Teaching
Supervisions

University of Surrey (UK), 2018—present
Principal PhD Supervisor
of Tommaso Macrelli with Alessandro Torrielli as Secondary PhD Supervisor 
University of Surrey (UK), 2017—present
Principal PhD Supervisor
of Lorenzo Raspollini with Jan Gutowski as Secondary PhD Supervisor 
University of Surrey (UK), 2015—present
Secondary PhD Supervisor
of Joakim Strömvall with Alessandro Torrielli as Principal PhD Supervisor 
University of Surrey (UK), 2014—present
BSc/MMath/MSc project supervisor 
University of Surrey (UK), 2013—2016
Collaborative PhD Supervisor
Alessandro Torrielli and I jointly supervised Antonio Pittelli 
University of Surrey (UK), 2012—2015
Secondary PhD Supervisor
of Fabrizio Nieri with Sara Pasquetti as Principal PhD Supervisor 
University of Surrey (UK), 2012
UG project supervisor
8week undergraduate research project supported by the Nuffield Foundation 
University of Surrey (UK), 2011—present
Tutor
This post involves academic and pastoral care of students. I also take part in the admissions process by means of interviews. 
Wolfson College (UK), 2010—2011
College tutor
This post involved pastoral care of students. I was responsible for about 80 students. It also included taking part in the admissions process by means of interviews.
Lecturer

University of Surrey (UK), 2018
Linear algebra
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Time & Place Lectures:
Tuesdays, 9:00—11:00 in LTL
Thursdays, 16:00—18:00 in LTL
Tutorials:
See your seminar group.Synopsis The aim of this module is to extend students' knowledge of matrices, vectors and systems of linear equations and to introduce the abstract concepts of vector spaces, linear maps, and inner products. The module is divided into four main parts: Part I: Systems of linear equations (rank and nullity of matrices, elementary row operations and rowechelon form, Gaussian elimination, solubility of linear equations); Part II: Vector spaces (axiomatic development of vector spaces, linear independence of vectors, basis representations of vectors, change of basis, dimension, vector subspaces); Part III: Linear operators (basic properties of linear operators, ranknullity theorem, matrix representation, CayleyHamilton theorem, eigenvalues and eigenvectors, eigenspaces, algebraic and geometric multiplicities, diagonalisation); Part IV: Inner product spaces (inner products, norms, CauchySchwarz inequality, orthogonality, orthogonal complement, GramSchmidt process, orthogonal and unitary changes of basis, isometries, selfadjoint operators). Prerequisites MAT1031 algebra Literature M Anthony & M Harvey Linear algebra: concepts and methods
H Anton & C Rorres Elementary linear algebra
D Poole Linear algebra: a modern introduction
Lecture Notes TBA Assessment Criteria Class test (25%), final exam (75%). An overall aggregate mark of 40% for the module is required to pass the module.
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University of Surrey (UK), 2017
Linear algebra
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Time & Place Lectures:
Mondays, 14:00—16:00 in LTL
Thursdays, 16:00—18:00 in LTL
Tutorials:
See your seminar group.Synopsis The aim of this module is to extend students' knowledge of matrices, vectors and systems of linear equations and to introduce the abstract concepts of vector spaces, linear maps, and inner products. The module is divided into four main parts: Part I: Systems of linear equations (rank and nullity of matrices, elementary row operations and rowechelon form, Gaussian elimination, solubility of linear equations); Part II: Vector spaces (axiomatic development of vector spaces, linear independence of vectors, basis representations of vectors, change of basis, dimension, vector subspaces); Part III: Linear operators (basic properties of linear operators, ranknullity theorem, matrix representation, CayleyHamilton theorem, eigenvalues and eigenvectors, eigenspaces, algebraic and geometric multiplicities, diagonalisation); Part IV: Inner product spaces (inner products, norms, CauchySchwarz inequality, orthogonality, orthogonal complement, GramSchmidt process, orthogonal and unitary changes of basis, isometries, selfadjoint operators). Prerequisites MAT1031 algebra Literature M Anthony & M Harvey Linear algebra: concepts and methods
H Anton & C Rorres Elementary linear algebra
D Poole Linear algebra: a modern introduction
Lecture Notes TBA Assessment Criteria Class test (25%), final exam (75%). An overall aggregate mark of 40% for the module is required to pass the module.
Facts Sheet PDF Print Module Specifications 
University of Surrey (UK), 2016
Relativity
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Time & Place Lectures:
Tuesdays, 16:00—17:00, TB11
Thursdays, 11:00—13:00, TB23
Tutorials:
Tuesdays, 17:00—18:00, TB11Synopsis This module introduces the basic concepts and techniques of special and general relativity. This module is divided into three parts: Part I: Basics of special relativity (such as Minkowski space, Lorentz transformations); Part II: Basics of (pseudo)Riemannian geometry (such as manifolds, metrics, connections, geodesics, torsion, and curvature); Part III: Basics of general relativity (such as Einstein's field equations, Einstein—Hilbert action, Schwarzschild solution and applications) Prerequisites MAT1005 vector calculus; MAT1036 classical dynamics; MAT2011 linear PDEs Literature R D'Inverno Introducing Einstein's relativity
M P do Carmo Riemannian geometry
L P Hughston & K P Tod An intorduction to general relativity
R Wald General relativity
Lecture Notes TBA Assessment Criteria Class test 1 (20%), class test 2 (20%), final exam (60%). An overall aggregate mark of 40% for the module is required to pass the module.
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University of Surrey (UK), 2016
Linear algebra
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Time & Place Lectures:
Mondays, 13:00—15:00 in LTL
Thursdays, 16:00—18:00 in LTL
Tutorials:
See your seminar group.Synopsis The aim of this module is to extend students' knowledge of matrices, vectors and systems of linear equations and to introduce the abstract concepts of vector spaces, linear maps, and inner products. The module is divided into four main parts: Part I: Systems of linear equations (rank and nullity of matrices, elementary row operations and rowechelon form, Gaussian elimination, solubility of linear equations); Part II: Vector spaces (axiomatic development of vector spaces, linear independence of vectors, basis representations of vectors, change of basis, dimension, vector subspaces); Part III: Linear operators (basic properties of linear operators, ranknullity theorem, matrix representation, CayleyHamilton theorem, eigenvalues and eigenvectors, eigenspaces, algebraic and geometric multiplicities, diagonalisation); Part IV: Inner product spaces (inner products, norms, CauchySchwarz inequality, orthogonality, orthogonal complement, GramSchmidt process, orthogonal and unitary changes of basis, isometries, selfadjoint operators). Prerequisites MAT1031 algebra Literature M Anthony & M Harvey Linear algebra: concepts and methods
H Anton & C Rorres Elementary linear algebra
D Poole Linear algebra: a modern introduction
Lecture Notes TBA Assessment Criteria Class test (25%), final exam (75%). An overall aggregate mark of 40% for the module is required to pass the module.
Facts Sheet PDF Print Module Specifications 
University of Surrey (UK), 2015
Linear algebra
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Time & Place Lectures:
Mondays, 14:00—16:00 in LTL
Fridays, 14:00—16:00 in LTL
Tutorials:
See your seminar group.Synopsis The aim of this module is to extend students' knowledge of matrices, vectors and systems of linear equations and to introduce the abstract concepts of vector spaces, linear maps, and inner products. The module is divided into four main parts: Part I: Systems of linear equations (rank and nullity of matrices, elementary row operations and rowechelon form, Gaussian elimination, solubility of linear equations); Part II: Vector spaces (axiomatic development of vector spaces, linear independence of vectors, basis representations of vectors, change of basis, dimension, vector subspaces); Part III: Linear operators (basic properties of linear operators, ranknullity theorem, matrix representation, CayleyHamilton theorem, eigenvalues and eigenvectors, eigenspaces, algebraic and geometric multiplicities, diagonalisation); Part IV: Inner product spaces (inner products, norms, CauchySchwarz inequality, orthogonality, orthogonal complement, GramSchmidt process, orthogonal and unitary changes of basis, isometries, selfadjoint operators). Prerequisites MAT1031 algebra Literature M Anthony & M Harvey Linear algebra: concepts and methods
H Anton & C Rorres Elementary linear algebra
D Poole Linear algebra: a modern introduction
Lecture Notes TBA Assessment Criteria Class test (25%), final exam (75%). An overall aggregate mark of 40% for the module is required to pass the module.
Facts Sheet PDF Print Module Specifications 
University of Surrey (UK), 2014/15
Operations research and optimisation
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Time & Place Lectures:
Mondays, 13:00—14:00 in AP1
Mondays, 16:00—17:00 in LTE
Fridays, 15:00—16:00 in LTD
Synopsis This module introduces a variety of commonly used techniques from operations research. The module leads to a deeper understanding of linear programming problems and the theory that underpins their solving. Tools such as the simplex method are presented and a basic introduction to nonlinear optimisation methods is also provided. This module is good preparation for elements of mathematical economics (MAT3002). However, this module is not a prerequisite for MAT3002. More generally, it supports and complements other modules where optimisation and constrained optimisation is considered. In particular, the module covers the following topics: problem formulation for linear programming problems; graphical method; simplex method and sensitivity analysis; duality and complementary slackness; theory and applications of the transportation algorithm; convex sets, convex functions, concave functions; nonlinear optimisation and conditions for local/global optima; Lagrange multipliers and Lagrange multiplier theory. Prerequisites Either MAT1037 linear algebra and vector calculus or both MAT1034 linear algebra and MAT1005 vector calculus Literature F S Hillier & G J Lieberman Introduction to operations research
H A Taha Operations research: an introduction
Lecture Notes TBA Assessment Criteria Class test (25%), final exam (75%). An overall aggregate mark of 40% for the module is required to pass the module.
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University of Surrey (UK), 2014
Linear algebra
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Time & Place Lectures:
Tuesdays, 10:00—11:00 in 01AC01
Thursdays, 09:00—11:00 in LTL
Fridays, 15:00—16:00, AP3 & AP4 (weeks 1—9 and 11) and LTL (weeks 10 and 12)
Tutorials:
See your seminar group.Synopsis The aim of this module is to extend students' knowledge of matrices, vectors and systems of linear equations and to introduce the abstract concepts of vector spaces, linear maps, and inner products. The module is divided into four main parts: Part I: Systems of linear equations (rank and nullity of matrices, elementary row operations and rowechelon form, Gaussian elimination, solubility of linear equations); Part II: Vector spaces (axiomatic development of vector spaces, linear independence of vectors, basis representations of vectors, change of basis, dimension, vector subspaces); Part III: Linear maps (basic properties of linear maps, ranknullity theorem, matrix representation, CayleyHamilton theorem, eigenvalues and eigenvectors, eigenspaces, algebraic and geometric multiplicities, diagonalisation); Part IV: Inner product spaces (inner products, norms, CauchySchwarz inequality, orthogonality, orthogonal complement, GramSchmidt process, orthogonal and unitary changes of basis, isometries, selfadjoint operators). Prerequisites MAT1031 algebra Literature M Anthony & M Harvey Linear algebra: concepts and methods
H Anton & C Rorres Elementary linear algebra
D Poole Linear algebra: a modern introduction
Lecture Notes TBA Assessment Criteria Class test (25%), final exam (75%). An overall aggregate mark of 40% for the module is required to pass the module.
Facts Sheet PDF Print Module Specifications 
University of Surrey (UK), 2014
Introduction to relativity
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Time & Place Lectures:
Tuesdays, 15:00—16:00, 24AA04
Thursdays, 16:00—18:00, TB21
Tutorials:
Wednesdays, 09:00—10:00, 01AC03Synopsis This module introduces the basic concepts and techniques of special and general relativity. This module is divided into three parts: Part I: Basics of special relativity (such as Minkowski space, Lorentz transformations); Part II: Basics of (pseudo)Riemannian geometry (such as manifolds, metrics, connections, geodesics, torsion, and curvature); Part III: Basics of general relativity (such as Einstein's field equations, Einstein—Hilbert action, Schwarzschild solution and applications) Prerequisites MAT1005 vector calculus; MAT1036 classical dynamics; MAT2011 linear PDEs Literature R D'Inverno Introducing Einstein's relativity
M P do Carmo Riemannian geometry
L P Hughston & K P Tod An intorduction to general relativity
R Wald General relativity
Lecture Notes TBA Assessment Criteria Class test 1 (20%), class test 2 (20%), final exam (60%). An overall aggregate mark of 40% for the module is required to pass the module.
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University of Surrey (UK), 2013
(Introduction to) Relativity
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Time & Place Lectures:
Tuesdays, 1:00pm—2:00pm, LTE (week 1—12)
Fridays, 10:00am—12:00pm, 24 AA 04 (week 1—6)
Fridays, 10:00am—12:00pm, 35 AC 04 (week 7—12)
Tutorials:
Thursdays: 4:00pm—5:00pm, 03 AZ 01 (week 11 in TB 18)Synopsis This module introduces the basic concepts and techniques of special and general relativity. This module is divided into three parts: Part I: Basics of special relativity (such as Minkowski space, Lorentz transformations); Part II: Basics of (pseudo)Riemannian geometry (such as manifolds, metrics, connections, geodesics, torsion, and curvature); Part III: Basics of general relativity (such as Einstein's field equations, Einstein—Hilbert action, Schwarzschild solution and applications) Prerequisites MAT1005 vector calculus; MAT1036 classical dynamics; MAT2011 linear PDEs Literature M P do Carmo Riemannian geometry
R Wald General relativity
Lecture Notes TBA Assessment Criteria MAT3038:
Class test 1 (10%), class test 2 (15%), final exam (75%). An overall aggregate mark of 40% for the module is required to pass the module.
MATM036:
Class test 1 (10%), class test 2 (15%), final exam (50%), 30min presentation (25%) An overall aggregate mark of 50% for the module is required to pass the module.Facts Sheet PDF Print Module Specifications 
University of Cambridge (UK), 2009
Gluon scattering amplitudes, twistors, and integrability
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Time & Place Tuesdays, 2:00pm—3:00pm, MR12 (CMS) Synopsis Traditionally, scattering amplitudes in quantum field theory are computed by means of Feynman rules obtained from a local Lagrangian description of the theory. However, as one increases the number and species of particles involved in the scattering processes, the complexity of such computations increases dramatically. On a more technical level, this is due to the fact that the Feynman rules involve (gaugedependent) offshell states. One therefore seeks for alternative formulations.
In this course, we consider certain supersymmetric gauge theories on fourdimensional spacetime and reformulate them on an auxiliary space called twistor space. We first discuss how these theories can be understood in terms of twistors at the classical level. In particular, we shall see how the field equations can be encoded in geometric data on the twistor space thereby making use of the spinor helicity formalism. We then move on and reformulate treelevel gluon scattering amplitudes in terms of twistor theory. The twistor approach makes manifest certain symmetries that are not straightforwardly seen in terms of the Feynman approach, thus explaining certain magic cancellations. If time permits, we shall also talk about loop amplitudes and string theory reinterpretations.Prerequisites The module is selfcontained. No previous knowledge of twistor theory is assumed. However, familiarity with topics from Advanced quantum field theory (L24) such as nonAbelian gauge fields and from Introduction to supersymmetry (M16) such as superspaces would be very helpful. On the mathematical side, a basic knowledge of complex calculus and differential geometry (e.g. the concepts of manifolds, differential forms, etc) would also be helpful. Literature R O Wells Complex manifolds and mathematical physics
R S Ward & R O Wells Twistor geometry and field theory
L J Mason & N M J Woodhouse Integrability, selfduality, and twistor theory
M Dunajski Solitons, instantons, and twistors
F Cachazo & P Svrcek Lectures on twistor strings and perturbative Yang—Mills theory
J Bedford On perturbative field theory and twistor string theoryLecture Notes The lecture notes for this module can be found on the arXiv (current version: v2). Alternatively, they can be downloaded from here (current version: v2). Assessment Criteria This is a nonexaminable module. Print Module Specifications 
Leibniz Universität Hannover (Germany), 2005/06
Quantum field theory
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Time & Place Lectures:
Tuesdays, 3:00pm—5:00pm, 268 (ITP)
Thursdays, 11:00am—1:00pm, 269 (ITP)
Tutorials:
Mondays, 5:00pm—7:00pm, 269 (ITP)Synopsis This is an introductory module on quantum field theory covering the following (main) topics: classical field theory, construction of quantum fields, perturbation theory, scattering theory and Feynman diagrams, renormalization theory, additional topics to be determined. Prerequisites Quantum theory I & II Literature M Peskin & D Schroeder An introduction to quantum field theory
S Weinberg The quantum theory of fields, volume I
J ZinnJustin Quantum field theory and critical phenomena
Assessment Criteria There will be weekly problem sets and a final exam (take home). Marks will be given according to the following weighted average: problem sets: 60%; final exam: 40%. In total you need at least 60%. Exercises Problem Set 01  Due 17 October
Problem Set 02  Due 24 October
Problem Set 03  Due 13 October
Problem Set 04  Due 07 November
Problem Set 05  Due 14 November
Problem Set 06  Due 21 November
Problem Set 07  Due 28 November
Problem Set 08  Due 05 December
Problem Set 09  Due 12 December
Problem Set 10  Due 19 December
Problem Set 11  Due 16 January
Problem Set 12  Due 23 January
Problem Set 13  Due 30 January
Problem Set 14  Due 06 FebruaryPrint Module Specifications 
Leibniz Universität Hannover (Germany), 2005
General relativity
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Time & Place Tuesdays, 11:00am—1:00pm, 269 (ITP)
Wednesdays, 5:00pm—7:00pm, 268 (ITP)
Thursdays, 9:00am—11:00am, 268 (ITP)Synopsis This module is essentially divided into three parts. In the first part we discuss (semi)Riemannian geometry by introducing the basic notions of differential geometry such as differentiable manifolds, tangent spaces and vector fields, metrics, affine connections, parallel transport, geodesics, curvature and torsion. The second part deals with the basics of general relativity. We will talk about the field equations, the Einstein—Hilbert action and the Schwarzschild solution. The latter includes a discussion on perihelion precession, light bending and redshift. Finally, in the last part we will focus on some advanced topic. Prerequisites Knowledge of multivariable calculus and linear algebra is needed. Furthermore, a well founded knowledge of classical mechanics and electromagnetism is essential, as well. Literature M P do Carmo Riemannian geometry
T Sakai Riemannian geometry
R Wald General relativity
N Dragon Geometrie der RelativitätstheorieAssessment Criteria To pass the module at least 25% of the maximally reachable coursework marks plus one talk (90 min + written summary) on one of the following topics are needed:
A summary of special relativity For more details, click here.
Lorentz transformations and SL(2,C)
Differential forms
Frenet—Serret equations, vielbeins and Maurer—Cartan equations
Gauge theory  from Maxwell to Yang—Mills and Einstein
Energy stress tensor
Graviational waves  theory and experiment
Perihelion precession and light bending in the Schwarzschild field
Parallel transport and precession
Black holes and Kruskal extension
Light propagation and Penrose diagrams
Robertson—Walker universe
Exercises Problem Set 01  Due 14 April
Problem Set 02  Due 21 April
Problem Set 03  Due 28 April
Problem Set 04  Due 03 May
Problem Set 05  Due 12 May
Problem Set 06  Due 26 May
Problem Set 07  Due 02 June
Problem Set 08  Due 09 June
Problem Set 09  Due 16 June
Problem Set 10  Due 30 June
Problem Set 11  Due 07 JulyPrint Module Specifications 
Leibniz Universität Hannover (Germany), 2004
General relativity
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Time & Place Mondays, 4:00pm—6:00pm, 368a (ITP)
Tuesdays, 5:00pm—7:00pm, 368a (ITP)Synopsis This module is essentially divided into three parts. In the first part we discuss (semi)Riemannian geometry by introducing differentiable manifolds, tangent spaces and vector fields, metrics, affine connections, parallel transport, geodesics and curvature. The second part deals with the basics of general relativity. We will talk about the field equations, the Einstein—Hilbert action and the Schwarzschild solution. The latter includes a discussion on perihelion precession, light bending and redshift. Finally, in the last part we will focus on KaluzaKlein reductions. Prerequisites Knowledge of multivariable calculus and linear algebra is needed. A basic knowledge of classical mechanics and electromagnetism is essential, as well. Literature M P do Carmo Riemannian geometry
T Sakai Riemannian geometry
R Wald General relativity
N Dragon Geometrie der RelativitätstheorieAssessment Criteria To get a Seminarschein at the end, one needs at least 25% of the maximally reachable coursework marks plus one talk (approximately half an hour). To get an ordinary Schein, one needs least 50% of the marks plus a talk. Exercises Problem Set 01  Due 26 April
Problem Set 02  Due 04 May
Problem Set 03  Due 11 May
Problem Set 04  Due 18 May
Problem Set 05  Due 25 May
Problem Set 06  Due 08 June
Problem Set 07  Due 15 June
Problem Set 08  Due 22 June
Problem Set 09  Due 06 July
Problem Set 10  Due 13 July
Problem Set 11  Due 20 JulyPrint Module Specifications
Teaching Assistant

Leibniz Universität Hannover (Germany), 2004/05
Theoretical mechanics and electrodynamics
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Time & Place Tuesdays, 1:00pm—3:00pm, 269 (ITP) Assessment Criteria 50% of the coursework to be admitted for the final exam and 50% (?) of the final exam to pass the module. You should also participate in the tutorials actively. Exercises Problem Set 01  Due 19 October
Problem Set 02  Due 26 October
Problem Set 03  Due 02 November
Problem Set 04  Due 09 November
Problem Set 05  Due 16 November
Problem Set 06  Due 23 November
Problem Set 07  Due 30 November
Problem Set 08  Due 07 December
Problem Set 09  Due 17 December
Problem Set 10  Due 21 December
Problem Set 11  Due 18 JanuaryPrint Module Specifications 
Leibniz Universität Hannover (Germany), 2003/04
Theoretical mechanics and electrodynamics
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Time & Place Tuesdays, 1:00pm—3:00pm, 368a (ITP) Assessment Criteria 50% of the coursework to be admitted for the final exam and 50% (?) of the final exam to pass the module. You should also participate in the tutorials actively. Exercises Problem Set 01  Due 21 October
Problem Set 02  Due 28 October
Problem Set 03  Due 04 November
Problem Set 04  Due 11 November
Problem Set 05  Due 18 November
Problem Set 06  Due 25 November
Problem Set 07  Due 02 December
Problem Set 08  Due 09 December
Problem Set 09  Due 16 December
Problem Set 10  Due 06 January
Problem Set 11  Due 13 January
Problem Set 12  Due 20 January
Problem Set 13  Due 27 JanuaryPrint Module Specifications
I have also tutored other modules which are not listed above. These include advanced quantum mechanics (Leibniz Universität Hannover) as well as advanced statistical mechanics and quantum field theory (Duke University).