**Content**

**Intended Learning Outcomes**

- Describe the problem of supervised learning from the point of view of function approximation, optimization, and statistics.
- Identify the most suitable optimization and modelling approach for a given machine learning problem.
- Analyse the performance of various optimization algorthms from the point of view of computational complexity (both space and time) and statistical accuracy.
- Implement a simple neural network architecture and apply it to a pattern recognition task.
- Summarize current developments in deep learning, including sequence models, generative models, robustness, and reinforcement learning.

**Lecturer: **Christian BÃ¶hning

**Term(s):** Term 2

**Status for Mathematics students:** List C

**Commitment:** 30 lectures plus assignments

**Assessment:** Assignments (15%), 3 hour written exam (85%).

**Prerequisites:**

A background in algebra (especially MA249 Algebra II) is essential. The module develops more specialised material in commutative algebra and in geometry from first principles, but MA3G6 Commutative Algebra will be useful. More than technical prerequisites, the main requirement is the sophistication to work simultaneously with ideas from several areas of mathematics, and to think algebraically and geometrically. Some familiarity with projective geometry (e.g. from MA243 Geometry) is helpful, though not essential.

**Leads To: **A first module in algebraic geometry is a basic requirement for study in geometry, number theory or many branches of algebra or mathematical physics at the MSc or PhD level. Many MA469 projects are on offer involving ideas from algebraic geometry.

**Content**:

Algebraic geometry studies solution sets of polynomial equations by geometric methods. This type of equations is ubiquitous in mathematics and much more versatile and flexible than one might as first expect (for example, every compact smooth manifold is diffeomorphic to the zero set of a certain number of real polynomials in R^N). On the other hand, polynomials show remarkable rigidity properties in other situations and can be defined over any ring, and this leads to important arithmetic ramifications of algebraic geometry.

Methodically, two contrasting cross-fertilizing aspects have pervaded the subject: one providing formidable abstract machinery and striving for maximum generality, the other experimental and computational, focusing on illuminating examples and forming the concrete geometric backbone of the first aspect, often uncovering fascinating phenomena overlooked from the bird's eye view of the abstract approach.

In the lectures, we will introduce the category of (quasi-projective) varieties, morphisms and rational maps between them, and then proceed to a study of some of the most basic geometric attributes of varieties: dimension, tangent spaces, regular and singular points, degree. Moreover, we will present many concrete examples, e.g., rational normal curves, Grassmannians, flag and Schubert varieties, surfaces in projective three-space and their lines, Veronese and Segre varieties etc.

**Books**:

- Atiyah M.& Macdonald I. G., Introduction to commutative algebra, Addison-Wesley, Reading MA (1969)
- Harris, J., Algebraic Geometry, A First Course, Graduate Texts in Mathematics 133, Springer-Verlag (1992)
- Mumford, D., Algebraic Geometry I: Complex Projective Varieties, Classics in Mathematics, reprint of the 1st ed. (1976); Springer-Verlag (1995)
- Reid, M., Undergraduate Algebraic Geometry, London Math. Soc. Student Texts 12, Cambridge University Press (2010)
- Shafarevich, I.R., Basic Algebraic Geometry 1, second edition, Springer-Verlag (1994)
- Zariski, O. & Samuel, P., Commutative algebra, Vol. II, Van Nos- trand, New York (1960)

### Overview

**Location: Warwick Mathematics Institute**

In this lecture will learn how to start the modelling process by thinking about the model's static structure, which then in a dynamic model gives rise to the choice of variables. Finally, with the dive into mathematical learning theories, the students will understand that a mathematical model is never finished, but needs recursive learning steps to improve its parametrisation and even structure.

A very important aspect of the lecture is the smooth transition from static to dynamic stochastic models with the help of rule-based system descriptions which have evolved from the modelling of chemical reactions.

### Weekly Overview

**Introduction:**

**Week 1:**Mathematical Modelling, Past, Present and Future

â€¢ What is Mathematical Modelling?

â€¢ Why Complex Systems?..

â€¢ Philosophy of Science, Empirical Data and Prediction.

â€¢ About this course.

Part I Structural Modelling

Part I Structural Modelling

**Week 2:**Relational Structures

â€¢ Relational family: hypergraphs, simplicial complexes and hierachical hypergraphs.

â€¢ Graph characteristics, examples from real world complex systems (social science, infrastructure, economy, biology, internet).

â€¢ Introduction to algebraic and computational graph theory.**Week 3:** Transformations of Relational Models

â€¢ Connections between graphs, hypergraphs, simplicial complexes and hierachical hypergraphs.

â€¢ Applications of hierachical hypergraphs.

â€¢ Stochastic processes of changing relational model topologies. **Part II Dynamic Modelling****Week 4:** Stochastic Processes

â€¢ Basic concepts, Poisson Process.

â€¢ Opinion formation: relations and correlations.

â€¢ Master eqation type-rule based stochastic collision processes.**Week 5:** Applications of type-rule based stochastic collision processes

â€¢ Chemical reactions and Biochemistry.

â€¢ Covid-19 Epidemiology.

â€¢ Economics and Sociology, Agent-based modelling.**Week 6:** Dynamical Systems (single compartment)

â€¢ Basic concepts, examples.

â€¢ Relation between type-rule-based stochastic collision processes in single compartments and ODE

â€¢ Applications, connections between dynamical systems and structural modelling (from Part I), the interaction graph, feedback loops.

â€¢ Time scales: evolutionary outlook.**Week 7:** Spatial processes and Partial Differential Equations:

â€¢ Type-rule-based multi-compartment models.

â€¢ Reaction-Diffusion Equations.

â€¢ Applications.**Part III Data Analysis and Machine Learning****Week 8:** Statistics and Mathematical Modelling

â€¢ Statistical Models and Data.

â€¢ Classification.

â€¢ Parametrisation.**Week 9:** Machine Learning and Mathematical Modelling:

â€¢ Mathematical Learning Theory.

â€¢ Bayesian Networks.

â€¢ Bayesian Model Selection. **Week 10:** Neural Networks and Deep Learning:

â€¢ Basic concepts.

â€¢ Neural Networks and Machine Learning.

â€¢ Discussion and outlook.

https://www.mathematical-modelling.science/index.php/lectures/warwick-2020-2021

This is the third module in the Warwick algebraic topology sequence.

The abelian group structure on homology is upgraded via dualization to a natural ring structure on cohomology. This additional structure makes cohomology a more powerful invariant than homology. Cohomology is also a more natural invariant in many contexts, and various cohomology theories play a key role in a number of fields, including differential geometry, mathematical physics, algebraic geometry, and number theory, among others.

PoincarÃ© duality is a relation
between the homology and cohomology of an oriented manifold. The ring
structure of cohomology plays a critical role in its proof.

The module MA4K0 Introduction to Uncertainty Quantification sits at the meeting point of mathematics, statistics, and many applications. We draw on techniques from functional analysis, numerical analysis, probability theory, Bayesian statistics, and computational mathematics to answer questions such as

- How do we represent random or uncertain quantities?
- How can we propagate uncertainty forwards through systems of interest in the real world to make predictions?
- How can we propagate uncertainty backwards through systems of interest in the real world to learn about them, e.g. initial conditions or governing parameters?

#### Overview

There is much active mathematical research into aeroacoustics (the study of sound in aircraft engines). This field is closely followed, and often contributed to (sometimes helpfully) by engineers in both academia and industry (e.g. Airbus, Boeing, NASA, etc). The aim of this course is to give an overview of the mathematical techniques needed to understand the current research problems, and read current papers in the area. This could lead on to several possible PhD projects, including in asymptotics, numerical analysis, and stability theory.

#### Aims

The application of wave theory to problems involving the generation, propagation and scattering of acoustic and other waves is of considerable relevance in many practical situations. These include, for example, underwater sound propagation, aircraft noise, remote sensing, the effect of noise in built-up areas, and a variety of medical diagnostic applications. This course would aim to provide the basic theory of wave generation, propagation and scattering, and an overview of the mathematical methods and approximations used to tackle these problems, with emphasis on applications to aeroacoustics. The ultimate aim is for students to understand the underlying mathematical tools of acoustics sufficiently to read current research publications on acoustics, and to be able to apply these techniques to current research questions within mathematics, engineering and industry.

#### Learning Outcomes

- Reproduce standard models and arguments for sound generation and propagation.
- Apply mathematical techniques to model sound generation and propagation in simple systems.
- Understand and apply Wiener-Hopf factorisation in the scalar case.

#### Approximate Syllabus

- Some general acoustic theory.
- Sound generation by turbulence and moving bodies (including the Lighthill and Ffowcs Williams Hawkings acoustic analogies).
- Scattering (including the scalar Wiener-Hopf technique applied to the Sommerfeld problem of scattering by a sharp edge)
- Long-distance sound propagation including nonlinear and viscous effects.
- Wave-guides.
- High frequencies and Ray Tracing.

#### Reading List

- D.G. Crighton, A.P. Dowling, J.E. Ffowcs Williams, et al, "Modern Methods in Analyticial Acoustics", Springer 1992.
- M. Howe, "Acoustics & Aerodynamic Sound", Cambridge 2015 (available online through Warwick Library).
- S.W. Rienstra & A. Hirschberg, "An Introduction to Acoustics", (available online).