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TOM LEINSTER
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TL.GRP.1
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- Advanced linear algebra
- 20 point group project
- Single honours mathematics or any joint honours degree
- Your first exposure to linear algebra was probably quite computational, involving matrices, row operations, determinants, etc. But there is a different approach to linear algebra, an abstract approach, which is much closer in feel to group and ring theory. Here, the roles of vector space and linear map are emphasized, and as much as possible is done without matrices or determinants. The contrast between the two approaches can be illustrated by two different proofs of the fact that a square matrix of complex numbers (or, more abstractly, an operator on a finite-dimensional vector space over C) always has at least one eigenvalue. The proof you probably know goes via the notions of determinant and characteristic polynomial, and uses the fact that a matrix is non-invertible if and only if its determinant is zero. But there is also a more direct, abstract proof, which you will learn in this project. The core material on eigenvalues and eigenvectors leads on to more advanced topics such as Jordan normal form and operators on inner product spaces.
- References:
- Linear Algebra Done Right, Sheldon Axler, Springer, 1996;
- Down with determinants!, Sheldon Axler, American Mathematical Monthly 102 (1995), 139-154.
- Requirements: You should have done well in Honours Algebra and have a taste for abstract algebra in general.
- Second marker: Sue Sierra, Antony Maciocia, David Jordan, Arend Bayer, Ivan Cheltsov or Harry Braden.
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TL.GRP.2
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- Information Theory
- 20 point group project
- Single honours mathematics or any joint honours degree
- Thsi sentnece demontrates taht, even whn errgors ocucr, relaible commjnication cna oftenbe achheved.
Lecture notes are another example: errors are made when the lecturer copies from their notes onto the board, and again when the student copies from the board onto paper, yet usually (we hope) it is possible to reconstruct what was intended.
A third example, taken from the book of Jones and Jones cited below, is that
space exploration has created a demand for accurate transmission of very weak signals through an extremely noisy channel: there is no point in sending a probe to Mars if one cannot hear and decode the messages it sends back.
In 1948, the American engineer Claude Shannon gave the world a new mathematical theory of communication, from which present-day information theory is descended.
Among other things, Shannon put forward a precise answer to the question what is information?
For example, suppose someone is dictating a word to you down the phone, and the first two letters are Q and U.
There is a sense in which the U does not give you very much information, because almost all words starting in Q have U as their second letter.
But if the second letter had been A, that would have conveyed more information, because there are relatively few words beginning with QA.
The fundamental quantity in information theory is entropy.
This is closely related to the entropy that occurs in physics, and is also used in theoretical ecology (although you will not need to know about either of those subjects).
For example, the entropy per letter of ordinary English prose is about 4.03, which means that we could get away with about 24=16 letters (instead of 26) if we used them more efficiently.
Shannon also proved the surprising and fundamental result that even when the environment is terribly noisy, communication can be achieved with arbitrarily
high accuracy and arbitrarily little waste. This is Shannon's Fundamental (or Second) Theorem.
Your task is to read, understand and present a significant amount of information theory, including at least one of Shannon's major theorems.
Information theory is used in many other branches of science (from physics to engineering to biology), and there may be time at the end of the project for you to address some of the aspects that interest you most.
- References:
- Elements of Information Theory, Thomas M. Cover and Joy A. Thomas, Wiley, 1991;
- Information and Coding Theory, Gareth A. Jones and J. Mary Jones, Springer, 2000.
- Requirements: The basic vocabulary of probability theory, e.g. as in the second-year courses Probability or Probability with Applications.
- Second marker: Michal Branicki or Joan Simon.
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TL.IND.1
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- Sheaf theory
- 20 point individual project
- Single honours mathematics or any joint honours degree
- Sheaves are a central concept of modern topology and geometry.
Roughly speaking, a sheaf on a space is a collection of functions that may be defined only locally (i.e. on a small patch of the space)
and may be required to satisfy some locally-defined property.
For instance, continuous real-valued functions on any given space form a sheaf,
because continuity is a local property, but bounded real-valued functions do not, because boundedness is not determined locally.
The totality of all sheaves on a given space forms a category,
and categories that arise in this way are called toposes. So, any space gives rise to a topos.
This topos contains a great deal of information about the original space;
indeed, almost everything about the space is determined by its topos of sheaves.
From this viewpoint, topology can be understood as the study of toposes.
But toposes are not only deeply connected to geometry and topology;
they are also deeply connected to logic and set theory.
Indeed, a topos can be regarded as a generalized universe of sets as well as a generalized space.
It is a universe of sets in which the usual laws of logic do not all apply;
for instance, proof by contradiction is not usually valid, and in general we can only use constructive methods.
Your role in this project would be to learn and explain some (definitely not all!) of this story,
taking a direction that suits your background and taste.
- References: Sheaves in Geometry and Logic by Saunders Mac Lane and Ieke Moerdijk, Springer, 1994.
- Requirements: You should be taking General Topology or General and Algebraic Topology, and have a taste for abstract algebra.
- Second marker: Antony Maciocia, David Jordan, Arend Bayer or Jon Pridham.
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TL.DISS.1
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- Integral geometry
- 40 point dissertation project
- Single honours mathematics
- What ways are there of "measuring" a convex subset of Rn?
For n=2, there are two obvious ways: take the perimeter or the area. For n = 3, there are also two obvious ways: take the surface area or the volume. But there are also some ways that are useful but less obvious: e.g. for a convex subset X of R3, take the length of the shadow that X casts on a line oriented at random.
These measures are called intrinsic volumes, and form part of a subject called integral geometry (a kind of partner to differential geometry). It turns out that using the intrinsic volumes, you can answer many very concrete geometric questions. For example, a 20p piece has the property that no matter how you orient it, it always has the same width w (an important property for vending machines), and by the end of this project you'll understand why this implies that it must have the same perimeter as a disc of diameter w. Some other similar questions are listed here: What is integral geometry?
Your task will be to learn some of the theory and its applications.
- References: Introduction to Geometric Probability, Daniel A. Klain and Gian-Carlo Rota, Lezioni Lincee, Cambridge University Press, 1997.
- Requirements:
You will need some measure theory, which means that you should be taking either Essentials in Analysis and Probability or Probability, Measure and Finance.
- Second marker: Jonas Azzam.
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