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Events

« February 24, 2010 - March 26, 2010 »
 
02 / 24
02 / 25
DG := <Morse Theory>
Start: 14:03

Morse Theory analyses manifolds by looking at the behaviour of differentiable functions on that manifold. We can gain a lot of insight into the topology of a manifold by looking at the critical points of a differentiable function on that manifold: different matrices of second partial derivatives (the Hessian) gives different local behaviours, like saddles, maxima or minima. Looking at what happens between different critical points, we can try to patch up what happens near each critical point to reconstruct our manifold somehow.

To do this, our smooth functions need to be sufficiently nice; the so called Morse functions.

We can in fact strengthen the approach by taking more care at what happens around critical points; we will then find a particularly neat way of packaging that information and passing from that information to topological information. In particular, a theorem of John Jones (with Graeme Seagal and Ralph Cohen) will make its appearance!

So make sure to come to MS.05 at 7:30 to learn about what Morse Theory is all about and why it's so amazing! After which we flow to the pub!
02 / 26
02 / 27
02 / 28
03 / 1
DG := <Riemann Surfaces, Algebraic K Theory>
Start: 19:00
End: 21:00

The theory of Riemann surfaces began (as one might expect) with Riemann wondering about the correct framework in which to study multivalued functions. Riemann realised he needed to consider 2-dimensional `domains' with some inscribed `geometry'; what we would now call Riemann surfaces and then of course the functions on these. As with complex analysis a number of remarkable theorems were then discovered with such beautiful examples as the Uniformization Theorem, Riemann-Roch Theorem and Riemann's Existence Theorem. This led naturally to wondering whether the analogues of these statements held for higher dimensional complex manifolds after which the theory was largely overhauled with the introduction of the language of sheaves, cohomology and the Serre Duality Theorem taking the place of the analytic tools previously developed with regards to harmonic functions and integrals.

If any of this has piqued your interest come along to MS.05 at the slightly unusual time of 7:00 to hear Callan McGill expound on this! After which, we will also have the opportunity to hear a bit about Algebraic K Theory from Joe Tait! Thereafter we will analytically continue ourselves to the pub!
03 / 2
03 / 3
03 / 4
03 / 5
03 / 6
03 / 7
03 / 8
DG := <Supersingular Elliptic Curves>
Start: 19:30
End: 21:00

Update: This Discussion Group is postponed to Monday, March 8th.

Elliptic curves are, as any regular attendee of discussion groups knows by now, one of the richest and most interesting topics in mathematics.

Of particular importance over the rational numbers are elliptic curves with complex multiplication: they are those that come with additional endomorphisms. Over finite fields however, all elliptic curves have more endomorphisms than just the usual multiplication by $ n $ maps: there is the beloved Frobenius that helps us count points and solve many other problems. But there are even more special elliptic curves over finite fields: supersingular curves. In this case, their ring of endomorphisms has the interesting structure of an order in a quaternion algebra, and so in particular is non-commutative.

All supersingular elliptic curves share many important properties, and it will be one of the aims of this talk to show the equivalence of many of these properties. These range from the description of the endomorphism ring to measuring $ p $-torsion, considerations of isogenies or just counting the number of points.

The talk will end with the consideration of the relation between elliptic curves with complex multiplication over $ \mathbb{Q} $ and supersingular elliptic curves over finite fields; in particular, we deduce many easy estimates for the size of the torsion of an elliptic curve over $ \mathbb{Q} $ (or over other number fields).

Come to MS.05 at 7:30 to hear about all this, and more! After which we reduce ourselves to the pub.
03 / 9
03 / 10
03 / 11
03 / 12
03 / 13
03 / 14
03 / 15
DG := <Clifford algebras and vector fields on spheres>
Start: 19:30
End: 21:00

The determination of how many independent vector fields there are on the $ n $ sphere $ S^n $ was one of the great successes of algebraic topology.

The story starts with the Hairy Ball Theorem: if $ n $ is even, there are no nowhere zero (continuous) vector fields on $ S^n $. We must then turn to odd dimensional spheres, and wonder how many linearly independent vector fields can there be? On our way to answering this question, we will encounter many familiar structures: $ \mathbb{R} $, $ \mathbb{C} $, $ \mathbb{H} $ and even more general algebraic structures known as Clifford algebras, which will allow us to give a lower bound on the number of linearly independent vecor fields on spheres.

We will find many relations with K-Theory, similarities and common periodicities; these have been exploited by Frank Adams to prove that the lower bound provided by Clifford algebras is actually an equality, although we will not delve into the details.

Come tonight at MS.05 to hear about all this, and more! After which we find our way to the pub.
03 / 16
03 / 17
03 / 18
03 / 19
03 / 20
03 / 21
03 / 22
03 / 23
03 / 24
03 / 25
03 / 26
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