Directions in Research
The Directions in Research seminars haven't been running lately (and, to my knowledge, won't be running this year!) -- sorry!
Directions in Research Seminar Series -- 2008
The Directions in Research seminars are intended to give Part III students a glimpse of what kind of thing research students in the departments are actually working on, and also to allow PhD students to see what their colleagues are doing. They take place every Monday afternoon from 4-5pm in MR5.
Here is this term's programme so far:
- Monday 12th October: The Story of Hodge Theory - Alexander Shannon (DPMMS)
This talk will start with an explanation of the celebrated theorem that the cohomology of a smooth projective complex algebraic variety breaks up in a canonical way - the so-called Hodge decomposition. I shall then go on to mention some of the ways this phenomenon generalises, including Deligne's notion of a mixed Hodge structure, which is carried by the cohomology of any complex algebraic variety and Griffiths' notion of a variation of Hodge structure, describing how the Hodge decomposition varies in families. Time-permitting, I shall also explain some ways in which Hodge structures appear in the more general setting of non-commutative algebraic geometry, and the efforts currently being made to study them.
- Monday 19th October: Explicit Local Reciprocity - Rachel Newton (DPMMS)
I will start by recalling some facts about local fields and their extensions and go on to define the local reciprocity map via central simple algebras and the Brauer group. Along the way, I will introduce cyclic algebras and the Hasse invariant.
I will describe the cases in which an explicit formula has been found for the local reciprocity map and explain why this is difficult in general for extensions with ramification.
I will finish by computing local reciprocity for a concrete example of a totally ramified extension.
- Monday 2nd November: Road traffic, the button model, and internet protocols - Elena Yudovina (Statslab)
Stochastic networks is a relatively young field that often seems more like disjointed examples than like a unified theory. However, one topic that comes through often is the relationship between a microscopic distributed description and a global optimization problem. I will illustrate this by talking about some or all of road traffic, the button model of electric circuits, and current research into internet protocols.
- Monday 9th November: Symplectic topology and Floer homology - Richard Harris (DPMMS)
Symplectic geometry grew out of attempts to understand questions in classical mechanics, but has undergone something of a revolution over the past 25 years and now touches many diverse areas of maths including knot theory and string theory.
I shall begin by defining the basic notions of symplectic geometry before giving a brief overview of classical Morse theory as motivation for Floer homology, a wonderfully powerful tool in the field. I will use this theory to sketch a proof of the Arnold conjecture on the existence of closed Hamiltonian orbits. If there is any time at the end, I may say some very brief words about other variants of Floer homology and their role in Homological Mirror Symmetry.
- Monday 16th November: From Quantum Field Theory to particle physics: the Standard Model - Hiro Funakoshi (DAMTP)
This talk will begin with a brief introduction about the link between quantum field theory and the standard model of particle physics. I will go on to talk about some of the key ideas in the standard model including particle representations, gauge fields and the Higgs effect, and then illustrate all these using the electroweak interaction as a single example. At the end, I shall make some closing remarks about why, despite its massive success, this isn't the end of theoretical physics and talk about what more there is to do.
Despite the title of the talk, no quantum field theory knowledge will be required.
For more information, or if you'd like to volunteer to give a talk, please email the organiser, Alexander Shannon (adjs2).