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

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.

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).

MB ChB Phase 3 spans year 3 AND 4.

2017 Cohort are supported through year 4 via the MD30X-19/20 module Moodle space.

The two laboratory mini-projects conclude your MSc year and prepare you for your PhD projects. MD979 is the first one. Due to the highly inter-disciplinary nature of the IBR DTP, you have a great variety of choice for these projects. You will have an 11-week period for each of the two mini-project modules (MD979 and the subsequent MD980). Normally there would be one experimental biology project and one either theoretical biology project (e.g., bioinformatics, computational biology) or experimental project in chemistry, physics or engineering. The supervisor pool is accordingly expanded to include colleagues across the departments of the Faculty of Science in addition to WMS.

The projects will either be developed by the you together with an academic from the supervisor pool, or initially by the supervisors alone, who will submit projects directly to the mini-project call for presentation to the student cohort at a mini-project “fair”. You should be aware that all projects will first be vetted in the IBR DTP management team before inclusion in the course.

This short course is designed to develop the skills required to manage patients with a range of chronic diseases such as diabetes, dementia, musculoskeletal conditions, depression and mental illness, and end of life care within primary care.

The course will introduce you to the knowledge and skills that are needed for the delivery and organisation of high quality care, responsive to the physical, emotional, psychological and social needs of people with chronic conditions within the multidisciplinary setting.

The course will provide a sound foundation and understanding of economic concepts and their relevance to decisions around the allocation of healthcare resources, allowing you to critically appraise health economics studies and work effectively with health economists in your team.  It will provide you with a toolkit of methods useful for research and healthcare management.

The Physical Biology of the Cell module is a core module of the MSc IBR, which underpins the MRC-funded IBR DTP.

The module aims to provide a physical sciences perspective to cellular biology and equip postgraduate students to begin a research career at the interface of biology and physics.

You will explore the basic physical concepts underlying the behaviour of biomolecules, dynamic cell processes, cellular structure and signalling events. You will learn how to estimate sizes, speed and energy requirements for a variety of biological processes and build simple explicit models to fit experimental data from cell biological experiments.

PBoC is about learning to ask and answer quantitative scientific questions in the realm of biophysical cell biology.

It is arguably possible to sask scientific questions that are not quantitative*, but in general, useful scientific ideas make quantitative predictions that can be tested by observation and experiment. And arguably again, the most powerful scientific ideas are those that make the firmest quantitative predictions, and can thereby be definitively disproved.

Our goal with this course is to equip you with a basic set of tools to think quantitatively about the biological world, design better (more incisive) experiments, and analyse and interpret your data in useful and formally correct ways.

On completing the module, you should be able to analyse and quantify physical biological properties and behaviours of living systems; formulate scientific questions by harnessing the core concepts of physical biology and design experiments that effectively address your scientific questions.

PBoC is designed to help you to think!  Your instructors will aim to make the material challenging, but accessible, and above all, interesting.

Cohort A01 (November 2016)

The aim of the material being taught during this module is to help you to gain a better understanding of research and critical appraisal. As such, this module will help prepare you for other modules, and
also increase your ability to use research as part of your clinical or research / management role.  While the material may sometimes be challenging, it is appropriate for a Masters level module, which aims to help students think deeply and develop their critical skills.  The course includes a large practical component in which you will have the opportunity to practice the new skills that you have been taught, and to undertake some of the tasks necessary for the course assignment.  

Dear All,

Welcome to Qualitative Research Methods in Health. 

This module aims to provide you with a) a critical perspective on the contribution of qualitative research methods to understanding and improving health and b) an introduction to qualitative research methods and their application in health related research.

Aims

Develop knowledge and understanding of qualitative methods as used in health related research and develop your skills in the use of these methods. Gain the capability to use these research methods appropriately for undertaking research and evaluation both as part of postgraduate study and in your working environment.

Learning Outcomes
By the end of the module you should be able to:
 Demonstrate a critical understanding of the origins and usage of qualitative research methods in relation to health.
 Demonstrate how to develop a research question and use appropriate qualitative and methods to answer it.
 Demonstrate an understanding of the range of research methods and when and how they should be used.
 Demonstrate a critical understanding of the use of qualitative methods in relation to other widely used research methods in health care.

Here and in the module guide, you will find the pre-course preparation task. We strongly encourage you to undertake the preparatory work, especially if you are new to qualitative research methods.
We look forward to meeting you.

Frances Griffiths and Bronwyn Harris
Module Co-leads

This course focuses on the development of analysis and integration strategies for mixed methods researchers. It follows on from the Mixed Methods Design course by introducing students to a range of mixed methods analysis approaches and techniques through seminar style presentations, hands on workshops and tutorials where students can work on their own mixed methods projects.

This course outlines Moodle's features by providing examples of activities and resources.

This course is used for demoing quizzes in student view.