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Events
11 / 3
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11 / 4
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11 / 5
Start: 19:30
Homotopy theory is the study of continuous deformation of topological spaces ("continuous" spaces). One especially important idea in that area is that of a homotopy group: it encodes all the ways to map a sphere continuously into any space, up to continuous deformation of such maps. Homotopy groups turn out to be particularly difficult to describe in general - even the homotopy groups of spheres (which just represent continuous maps from spheres to spheres) are very complicated and their structure is still not entirely explained. Several patterns emerge however, some quite obvious but many more subtle. Of particular interest is that some homotopy groups eventually stabilise, giving rise to stable homotopy groups. These are also very mysterious, and are only entirely calculated up to Join us in MS.05 at 7:30pm as Sam Derbyshire explains this - starting from the basic definitions of homotopy groups, which just requires an intuitive understanding of continuity. After this, we homotopy ourselves to the pub! | ||
11 / 6
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11 / 7
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11 / 8
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11 / 9
Start: 19:30
The theory of elliptic curves is one of the most active but also one of the most mysterious in modern number theory. At its most basic level, it originates from one of the oldest problems in mathematics, that of finding integer solutions to various polynomial equations, and yet, despite this, it lies at the heart of many current advances of which Andrew Wiles' recent proof of Fermat's Last Theorem is a notable example. Their definition is a deceptively simple one: an elliptic curve can generally be defined as the set of solutions to certain cubic equations of the form | ||
11 / 10
Start: 19:30
Start: 10 Nov 2009 - 7:30pm
End: 11 Nov 2009 - 4:00am
Term is well on its way now, so what better way to break up the monotony of lectures than drunkenly crawling through Leamington! We'll be starting in the Duck at 7:30 (aiming to be in the White Horse around 8:30 for those who live in Leam), before moving down the parade and finishing the night in Kelseys. If th...e carnage which was 'The Integrating Factor' was anything to go by, this is one you'd be crazy to miss! | ||
11 / 11
End: 04:00
Start: 10 Nov 2009 - 7:30pm
End: 11 Nov 2009 - 4:00am
Term is well on its way now, so what better way to break up the monotony of lectures than drunkenly crawling through Leamington! We'll be starting in the Duck at 7:30 (aiming to be in the White Horse around 8:30 for those who live in Leam), before moving down the parade and finishing the night in Kelseys. If th...e carnage which was 'The Integrating Factor' was anything to go by, this is one you'd be crazy to miss! | ||
11 / 12
Start: 19:30
End: 21:00
Few theorems can be said to have opened up entire new areas of mathematical research - but this is what the Riemann Roch theorem managed. Initially considered by Riemann as a helpful way of approximating the dimension of the space of certain meromorphic functions on a Riemann surface; this was then refined by his student Roch to give a precise formula for the dimension. The applications of this single formula are numerous; but this formula also lead to many deep generalisations, starting with Hirzebruch's interpretation of the theorem in terms of vector bundles (and his use of algebraic topology, in particular the theory of characteristic classes, to express this). A few years later followed Grothendieck's interpretation of the theorem, changing the absolute viewpoint to a relative one, something new for the time, and introducing ideas that were going to lead to K-Theory. Meet us in MS.05 at 7:30pm for an introduction to Riemann surfaces and the Riemann-Roch Theorem (not the generalizations!). Then we invert some line bundles and appear in the pub. | ||
11 / 13
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11 / 14
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11 / 15
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11 / 16
Start: 19:30
End: 21:00
Tonight will be Christian Richard telling us about the Poincaré conjecture, the first (and only, so far) Clay Math problem to have been solved. Join us in MS.05 at 7:30pm for an historical overview of the developments in algebraic topology, leading on from Poincaré's revolutionary ideas, and an explanation of their relevance to the Poincaré conjecture. After which we go to the pub. (Sorry about the late notice, was waiting on Christian to provide a description which he did not do.) | ||
11 / 17
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11 / 18
Start: 13:00
End: 15:00
Ever seen a fractal on a poster? Would you buy one if you could? Next Wednesday during Maths Café the WMS will be selling posters. Posters currently available are here Prices are You will only be able to order them on Wednesday. We will send them to print that day, and they will be ready before next Wednesday when they can be collected or delivered. | ||
11 / 19
Start: 19:30
End: 21:00
Modern mathematics is often seen to be built from the ground up from a foundation of set theory axioms, and we are all used to seeing maths done in terms of sets; however, category theory, designed primarily as a tool for algebraic topology, can be also be used as a foundation for mathematics. Come and hear Ben Simpson explain how one can think of sets in terms categories, and how this makes several areas of maths that little bit shinier. After which, we naturally transform ourselves towards the pub. Update: Unfortunatley, it looks like Ben won't be able to present his talk this evening but we will nevertheless hold a "free-for-all" DG instead (same time and place, of course). | ||
11 / 20
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11 / 21
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11 / 22
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11 / 23
Start: 19:30
End: 21:00
Just a short message to inform everyone that tonight's discussion group will run normally but will be a free for all: whether you've recently found some tidbit of interesting maths which you want to tell us about or just want listen to a few varied "mini-DGs", this is a great opportunity to do so. The fun starts in MS.05 at 7:30 PM and once it's finished we'll head on to the pub as usual. | ||
11 / 24
Start: 12:00
If you've taken a look at any recently published book or mathematical paper, chances are you've already seen what LaTeX can do. LaTeX is, simply put, an elegant and convenient way of typing mathematics on a computer and particularly useful if you would like to typeset any type of mathematical essay or project (second year essays in particular). Unlike word processors such as Microsoft Word and Open Office Writer, you do not work directly on something which looks like the finished document. Instead, it is much more similar to a programming language (albeit a very simple one): you have to type in text along with a series of commands which determine the mathematical equations, alignment, font size and almost everything else. This may sound daunting at first but is in fact much easier than it sounds and, as usual, the WMS is here to help you: we will be running two LaTeX classes per week during the final three weeks of term, Tuesdays 12-1 and Wednesdays 11-12 (the two classes will have exactly the same content). By the end of these, you should be able to write mathematics much better and faster than you would on a conventional word processor and be able to typeset a professional looking second year essay. | ||
11 / 25
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11 / 26
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11 / 27
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11 / 28
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11 / 29
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11 / 30
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12 / 1
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12 / 2
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12 / 3
Start: 19:30
End: 21:00
Classification of manifolds up to diffeomorphism is a hard problem (deciding whether two given 4-manifolds are diffeomorphic is at least as hard as the word problem for groups). Thom's brilliant idea (which had been considered, before him, by Poincaré and Pontryagin) was to introduce a weaker equivalence relation than diffeomorphism: the equivalence relation of cobordism. Two There are other types of cobordism, that arise by putting structure on our manifolds and requiring that the cobordisms preserve this, for example oriented cobordism: we only study oriented manifolds, and the oriented boundary of the Pontryagin's success was to realise that certain cobordism groups end up being equal to stable homotopy groups of spheres, helping the calculation of the latter. Thom largely reversed the process and generalised Pontryagin's idea, making an explicit correspondence between cobordism groups and homotopy groups of certain spaces, now called Thom spaces. Another useful way to study cobordism is through characteristic classes. Many cobordism theories have particular characteristic classes associated to them, and it turns out that two manifolds are cobordant if and only if all the characteristic numbers of these two manifolds agree. This lead to great insights into the theory of characteristic classes, for example the idea of a generalised genus (a topological invariant) as a homomorphism from the cobordism ring to some other ring, for example the rationals. This is then of great interest in classifying manifolds, and allowed things such as Milnor's construction of exotic If you're already impatient to hear more, then you should come to MS.05 this thursday at 7:30 pm; you'll hear about all this and then much more (such as topological quantum field theories!). After which we realise that we are cobordant to our future selves that are in the pub. | ||
. Many interest patterns emerge here too, especially some glimpse of a periodicity. To explain this, we must first study the homotopy groups of some matrix groups, which are of particular importance in algebraic topology. This reveals a beautiful pattern known as Bott Periodicity, which will turn out to explain many of the features we have encountered so far.
. Things start to get interesting when we define a very elementary operation on this set which turns it into a group: the resulting structure is one of the deepest and richest in mathematics. So, join us this Monday at 7:30 in 
-manifolds are (unoriented) cobordant if their disjoint union is the boundary of an 
-manifold with boundary. We can see that we can compose cobordisms end to end, and every manifold is cobordant to itself as we have 
, so it is easy to check it is indeed an equivalence relation. In each dimension we can form an abelian group, with operation disjoint union, the 
-spheres.