questions that onecan ask about them, and various methods used to answer these questions; these involve among otherthings rings and trees.

Synopsis Free groups and their subgroups; finitely generated groups: counting finite-index sub-groups; finite presentations and decision problems; Linear groups: residual finiteness; structure ofsoluble linear groups; Nilpotency and solubility: lower central series and derived series; structuraland residual properties of finitely generated nilpotent groups and polycyclic groups; characterizationof polycyclic groups as soluble Z-linear groups; Torsion groups and the General Burnside Problem.

Reading

1. D. J. S. Robinson, A course in the theory of groups, 2nd ed., Graduate texts in Mathematics,(Springer-Verlag, 1995). Chapters 2, 5, 6, 15.

2. D. Segal, Polycyclic groups, (CUP, 2005) Chapters 1 and 2.

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2.19 Introduction to Schemes — Prof Damian Rossler — 16HT

Recommended Prerequisites Commutative Algebra is essential. Homological Algebra is highlyrecommended and Category Theory is recommended but the necessary material from both coursescan be learnt during the course (see the beginning of the lecture notes for precise references). Alge-braic Geometry is recommended but not technically necessary. Algebraic Topology contains manyhomological techniques also used in C2.6.

Overview Scheme theory is the foundation of modern algebraic geometry. It unifies algebraicgeometry with algebraic number theory. This unification has led to proofs of important conjecturesin number theory such as the Weil conjecture by Deligne and the Mordell conjecture by Faltings.

This course will cover the basics of the theory of schemes, with an emphasis on cohomologicaltechniques.

Learning Outcomes Students will have developed a thorough understanding of the basic conceptsand methods of scheme theory. They will be able to work with affine and projective schemes, aswell as with coherent sheaves and their cohomology groups.

Synopsis Sheaves and cohomology of sheaves.

Affine schemes: points, topology, structure sheaf. Schemes: definition, subschemes, morphisms,glueing. Relative schemes: fibred products, Cohomological characterisation of affine schemes.

Projective schemes, morphisms to projective space. Ample line bundles. Cohomological characteri-sation of ampleness.

Flat morphisms, semicontinuity, Hilbert polynomials. Cohomological characterisation of flatness.

Constructibility and irreducibility. Images of constructible sets.

Separatedness, properness and valuative criteria. Hilbert and Quot schemes.

Reading

1. Robin Hartshorne, Algebraic Geometry.

2. Ravi Vakil, Foundations of Algebraic Geometry, online notes on the website of Stanford Uni-versity (open access).

Further reading

1. David Mumford, The Red Book of Varieties and Schemes.

2. David Eisenbud and Joe Harris, The Geometry of Schemes.

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2.20 Non-Commutative Rings — Dr Thomas Bitaun — 16HT

Recommended Prerequisites Prerequisites: Part A Rings and Modules.Recommended background: Introduction to Representation Theory B2.1, Part B CommutativeAlgebra (from 2016 onwards).

Overview This course builds on Algebra 2 from the second year. We will look at several classesof non-commutative rings and try to explain the idea that they should be thought of as functionson ”non-commutative spaces”. Along the way, we will prove several beautiful structure theoremsfor Noetherian rings and their modules.

Learning Outcomes Students will be able to appreciate powerful structure theorems, and befamiliar with examples of non-commutative rings arising from various parts of mathematics.

Synopsis 1. Examples of non-commutative Noetherian rings: enveloping algebras, rings of differ-ential operators, group rings of polycyclic groups. Filtered and graded rings. (3 hours)

2. Jacobson radical in general rings. Jacobson’s density theorem. Artin-Wedderburn. (3 hours)

3. Ore localisation. Goldie’s Theorem on Noetherian domains. (3 hours)

4. Minimal prime ideals and dimension functions. Rees rings and good filtrations. (3 hours)

5. Bernstein’s Inequality and Gabber’s Theorem on the integrability of the characteristic variety. (4hours)

Reading

1. K.R. Goodearl and R.B. Warfield, An Introduction to Noncommutative Noetherian Rings(CUP, 2004).

Further reading

1. M. Atiyah and I. MacDonald, Introduction to Commutative Algebra (Westview Press, 1994).

2. S.C. Coutinho, A Primer of Algebraic D-modules (CUP, 1995).

3. J. Bjork, Analytic D-Modules and Applications (Springer, 1993).

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2.21 Geometric Group Theory — Prof. Andre Henriques — 16HT

Recommended Prerequisites. The Topology & Groups course is a helpful, though not essentialprerequisite.

Overview. The aim of this course is to introduce the fundamental methods and problems ofgeometric group theory and discuss their relationship to topology and geometry.

The first part of the course begins with an introduction to presentations and the list of problemsof M. Dehn. It continues with the theory of group actions on trees and the structural study offundamental groups of graphs of groups.

The second part of the course focuses on modern geometric techniques and it provides an introductionto the theory of Gromov hyperbolic groups.

Synopsis. Free groups. Group presentations. Dehn’s problems. Residually finite groups.

Group actions on trees. Amalgams, HNN-extensions, graphs of groups, subgroup theorems forgroups acting on trees.

Quasi-isometries. Hyperbolic groups. Solution of the word and conjugacy problem for hyperbolicgroups.

If time allows: Small Cancellation Groups, Stallings Theorem, Boundaries.

Reading.

1. J.P. Serre, Trees (Springer Verlag 1978).

2. M. Bridson, A. Haefliger, Metric Spaces of Non-positive Curvature, Part III (Springer, 1999),Chapters I.8, III.H.1, III. Gamma 5.

3. H. Short et al., ‘Notes on word hyperbolic groups’, Group Theory from a Geometrical View-point, Proc. ICTP Trieste (eds E. Ghys, A. Haefliger, A. Verjovsky, World Scientific 1990)

available online at: http://www.cmi.univ-mrs.fr/ hamish/

4. C.F. Miller, Combinatorial Group Theory, notes:http://www.ms.unimelb.edu.au/ cfm/notes/cgt-notes.pdf.

Further Reading.

1. G. Baumslag, Topics in Combinatorial Group Theory (Birkhauser, 1993).

2. O. Bogopolski, Introduction to Group Theory (EMS Textbooks in Mathematics, 2008).

3. R. Lyndon, P. Schupp, Combinatorial Group Theory (Springer, 2001).

4. W. Magnus, A. Karass, D. Solitar,Combinatorial Group Theory: Presentations of Groups inTerms of Generators and Relations (Dover Publications, 2004).

5. P. de la Harpe, Topics in Geometric Group Theory, (University of Chicago Press, 2000).

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2.22 Representation Theory of Semisimple Lie Algebra —Prof Dan Ciub-otaru —HT

Recommended Prerequisites Past attendance at Lie algebras is recommended, but not re-quired. Past attendance at Introduction to Representation Theory is recommended as well, but notrequired.

Overview The representation theory of semisimple Lie algebras plays a central role in modernmathematics with motivation coming from many areas of mathematics and physics, for example, theLanglands program. The methods involved in the theory are diverse and include remarkable interac-tions with algebraic geometry, as in the proofs of the Kazhdan-Lusztig and Jantzen conjectures. Thecourse will cover the basics of finite dimensional representations of semisimple Lie algebras (e.g., theCartan-Weyl highest weight classification) in the framework of the largerBernstein-Gelfand-Gelfandcategory O.

Learning Outcomes The students will have developed a comprehensive understanding of the ba-sic concepts and modern methods in the representation theory of semisimple Lie algebras, includingthe classification of finite dimensional modules, the classification of objects in category O, characterformulas, Lie algebra cohomology and resolutions of finite dimensional modules.

Synopsis Universal enveloping algebra of a Lie algebra, Poincare-Birkhoff-Witt theorem, basicdefinitions and properties of representations of Lie algebras, tensor products. The example of sl(2):finite dimensional modules, highest weights. Category O: Verma modules, highest weight modules,infinitesimal characters and HarishChandras isomorphism, formal characters, contravariant (Shapo-valov) forms.

Finite dimensional modules of a semisimple Lie algebra: the Cartan-Weyl classification, Weyl char-acter formula, dimension formula, Kostants multiplicity formula, examples.

Homological algebra: Lie algebra cohomology, Bernstein-Gelfand-Gelfand resolution of finite dimen-sional modules, Ext groups in category O. Topics: applications, Botts dimension formula for Liealgebra cohomology groups, characters of the symmetric group (via Zelevinskys application of theBGG resolution to SchurWeyl duality).

Reading

• Lecture Notes.

• J. Bernstein, Lectures on Lie algebras, in Representation Theory, Complex Analysis, andIntegral Geometry (Springer 2012).

Further reading

• J. Humphreys, Representations of semisimple Lie algebras in the BGG category O(AMS, 2008).

• J. Humphreys, Introduction to Lie algebras and representation theory (Springer, 1997).

• W. Fulton, J. Harris, Representation Theory (Springer 1991).

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3 SECTION B: APPLICABLE THEORIES

3.1 Schedule I

3.2 Advanced Machine Learning — Dr Varun Kanade — 16 HT

Recommended Prerequisites: These are not hard pre-requisites. However, if you have not takena majority of the following courses at Oxford (or equivalent elsewhere), you will find the coursechallenging.

• Algorithms and Data Structures

• Machine Learning

• At least one of Probability and Computing, Computational Complextiy, Foundations of Com-puter Science

If you have any doubt about your background, you are encouraged to discuss with the instructorbefore the beginning of term or as soon thereafter as possible. Extra lectures will cover some basiccomputer science background for mathematics students and basic inequalities from probability thatwe will encounter frequently.

Overview Machine learning studies automatic methods for identifying patterns in complex dataand for making accurate predictions based on past observations. In this course, we develop rigorousmathematical foundations of machine learning, in order to provide guarantees about the behaviourof learning algorithms and also to understand the inherent difficulty of learning problems.

The course will begin by providing a statistical and computational toolkit, such as generalisationguarantees, fundamental algorithms, and methods to analyse learning algorithms. We will coverquestions such as when can we generalise well from limited amounts of data, how can we developalgorithms that are computationally efficient, and understand statistical and computational trade-offs in learning algorithms. We will also discuss new models designed to address relevant practicalquestions of the day, such as learning with limited memory, communication, privacy, and labelledand unlabelled data. In addition to core concepts from machine learning, we will make connectionsto principal ideas from information theory, game theory and optimisation.

This is an advanced course requiring a high level of mathematical maturity. It is expected thatstudents will have taken prior courses on machine learning, algorithms, and complexity.

Synopsis (Background) Probability (Background) Computational Complexity PAC learning, sam-ple complexity, computational complexity Proper vs improper learning Growth function, VC di-mension, sample complexity lower bounds Learning with membership queries, Angluin’s algorithmBoosting, weak learning, Adaboost, Margin Bounds Learning in the presense of noise, SQ Learning,Agnostic Learning Learning Parities with noise, Learning Juntas Complexity-theoretic and crypto-graphic hardness of learning Online learning, perceptron and winnow, attribute efficient learningLearning from expert advice, multi-armed bandit problems (Time permitting) Learning discretedistributions (Time permitting) Privacy preserving learning, adaptive data analysis

Reading

1. Trevor Hastie, Robert Tibshirani, Jerome Friedman. The Elements of Statistical Learning.Springer 2013

2. Shai Shalev-Shwartz and Shai Ben-David. Understanding Machine Learning: From Theory toAlgorithms. Cambridge University Press, 2014.

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3.3 Applied Probability — Prof Chleboun — MT

This course is intended to show the power and range of probability by considering real examplesin which probabilistic modelling is inescapable and useful. Theory will be developed as required todeal with the examples.

Synopsis Poisson processes and birth processes. Continuous-time Markov chains. Transitionrates, jump chains and holding times. Forward and backward equations. Class structure, hittingtimes and absorption probabilities. Recurrence and transience. Invariant distributions and limitingbehaviour. Time reversal. Renewal theory. Limit theorems: strong law of large numbers, stronglaw and central limit theorem of renewal theory, elementary renewal theorem, renewal theorem, keyrenewal theorem. Excess life, inspection paradox.

Applications in areas such as: queues and queueing networks - M/M/s queue, Erlang’s formula,queues in tandem and networks of queues, M/G/1 and G/M/1 queues; insurance ruin models;applications in applied sciences.

Reading

1. J. R. Norris, Markov Chains (Cambridge University Press, 1997).

2. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes (3rd edition, OxfordUniversity Press, 2001).

3. G. R. Grimmett and D. R. Stirzaker, One Thousand Exercises in Probability (Oxford Univer-sity Press, 2001).

4. S. M. Ross, Introduction to Probability Models (4th edition, Academic Press, 1989).

5. D. R. Stirzaker: Elementary Probability (2nd edition, Cambridge University Press, 2003).

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3.4 Categories, Proofs and Processes — Dr Kishida — 20 lectures +extra reading MT

Recommended Prerequisites Some familiarity with basic discrete mathematics: sets, functions,relations, mathematical induction. Basic familiarity with logic: propositional and predicate calculus.Some first acquaintance with abstract algebra: vector spaces and linear maps, and/or groups andgroup homomorphisms. Some familiarity with programming, particularly functional programming,would be useful but is not essential.

Overview Category Theory is a powerful mathematical formalism which has become an importanttool in modern mathematics, logic and computer science. One main idea of Category Theory is tostudy mathematical ‘universes’, collections of mathematical structures and their structure-preservingtransformations, as mathematical structures in their own right, i.e. categories - which have theirown structure-preserving transformations (functors). This is a very powerful perspective, whichallows many important structural concepts of mathematics to be studied at the appropriate levelof generality, and brings many common underlying structures to light, yielding new connectionsbetween apparently different situations.Another important aspect is that set-theoretic reasoning with elements is replaced by reasoning interms of arrows. This is more general, more robust, and reveals more about the intrinsic structureunderlying particular set-theoretic representations.Category theory has many important connections to logic. We shall in particular show how itilluminates the study of formal proofs as mathematical objects in their own right. This will involvelooking at the Curry-Howard isomorphism between proofs and programs, and at Linear Logic, aresource-sensitive logic. Both of these topics have many important applications in Computer Science.Category theory has also deeply influenced the design of modern (especially functional) programminglanguages, and the study of program transformations. One exciting recent development we will lookat will be the development of the idea of coalgebra, which allows the formulation of a notion ofcoinduction, dual to that of mathematical induction, which provides powerful principles for definingand reasoning about infinite objects.This course will develop the basic ideas of Category Theory, and explore its applications to the studyof proofs in logic, and to the algebraic structure of programs and programming languages.Remark: It is recommended that students who intend to write their MSc thesis in the QuantumGroup at the Department of Computer Science take this course and additionally the QuantumComputer Science course in Hilary term.

Learning Outcomes

• To master the basic concepts and methods of categories.

• To understand how category-theory can be used to structure mathematical ideas, with the con-cepts of functoriality, naturality and universality; and how reasoning with objects and arrowscan replace reasoning with sets and elements. To learn the basic ideas of using commutativediagrams and unique existence properties.

• To understand the connections between categories and logic, focussing on structural prooftheory and the Curry-Howard isomorphism.

• To understand how some basic forms of computational processes can be modelled with cate-gories.

Synopsis

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• Introduction to category theory. Categories, functors, natural transformations. Isomorphisms.monics and epics. Products and coproducts. Universal constructions. Cartesian closed cate-gories. Symmetric monoidal closed categories. The ideas will be illustrated with many exam-ples, from both mathematics and Computer Science.

• Introduction to structural proof theory. Natural deduction, simply typed lambda calculus, theCurry-Howard correspondence. Introduction to Linear Logic. The connection between logicand categories.

• Further topics in category theory. Algebras and coalgebras. Connections to programming(structural recursion and corecursion).

A. Categories

• Background and definition

• Monics, epics, isomorphisms

• Products and coproducts

• Limits and colimits

• Functors and natural transformations

• Universal arrows and adjunctions

• Cartesian closed categories

B. Connections with logic

• Natural deduction, lambda calculus and Curry-Howard isomorphism

• Gentzen sequent calculus and linear logic

• Symmetric monoidal closed categories and categorical semantics of linear logic

• Algebras

• Coalgebras

• Topoi

Reading List Slides will be provided. The standard reference is

• Abramsky and Tzevelekos, ”Introduction to Categories and Categorical Logic”,http://web.comlab.ox.ac.uk/people/Bob.Coecke/AbrNikos.pdf

The following books provide useful background reading.

• B.C. Pierce, Basic Category Theory for Computer Science, MIT Press (1991)

• F.W. Lawvere, S.H. Schanuel, Conceptual Mathematics, Cambridge University Press (1997)

• S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer (1998)

• M. Barr, C. Wells, Category Theory for Computer Science, 2nd ed., Prentice Hall (1995)

• J.-Y. Girard, Y. Lafont, P. Taylor, Proofs and Types, http://www.paultaylor.eu/stable/prot.pdf

Of these, the book by Pierce provides a very accessible and user-friendly first introduction tothe subject (though we will cover more topics in the course).

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3.5 Communication Theory — Prof Harold Oberhauser — 16 MT

Recommended Prerequisites: Part A Probability would be helpful, but not essential.

Overview The aim of the course is to investigate methods for the communication of informationfrom a sender, along a channel of some kind, to a receiver. If errors are not a concern we are interestedin codes that yield fast communication; if the channel is noisy we are interested in achieving bothspeed and reliability. A key concept is that of information as reduction in uncertainty. The highlightof the course is Shannon’s Noisy Coding Theorem.

Learning Outcomes

(i) Know what the various forms of entropy are, and be able to manipulate them.

(ii) Know what data compression and source coding are, and be able to do it.

(iii) Know what channel coding and channel capacity are, and be able to use that.

Synopsis Uncertainty (entropy); conditional uncertainty; information. Chain rules; relative en-tropy; Gibbs’ inequality; asymptotic equipartition and typical sequences. Instantaneous and uniquelydecipherable codes; the noiseless coding theorem for discrete memoryless sources; constructing com-pact codes.

The discrete memoryless channel; decoding rules; the capacity of a channel. The noisy codingtheorem for discrete memoryless sources and binary symmetric channels.

Extensions to more general sources and channels.

Reading

1. D. MacKay, Information Theory, Inference, and Learning Algorithms (Cambridge, 2003). [Canbe seen at: http://www.inference.phy.cam.ac.uk/mackay/itila. Do not infringe the copyright!]

2. T. Cover and J. Thomas, Elements of Information Theory (Wiley, 1991), Chapters 1–5, 8.

Further Reading

1. R. B. Ash, Information Theory (Dover, 1990).

2. D. J. A. Welsh, Codes and Cryptography (Oxford University Press, 1988), Chapters 1–3, 5.

3. G. Jones and J. M. Jones, Information and Coding Theory (Springer, 2000), Chapters 1–5.

4. Y. Suhov & M. Kelbert, Information Theory and Coding by Example (Cambridge UniversityPress, not yet published - available at the end of 2013), Relevant examples.

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3.6 Computer-Aided Formal Verification — Dr Alessandro Abate —16MT

Overview This course introduces the fundamentals of computer-aided formal verification. Computer-aided formal verification aims to improve the quality of digital systems by using logical reasoning,supported by software tools, to analyse their designs. The idea is to build a mathematical model ofa system and then try to prove properties of it that validate the system’s correctness ? or at leasthelp discover subtle bugs. The proofs can be millions of lines long, so specially-designed computeralgorithms are used to search for and check them.

Learning Outcomes This course provides a survey of several major software-assisted verificationmethods, covering both theory and practical applications. The aim is to familiarise students withthe mathematical principles behind current verification technologies and give them an appreciationof how these technologies are used in industrial system design today.

Synopsis

1. Introduction.

2. Modelling sequential systems, Kripke structures.

3. Temporal logic: LTL, CTL*, and CTL.

4. Specifying systems with temporal logic.

5. Reachability calculations, model checking.

6. Binary Decision Diagrams (BDDs).

7. Algorithms over BDDs.

8. Combinational equivalence checking.

9. Symbolic model checking.

10. Propositional SAT.

11. Model Checking with SAT.

12. Abstraction Refinement.

13. Decision procedures.

14. Decision procedures in Model Checking.

15. Practical, industrial-scale hardware verification.

16. Computer-aided software verification.

Syllabus Introduction to formal hardware verification. Binary Decision Diagrams and their usein combinational equivalence checking. Modelling sequential systems; Kripke structures. Specifyingsystems with temporal logic; CTL*, CTL and LTL. Reachability and symbolic model checking. Newmodel checking approaches based on algorithms for Boolean satisfiability. Automatic abstractionrefinement. Decision procedures and their use in combination with model checking. Practical,industrial-scale hardware verification. Current approaches to computer-aided software verification.

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Reading The lectures will be supplemented with notes and pointers to published articles in thefield. The following may be helpful for reference or further reading on specific topics.

Surveys

1. Formal Verification in Hardware Design: A Survey, by C. Kern and M. R. Greenstreet, ACMTransactions on Design Automation of Systems , vol. 4 (April 1999), pp. 123-193.

2. A Survey of Automated Techniques for Formal Software Verification , by D’Silva et al., IEEETransactions on Computer-Aided Design of Integrated Circuits and Systems (TCAD), 2008(http://www.kroening.com/publications/view−publications−dkw2008.html)

Temporal Logic and Model Checking

1. From Philosophical to Industrial Logics, by M. Vardi, ICLA 2009 (http://www.cs.ox.ac.uk/teaching/courses/2014-2015/computeraidedverification/),

Binary Decision Diagrams and SAT

1. An Introduction to Binary Decision Diagrams, by Henrik Reif Andersen, Lecture Notes (Tech-nical University of Denmark, October 1997)(https://www.cs.ox.ac.uk/files/4298/bdd98.pdf).

2. Formal Hardware Verification with BDDs: An Introduction, by Alan J. Hu, IEEE PacificRim Conference on Communications, Computers, and Signal Processing (1997), pp. 677-682.(https://www.cs.ox.ac.uk/files/4309/97H1.pdf)

3. Chapter 2 in Decision Procedures, by Daniel Kroening and Ofer Strichman, Springer, 2008(http://www.decision-procedures.org/)

4. Handbook of Satisfiability, Biere, Heule, Van Maaren, Walsh, IOS Press 2009.

BDDs and Model Checking

1. Logic in Computer Science: Modelling and reasoning about systems, by Michael Huth andMark Ryan (Cambridge University Press, 2000).

2. Model Checking, by Edmund M. Clarke, Jr., Orna Grumberg, and Doron A. Peled, Secondprinting (The MIT Press, 2000).

3. Concepts, Algorithms, and Tools for Model Checking, Unpublished lecture notes by J.-P. Ka-toen, 1998. (https://www.cs.ox.ac.uk/files/4297/katoen.pdf)

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3.7 Concurrency — Dr Thomas Gibson-Robson — 16 lectures + extrareading HT

Overview Computer networks, multiprocessors and parallel algorithms, though radically differ-ent, all provide examples of processes acting in parallel to achieve some goal. All benefit fromthe efficiency of concurrency yet require careful design to ensure that they function correctly. Theconcurrency course introduces the fundamental concepts of concurrency using the notation of Com-municating Sequential Processes. By introducing communication, parallelism, deadlock, live-lock,etc., it shows how CSP represents, and can be used to reason about, concurrent systems. Studentsare taught how to design interactive processes and how to modularise them using synchronisation.One important feature of the module is its use of both algebraic laws and semantic models to reasonabout reactive and concurrent designs. Another is its use of FDR to animate designs and verify thatthey meet their specifications.

Learning Outcomes At the end of the course the student should:

• understand some of the issues and difficulties involved in Concurrency

• be able to specify and model concurrent systems using CSP

• be able to reason about CSP models of systems using both algebraic laws and semantic models

• be able to analyse CSP models of systems using the model checker FDR

Syllabus Deterministic processes: traces, operational semantics; prefixing, choice, concurrencyand communication. Nondeterminism: failures and divergences; nondeterministic choice, hiding andinterleaving. Further operators: pipes and (time permitting) sequential composition. Refinement,specification and proof. Process algebra: equational and inequational reasoning.

Synopsis

• Processes and observations of processes; point synchronisation, events, alphabets. Sequentialprocesses: prefixing, choice, nondeterminism. Operational semantics; traces; algebraic laws.[3]

• Recursion. Complete partial orders and fixed points as a means of explaining recursion; ap-proximation, limits, least fixed points; guardedness and unique fixed points. [1]

• Concurrency. Hiding. Renaming. [3]

• Non-deterministic behaviours, refusals, failures; the determinism ordering. [2]

• Hiding and divergence, the failures-divergences model. [1]

• Specification and correctness. [2]

• Communication, pipes, buffers. Sequential composition. [2]

• Case study. [2]

• + extra reading: Chapters 9 and 10 of “Understanding Concurrent Systems”

Reading List

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Course Text

• A. W. Roscoe, Understanding Concurrent Systems, Chapters 1-8, Springer 2010

Alternatives

• A. W. Roscoe, The Theory and Practice of Concurrency, Chapters 1-7, Prentice-Hall Interna-tional, 1997. (http://www.cs.ox.ac.uk/oucl/work/bill.roscoe/publications/68b.pdf.)

• C. A. R. Hoare, Communicating Sequential Processes, Prentice-Hall International, 1985, http://www.usingcsp.com.(http://www.usingcsp.com)

• S. A. Schneider, Concurrent and Real-time Systems, Chapters 1-8, Wiley, 2000. (http://www.computing.surrey.ac.uk/personal/st/S.Schneider/books/CRTS.pdf)

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3.8 Foundations of Computer Science — Prof Paul Goldberg — 16MT

Overview Computer scientists need to understand what it means for a problem to be determinableby a computer, what it means for a problem to be efficiently determinable by a computer, and howto reason in a semi-automated and automated fashion about computer programs and the structuresthey manipulate. The purpose of this course is to introduce students to the theoretical foundationsof computer science. It is intended both for students who have a degree in computer science and alsofor students with a good theoretical background (e.g. a degree in mathematics) but no exposure totheoretical computer science.

Students taking this course will gain background knowledge that will be useful in the course on:

• Theory of Data & Knowledge Bases

• Automata, Logics & Games

• Software Verification

• Categories, Proofs & Processes

• Game Semantics

• Computer-Aided Formal Verification

• Lambda Calculus & Types

• Logic of Multi-Agent Information Flow

Learning Outcomes At the end of this course, the student should be able to:

1. Describe in detail what is meant by a finite state automaton, a context-free grammar, and aTuring machine, and calculate the behaviour of simple examples of these devices.

2. Design machines of these types to carry out simple computational tasks.

3. Reason about the capabilities of standard machines, and demonstrate that they have limita-tions.

4. Describe precisely what it means for a problem to be in the classes P,NP, and PSPACE, andwhat it means to be complete for a class

5. Classify problems into appropriate complexity classes, including P, NP and PSPACE, and usethis information effectively.

6. Understand the syntax and semantics of propositional logic.

7. Understand the satisfiability problem for propositional logic and its connection with NP hard-ness.

8. Understand first-order predicate logic, along with the complexity/computability of the associ-ated satisfaction and satisfiability problems.

Syllabus Finite state machines. Reduction of non-deterministic finite automata to determinis-tic finite automata. Regular languges and their closure properties. Regular expressions. Inter-translations between regular expressions and NFA. Context-free grammars and pushdown automata.Intuitive notion of computability. Church’s Thesis. Turing machines and its expressive power. Uni-versal Turing machines. Undecidable problems. Diagonalization and the Halting Problem. Deter-ministic complexity classes. P, EXPTIME and the Hierarchy Theorem. NP and NP-completeness.Space complexity. Propositional logic. Truth tables. Propositional Logic and NP-completeness.Proof systems for Propositional Logic. Syntax and semantics of first-order logic. Complexity offirst-order logic.

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Reading List

1. M. Sipser, Introduction to the Theory of Computation, PWS Publishing Company, January1997. (Primary text).

2. M. Huth and M. Ryan, Logic in Computer Science: Modelling and Reasoning about Systems,2nd Editions. Cambridge University Press, 2004.

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3.9 Graph Theory — Prof. Oliver Riordan — 16 MT

Recommended Prerequisites: Part A Graph Theory is recommended.

Overview Graphs (abstract networks) are among the simplest mathematical structures, but nev-ertheless have a very rich and well-developed structural theory. Since graphs arise naturally in manycontexts within and outside mathematics, Graph Theory is an important area of mathematics, andalso has many applications in other fields such as computer science.

The main aim of the course is to introduce the fundamental ideas of Graph Theory, and some of thebasic techniques of combinatorics.

Learning Outcomes The student will have developed a basic understanding of the properties ofgraphs, and an appreciation of the combinatorial methods used to analyze discrete structures.

Synopsis Introduction: basic definitions and examples. Trees and their characterization. Eulercircuits; long paths and cycles. Vertex colourings: Brooks’ theorem, chromatic polynomial. Edgecolourings: Vizing’s theorem. Planar graphs, including Euler’s formula, dual graphs. Maximumflow - minimum cut theorem: applications including Menger’s theorem and Hall’s theorem. Tutte’stheorem on matchings. Extremal Problems: Turan’s theorem, Zarankiewicz problem, Erdos-Stonetheorem.

Reading

1. B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics 184 (Springer-Verlag,1998)

Further Reading

1. J. A. Bondy and U. S. R. Murty, Graph Theory: An Advanced Course, Graduate Texts inMathematics 244 (SpringerVerlag, 2007).

2. R. Diestel, Graph Theory, Graduate Texts in Mathematics 173 (third edition, Springer-Verlag,2005).

3. D. West, Introduction to Graph Theory, Second edition, (PrenticeHall, 2001).

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3.10 Introduction to Cryptology — Dr Ali El Kaafarani— MT

Prerequisites Elementary number theory, introductory probability, basic concepts of computerscience, and algebra 1.

Synopsis This course is an introduction to the foundations of modern cryptography. It will coverdifferent existing security definitions, proofs by reductions, private/public key cryptosystems andtheir underlying computational hardness assumptions.

Reading List

• Katz, Jonathan, and Yehuda Lindell. Introduction to modern cryptography. CRC Press, 2014.

• Smart, Nigel Paul. Cryptography: an introduction. New York: McGrawHill, 2003.

• Galbraith, Steven D. Mathematics of public key cryptography. Cambridge University Press,2012.

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3.11 Quantum Computer Science—Prof Bob Coecke —24 Lectures MT

Prerequisites We do not assume any prior knowledge of quantum mechanics. However, a solidunderstanding of basic linear algebra (finite-dimensional vector spaces, matrices, eigenvectors andeigenvalues, linear maps etc.) is required as a pre-requisite. The course notes and the slides containan overview of this material, so we advise students with a limited background in linear algebra toconsult the course notes before the course starts.

Overview Both physics and computer science have been very dominant scientific and technologi-cal disciplines in the previous century. Quantum Computer Science aims at combining both and maycome to play a similarly important role in this century. Combining the existing expertise in bothfields proves to be a non-trivial but very exciting interdisciplinary journey. Besides the actual issueof building a quantum computer or realising quantum protocols it involves a fascinating encounterof concepts and formal tools which arose in distinct disciplines.

This course provides an interdisciplinary introduction to the emerging field of quantum computerscience, explaining basic quantum mechanics (including finite dimensional Hilbert spaces and theirtensor products), quantum entanglement, its structure and its physical consequences (e.g. non-locality, no-cloning principle), and introduces qubits. We give detailed discussions of some key algo-rithms and protocols such as Grover’s search algorithm and Shor’s factorisation algorithm, quantumteleportation and quantum key exchange.

At the same time, this course provides a introduction to diagrammatic reasoning. As an entirelydiagrammatic presentation of quantum theory and its applications, this course is the first of its kind.

Learning outcomes Both physics and computer science have been very dominant scientific andtechnological disciplines in the previous century. Quantum Computer Science aims at combiningboth and may come to play a similarly important role in this century. Combining the existing exper-tise in both fields proves to be a non-trivial but very exciting interdisciplinary journey. Besides theactual issue of building a quantum computer or realising quantum protocols it involves a fascinatingencounter of concepts and formal tools which arose in distinct disciplines.

This course provides an interdisciplinary introduction to the emerging field of quantum computerscience, explaining basic quantum mechanics (including finite dimensional Hilbert spaces and theirtensor products), quantum entanglement, its structure and its physical consequences (e.g. non-locality, no-cloning principle), and introduces qubits. We give detailed discussions of some key algo-rithms and protocols such as Grover’s search algorithm and Shor’s factorisation algorithm, quantumteleportation and quantum key exchange.

At the same time, this course provides a introduction to diagrammatic reasoning. As an entirelydiagrammatic presentation of quantum theory and its applications, this course is the first of its kind.

Syllabus TBC

Reading list Lecture notes, slides and additional handouts will be provided as the course pro-gresses.

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3.12 Schedule II

3.13 Automata, Logic and Games — Prof Luke Ong —24 lectures +reading MT

Recommended Prerequisites: Knowledge of first-order predicate calculus will be assumed. Fa-miliarity with the basics of Finite Automata Theory and Computability (for example, as coveredby the course, Models of Computation) and Complexity Theory would be very helpful. Candidateswho do not have the required background are expected to have taken the course, Introduction tothe Foundations of Computer Science.

Overview To introduce the mathematical theory underpinning the Computer-Aided Verificationof computing systems. The main ingredients are:

The main ingredients are:

• Automata (on infinite words and trees) as a computational model of state-based systems.

• Logical systems (such as temporal and modal logics) for specifying operational behaviour.

• Two-person games as a conceptual basis for understanding interactions between a system andits environment.

Learning Outcomes At the end of the course students will be able to:

1. Describe in detail what is meant by a Buchi automaton, and the languages recognised bysimple examples of Buchi automata.

2. Use linear-time temporal logic to describe behaviourial properties such as recurrence and pe-riodicity, and translate LTL formulas to Buchi automata.

3. Use S1S to define omega-regular languages, and give an account of the equivalence betweenS1S definability and Buchi recognisability.

4. Explain the intuitive meaning of simple modal mu-calculus formulas, and describe the corre-spondence between property-checking games and modal mu-calculus model checking.

Synopsis/Syllabus

• Automata on infinite words. Buchi automata: Closure properties. Determinization and Rabinautomata.

• Nonemptiness and Nonuniversality problems for Buchi automata.

• Linear temporal logic and alternating Buchi automata.

• Modal mu-calculus: Fundamental Theorem, decidability and finite model property. ParityGames and the Model-Checking Problem: memoryless determinacy, algorithmic issues.

• Monadic Second-order Logic and its relationship with the modal mu-calculus.

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Reading List Selected parts from:

• J. Bradfield and C. P. Stirling. Modal logics and mu-calculi. In J. Bergstra, A. Ponse, and S.Smolka, editors, Handbook of Process Algebra, pages 293-332. Elsevier, North-Holland, 2001.

• B. Khoussainov and A. Nerode. Automata Theory and its Applications. Progress in ComputerScience and Applied Logic, Volume 21. Birkhauser, 2001.

• C. P. Stirling. Modal and Temporal Properties of Processes. Texts in Computer Science.Springer-Verlag, 2001.

• W. Thomas. Languages, Automata and Logic. In G. Rozenberg and A. Salomaa, editors,Handbook of Formal Languages, volume 3. Springer-Verlag, 1997.

• M. Y. Vardi. An automata — theoretic approach to linear temporal logic. In Logics for Con-currency: Structure versus Automata, ed. F. Moller and G. Birtwistle, LNCS vol. 1043, pp.238–266, Springer–Verlag, 1996.

The only copy of the Vardi book is in the RSL on open shelves. However, the article in questionis available in pdf form online at this address:http://folli.loria.fr/cds/1998/pdf/degiacomo-nardi/vardi.pdf

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3.14 Advanced Cryptology — Dr Christophe Petit — HT

Prerequisites The introductory Cryptography Course of Part A, and all its pre-requisites. Awillingness to write some simple programs in Sage, Magma or any computer algebra language ishighly desirable

Overview/Synopsis We will build on the introductory course on Cryptography and cover

• Advanced cryptographic primitives and protocols, such as zero knowledge proofs of knowledgeor secure multi-party computation

• Alternative ways to build standard primitives and protocols based on elliptic curves, latticeproblems, syndrome decoding, computational group theory problems or polynomial systemsolving problems.

• Advanced cryptanalysis techniques, such as the function field sieve, side-channel attacks orquantum attacks

Reading List

• Antoine Joux. Algorithmic Cryptanalysis.

• Ian Blake and Gadiel Serousi and Nigel Smart. Elliptic Curve Cryptography.

• Steven Galbraith. The Mathematics of Public Key Cryptography.

• Daniele Micciancio and Shafi Goldwasser. Complexity of Lattice Problems: A CryptographicPerspective.

• Stefan Mangard and Elisabeth Oswald and Thomas Popp. Power Analysis Attacks - Revealingthe Secrets of Smartcards.

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3.15 Categorical Quantum Mechanics–Dr Dan Marsden & Jamie Vicary—16 Lectures HT

Prerequisites Ideal foundations for this course are ‘Categories, Proofs and Processes”, and “Quan-tum Computer Science”. Students who have not taken these courses will need to be familiar withbasic topics from category theory and linear algebra, including categories, functors, natural trans-formations, vector spaces, Hilbert spaces and the tensor product. Chapter zero in the lecture notesbriefly recall this background information

Students wishing to do their dissertation with the Quantum Group are expected to sit this course,as well as the two mentioned above.

Overview This course gives an introduction to the theory of monoidal categories, and investigatestheir application to quantum computer science. We will cover the following topics, illustratingapplications throughout to quantum computation:

• Monoidal categories, the graphical calculus, coherence

• Linear structure on categories, biproducts, dagger-categories

• Dual objects, traces, entangled states

• Monoids and comonoids, copying and deleting

• Frobenius structures, normal forms, characterizing bases

• Complementarity, bialgebras, Hopf algebras

• Completely positive maps, the CP construction

• Bicategories and their graphical calculus

To complement the theoretical side of the course, we will also learn about the proof assistant Globu-lar, and use it to formalize some of the results. There will be a practical session using the tool laterin the term (date TBA).//

This course can currently only be taken by students enrolled on the DPhil, MFoCS or MSc incomputer science programmes. However, everyone is welcome to sit in and follow the lectures.//

Reading list The notes and slides for the entire course can be downloaded using the followinglink: http://http://www.cs.ox.ac.uk/teaching/courses/2016-2017/cqm/

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3.16 Combinatorics — Prof. Alex Scott — 16MT

Recommended Prerequisites Part B Graph Theory is helpful, but not required.

Overview An important branch of discrete mathematics concerns properties of collections of sub-sets of a finite set. There are many beautiful and fundamental results, and there are still many basicopen questions. The aim of the course is to introduce this very active area of mathematics, withmany connections to other fields.

Learning Outcomes The student will have developed an appreciation of the combinatorics offinite sets.

Synopsis Chains and antichains. Sperner’s Lemma. LYM inequality. Dilworth’s Theorem.

Shadows. Kruskal-Katona Theorem.

Intersecting families. Erdos-Ko-Rado Theorem. Cross-intersecting families.

VC-dimension. Sauer-Shelah Theorem.

t-intersecting families. Fisher’s Inequality. Frankl-Wilson Theorem. Application to Borsuk’s Con-jecture.

Combinatorial Nullstellensatz.

Reading

1. Bela Bollobas, Combinatorics, CUP, 1986.

2. Stasys Jukna, Extremal Combinatorics, Springer, 2007

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3.17 Computational Algebraic Topology — Prof Ulrike Tillmann & ProfSamson Abramsky 16HT

Prerequisites Some familiarity with the main concepts from algebraic topology, homological al-gebra and category theory will be helpful.

Overview Ideas and tools from algebraic topology have become more and more important incomputational and applied areas of mathematics. This course will provide at the masters level anintroduction to the main concepts of (co)homology theory, and explore areas of applications in dataanalysis and in foundations of quantum mechanics and quantum information.

Learning outcomes Students should gain a working knowledge of homology and cohomology ofsimplicial sets and sheaves, and improve their geometric intuition. Furthermore, they should gainan awareness of a variety of application in rather different, research active fields of applications withan emphasis on data analysis and contextuality.

Synopsis The course has two parts. The first part will introduce students to the basic conceptsand results of (co)homology, including sheaf cohomology. In the second part applied topics are in-troduced and explored.

Core: Homology and cohomology of chain complexes. Algorithmic computation of boundary maps(with a view of the classification theorm for finitely generated modules over a PID). Chain homotopy.Snake Lemma. Simplicial complexes. Other complexes (Delaunay, Cech). Mayer-Vietoris sequence.Poincare duality. Alexander duality. Acyclic carriers. Discrete Morse theory. (6 lectures)

Topic A: Persistent homology: barcodes and stability, applications todata analysis, generalisations.(4 lectures)

Topic B:Sheaf cohomology and applications to quantum non-locality and contextuality.Sheaf-theoreticrepresentation of quantum non-locality and contextuality asobstructions to global sections. Coho-mological characterizations and proofs of contextuality.(6 lectures)

Reading List H. Edelsbrunner and J.L. Harer, Computational Topology -An Introduction,AMS(2010).

See also, U. Tillmann, Lecture notes for CAT 2012, in http://people.maths.ox.ac.uk/tillmann/CAT.html

Topic A:

G. Carlsson, Topology and data, Bulletin A.M.S.46 (2009), 255-308.

H. Edelsbrunner, J.L. Harer, Persistent homology: A survey, Contemporary Mathematics 452A.M.S. (2008), 257-282.

S. Weinberger, What is ... Persistent Homology?, Notices A.M.S. 58 (2011), 36-39.

P. Bubenik, J. Scott, Categorification of Persistent Homology, Discrete Comput. Geom. (2014),600–627.

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Topic B:

S. Abramsky and Adam Brandenburger, The Sheaf-Theoretic Structure Of Non-Locality and Con-textuality. In New Journal of Physics, 13(2011), 113036, 2011.

S. Abramsky and L. Hardy, Logical Bell Inequalities, Phys. Rev. A 85, 062114 (2012).

S. Abramsky, S. Mansfield and R. Soares Barbosa, The Cohomology of Non-Locality and Contex-tuality, in Proceedings of Quantum Physics and Logic 2011, Electronic Proceedings in TheoreticalComputer Science, vol. 95, pages 1–15, 2012.

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3.18 Computational Number Theory — Prof Richard Pinch — Readingcourse TT

Prerequisites Despite the sophistication of this course the only pre-requisites are parts of a stan-dard elementary number theory course: Euclid’s algorithm, Quadratic residues, The law of reci-procity for Legendre and Jacobi symbols, Fermat’s theorem, primitive roots.

Aims and Synopsis This course aims to describe the algorithms used for efficient practical com-putations in number theory. It is based on recent research papers, along with parts of the text byCohen.

The course covers: The Euclidean Algorithm, computation of powers and square roots moduloprimes; the arithmetic of elliptic curves over finite fields; lattices and the LLL reduction algorithm;factorization algorithms.

Reading Henri Cohen, A course in computational algebraic number theory, Springer-Verlag (1993).

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3.19 Computational Game Theory — Prof Edith Elkind & Prof MikeAldridge — MT

Overview Game theory is the mathematical theory of strategic interactions between self-interestedagents. Game theory provides a range of models for representing strategic interactions, and associ-ated with these, a family of solution concepts, which attempt to characterise the rational outcomesof games. Game theory is important to computer science for several reasons: First, interaction isa fundamental topic in computer science, and if it is assumed that system components are self-interested, then the models and solution concepts of game theory seems to provide an appropriateframework with which to model such systems. Second, the problem of computing with the solutionconcepts proposed by game theory raises important challenges for computer science, which test theboundaries of current algorithmic techniques. This course aims to introduce the key concepts ofgame theory for a computer science audience, emphasising both the applicability of game theoreticconcepts in a computational setting, and the role of computation in game theoretic problems. Thecourse assumes no prior knowledge of game theory.

Aims The aims of this module are threefold:

• to introduce the key models and solution concepts of non-cooperative and cooperative gametheory;

• to introduce the issues that arise when computing with game theoretic solution concepts, andthe main approaches to overcoming these issues, and to illustrate the role that computationplays in game theory;

• to introduce a research-level topic in computational game theory.

Outline Syllabus

• 1. Preferences, Utility, and Goals:

• Preference relations and their interpretation; utility as a numeric model of preference.

• Decision-making under uncertainty: preferences over lotteries; von Neumann and Morgensternutility functions; expected utility and expected utility maximisation.

• Paradoxes of expected utility maximisation; framing effects and prospect theory.

• Compact representations for preference relations (e.g., CP-NETS).

• Dichotomous preferences and goals. Representations for specifying goals (e.g., weighted for-mula representations for combinatorial domains); expressiveness and computational issues.

• 2. Strategic Form Non-Cooperative Games:

• The basic model; solution concepts: pure strategy Nash equilibrium; dominant strategies;notable games (e.g., Prisoner’s Dilemma; Game of Chicken; Stag Hunt); coordination gamesand focal points; complexity of pure strategy Nash equilibrium.

• Measuring social welfare; utilitarian social welfare; egalitarian social welfare.

• Mixed strategies; Nash’s theorem; -Nash equilibrium.

• Computing mixed strategy Nash equilibria: the Lemke-Howson algorithm.

• Zero sum games; the Minimax Theorem.

• Compact representations for strategic form games; Boolean games; congestion games.

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• 3. Iterated Games:

• Finitely repeated games and backward induction.

• Infinitely repeated games; measuring utility over infinite plays; modelling strategies as fi-nite state machines with output (Moore machines); the folk theorems; implications of usingbounded automata to model strategies.

• Iterated Boolean games.

• Axelrod’s tournament; the Hawk-Dove game; evolutionary game theory; evolutionarily stablestrategies.

• 4. Extensive Form Non-Cooperative Games:

• Extensive form games of perfect information; Zermelo’s algorithm and backward induction;P-completeness of Zermelo’s algorithm; subgame perfect equilibrium.

• Win-lose games; Zermelo’s theorem.

• Compact representations for extensive form games; the PEEK games and EXPTIME-completenessresults; the Game Description Language (GDL).

• Imperfect information games; information sets; solution concepts for imperfect informationgames.

• Compact representations for imperfect information games; PEEK games with incomplete in-formation; undecidability results.

• 5. Cooperative Games:

• Transferable utility (TU) characteristic function games, the basic model, stability & fairnesssolution concepts, the core, the kernel, the Nucleolus, the cost of stability, the Shapley value,the Banzhaf index.

• Compact representations for TU games; induced subgraph representation; marginal contribu-tion nets.

• Simple TU games; swap and trade robustness; weighted voting games; vector weighted votinggames; network flow games.

• NTU games and representations for them; hedonic games.

• Coalition structure formation; exact and approximation algorithms.

• 6. Social Choice:

• social choice and social welfare functions;

• Condorcet’s paradox;

• desirable properties of social choice procedures (Pareto condition, independence of irrelevantalternatives);

• popular voting procedures (Borda, etc);

• Arrow’s theorem;

• strategic manipulation of voting procedures and associated impossibility results (Gibbard-Satterthwaite theorem);

• complexity of manipulation for voting protocols.

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• 7. Research Topics: In the final part of the course we will focus on a research topic from thecontemporary literature; possible examples include:

• algorithmic mechanism design & auctions;

• selfish routing, the price of anarchy;

• security games.

Learning Outcomes Upon completing this module, a student will:

• 1. understand the key concepts of preferences, utility, and decision-making under certainty anduncertainty, and the key computational issues in representing and manipulating representationsof preferences and utility;

• 2. understand and be able to apply the key models and solution concepts of non-cooperativegame theory, including both strategic form and extensive form games, and the key computa-tional issues that arise when applying these models;

• 3. understand and be able to apply the key models and solution concepts of cooperative gametheory, including TU and NTU games, and the

• 4. understand a contemporary research-level topic at the intersection between game theoryand computer science

Recommended Reading

1. Michael Maschler, Eilon Solan, Shmuel Zamir. Game Theory, Cambridge UP, 2013. The bestcontemporary overview of game theory.

2. Martin J. Osborne and Ariel Rubinstein. A Course in Game Theory. MIT Press, 1994. Anexcellent introduction to game theory, freely available from:

http://books.osborne.economics.utoronto.ca

3. Y. Shoham and K. Leyton-Brown. Multiagent Systems. Cambridge UP, 2009. Freely availablefrom: http://www.masfoundations.org/

4. G. Chalkiadakis, E. Elkind, and M. Wooldridge. Computational Aspects of Cooperative GameTheory. Morgan & Claypool, 2011. The book for cooperative games.

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3.20 Distributional Models of Meaning — Prof Bob Coecke— Readingcourse HT

Prerequisites This course will make heavy use of linear algebra, so students are expected to becomfortable with vectors and matrices. Some knowledge of discrete mathematics (sets, groups), ofcategory theory (categories, monoidal categories, functors), and of computational linguistics (phrase-structure grammar, parsing, semantics) is highly desirable, but can be acquired with supplementaryreading, and through available course notes for other courses in the department.

Overview Modelling the meaning of natural (as opposed to computer) languages is one of thehardest problems in artificial intelligence. Solving this problem has the potential to dramaticallyimprove the quality and impact of a wide range of text and language processing applications suchas text summarisation, search, machine translation, language generation, question answering, etc.

A host of different approaches to this problem have been devised throughout the years. One notableapproach is Formal Semantics, which treats natural languages as programming languages which‘compile’ to higher order logics. Another is Distributional Semantics, which models the meanings ofwords as points in high dimensional semantic spaces, determined by the contexts of occurrence.

Recent research has attacked the task or reconciling the strengths of both of these approaches toproduce compositional distributed (i.e. predominantly vector-based) models of meaning. This courseserves as an introduction to the theoretical end of this new and rapidly growing field. During it, wewill discuss algebraic approaches to reasoning about vectors and their compositionality, using toolsfrom Category Theory.

Learning Outcomes Students will be expected to:

• Strengthen background knowledge of category theory and their applications.

• Become familiar with the use of monoidal categories and other algebras (e.g. pregroup gram-mars) to represent and bring into interaction syntactic and semantic aspects of language.

• Have a reasonably complete grasp of core concepts present in the literature on distributionalmodels of semantics and their compositionality.

• Understand the logical properties of tensor-based models of semantics.

Synopsis QPL followed by a number refers to chapters from Quantum Physics and Linguistics.Chapter preprints will be made available on the course website.

Topics:

• Introduction to formal and distributional semantics (QPL 12)

• Overview of compositional distributional semantics (QPL 13)

• A diagrammatic language for processes (Coecke and Paquette 2009)

• Compact closed categories for composition (Coecke et al. 2010)

• Abstract algebra and compositional structure (QPL 1)

• Tensors and logic (Grefenstette 2013)

• Reasoning about function words with Frobenius algebras (Sadrzadeh et al. 2013)

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Reading

1. S. Clark, Type-driven syntax and semantics for composing meaning vectors (2013).

2. B. Coecke, An alternative gospel of structure: order, composition, processes, arXiv preprintarXiv:1307.4038, (2013).

3. B. Coecke and E. Paquette, Categories for the practising physicist, arXiv preprint or arXiv:0905.3010(2009).

4. B. Coecke, M. Sadrzadeh, and S. Clark Mathematical Foundations for a Compositional Dis-tributional Model of Meaning, (March, 2010).

5. E. Grefenstette, Towards a formal distribution semantics: Simulating logical calculi with ten-sors. arXiv preprint or arXiv:1304.5823 (2013).

6. C. Heunen, M.Sadrzadeh, and E. Grefenstette, editors. Quantum Physics and Linguistics: ACompositional, Diagrammatic Discourse. (Oxford University Press, 2013).

7. S. Pulman, Distributional Semantic Models. (Oxford University Press, 2013).

8. M. Sadrzadeh, S. Clark, and B. Coecke, The frobenius anatomy or word meanings i: subjectand object relative pronouns. (Journal of Logic and Computation, page ext044, 2013).

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3.21 Elliptic Curves — Prof Victor Flynn — 16HT

Recommended Prerequisites It is helpful, but not essential, if students have already taken astandard introduction to algebraic curves and algebraic number theory. For those students who mayhave gaps in their background, I have placed the file “Preliminary Reading” permanently on theElliptic Curves webpage, which gives in detail (about 30 pages) the main prerequisite knowledge forthe course. Go first to: http://www.maths.ox.ac.uk/courses/material then click on “C3.7 EllipticCurves” and then click on the pdf file “Preliminary Reading”.

Overview Elliptic curves give the simplest examples of many of the most interesting phenomenawhich can occur in algebraic curves; they have an incredibly rich structure and have been the testingground for many developments in algebraic geometry whilst the theory is still full of deep unsolvedconjectures, some of which are amongst the oldest unsolved problems in mathematics. The coursewill concentrate on arithmetic aspects of elliptic curves defined over the rationals, with the study ofthe group of rational points, and explicit determination of the rank, being the primary focus. Usingelliptic curves over the rationals as an example, we will be able to introduce many of the basic toolsfor studying arithmetic properties of algebraic varieties.

Learning Outcomes On completing the course, students should be able to understand and useproperties of elliptic curves, such as the group law, the torsion group of rational points, and 2-isogenies between elliptic curves. They should be able to understand and apply the theory of fieldswith valuations, emphasising the p-adic numbers, and be able to prove and apply Hensel’s Lemmain problem solving. They should be able to understand the proof of the Mordell–Weil Theorem forthe case when an elliptic curve has a rational point of order 2, and compute ranks in such cases, forexamples where all homogeneous spaces for descent-via-2-isogeny satisfy the Hasse principle. Theyshould also be able to apply the elliptic curve method for the factorisation of integers.

Synopsis Non-singular cubics and the group law; Weierstrass equations.Elliptic curves over finite fields; Hasse estimate (stated without proof).p-adic fields (basic definitions and properties).1-dimensional formal groups (basic definitions and properties).Curves over p-adic fields and reduction mod p.Computation of torsion groups over Q; the Nagell–Lutz theorem.2-isogenies on elliptic curves defined over Q, with a Q-rational point of order 2.Weak Mordell–Weil Theorem for elliptic curves defined over Q, with a Q-rational point of order 2.Height functions on Abelian groups and basic properties.Heights of points on elliptic curves defined over Q; statement (without proof) that this gives a heightfunction on the Mordell–Weil group.Mordell–Weil Theorem for elliptic curves defined over Q, with a Q-rational point of order 2.Explicit computation of rank using descent via 2-isogeny.Public keys in cryptography; Pollard’s (p−1) method and the elliptic curve method of factorisation.

Reading

1. J.W.S. Cassels, Lectures on Elliptic Curves, LMS Student Texts 24 (Cambridge UniversityPress, 1991).

2. N. Koblitz, A Course in Number Theory and Cryptography, Graduate Texts in Mathematics114 (Springer, 1987).

3. J.H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Math-ematics (Springer, 1992).

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4. J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106 (Springer,1986).

Further Reading

1. A. Knapp, Elliptic Curves, Mathematical Notes 40 (Princeton University Press, 1992).

2. G, Cornell, J.H. Silverman and G. Stevans (editors), Modular Forms and Fermat’s Last The-orem (Springer, 1997).

3. J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts inMathematics 151 (Springer, 1994).

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3.22 Networks — Dr Heather Harrington — 16HT

Recommended Prerequisites None [in particular, C5.3 (Statistical Mechanics) is not¯

required],though some intuition from modules like C5.3, the Part B graph theory course, and probabilitycourses (at the level that everybody has to take anyway) can be useful. However, everything is self-contained, and none of these courses are required. Some computational experience is also helpful,and ideas from linear algebra will certainly be helpful.

Overview This course aims to provide an introduction to network science, which can be used tostudy complex systems of interacting agents. Networks are interesting both mathematically andcomputationally, and they are pervasive in physics, biology, sociology, information science, andmyriad other fields. The study of networks is one of the ”rising stars” of scientific endeavors, andnetworks have become among the most important subjects for applied mathematicians to study.Most of the topics to be considered are active modern research areas.

Learning Outcomes Students will have developed a sound knowledge and appreciation of someof the tools, concepts, models, and computations used in the study of networks. The study ofnetworks is predominantly a modern subject, so the students will also be expected to develop theability to read and understand current (2016) research papers in the field.

Synopsis

1. Introduction and Basic Concepts (1-2 lectures): nodes, edges, adjacencies, weighted networks,unweighted networks, degree and strength, degree distribution, other types of networks

2. Small Worlds (2 lectures): clustering coefficients, paths and geodesic paths, Watts-Strogatznetworks [focus is on modelling and heuristic calculations]

3. Toy Models of Network Formation (2 lectures): preferential attachment, generalizations ofpreferential attachment, network optimization

4. Additional Summary Statistics and Other Useful Concepts (2 lectures): modularity and as-sortativity, degree-degree correlations, centrality measures, communicability, reciprocity andstructural balance

5. Random Graphs (2 lectures): Erds-Rnyi graphs, configuration model, random graphs withclustering, other models of random graphs or hypergraphs; application of generating-functionmethods [focus is on modelling and heuristic calculations; material in this section forms animportant basis for sections 6 and 7]

6. Community Structure and Mesoscopic Structure (2 lectures): linkage clustering, optimiza-tion of modularity and other quality functions, overlapping communities, other methods andgeneralizations

7. Dynamics on (and of) Networks (3-4 lectures): general ideas, models of biological and socialcontagions, percolation, voter and opinion models, temporal networks, other topics

8. Additional Topics (0-2 lectures): games on networks, exponential random graphs, networkinference, other topics of special interest to students [depending on how much room there isand interest of current students]

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3.23 Probabilistic Combinatorics — Prof. Oliver Riordan — 16MT

Recommended Prerequisites Part B Graph Theory and Part A Probability. C8.3 Combina-torics is not as essential prerequisite for this course, though it is a natural companion for it.

Overview Probabilistic combinatorics is a very active field of mathematics, with connections toother areas such as computer science and statistical physics. Probabilistic methods are essential forthe study of random discrete structures and for the analysis of algorithms, but they can also providea powerful and beautiful approach for answering deterministic questions. The aim of this course isto introduce some fundamental probabilistic tools and present a few applications.

Learning Outcomes The student will have developed an appreciation of probabilistic methodsin discrete mathematics.

Synopsis First-moment method, with applications to Ramsey numbers, and to graphs of highgirth and high chromatic number.Second-moment method, threshold functions for random graphs.Lovasz Local Lemma, with applications to two-colourings of hypergraphs, and to Ramsey numbers.Chernoff bounds, concentration of measure, Janson’s inequality.Branching processes and the phase transition in random graphs.Clique and chromatic numbers of random graphs.

Reading

1. N. Alon and J.H. Spencer, The Probabilistic Method (third edition, Wiley, 2008).

Further Reading

1. B. Bollobas, Random Graphs (second edition, Cambridge University Press, 2001).

2. M. Habib, C. McDiarmid, J. Ramirez-Alfonsin, B. Reed, ed., Probabilistic Methods for Algo-rithmic Discrete Mathematics (Springer, 1998).

3. S. Janson, T. Luczak and A. Rucinski, Random Graphs (John Wiley and Sons, 2000).

4. M. Mitzenmacher and E. Upfal, Probability and Computing : Randomized Algorithms andProbabilistic Analysis (Cambridge University Press, New York (NY), 2005).

5. M. Molloy and B. Reed, Graph Colouring and the Probabilistic Method (Springer, 2002).

6. R. Motwani and P. Raghavan, Randomized Algorithms (Cambridge University Press, 1995).

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3.24 Probability and Computing— Prof Stefan Kiefer& Prof StanislavZivny — 20MT

Recommended prerequisites Exposure to probability theory, discrete mathematics, algorithmsand linear algebra will be assumed. Specific concepts that will be assumed include discrete prob-ability spaces, the principle of inclusion-exclusion, conditional probability, random variables, ex-pectation, basic graph theory, the binomial theorem, finite fields, power series, sorting algorithms,asymptotic notation, vector spaces, linear transformations, matrices and determinants. Graduatestudents who have not taken a course in algorithms will not be at a significant disadvantage. Diligentstudents lacking some of the required background but willing to acquaint themselves with the nec-essary material are encouraged to contact the lecturer to discuss their situation prior to registeringfor this course.

Overview In this course we study applications of probabilistic techniques to computer science,focusing on randomised algorithms and the probabilistic analysis of algorithms. Randomisation andprobabilistic techniques have applications ranging from communication protocols, combinatorial op-timisation, computational geometry, data structures, networks and machine learning. Randomisedalgorithms, which typically guarantee a correct result only with high probability, are often simplerand faster than corresponding deterministic algorithms. Randomisation can also be used to breaksymmetry and achieve load balancing in parallel and distributed computing. This course intro-duces students to those ideas in probability that most are most relevant to computer science. Thisbackground theory is motivated and illustrated by a wide-ranging series of applications.

Synopsis The material will be mostly drawn from Chapters the course text ”Probability andComputing”, by Mitzenmacher and Upfal. Supplementary material is also taken from the book”Randomized Algorithms” by Motwani and Raghavan. It is expected that the following will becovered:

• Events and Probability (1 lectures): Verifying matrix multiplication, long-path algorithm.

• Polynomial Identity Testing (1 lecture): Schwartz-Zippel lemma, isolating lemma, bipartitematching.

• Min-Cuts in Graphs (1 lecture).

• Random Variables and Expectation (1 lecture): Coupon collector, randomised Quicksort.

• Moments and Deviations (1 lecture): Chebyshev, median finding.

• Stable Marriage Problem (1 lecture).

• Chernoff Bounds (2 lectures): Las Vegas and Monte Carlo algorithms, set balancing, facilitylocation, random rounding, permutation routing.

• Pairwise Independence (1 lecture): finding cuts, derandomisation, sampling.

• Markov Chains (3 lectures): Random-walk algorithms for 2-SAT, 3-SAT, random-walks ongraphs.

• Balls and Bins (2 lectures): Poisson approximation, occupancy problems, Bloom filters.

• The Monte Carlo Method (1 lecture): DNF counting, importance sampling.

• Pairwise Independence (1 lecture): finding cuts, derandomisation, sampling.

• Universal Hashing (1 lecture): universal hash functions, perfect hashing, count-min filters.

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• The Probabilistic Method (1 lectures): Derandomisation, lower bounds for set balancing,Lovasz local lemma.

• Markov Chain Coupling (2 lectures): Metropolis Algorithm, variation distance, mixing times.

Learning Outcomes At the end of the course students are expected to

• Understand basic concepts and tools in probability theory that are relevant to computing, in-cluding random variables, independence, moments and deviations, tail inequalities, occupancyproblems, the probabilistic method, derandomisation and Markov chains.

• Be able to use the above tools to devise and analyse randomised algorithms and carry out theprobabilistic analysis of deterministic algorithms.

• Understand some of the main paradigms in the design of randomised algorithms, includingrandom sampling, random walks, random rounding, algebraic techniques, foiling the adversaryand amplification.

Syllabus Tools and Techniques: random variables; conditional expectation; moment bounds; tailinequalities; independence; coupon collection and occupancy problems; the probabilistic method;Markov chains and random walks; algebraic techniques; probability amplification and derandomisa-tion.

Applications: Sorting and searching; graph algorithms; combinatorial optimisation; propositionalsatisfiability; hashing; routing; random sampling; DNF-counting; bloom filters, count-min filters.

Reading list Required text:

• Probability and Computing, by Michael Mitzenmacher and Eli Upfal, Cambridge UniversityPress.

Also useful:

• Randomized Algorithms, by Rajeev Motwani and Prabhakar Raghavan, Cambridge UniversityPress.

• Design and Analysis of Randomized Algorithms, by Juraj Hromkovic, Springer.

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