Moodle@Warwick

Lecturer: Christian Böhning

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)

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

This module, MA4J7 (cohomology and Poincaré duality), has the prerequisites MA3F1 (introduction to topology) and MA3H6 (algebraic topology).

Cohomology is a dual theory to homology; it continues our development of algebraic tools for the study of topological spaces. Cohomology is a richer, more algebraic, theory than homology is because it has a naturally defined ring structure coming from the cup product. It is also a key tool in the modern proof of Poincaré duality for manifolds.

The material covered in this module is essentially required for advanced study in the fields of topology, differential geometry, algebraic geometry, algebraic number theory, and others.

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).

Topological Data Analysis (TDA) is an approach to data analysis based on techniques from algebraic topology. Topology is the study of properties of sets that are invariant under continuous deformations; it is concerned with concepts such as ``nearness'', ``neighbourhood'', and ``convergence''. Nowadays, topological ideas are an indispensable part of many fields of mathematics, ranging from number theory to partial differential equations. Algebraic topology, in particular, aims to understand topological properties of spaces through algebraic invariants. The premise of topological data analysis is that data there is an underlying topological structure to data. Familiar examples include clustering, where the aim is to subdivide data into different clusters, or ``connected components'', and connectivity in networks. In this module we introduce persistent homology, a powerful method for studying the topology of data. We discuss the theoretical foundations, as well as computational and algorithmic aspects and various applications. While the course is mainly theoretical in nature, you are encouraged to experiment using a range of available software and applications. The lecture material will be available as video recording and slides, and exercises will be published semi-regularly.

Intended Learning Outcomes

Upon completion of this module you should be able to:

• understand how topological information can be extracted from discrete data;
• use persistent homology to compute persistence diagrams and barcodes;
• explain the different parts of the persistent homology pipeline and the computational challenges involved;
• evaluate the stability and robustness of persistent homology computations;
• summarize different approaches to the topology of data and discuss applications

Literature

1. Steve Oudot. Persistence Theory: From Quiver Representations to Data Analysis. AMS 2015
2. Herbert Edelsbrunner and John Harer. Computational Topology, An Introduction. AMS 2010
3. Nina Otter, Mason A Porter, Ulrike Tillmann, Peter Grindrod & Heather A Harrington. A roadmap for the computation of persistent homology. 2017

More specialised sources and papers will be made available in time.

The purpose of this module is to provide rigorous training in probability theory for students who plan to specialise in this area or expect probability to feature as an essential tool in their subsequent research. It will also be accessible to students who never got into probability theory beyond core-module level taught in the first year and who are eager to get acquainted with basic probability theory, in particular, the aim is to appeal to but not limited to students working in analysis, dynamical systems, combinatorics & discrete mathematics, and statistical mechanics. To include these two different groups of students and to accommodate their needs and various background the module will cover in the first two weeks a steep learning curve into basic probability theory (see part I below). Secondly, the written assessment, 50 % essay with 16 pages, can be chosen either from a list of basic probability theory (standard textbooks in probability and graduate lecture notes on probability theory) or from a list of high-level hot research topics including original research papers and reviews and lecture notes (see below). List of possible essay topics see attached pdf - file.

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).

Throughout this module, we hope to show you that many of the challenges health care systems face are fundamentally economic. With this in mind, this module aims to offer an informative introduction to economic concepts and tools, with a view to understanding how these concepts can be used to answer various questions in health care.

Over the 5 days of the module, we will be looking at various issues, topics and questions that modern health care systems (such as our NHS) are called to answer, including the extent of government intervention in health care provision, contemporary ways of financing health care, methods for estimating the inputs and outputs of health care programmes, and, importantly, optimal ways of allocating our limited resources to existing and new interventions and technologies.We look forward to welcoming you to our module in early March.