## Trinity College Dublin, The University of Dublin

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# Mathematics

Visiting student co-ordinator: Professor Dmitri Zaitsev, zaitsev@maths.tcd.ie
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA1111 Linear algebra I)

(05 ECTS credits) Michaelmas term NA This module will be examined in a 2 hour examination in Trinity term. Homework assignments will be due every Thursday. 20% homework, 80% final exam (based on homework and tutorials). 11 weeks, 3 lectures including tutorials per week Vladimir Dotsenko (vdots@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• operate with vectors in dimensions 2 and 3, and apply vectors to solve basic geometric problems;
• apply various standard methods (Gauss-Jordan elimination, inverse matrices, Cramer's rule) to solve systems of simultaneous linear equations;
• compute the sign of a given permutation, and apply theorems from the module to compute determinants of square matrices;
• demonstrate that a system of vectors forms a basis of the given vector space, compute coordinates of given vectors relative to the given basis, and calculate the matrix of a linear operator relative to the given bases;
• give examples of sets where some of the defining properties of vectors, matrices, vector spaces, subspaces, and linear operators fail;
• identify the above linear algebra problems in various settings (e.g. in the case of the vector space of polynomials, or the vector space of matrices of given size), and apply methods of the module to solve those problems.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA1125 Single-variable calculus and introductory analysis)

(10 ECTS credits) Michaelmas term NA This module will be examined in a 3 hour examination in Trinity term. Continuous assessment will contribute 15% to the final grade for the module at the annual examination session. 11 weeks, 5 lectures plus tutorials per week Paschalis Karageoris (pete@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Determine whether a given relation is a function or not, and whether an inverse function exists.
• Find limits and determine whether given functions are continuous or not
• Differentiate functions and use derivatives to graph functions, solve extremal problems and related rates problems.
• Integrate functions using substitution, integration by parts, partial fractions and reduction formulae.
• Find areas, volumes, length of curves, averages and work done.
• Solve simple first order differential equations and higher order linear homogeneous differential equations.
• Determine whether a given sequence or series converges or not.
• Determine where a given power series converges absolutely, converges conditionally or diverges.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA1126 Introduction to set theory and general topology)

(05 ECTS credits) Hilary term MA1125 This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 15% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Donal O'Donovan (don@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Prove or disprove logical equivalences.
• Use the predicate calculus.
• Prove or disprove set equivalences.
• Test the properties of relations.
• Prove and apply the theorems that are covered.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

(05 ECTS credits) Hilary term MA1111, MA1125 This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Vladimir Dotsenko (vdots@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Analyse the behaviour of functions of several variables, present the result graphically and efficiently calculate partial derivatives of functions of several variables (also for functions given implicitly);
• Obtain equations for tangent lines to plane curves and tangent planes to space surfaces;
• compute the sign of a given permutation, and apply theorems from the module to compute determinants of square matrices;
• Apply derivative tests and the method of Lagrange multipliers to find maxima and minima of functions of several variables, local and global;
• Effectively calculate multiple integrals, in Cartesian and polar coordinates, in particular, to find areas, volumes and centres of mass
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA1212 Linear algebra II)

(05 ECTS credits) Hilary term MA1111 This module will be examined in a 2 hour examination in Trinity term. Homework assignments will be due every Thursday. 20% homework, 20% midterm exam, 60% final exam (based on homework and tutorials). 11 weeks, 3 lectures including tutorials per week Paschalis Karageorgis (pete@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Compute the rank of a given linear operator, and use proofs of theoretical results on ranks explained in the course to derive similar theoretical results;
• Compute the dimension and determine a basis for the intersection and the sum of two subspaces of a given space, determine a basis of a given vector space relative to a given subspace;
• Calculate the basis consisting of eigenvectors for a given matrix with different eigenvalues and, more generally, calculate the Jordan normal form and a Jordan basis for a given matrix;
• Apply Gram-Schmidt orthogonalisation to obtain an orthonormal basis of a given Euclidean space;
• Apply various methods (completing the squares, Sylvester's criterion, eigenvalues) to determine the signature of a given symmetric bilinear form;
• Identify the above linear algebra problems in various settings (e.g. in the case of the vector space of polynomials, or the vector space of matrices of given size), and apply methods of the course to solve those problems.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA1213 Introduction to group theory)

(05 ECTS credits) Michaelmas term NA This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination. 11 weeks, 2 lectures plus 1 tutorial per week Dmitri Zaitsev (zaitsev@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Apply the notions: map/function, surjective/injective/bijective/invertible map, equivalence relation, partition. Give the definition of: group, abelian group,subgroup, normal subgroup, quotient group, direct product of groups, homomorphism, isomorphism, kernel of a homomorphism, cyclic group, order of a group element.
• Apply group theory to integer arithmetic: define what the greatest common divisor of two nonzero integers m and n is, compute it and express it as a linear combination of n and m using the extended Euclidean algorithm; Write down the Cayley table of a cyclic group Zn or of the multiplicative group (Zn)x for small n; determine the order of an element of such a group.
• Define what a group action is and be able to verify that something is a group action. Apply group theory to describe symmetry; know the three types of rotation symmetry axes of the cube (their 'order' and how may there are of each type); describe the elements of symmetry group of the regular n-gon (the dihedral group D2n) for small values of n and know how to multiply them.
• Compute with the symmetric group; determine disjoint cycle form, sign and order of a permutation; multiply two permutations.
• Know how to show that a subset of a group is a subgroup or a normal subgroup. State and apply Lagrange's theorem. State and prove the first isomorphism theorem.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA1241 Mechanics I)

(05 ECTS credits) Michaelmas term NA This module will be examined in a 2-hour examination in Trinity term. The final grade will consist of 80% exam and 20% continuous assessment. 11 weeks, 3 lectures including tutorials per week Jan Manschot (manschot@maths.tcd.ie)

## Description

Module Content

• Mathematical preliminaries (vectors and their role in mechanics, elements of vector algebra.);
• Kinematics;
• Newton's Laws: the foundations of classical mechanics;
• Linear momentum (dynamics of multi-particle systems, centre of mass, conservation of momentum, impulse);
• Work and energy (definition of work and the work-energy theorem, potential and kinetic energy);
• Potential and kinetic energy (conservative and non-conservative forces, conservation of energy);
• Angular momentum (torque, conservation of angular momentum);
• Moment of inertia (motion involving translation and rotation).

Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA1242 Mechanics II)

(05 ECTS credits) Hilary term MA1241 This module will be examined 2-hour examination in Trinity term. The final grade will consist of 80% exam and 20% continuous assessment. 11 weeks, 3 lectures including tutorials per week Jan Manschot (manschot@maths.tcd.ie)

## Description

Module Content

• Collisions (elastic and inelastic);
• Rigid body motion (precession, tensor of inertia);
• Non-inertial systems & ficticious forces (centrifugal and coriolis forces) ;
• Central forces (two-body problem, general properties of central force motion);
• Harmonic oscillator (driven and damped oscillations).
• Non-inertial frames and fictitious forces (accelerating non-rotating frames, rotating coordinate systems, centrifugal and Coriolis forces, tidal forces, rotating bucket and Mach's principle, the equivalence principle and origins of General Relativity, Galilean transformations, principle of Relativity);

Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA1E01 Engineering mathematics I)

(05 ECTS credits) Michaelmas term NA Weekly continuous assessment contributes 20% towards the final grade with the end-of-year final written two-hour examination contributing 80%. 11 weeks, 3 lectures including tutorials per week Alberto Ramos (alberto.ramos@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Recognise mathematical structures in practical problems, translate problems into mathematical language, and analyse problems using methods from one-dimensional calculus;
• Solve problems involving concepts of calculus;
• Apply differentiation to solve practical problems and to graph a wide range of functions of one real variable;
• Apply integration to solve geometrical problems such as computing the area or volume of solids of revolution;
• Use standard computer input for mathematical expressions.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA1E02 Engineering mathematics II)

(05 ECTS credits) Hilary term MA1E01 Assessment is by means of assignments and a two-hour end-of-year written examination. The end-of-year examination contributes 80% towards the final grade and the weekly assignments contribute 20%. 11 weeks, 3 lectures including tutorials per week Paschalis Karageorgis (pete@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Apply vectors to geometrical problems in space
• Integrate by parts;
• Integrate trigonometric and rational functions;
• Formulate and solve a first order differential equation;
• Determine if a sequence converges or not;
• Test a series for convergence;
• Approximate a function by polynomials;
• Calculate solutions to systems of linear equations and find inverse matrices, by different methods and describe why some methods are more efficient than others.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MAU1101 Mathematical methods)

(10 ECTS credits) Michaelmas term NA This module will be examined in a 3 hour examination in Trinity term. The exam will count 75% of the final grade with the remaining 25% for continuous assessment. 11 weeks; 8 hours per week, including 5 lectures, 2 tutorials and 1 computer practical. • 1 or 2 lecturers from the school of mathematics, (nomen nominandum) • 1 lecturer from the department of statistics, (nomen nominandum) • teaching assistants/demonstrators for tutorial groups and practicals. 4 lectures + 2 tutorials per week will be covered by the school of maths; 1 lecture + 1 computer practical per week will be covered by the department of statistics Sinead Ryan ryan@maths.tcd.ie And Alberto Ramos Alberto.ramos@maths.tcd.ie

## Description

On successful completion of this module students will be able to • Manipulate vectors to perform alegebraic operations on them such as dot products and orthogonal projections and apply vector concepts to manipulate lines and planes in space R3 or in Rn with n≥4. • Use Gaussian elimination techniques to solve systems of linear equations, find inverses of matrices and solve problems which can be reduced to such systems of linear equations. • Manipulate matrices algebraically and use concepts related to matrices such as invertibility, symmetry, triangularity, nilpotence. • Manipulate numbers in different bases and explain the usefulness of the ideas in computing. • Use computer algebra and spreadsheets for elementary applications. • Explain basic ideas relating to functions of a single variable and their graphs such as limits, continuity, invertibility, even/odd, differentiabilty and solve basic problems involving these concepts. • Give basic properties and compute with a range of rational and standard transcendental functions, for instance to find derivatives, antiderivatives, critical points and to identify key features of their graphs. • Use a range of basic techniques of integration to find definite and indefinite integrals. • Apply techniques from calculus to a variety of applied problems. Module content The module is divided into a maths and a statistics part, with maths further divided into calculus and linear algebra/discrete mathematics. Mathematics: a) Calculus: 3 lectures plus one tutorial per week. The syllabus is largely based on the text book [Stewart-Day], and will cover most of Chapters 1-6 along with the beginning of Chapter 7 on differential equations: • Functions and graphs. Lines, polynomials, rational functions, exponential and logarithmic functions, trigonometric functions and the unit circle. • Limits, continuity, average rate of change, first principles definition of derivative, basic rules for differentiation • Graphical interpretation of derivatives, optimization problems • Exponential and log functions. Growth and decay applications. semilog and log-log plots. • Integration (definite and indefinite). Techniques of substitution and integration by parts. Applications. • Differential equations and initial value problems, solving first order linear equations. Applications in biology or ecology. b) Linear algebra/discrete mathematics: 1 lecture and 1 tutorial per week. The syllabus will cover parts of chapter 1 on sequences, limits of sequences and difference equations and then chapter 8 of [Stewart-Day] on linear algebra. The syllabus is approximately: • Sequences, limits of sequences, difference equations, discrete time models • Vectors and matrices , matrix algebra • inverse matrices, determinants. • systems of difference equations, systems of linear equations, eigenvalues and eigenvectors. Leslie matrices, matrix models. Statistics: There will be 1 lecture per week and 1 computer practical. The syllabus will cover much of chapters 11-13 of [Stewart-Day] and use [Bekerman-et-al] as main reference for R in the computer practicals. The syllabus is approximately: • Numerical and Graphical Descriptions of Data • Relationships and linear regression • Populations, Samples and Inference • Probability, Conditional Probability and Bayes’ Rule • Discrete and Continuous Random Variables • The Sampling Distribution Confidence Intervals
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA1S11 Mathematics for scientists)

(10 ECTS credits) Michaelmas term NA This module will be examined in a 3 hour examination in Trinity term. Continuous assessment in the form of weekly tutorial work will contribute 20% to the final grade at the annual examinations, with the examination counting for the remaining 80%. 11 weeks, 6 lectures including tutorials per week Sergey Mozgovoy (mozgovoy@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Manipulate vectors to perform algebraic operations on them such as dot products and orthogonal projections and apply vector concepts to manipulate lines and planes in space.
• Use Gaussian elimination techniques to solve systems of linear equations, find inverses of matrices and solve problems which can be reduced to such systems of linear equations.
• Manipulate matrices algebraically and use concepts related to matrices such as invertibility, symmetry, triangularity, nilpotence.
• Manipulate numbers in different bases and explain the usefulness of the ideas in computing.
• Use computer algebra and spreadsheets for elementary applications.
• Explain basic ideas relating to functions of a single variable and their graphs such as limits, continuity, invertibility, even/odd, differentiabilty and solve basic problems involving these concepts.
• Give basic properties and compute with a range of rational and standard transcendental functions, for instance to find derivatives, antiderivatives, critical points and to identify key features of their graphs.
• Use a range of basic techniques of integration to find definite and indefinite integrals.
• Apply techniques from calculus to a variety of applied problems.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2223 Metric spaces)

(05 ECTS credits) Michaelmas term MA1126 This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 15% to the final grade for the module at the annual examination session, with the examination counting for the remaining 85%. 11 weeks, 3 lectures including tutorials per week Sergey Mozgovoy (mozgovoy@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Accurately recall definitions, state theorems and produce proofs on topics in metric spaces normed vector spaces and topological spaces;
• Construct rigourous mathematical arguments using appropriate concepts and terminology from the module, including open, closed and bounded sets, convergence, continuity, norm equivalence, operator norms, completeness, compactness and connectedness;
• Solve problems by identifying and interpreting appropriate concepts and results from the module in specific examples involving metric, topological and /or normed vector spaces;
• Construct examples and counterexamples related to concepts from the module which illustrate the validity of some prescribed combination of properties;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2224 Lebesgue integral)

(05 ECTS credits) Hilary term MA2223 This module will be examined in a 2-hour examination in Trinity term. Continuous assessment in the form of weekly tutorial work will contribute 20% to the final grade at the annual examinations, with the examination counting for the remaining 80%. 11 weeks, 3 lectures including tutorials per week Richard Timoney (richardt@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Discuss countable sets, characteristic functions and boolean algebras;
• State and prove properties of length measure, outer measure and Lebesgue measure for subsets of the real line and establish measurability for a range of functions and sets;
• Define the Lebesgue integral on the real line and apply basic results including convergence theorems.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA22S1 Multi-variable calculus for science)

(05 ECTS credits) Michaelmas term MA1S12 This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination. 11 weeks, 3 lectures including tutorials per week John Stalker (stalker@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Write equations of planes, lines and quadric surfaces in the 3-space;
• Determine the type of conic section and write change of coordinates turning a quadratic equation into its standard form;
• Use cylindrical and spherical coordinate systems;
• Write equations of a tangent line, compute unit tangent, normal and binormal vectors and curvature at a given point on a parametric curve; compute the length of a portion of a curve;
• Apply above concepts to describe motion of a particle in the space;
• Calculate limits and partial derivatives of functions of several variables
• Write local linear and quadratic approximations of a function of several variables, write equation of the plane tangent to its graph at a given point;
• Compute directional derivatives and determine the direction of maximal growth of a function using its gradient vector;
• Use the method of Lagrange multipliers to find local maxima and minima of a function;
• Compute double and triple integrals by application of Fubini's theorem or use change of variables;
• Use integrals to find quantities defined via integration in a number of contexts (such as average, area, volume, mass)
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA22S2 Vector calculus for science)

(05 ECTS credits) Hilary term MA22S1, MA22S3 This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Joe O Hogain (johog@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Manipulate vectors in R^3 to evaluate dot products and cross products and investigate if vectors are linearly independent;
• Understand the concepts of vector fields, conservative vector fields, curves and surfaces in R^3;
• Find the equation of normal lines and tangent planes to surfaces in R^3;
• Evaluate line integrals and surface integrals from the definitions;
• Use Green's Theorem to evaluate line integrals in the plane and use the Divergence Theorem (Gauss's Theorem) to evaluate surface integrals;
• Apply Stokes's Theorem to evaluate line integrals and surface integrals;
• Solve first order PDEs using the method of characteristics and solve second order PDEs using separation of variables
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA22S3 Fourier analysis for science)

(05 ECTS credits) Michaelmas term MA1S12 This module will be examined in a 2 hour examination in Trinity term. Continuous Assessment will contribute 20% to the final annual grade, with the examination counting for the remaining 80%. 11 weeks, 3 lectures including tutorials per week Ruth Britto (britto@maths.tcd.ie)

## Description

• Calculate the real and complex Fourier series of a given periodic function;
• Obtain the Fourier transform of non-periodic functions;
• Evaluate integrals containing the Dirac Delta;
• Solve ordinary differential equations with constant coefficients of first or second order, both homogenous and inhomogenous;
• Obtain series solutions (including Frobenius method) to ordinary differential equations of first or second order;
• apply their knowledge to the sciences where relevant.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA22S4 Mechanics)

(05 ECTS credits) Hilary term MA22S1 This module will be examined in a 2 hour examination in Trinity term. Continuous Assessment will contribute 10% to the final annual grade. 11 weeks, 3 lectures including tutorials per week Paschalis Karageorgis (pete@maths.tcd.ie)

## Description

### This is a provisional syllabus.

• Introduction

Scalar and vector products, differentiation and integration of vectors, velocity and acceleration, Newton Laws.

• Motion in Plane Polar Coordinates

Derivation of velocity and acceleration in polar coordinates and applications to circular and elliptical motion of a particle.

• Central Force Motion

Equations of motion for a particle in a central force field, derivation of the orbit equation, conservation of angular momentum, potential energy, conservation of energy, solution of the orbit equation for different force fields, apsides and apsidal angles, calculation of maximum and minimum distance of a particle from the origin of a force, inverse square law of attraction and conic sections, properties of the ellipse. Planetary motion, Newton's Universal Law of Gravitation, proof of Kepler's Laws, examples involving calculating eccentricity, periodic time, velocity at aphelion and perihelion of planets and related problems.

• Work and Energy

Evaluation of work done by a force on a particle using line integrals, work as related to kinetic and potential energy, conservative forces, path independence, conservation of energy. Energy diagrams: use of energy diagrams to analyse the motion of a particle qualitatively, positions of stable and unstable equilibrium, small oscillations in a bound system.

• Rotating Frames

Non-inertial coordinate systems, velocity and acceleration in rotating systems, centrifugal and coriolis forces, derivation of the equation of motion for a particle moving in the vicinity of the rotating earth and related examples.

Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA22S6 Numerical and data analysis techniques)

(05 ECTS credits) Hilary term MA1S12 This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Stefan Sint (sint@maths.tcd.ie)

## Description

The students will learn in a practical way the main numerical techniques used in different areas of science. They will learn the mathematical background of probability and statistics and most importantly the practical aspects.

On successful completion of this module, students will be able to:

• Use discrete and continuous random variables to describe phenomena observed in nature (science experiments, population statistics, ...) and to quantify how well a model works;
• Find a simple model for a given dataset, such as the output of an experiment;
• Perform a chi^2 analysis to estimate the model parameters and their standard deviations;
• Use Markov processes to describe stochastic phenomena;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2214 Fields, rings and modules)

(05 ECTS credits) Hilary term MA1213 This module will be examined in a 2 hour examination in Trinity term. Continuous assessment is via weekly home assignments and it will contribute to the final grade for the module at the annual examination session. The final mark is 85% of the exam mark plus 15% of continuous assessment. 11 weeks, 3 lectures including tutorials per week Sergey Mozgovoy (mozgovoy@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• State definitions of concepts used in the module, and prove their simple properties;
• Describe rings and fields commonly used in the module, and perform computations in them;
• State theoretical results of the module, demonstrate how one can apply them, and outline proofs of some of them (e.g. first isomorphism theorem, or ``an Euclidean domain is a principal ideal domain'', or ``a principal ideal domain is a unique factorisation domain'');
• Perform and apply the Euclidean algorithm in a Euclidean domain;
• Give examples of sets where some of the defining properties of fields, rings and modules fail, and give examples of fields, rings and modules satisfying some additional constraints;
• State and prove the tower law, and use it to prove the impossibility of some classical ruler and compass geometric constructions;
• Identify concepts introduced in other modules as particular cases of fields, rings and modules (e.g. functions on the real line as a ring, abelian groups and vector spaces as modules).
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2321 Analysis in several real variables)

(05 ECTS credits) Michaelmas term MA1111, MA1123, MA1132 This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 10% to the final grade for the module at the annual examination session, with the examination counting for the remaining 90%. 11 weeks, 3 lectures including tutorials per week David Wilkins (dwilkins@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• justify with logical argument basic results concerning the topology of Euclidean spaces, convergence of sequences in Euclidean spaces, limits of vector-valued functions defined over subsets of Euclidean spaces and the continuity of such functions;
• specify accurately the concepts of differentiability and (total) derivative for functions of several real variable;
• justify with logical argument basic properties of differentiable functions of several real variables including the Product Rule, the Chain Rule, and the result that a function of several real variables is differentiable if its first order partial derivatives are continuous;
• determine whether or not specified functions of several real variables satisfy differentiability requirements.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2322 Calculus on manifolds)

(05 ECTS credits) Hilary term MA2321 This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 10% to the final grade for the module at the annual examination session, with the examination counting for the remaining 90%. 11 weeks, 3 lectures including tutorials per week Jan Manschot (manschot@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• proof theorems about manifolds in Euclidean space,
• proof theorems about differential forms and perform calculations with them,
• carry out integration on manifolds in Euclidean space,
• explain the relation between scalar, vector & tensor fields and differential forms,
• explain, proof and apply Stokes' theorem for differential forms,
• explain and apply the Poincare lemma.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2327 Ordinary differential equations)

(05 ECTS credits) Michaelmas term MA1126, MA1212 This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Miriam Logan (will advise email )

## Description

On successful completion of this module, students will be able to:

• Apply various standard methods (separation of variables, integrating factors, reduction of order, undetermined coefficients) to solve certain types of differential equations (separable, 1st-order linear, linear with constant coefficients;
• Give examples of differential equations for which either existence or uniqueness of solution fails;
• Compute the exponential of a square matrix;
• Apply standard methods (linearization, Lyapunov theorems) to check the stability of critical points for autonomous systems
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2328 Complex analysis)

(05 ECTS credits) Hilary term MA1124 or MA2321 This module will be examined in a 2 hour examination in Trinity term. 11 weeks, 3 lectures per week Marius de Leeuw - mdeleeuw@maths.tcd.ie

## Description

On successful completion of this module, students will be able to:

• Use basic theorems on complex sequences and series, with a particular emphasis on power series. Calculate coefficients and radii of convergence of power series using these theorems.
• Demonstrate a familiarity with the basic properties of analytic functions. Apply these theorems to simple examples.
• State correctly the theorems of Cauchy and Morera. Calculate, using Cauchy's theorem and its corollaries, the values of contour integrals.
• Prove and apply properties of important examples of analytic functions, including rational functions, the exponential and logarithmic functions, trigonometric and hyperbolic functions and elliptic functions.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2331 Equations of mathematical physics I)

(05 ECTS credits) Michaelmas term MA1132 This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 10% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Manuela Kulaxizi nmanuela@maths.tcd.ie

## Description

On successful completion of this module, students will be able to:

• Compute the real and complex Fourier series of a given periodic function;
• Evaluate the Fourier transform of a given non-periodic function;
• Evaluate integrals containing the Dirac delta distribution;
• Compute the gradient of a given scalar field and the divergence and curl of a given vector field;
• Calculate line and surface integrals;
• Apply their knowledge to relevant problems in mathematics and physics;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2332 Equations of mathematical physics II)

(05 ECTS credits) Hilary term MA2331 This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 10% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Anthony Brown Anthony.brown@ucd.ie

## Description

On successful completion of this module, students will be able to:

• State and prove the Green's, Stokes' and Gauss' integral theorems;
• Determine series solutions (including Frobenius method) of first and second order ordinary differential equations with non-constant coefficients;
• Determine series solutions (including Frobenius method) of first and second order ordinary differential equations with non-constant coefficients;
• Apply their knowledge in mathematical and physical domains where relevant
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2341 Advanced classical mechanics I)

(05 ECTS credits) Michaelmas term MA1242 This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Sergey Frolov (frolovs@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• have a knowledge of the Lagrangian mechanics including a familiarity with Lagrangians describing various important mechanic systems;
• derive the equations of motion of a mechanical system with several degrees of freedom from its Lagrangian by using Hamilton's principle;
• use Noether's theorem to derive conservation laws;
• analyse small free oscillations, forced oscillations and damped oscillations of a system with any number of degrees of freedom, compute its characteristic frequencies and normal coordinates;
• use Euler angles and Euler equations to analyze the motion of a rigid body;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2342 Advanced classical mechanics II)

(05 ECTS credits) Hilary term MA2341 This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 20% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Tristan Mc Loughlin tristan@maths.tcd.ie

## Description

• Hamilton formalism: Legendre transform, Hamilton equations, Liouville theorem;
• Canonical transformations;
• Hamilton-Jacobi equations, action-angle variables;
• Special theory of relativity;
• Mechanics of continuous systems and fields;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2C03 Discrete mathematics)

(05 ECTS credits) Michaelmas term and Hilary term NA This module will be examined in a 3 hour examination in Trinity term. Also students should complete a small number of assignments during the academic year. The final grade at the annual examination session will be a weighted average over the examination mark (90%) and the continuous assessment mark (10%). It is possible for visiting students to take half of the module for 5 credits. 22 weeks, 3 lectures including tutorials per week Andreea Nicoara (anicoara@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Construct reasoned logical arguments to identify and justify basic properties of mathematical objects that are specified as sets, relations on sets, functions between sets, and/or monoids.
• Identify formal languages generated by simple context-free grammars, and construct specifications of context-free grammars and finite state machines that generate and/or determine formal languages, given specifications of such formal languages.
• Recognize and identify properties of undirected graphs that are networks consisting of vertices together with edges joining pairs of vertices, and find examples of isomorphisms between such graphs satisfying given criteria
• Find solutions to certain types of homogeneous and inhomogeneous linear ordinary differential equations of degree at least two, using methods based on the use of power series, and also methods based on the identification of particular integrals and complementary functions, where the coefficients of the differential equation are constants and the forcing function is typically constructed from polynomial, exponential and trigonometric functions.
• Expound and apply basic properties of exponential and trigonometric functions, where the arguments of those functions are complex numbers and variables, and thereby obtain results that are relevant to the basic implementation of the Discrete Fourier Transform.
• Perform calculations within the algebra of vectors in three-dimensional space, and the algebra of quaternions, and apply the results of such calculations to the solution of simple geometrical problems.
• Perform calculations in basic number theory, justified on the basis of theorems explicitly presented and proved within the module, that have relevance to the implementation of public key cryptographic systems such as the Rivest-Shamir-Adelman (RSA) public key cryptosystem.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2E01 Engineering mathematics III)

(05 ECTS credits) Michaelmas term MA1E02 Assessment is by means of tutorial assignments and a two-hour end-of-year written examination. The overall grade is calculated using the maximum of 90% end-of-year examination + 10% assessment. 11 weeks, 3 lectures including tutorials per week Dmitri Zaitsev (zaitsev@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Pass effectively between linear systems, linear transformations and their matrices.
• Analyse a system of vectors for linear dependence and for being a basis.
• Calculate dimension of a subspace.
• Calculate the rank and nullity of a matrix and understand their importance.
• Construct a basis for row, column, and null spaces of a matrix.
• Calculate eigenvalues and eigenvectors of matrices.
• Apply the Gram-Schmidt process to transform a given basis into orthogonal one.
• Apply methods of general and particular solutions to ordinary differential equations.
• Calculate the Fourier series of a given function and analyse its behaviour.
• Apply Fourier series to solving ordinary differential equations.
• Calculate the Fourier transformation.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2E02 Engineering mathematics IV)

(05 ECTS credits) Hilary term MA2E01 Tutorial assignments contribute 10% towards the final grade with the end-of-year final written two-hour examination contributing 90%. 11 weeks, 3 lectures including tutorials per week Anthony Brown - Anthony.brown@ucd.ie

## Description

On successful completion of this module, students will be able to:

• analyse the behaviour of functions of several variables, present the result graphically and efficiently calculate partial derivatives of functions of several variables (also for functions given implicitly);
• obtain equations for tangent lines to plane curves and tangent planes to space surfaces;
• apply derivative tests to find maxima and minima of functions of several variables, local and global;
• effectively calculate multiple integrals, in Cartesian, polar, cylindrical and spherical coordinates, in particular, to find areas, volumes, masses and centres of gravity of two- and three-dimensional objects;
• determine whether a vector field is conservative, find a potential function for a conservative field, and use it to calculate line integral along the field;
• use Green's, Stokes' and divergence theorems to calculate double, surface and flux integrals;
• solve differential equations by applying Laplace transforms
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3416 Group representations)

(05 ECTS credits) Hilary term MA1212, MA1213 This module will be examined in a 2 hour examination in Trinity term. Continuous assessment is via bi-weekly home assignments. The final mark is 80% of the exam mark plus 20% of continuous assessment. 11 weeks, 3 lectures including tutorials per week Paschalis Karageorgis (pete@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• construct complex irreducible representations for various finite groups of small orders;
• reproduce proofs of basic results that create theoretical background for dealing with group representations;
• apply orthogonality relations for characters of finite groups to find multiplicities of irreducible constituents of a representation;
• apply representation theoretic methods to simplify problems from other areas that "admit symmetries";
• identify group theoretic questions arising in representation theoretic problems, and use results in group theory to solve problems on group representations.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA341F Introduction to algebraic geometry)

(05 ECTS credits) Hilary term MA2314, MA2322 This module will be examined in a 2-hour examination in Trinity term. The final mark is 80% of the exam mark plus 20% continuous assessment consisting of a certain number of homework sets assigned throughout the term. 11 weeks, 3 lectures including tutorials per week Andreea Nicoara (anicoara@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Work with curves, surfaces, projective and affine varieties.
• Understand the relationship between commutative algebra and geometry that underlies this field as well as its connections to number theory and complex analysis.
• Define concepts, prove theorems, and write down examples and counterexamples.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3421 Functional analysis I)

(05 ECTS credits) Michaelmas term MA2223, MA2224 are desirable This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 15% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week John Stalker stalker@maths.tcd.ie

## Description

On successful completion of this module, students will be able to:

• Give the appropriate definitions, theorems and proofs concerning the syllabus topics, including topics in general topology, elementary theory of Banach spaces and of linear operators;
• Solve problems requiring manipulation or application of one or more of the concepts and results studied;
• Formulate mathematical arguments in appropriately precise terms for the subject matter;
• Apply their knowledge in mathematical domains where functional analytic techniques are relevant;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3422 Functional analysis II)

(05 ECTS credits) Hilary term MA3421 This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 15% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Richard Timoney (richardt@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• give the appropriate definitions, theorems and proofs concerning the syllabus topics, including topics related to weak toploogies, compactness, Hahn-Banach theorem, reflexivity;
• solve problems requiring manipulation or application of one or more of the concepts and results studied;
• formulate mathematical arguments in appropriately precise terms for the subject matter;
• apply their knowledge in mathematical domains where functional analytic techniques are relevant.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3429 Differential geometry)

(05 ECTS credits) Michaelmas term MA2322 This module will be examined in a 2-hour examination in Trinity term. 11 weeks, 3 lectures including tutorials per week Sergey Frolov (frolovs@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Obtain a coordinate-induced basis for the tangent space and cotangent space at points of a differentiable manifold, construct a coordinate induced basis for arbitrary tensors and obtain the components of tensors in this basis;
• Determine whether a particular map is a tensor by either checking multi-linearity or by showing that the components transform according to the tensor transformation law;
• Construct manifestly chart-free definitions of the Lie derivative of a function and a vector, to compute these derivatives in a particular chart and hence compute the Lie derivative of an arbitrary tensor;
• Compute, explicitly, the covariant derivative of an arbitrary tensor;
• Define parallel transport, derive the geodesic equation and solve problems involving parallel transport of tensors;
• Obtain an expression for the Riemann curvature tensor in an arbitrary basis for a manifold with vanishing torsion, provide a geometric interpretation of what this tensor measures, derive various symmetries and results involving the curvature tensor;
• Define the metric, the Levi-Civita connection and the metric curvature tensor and compute the components of each of these tensors given a particular line-element;
• Define tensor densities, construct chart-invariant volume and surface elements for curved Lorentzian manifolds and hence construct well-defined covariant volume and surface integrals for such manifolds;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA342T Partial differential equations)

(05 ECTS credits) Hilary term MA2327 This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 10% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week John Stalker (stalker@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Solve, in a higher number of dimensions, problems for the Wave, Heat, and Laplace Equations. In addition to those, students should be able to use Young's inequality to obtain estimates on solutions in terms of data;
• Demonstrate a familiarity with the definition and main properties of distributions and the principal operations on distributions: addition, multiplication by smooth functions, differentiation and convolution. Give the definition of the term ''fundamental solution'' and verify that a given distribution is a fundamental solution for a given differential equation;
• Solve, by the method of characteristics, first order linear scalar partial differential equations. Students should also be able to determine when the initial value problem for such an equation has a unique global solution.
• Solve the initial value problem for Burgers' equation, including cases where shocks are present initially or develop later. Give the definitions of ''weak solution'' and ''shock'' and determine whether the singularity of a given weak solution are shocks.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3431 Classical field theory)

(05 ECTS credits) Michaelmas term MA2342 This module will be examined in a 2-hour examination in Trinity term. Assignments will contribute 15% to the final result. 11 weeks, 3 lectures including tutorials per week Alberto Ramos Alberto.ramos@maths.tcd.ie

## Description

On successful completion of this module, students will be able to:

• Apply standard methods, such as orthogonal functions, to solve problems in electro- and magneto-statics;
• Describe how to find the equation of motion for a scalar field using a given Lagrangian density;
• Calculate the stress tensor and evaluate its four divergence, relating it to a conservation law;
• Employ a variational principle to find the relativistic dynamics of a charged particle interacting with an electromagnetic potential;
• Use the Euler-Lagrange equation to show how a Lorentz scalar Lagrangian density with an interaction term leads to the Maxwell equations;
• Explain the concepts of guage invariance and tracelessness in the context of the stress tensor of a vector field;
• Demonstrate how the divergence of the symmetric stress tensor is related to the four current density of an external source;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3432 Classical electrodynamics)

(05 ECTS credits) Hilary term MA3431 This module will be examined in a 2-hour examination in Trinity term. Assignments will contribute 15% to the final annual grade. 11 weeks, 3 lectures including tutorials per week Tristan McLoughlin (tristan@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Describe how to find the Fourier transform of a Green function and hence evaluate it for the equation of d'Alembert.
• Use the retarded Green function to solve the Maxwell equations for electromagnetic fields.
• Describe electromagnetic radiation, including plane-wave and spherical vector waves.
• Explain the concepts of electromagnetic potential and that of retarded time for charges undergoing acceleration.
• Analyse simple radiating systems, in which the electric dipole, magnetic dipole or electric quadrupole dominate.
• Show how the orthogonality and magnitude of electric and magnetic radiative fields may be established.
• Use expressions for the fields to evaluate the differential power radiated in a particular direction, and hence find the total power.
• Determine the motion of a radiating charged particle in the electric field of another charged particle or in a constant magnetic field.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3441 Quantum mechanics I)

(05 ECTS credits) Michaelmas term MA2342 This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 10% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Tristan McLoughlin tristan@maths.tcd.ie

## Description

On successful completion of this module, students will be able to:

• State the basic postulates of quantum mechanics;
• Derive the general Schroedinger and Heisenberg equations of motion;
• Apply quantum theoretical techniques to complex problems;
• Demonstrate understanding-at and entry level-of 20th/21st century physics;
• Solve problems in assigned and graded weekly problem sets;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3442 Quantum mechanics II)

(05 ECTS credits) Hilary term MA3441 This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 10% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Tristan McLoughlin tristan@maths.tcd.ie

## Description

On successful completion of this module, students will be able to:

• Demonstrate understanding - at an entry level- of 20th/21st century physics;
• Formulate solutions to complex problems;
• Apply quantum theoretical techniques to complex problems;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3443 Statistical physics I )

(05 ECTS credits) Michaelmas term MA2342 This module will be examined in a 2-hour examination in Trinity term. Two homework assignments will be given. One is devoted to solving exercises, another one is to perform a mini-research and to demonstrate in this way your comprehension of the subject. The assignments contribute 25% to the final annual mark. 11 weeks, 3 lectures including tutorials per week Manuela Kulaxizi -manuela@maths.tcd.ie

## Description

On successful completion of this module, students will be able to:

• Explain the ideas of equilibrium thermodynamics and apply them to various systems.
• Demonstrate an understanding of how macroscopic equilibrium properties arise from the underlying microscopic physics.
• Show familiarity with the notion of ensembles and use the formalism of statistical physics.
• Include interactions systematically through appropriate expansions.
• Explain the concept of a first order phase transition.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3444 Statistical physics II )

(05 ECTS credits) Hilary term MA3443 This module will be examined in a 2-hour examination in Trinity term. The continuous assessment percentage contribution to the annual results is 20%. 11 weeks, 3 lectures including tutorials per week Manuela Kulaxizi (manuela@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Explain the difference between fermions and bosons and work out the consequences for N-particle systems at low temperatures;
• Make content with the classical regime at high temperatures and or low particle densities;
• Apply the formalism of statistical physics to systems without particle number conservation (e.g. photons, phonons);
• Apply the formalism of quantum statistical physics to simple model systems;
• Apply the formalism of thermodynamics to magnetic/spin systems;
• Do a mean field analysis of spin systems;
• Solve the 1-dimensional Ising model, and show familiarity with Peierl's argument in 2 dimensions
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3463 Computation theory and logic)

(05 ECTS credits) Michaelmas term NA This module will be examined in a 2-hour examination in Trinity term. Fortnightly written assignments will count 10%, and 90% for the final. 10 weeks, 3 lectures including tutorials per week Colm O Dunlaing (odunlain@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Construct very simple Turing machine programs.
• Construct proofs of formulae in propositional and first-order logic, including resolution, the Deduction Theorem, and derived rules.
• Determine the solvability or otherwise of various computational problems.
• Extend their knowledge of mathematical logic or proceed to further study of the subject.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3484 Methods of mathematical economics )

(05 ECTS credits) Hilary term MA1212 This module will be examined in a 2-hour examination in Trinity term. 11 weeks, 3 lectures including tutorials per week David Wilkins (dwilkins@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• determine optimal solutions of simple linear programming problems using the simplex method;
• justify with reasoned logical argument the basic relationships between feasible and optimal solutions of a primal linear programming problem and those of the corresponding dual programme;
• explain why the simplex method provides effective algorithms for solving linear programming problems;
• explain applications of linear algebra and linear programming in contexts relevant to mathematical economics;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3496 Mathematics education)

(10 ECTS credits) Michaelmas term and Hilary term NA This module is assessed by means of coursework: tutorial exercises involving postings to the module's discussion forum (5%) a mathematical autobiography (5%); and a project and report related to students' experience in Schools (90%). Initially 3 per week (including tutorials); later, time in schools and some lectures/tutorials. Elizabeth Oldham (eoldham@tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Describe and critique major theories about mathematics teaching and learning;
• Outline and critique the context, aims, objectives, content, resource implications and assessment procedures of Irish school mathematics curricula;
• Identify and describe their own current beliefs about the nature of mathematics and their philosophies of mathematics education;
• Report on their classroom experience in the light of the theories and topics addressed: describing and analysing their own reactions to the school experience; describing the teaching approaches and analyzing the student behaviours observed; and (in conjunction with this)
• Research and present a project on a topic in mathematics education
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3E1 Engineering mathematics V)

(05 ECTS credits) Michaelmas term MA2E02 Assessment for this module is carried out by means of a written two-hour examination at the end of the academic year. The subject mark is based entirely on the result of this written examination. 11 weeks, 3 lectures including tutorials per week Joe O Hogain (johog@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Calculate the coefficients of both the complex and the real Fourier series for a variety functions, and to use them to solve some ordinary differential equations.
• Calculate Fourier transforms, discrete or continuous, for a variety of simple functions - students will then be able to use these to compute convolutions in simple cases.
• Solve the Laplace, heat and wave equations for a variety of boundary conditions in domains of simple geometry and with simple boundary conditions; the techniques available will include, separation of variables, Laplace and Fourier Transform methods.
• Apply various probability distributions to solve practical problems.
• Construct confidence intervals using sampling analysis.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA4445 Quantum field theory I)

(05 ECTS credits) Michaelmas term MA3442 (MA3444 recommended) This module will be examined in a 2-hour examination in Trinity term. 11 weeks, 3 lectures including tutorials per week Samson Shatashvili (samson@maths.tcd.ie)

## Description

• Noether's theorem, the Klein-Gordon field and its quantisation;
• The Dirac field and its quantisation;
• Quantisation of constrained systems;
• The Maxwell field and its quantisation;
• Perturbation theory, Wick's theorem, Feynman diagrams, S-matrix
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA4446 Quantum field theory II)

(05 ECTS credits) Hilary term MA4445 This module will be examined in a 2-hour examination in Trinity term. 11 weeks, 3 lectures including tutorials per week Samson Shatashvili (samson@maths.tcd.ie)

## Description

• Feynman diagram formalism for scalar ?4 theory;
• Feynman rules for Quantum Electrodynamics (QED);
• Elementary processes of QED;
• S-matrix: Scattering and decay;
• Trace technology;
• Crossing symmetry;
• Radiative corrections: Infrared and Ultraviolet divergencies, Loop computations, LSZ reduction formula, Optical theorem, Ward-Takahashi identities;
• Renormalization of electric charge;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA4448 General relativity)

(05 ECTS credits) Hilary term MA3429, MA3432 This module will be examined in a 2 hour examination at the annual session. Continuous assessment will contribute 15% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Andrei Parnachev (parnachev@maths.tcd.ie)

## Description

On successful completion of this module students will be able to

• Define the Einstein-Hilbert action and derive Einstein's equations from an action principle;
• Define the stress-energy-momentum tensor, obtain its components in an orthonormal tetrad, and obtain explicit expressions for the stress-energy-momentum tensor describing a perfect fluid matter distribution;
• Derive the canonical form of the Schwarzschild solution to the vacuum field equations under the sole assumption of spherical symmetry, and hence state Birkhoff's Theorem;
• Derive expressions for the gravitational redshift, perihelion advance of the planets, and light deflection in the Schwarzschild space-time and hence discuss solar system tests of General Relativity;
• Obtain the geodesic equations in arbitrary space-times and hence describe various trajectories such as radially in-falling particles or circular geodesics etc.;
• Obtain the maximal extension of the Schwarzschild solution in Kruskal coordinates and hence discuss the Schwarzschild black hole;
• Define spatial isotropy with respect to a universe filled with a congruence of time-like world-lines, discuss the consequences of global isotropy on the shear, vorticity and expansion of the congruence and hence construct the Friedmann-Robertson-Walker metric;
• Obtain the Friedmann and Raychaudhuri equations from the Einstein field equations, solve these equations for the scale factor and discuss the cosmogonical and eschatological consequences of the solutions;
• Derive the Einstein equations in the linear approximation and discuss the Newtonian limit in the weak-field, slow-moving approximation;
• Use the gauge freedom to show that, in the Einstein-deDonder gauge, the perturbations satisfy an inhomogeneous wave-equation, to solve in terms of plane-waves, and to use the residual gauge freedom to show that for waves propagating in ` the positive z-direction there are only two linearly independent non-zero components;
• Derive the Reissner-Nordstrom solution of the Einstein-Maxwell field equations, obtain its maximal extension and discuss the Reissner-Nordstrom black hole solution;
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA444D The standard model of elementary particle physics)

(05 ECTS credits) Hilary term MA3432, MA3442 This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 10% to the final grade for the module at the annual examination session. 11 weeks, 3 lectures including tutorials per week Stefan Sint (sint@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• Verify the Lorentz covariance of the Dirac and Klein-Gordon equations;
• Perform Lorentz transformations on spinors;
• Apply the gauge principle both in the abelian and non-abelian case;
• Apply the Goldstone theorem;
• Work out the consequences of the Higgs mechanism;
• Identify the basic interaction vertices between the fields in the Standard Model Lagrangian
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA4492 Project)

(10 ECTS credits) Michaelmas term and Hilary term Students must find a member of staff in College willing to supervise their work. This module will be 100% continuous assessment. Written thesis (which should normally exceed 35 pages) and presentations including a poster presentation. Academic year long module (2 terms), meetings with supervisor by arrangement. Significant independent work is required. Richard Timoney (richardt@maths.tcd.ie)

## Description

On successful completion of this module, students will be able to:

• demonstrate competence in independent study at a high mathematical level, at the forefront of knowledge in a specifically chosen topic
• demonstrate skills in scientific writing
• demonstrate presentation skills
• synthesise and apply materials used.
Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA1266 Introduction to Programming in C)

(5 ECTS credits) Hilary term Colm Ó Dúnlaing - odunlain@maths.tcd.ie

## Description

Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA2361 Computation Theory and Logic)

(5 ECTS credits) Michaelmas term Colm Ó Dúnlaing - odunlain@maths.tcd.ie

## Description

Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA346J Set Theory and Mathematical Logic)

(5 ECTS credits) Colm Ó Dúnlaing - odunlain@maths.tcd.ie

## Description

Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MA3469 Practical Numerical Simulations)

(5 ECTS credits) Michaelmas Term MA1241 Mechanics I and either MA2327 Ordinary Differential Equations or MA2332 Equations of Mathematic Physics II This module will be examined in 2 hour examination in Trinity term (60%). Four continuous assessment assignments will contribute 40% to the final grade for the module at the annual examination session. Supplemental if required will consist of 100% exam 11 weeks, 3 lectures including tutorials per week. Mike Peardon mjp@maths.tcd.ie

## Description

Learning Outcomes On successful completion of this module, students will be able to:
• • Write and compile numerical software in C++
• • Find an appropriate numerical technique to solve common problems in applied mathematics and theoretical physics and to recognise its limitations.
• • Construct numerical solutions to mathematical and physical problems in C++
• • Describe the output of their numerical software and interpret results reliably.
• Module Content The module aims to introduce the C++ programming language and object-oriented software concepts by getting students to construct numerical solutions to common problems in applied mathematics and theoretical physics. C++:
• • Language basics: the compiler, variables, functions.
• • Conditions and loops.
• • Arrays, pointers and references
• • C++ classes
• Numerical analysis:
• • Ordinary differential equations - solving initial-value problems with Euler and Runge-Kutta methods.
• • Ordinary differential equations - boundary-value problems and the shooting method.
• • Hamiltonian dynamics - symplectic integrators and the leap-frog method.
• • Partial differential equations - solving the M2d Laplace equation using simple iterative schemes (Jacobi, Gauss-Seidel and SOR) with Dirichlet and Von Neumann boundary data.
• • Introduction to Monte Carlo methods in statistical physics.
• Module Code & Name ECTs credits Duration and semester Prerequisite Subjects Assessment Contact Hours Contact Details

## (MAU11002; Mathematics Statistics and Computation Semester 2)

(10 ECTS credits) Hilary Term 11 weeks; 8 hours per week, including 5 lectures, 2 tutorials and 1 computer practical. • 1 or 2 lecturers from the school of mathematics, (nomen nominandum) • 1 lecturer from the department of statistics, (nomen nominandum) • teaching assistants/demonstrators for tutorial groups and practicals. 4 lectures + 2 tutorials per week will be covered by the school of maths; 1 lecture + 1 computer practical per week will be covered by the department of statistics Sinead Ryan ryan@maths.tcd.ie And Alberto Ramos Alberto.ramos@maths.tcd.ie

## Description

On successful completion of this module students will be able to • Manipulate vectors to perform alegebraic operations on them such as dot products and orthogonal projections and apply vector concepts to manipulate lines and planes in space R3 or in Rn with n≥4. • Use Gaussian elimination techniques to solve systems of linear equations, find inverses of matrices and solve problems which can be reduced to such systems of linear equations. • Manipulate matrices algebraically and use concepts related to matrices such as invertibility, symmetry, triangularity, nilpotence. • Manipulate numbers in different bases and explain the usefulness of the ideas in computing. • Use computer algebra and spreadsheets for elementary applications. • Explain basic ideas relating to functions of a single variable and their graphs such as limits, continuity, invertibility, even/odd, differentiabilty and solve basic problems involving these concepts. • Give basic properties and compute with a range of rational and standard transcendental functions, for instance to find derivatives, antiderivatives, critical points and to identify key features of their graphs. • Use a range of basic techniques of integration to find definite and indefinite integrals. • Apply techniques from calculus to a variety of applied problems. Module content The module is divided into a maths and a statistics part, with maths further divided into calculus and linear algebra/discrete mathematics. Mathematics: a) Calculus: 3 lectures plus one tutorial per week. The syllabus is largely based on the text book [Stewart-Day], and will cover most of Chapters 1-6 along with the beginning of Chapter 7 on differential equations: • Functions and graphs. Lines, polynomials, rational functions, exponential and logarithmic functions, trigonometric functions and the unit circle. • Limits, continuity, average rate of change, first principles definition of derivative, basic rules for differentiation • Graphical interpretation of derivatives, optimization problems • Exponential and log functions. Growth and decay applications. semilog and log-log plots. • Integration (definite and indefinite). Techniques of substitution and integration by parts. Applications. • Differential equations and initial value problems, solving first order linear equations. Applications in biology or ecology. b) Linear algebra/discrete mathematics: 1 lecture and 1 tutorial per week. The syllabus will cover parts of chapter 1 on sequences, limits of sequences and difference equations and then chapter 8 of [Stewart-Day] on linear algebra. The syllabus is approximately: • Sequences, limits of sequences, difference equations, discrete time models • Vectors and matrices , matrix algebra • inverse matrices, determinants. • systems of difference equations, systems of linear equations, eigenvalues and eigenvectors. Leslie matrices, matrix models. Statistics: There will be 1 lecture per week and 1 computer practical. The syllabus will cover much of chapters 11-13 of [Stewart-Day] and use [Bekerman-et-al] as main reference for R in the computer practicals. The syllabus is approximately: • Numerical and Graphical Descriptions of Data • Relationships and linear regression • Populations, Samples and Inference • Probability, Conditional Probability and Bayes’ Rule • Discrete and Continuous Random Variables • The Sampling Distribution • Confidence Intervals