# Courses

**Information for students starting in October 2024**

**Year 1**

Students are required to select **four courses** from the courses listed below.

The courses are based at the University of Oxford Mathematical Institute, Department of Computer Science and Department of Statistics.

## Oxford Mathematical Institute

**Course Overview:**

This course will serve as an introduction to optimal transportation theory, its application in the analysis of PDE, and its connections to the macroscopic description of interacting particle systems.

**Learning Outcomes:**

Getting familar with the Monge-Kantorovich problem and transport distances. Derivation of macroscopic models via the mean-field limit and their analysis based on contractivity of transport distances. Dynamic Interpretation and Geodesic convexity. A brief introduction to gradient flows and examples.

**Course Synopsis:**

- Interacting Particle Systems & PDE (2 hours)
- Granular Flow Models and McKean-Vlasov Equations.
- Nonlinear Diffusion and Aggregation-Diffusion Equations.

- Optimal Transportation: The metric side (4 hours)
- Functional Analysis tools: weak convergence of measures. Prokhorov’s Theorem. Direct Method of Calculus of Variations. (1 hour)
- Monge Problem. Kantorovich Duality. (1.5 hours)
- Transport distances between measures: properties. The real line. Probabilistic Interpretation: couplings.(1.5 hours)

- Mean Field Limit & Couplings (4 hours)
- Dobrushin approach: derivation of the Aggregation Equation. (1.5 hour)
- Sznitmann Coupling Method for the McKean-Vlasov equation. (1.5 hour)
- Boltzmann Equation for Maxwellian molecules: Tanaka Theorem. (1 hour)

- Gradient Flows: Aggregation-Diffusion Equations (6 hours)
- Brenier’s Theorem and Dynamic Interpretation of optimal tranport. Otto’s calculus. (2 hours)
- McCann’s Displacement Convexity: Internal, Interaction and Confinement Energies. (2 hours)

- Gradient Flow approach: Minimizing movements for the (McKean)-Vlasov equation. Properties of the variational scheme. Connection to mean-field limits. (2 hours)

**Reading List:**

- F. Golse,
*On the Dynamics of Large Particle Systems in the Mean Field Limit, Lecture Notes in Applied Mathematics and Mechanics 3.*Springer, 2016. - L. C. Evans,
*Weak convergence methods for nonlinear partial differential equations. CBMS Regional Conference Series in Mathematics 74*, AMS, 1990. - F. Santambrogio,
*Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and Their Applications*, Birkhauser 2015. - C. Villani,
*Topics in Optimal Transportation*, AMS Graduate Studies in Mathematics, 2003

*Please note that e-book versions of many books in the reading lists can be found on SOLO*

**Further Reading:**

- L. Ambrosio, G. Savare,
*Handbook of Differential Equations: Evolutionary Equations*, Volume 3-1, 2007. - C. Villani,
*Optimal Transport: Old and New*, Springer 2009

**More details: **Course: C4.9 Optimal Transport & Partial Differential Equations (2023-24) | Mathematical Institute (ox.ac.uk)

**General Prerequisites:** Basic linear algebra (such as eigenvalues and eigenvectors of real matrices), multivariate real analysis (such as norms, inner products, multivariate linear and quadratic functions, basis) and multivariable calculus (such as Taylor expansions, multivariate differentiation, gradients).

**Course Overview: **The solution of optimal decision-making and engineering design problems in which the objective and constraints are nonlinear functions of potentially (very) many variables is required on an everyday basis in the commercial and academic worlds. A closely-related subject is the solution of nonlinear systems of equations, also referred to as least-squares or data fitting problems that occur in almost every instance where observations or measurements are available for modelling a continuous process or phenomenon, such as in weather forecasting. The mathematical analysis of such optimization problems and of classical and modern methods for their solution are fundamental for understanding existing software and for developing new techniques for practical optimization problems at hand.

**More details:** Course: C6.2 Continuous Optimisation (2023-24) | Mathematical Institute (ox.ac.uk)

**Learning Outcomes:**

Students will learn how some of the various different ensembles of random matrices are defined. They will encounter some examples of the applications these have in Data Science, modelling Complex Quantum Systems, Mathematical Finance, Network Models, Numerical Linear Algebra, and Population Dynamics. They will learn how to analyse eigenvalue statistics, and see connections with other areas of mathematics and physics, including combinatorics, number theory, and statistical mechanics

**Course Synopsis:**

Introduction to matrix ensembles, including Wigner and Wishart random matrices, and the Gaussian and Circular Ensembles. Overview of connections with Data Science, Complex Quantum Systems, Mathematical Finance, Network Models, Numerical Linear Algebra, and Population Dynamics (1 Lecture)

Statement and proof of Wigner’s Semicircle Law; statement of Girko’s Circular Law; applications to Population Dynamics (May’s model). (3 lectures)

Statement and proof of the Marchenko-Pastur Law using the Stieltjes and R-transforms; applications to Data Science and Mathematical Finance. (3 lectures)

Derivation of the Joint Eigenvalue Probability Density for the Gaussian and Circular Ensembles;

method of orthogonal polynomials; applications to eigenvalue statistics in the large-matric limit;

behaviour in the bulk and at the edge of the spectrum; universality; applications to Numerical Linear

Algebra and Complex Quantum Systems (5 lectures)

Dyson Brownian Motion (2 lectures)

Connections to other problems in mathematics, including the longest increasing subsequence

problem; distribution of zeros of the Riemann zeta-function; topological genus expansions. (2

lectures)

**Reading List:**

- ML Mehta,
*Random Matrices*(Elsevier, Pure and Applied Mathematics Series) - GW Anderson, A Guionnet, O Zeitouni,
*An Introduction to Random Matrices*(Cambridge Studies in Advanced Mathematics) - ES Meckes,
*The Random Matrix Theory of the Classical Compact Groups*(Cambridge University Press) - G. Akemann, J. Baik & P. Di Francesco,
*The Oxford Handbook of Random Matrix Theory*(Oxford University Press) - G. Livan, M. Novaes & P. Vivo,
*Introduction to Random Matrices*(Springer Briefs in Mathematical Physics)

*Please note that e-book versions of many books in the reading lists can be found on SOLO*

**Further Reading:**

- T. Tao,
*Topics in Random Matrix Theory*(AMS Graduate Studies in Mathematics)

**More details: **Course: C7.7 Random Matrix Theory (2023-24) | Mathematical Institute (ox.ac.uk)

C8.1 Stochastic Differential Equations

**General Prerequisites:** Integration and measure theory, martingales in discrete and continuous time, stochastic calculus. Functional analysis is useful but not essential.

**Course Overview: **Stochastic analysis and partial differential equations are intricately connected. This is exemplified by the celebrated deep connections between Brownian motion and the classical heat equation, but this is only a very special case of a general phenomenon. We explore some of these connections, illustrating the benefits to both analysis and probability.

**Course Synopsis: **Feller processes and semigroups. Resolvents and generators. Hille-Yosida Theorem (without proof). Diffusions and elliptic operators, convergence and approximation. Stochastic differential equations and martingale problems. Duality. Speed and scale for one dimensional diffusions. Green's functions as occupation densities. The Dirichlet and Poisson problems. Feynman-Kac formula.

**More details: **Course: C8.2 Stochastic Analysis and PDEs (2023-24) | Mathematical Institute (ox.ac.uk)

**General Prerequisites: **Part B Graph Theory and Part A Probability. C8.3 Combinatorics is not as essential prerequisite for this course, though it is a natural companion for it.

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

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

**More details: **Course: C8.4 Probabilistic Combinatorics (2023-24) | Mathematical Institute (ox.ac.uk)

Course Overview:

The convergence theory of probability distributions on path space is an essential part of modern probability and stochastic analysis allowing the development of diffusion approximations and the study of scaling limits in many settings. The theory of large deviation is an important aspect of limit theory in probability as it enables a description of the probabilities of rare events. The emphasis of the course will be on the development of the necessary tools for proving various limit results and the analysis of large deviations which have universal value. These topics are fundamental within probability and stochastic analysis and have extensive applications in current research in the study of random systems, statistical mechanics, functional analysis, PDEs, quantum mechanics, quantitative finance and other applications.

Learning Outcomes:

The students will understand the notions of convergence of probability laws, and the tools for proving associated limit theorems. They will have developed the basic techniques for the establishing large deviation principles and be able to analyze some fundamental examples.

Course Synopsis:

1) (2 lectures) We will recall metric spaces, and introduce Polish spaces, and probability measures on metric spaces. Weak convergence of probability measures and tightness, Prohorov's theorem on tightness of probability measures, Skorohod's representation theorem for weak convergence.

2) (2 lectures) The criterion of pre-compactness for distributions on continuous path spaces, martingales and compactness.

3) (4 hours) Skorohod's topology and metric on the space D[0,∞)[0,∞) of right-continuous paths with left limits, basic properties such as completeness and separability, weak convergence and pre-compacness of distributions on D[0,∞)[0,∞). D. Aldous' pre-compactness criterion via stopping times.

4) (4 lectures) First examples - Cramér's theorem for finite dimensional distributions, Sanov's theorem. Schilder's theorem for the large deviation principle for Brownian motion in small time, law of the iterated logarithm for Brownian motion.

5) (4 lectures) General tools in large deviations. Rate functions, good rate functions, large deviation principles, weak large deviation principles and exponential tightness. Varadhan's contraction principle, functional limit theorems.

Decentralised Finance (2023-24)

Advanced Monte Carlo Methods (2023-24)

Advanced Numerical Methods (2023-24)

Financial Computing with C++ II (2023-24)

Fixed Income and Credit (2023-24)

Market Microstructure and Algorithmic Trading (2023-24)

Quantitative Risk Management (2023-24)

Advanced Volatility Modelling (2023-24)

## Department of Statistics, University of Oxford

This course runs Oct-Dec, but it may be possible to follow the course via pre-recorded videos and prepare an assessment.

**Aims and Objectives: **Many data come in the form of networks, for example friendship data and protein-protein interaction data. As the data usually cannot be modelled using simple independence assumptions, their statistical analysis provides many challenges. The course will give an introduction to the main problems and the main statistical techniques used in this field. The techniques are applicable to a wide range of complex problems. The statistical analysis benefits from insights which stem from probabilistic modelling, and the course will combine both aspects.

**Synopsis: **

Exploratory analysis of networks. The need for network summaries. Degree distribution, clustering coefficient, shortest path length. Motifs.

Probabilistic models: Bernoulli random graphs, geometric random graphs, preferential attachment models, small world networks, inhomogeneous random graphs, exponential random graphs.

Small subgraphs: Stein’s method for normal and Poisson approximation. Branching process approximations, threshold behaviour, shortest path between two vertices.

Statistical analysis of networks: Sampling from networks. Parameter estimation for models. Inference from networks: vertex characteristics and missing edges. Nonparametric graph comparison: subgraph counts, subsampling schemes, MCMC methods. A brief look at community detection.

**Reading: **R. Durrett, Random Graph Dynamics, Cambridge University Press,2007

E.D Kolaczyk and G. Csádi, Statistical Analysis of Network Data with R, Springer, 2014

M. Newman, Networks: An Introduction. Oxford University Press, 2010

**Recommended Prerequisites: **The course requires a good level of mathematical maturity. Students are expected to be familiar with core concepts in statistics (regression models, bias-variance tradeoff, Bayesian inference), probability (multivariate distributions, conditioning) and linear algebra (matrix-vector operations, eigenvalues and eigenvectors). Previous exposure to machine learning (empirical risk minimisation, dimensionality reduction, overfitting, regularisation) is highly recommended. Students would also benefit from being familiar with the material covered in the following courses offered in the Statistics department: SB2.1 (formerly SB2a) Foundations of Statistical Inference and in SB2.2 (formerly SB2b) Statistical Machine Learning.

**Aims and Objectives: **Machine learning is widely used across many scientific and engineering disciplines, to construct methods to find interesting patterns and to predict accurately in large datasets. This course introduces several widely used data machine learning techniques and describes their underpinning statistical principles and properties. The course studies both unsupervised and supervised learning and several advanced topics are covered in detail, including some state-of-the-art machine learning techniques. The course will also cover computational considerations of machine learning algorithms and how they can scale to large datasets.

**More details: **SC4 Advanced Topics in Statistical Machine Learning

**Synopsis: **Empirical risk minimisation. Loss functions. Generalization. Over- and under-fitting. Bias and variance. Regularisation.

Support vector machines.

Kernel methods and reproducing kernel Hilbert spaces. Representer theorem. Representation of probabilities in RKHS.

Deep learning: Neural networks. Computation graphs. Automatic differentiation. Stochastic gradient descent.

Probabilistic and Bayesian machine learning: Fundamentals of the Bayesian approach. EM algorithm. Variational inference. Latent variable models.

Deep generative models. Variational auto-encoders.

Gaussian processes. Bayesian optimisation.

**Software: **Knowledge of Python is not required for this course, but some examples may be done in Python. Students interested in learning Python are referred to the following free University IT online course, which should ideally be taken before the beginning of this course: https://skills.it.ox.ac.uk/whats-on#/course/LY046

**Reading: **C. Bishop, Pattern Recognition and Machine Learning, Springer,2007

K. Murphy, Machine Learning: a Probabilistic Perspective, MIT Press, 2012

**Further Reading: **T. Hastie, R. Tibshirani, J Friedman, Elements of Statistical Learning, Springer, 2009

Scikit-learn: Machine Learning in Python, Pedregosa et al., JMLR 12, pp. 2825-2830, 2011, http://scikit-learn.org/stable/tutorial/

SC5 Advanced Simulation Methods

## Department of Computer Science

Machine Learning

Geometric Deep Learning

Uncertainty in Deep Learning