In mathematics, knots are usually described by a continuous injective map from the circle into three dimensional Euclidean space; we sometimes require these maps are better behaved, for example piecewise linear, to avoid particularly pathological cases. This corresponds to what we would usually call knots, except that we require both ends to be attached so that there is no way of untying it without breaking it or passing it through itself.

Once this is defined, we need to get an idea of how to compare different knots - there are many ways of doing this and saying which knots are equivalent, for example we can use a homeomorphism of onto ; or we can alternatively use the Reidemeister moves on the knot diagrams.

One of the main problems in the study of knots was that of classifying knots, often by the minimum number of crossings one would need to have in any diagram of a knot (a well behaved projection of the knot onto a plane with under and over crossings). The hardest thing in making up these lists often comes down to figuring out if two different diagrams give the same knot - a nice way of telling the difference is by computing particular invariants attached to the knot - it might be a group (the knot group), a polynomial (the Alexander polynomial, Jones polynomial and many others), a number (genus, crossing number, ...).

A related area is the study of links - a link is like a knot except it might be constructed out of more than one circle. A basic invariant in this case is the linking number, telling us how much the different components are linked together.

Join us in MS.04 at 19:30 for an overview of knot theory, including descriptions of all the knot invariants described above. Then we untie ourselves and go to the pub!

(PS. The next discussion group is going to be David Mond's talk on Wednesday 13th, then no discussion group on Thursday 14th)

This term only, for all your revision and coffee based needs, Maths Café will be running twice a week under the guise of Revision Café. Every Wednesday and Friday from 1 PM to 3 PM, feel free to come have a cup of tea/coffee and get help with any course you need to revise for.

Don't miss out on the special guest Discussion Group held by the maths institute's very own Professor David Mond: one of the departments most popular lecturers (as attested to by over 300 first-year students each year)!

The talk, titled ‘The Mathematics of Climate Change' (or 'An Inconvenient Proof'), takes a new slant on the often over labored topic of the future of our planet: those expecting to hear about how to model the earth's temperature using complicated equations will be left disappointed.

Rather, Professor Mond is interested in how we can use mathematics to model social attitudes towards global warming. We are constantly told that NOW is the time to make a change before it is too late. But what change do we want to make? If each country is left to their own devices, will we do enough to save the planet? And what is the best strategy for us to adopt in Britain?

Join us on Wednesday 13th May at 6pm in MS.01 (Mathematics Institute) for what promises to be an exciting talk about an under-developed topic lying on the boarders of mathematics, climate politics and social science.

The talk will be followed by refreshments and a chance for further debate.

This term only, for all your revision and coffee based needs, Maths Café will be running twice a week under the guise of Revision Café. Every Wednesday and Friday from 1 PM to 3 PM, feel free to come have a cup of tea/coffee and get help with any course you need to revise for.

Elliptic Curves have motivated a large part of the mathematics of the last century: in particular, they have allowed for the famous proof of Fermat's Last Theorem. They still are an area of active research, in pure mathematics (for example, motivated by the Birch and Swinnerton-Dyer conjecture, one of the seven Clay Math problems) but also in applied mathematics in relation to many applications in cryptography.

Over the complex numbers, elliptic curves have a particularly simply description: an elliptic curve is just a complex torus. But this conceals a lot of their inner structure! Trying to classify them up to isomorphism leads into many interesting problems, such as the idea of moduli spaces, the mysterious j-invariant and doubly periodic functions on the complex plane.

These elliptic curves come with a group structure (either coming from their description as a torus or by considering a chord-tangent group law, the two being related by the Weierstrass function), and we can thus consider their endomorphisms (the group homomorphisms from the elliptic curve to itself). Every elliptic curve over the complex numbers comes with a multiplication by endomorphism for every integer , but for most of them, that's the end of the story. Special elliptic curves, with more endomorphisms, are said to have complex multiplication.

The study of elliptic curves with complex multiplication relates perhaps surprisingly with class field theory and the study of quadratic imaginary fields (fields of the form for positive integers ); there is a direct relation between isomorphism classes of elliptic curves with given endomorphisms and corresponding quadratic imaginary fields.

Join us in MS.04 at 7:30 to learn about all this and more! Don't miss it - this is probably the last Discussion Group for a while due to exams starting for many of us. We will of course translate ourselves to the pub afterwards.

This term only, for all your revision and coffee based needs, Maths Café will be running twice a week under the guise of Revision Café. Every Wednesday and Friday from 1 PM to 3 PM, feel free to come have a cup of tea/coffee and get help with any course you need to revise for.

This term only, for all your revision and coffee based needs, Maths Café will be running twice a week under the guise of Revision Café. Every Wednesday and Friday from 1 PM to 3 PM, feel free to come have a cup of tea/coffee and get help with any course you need to revise for.

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

1st Year Linear Algebra Crash Course.

Run by Ben Simpson and Colin Dold.

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

1st Year Differential Equations Crash Course.

Run by Sam Derbyshire.

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

2nd Year Differentiation Crash Course.

Run by Praveen Amarasinghe

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

2nd Year Algebra II Crash Course.

Run by Cosmin Davidescu and Richard Howell-Peak.

WMS Module Crash Courses for first and second year modules are already underway! Details are as follows:

1st Year:

Mon 25th May: GEOMETRY AND MOTION (MS.01, 3pm-5pm)

Tues 26th: ANALYSIS 2 (MS.01, 3pm-5pm)

Wed 27th: LINEAR ALGEBRA (MS.01, 3pm-5pm)

Thurs 28th: DIFFERENTIAL EQUATIONS (MS.01, 3pm-5pm)

Second Year:

Fri 29th: DIFFERENTIATION ( MS.04, 3pm-5pm )

Thurs 4th: ALGEBRA 2 (MS.01, 3pm-5pm)

Fri 5th: PDE'S (MS.01, 3pm-5pm)

Mon 8th: METRIC SPACES ( MS.04, 3pm-5pm )

Sorry for the delay!

(Updated places in red.)

2nd/3rd Year Introduction to Partial Differential Equations Crash Course.

Run by Mike Scott.

2nd/3rd Year Metric Spaces Crash Course.

Run by Dave McCormick and Justin Millette.

NB: Time Subject to Change.