Algebra Kinderseminar
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Wed, 13/10/2010 11:30 |
David Craven (University of Oxford) |
Algebra Kinderseminar  |
ChCh, Tom Gate, Room 2 |
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In this talk we will survey some aspects of social choice theory: in particular, various impossibility theorems about voting systems and strategies. We begin with the famous Arrow's impossibility theorem -- proving the non-existence of a 'fair' voting system -- before moving on to later developments, such as the Gibbard–Satterthwaite theorem, which states that all 'reasonable' voting systems are subject to tactical voting. Given time, we will study extensions of impossibility theorems to micro-economic situations, and common strategies in game theory given the non-existence of optimal solutions. |
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Wed, 20/10/2010 11:30 |
Ben Davison (University of Oxford) |
Algebra Kinderseminar |
ChCh, Tom Gate, Room 2 |
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Wed, 27/10/2010 11:30 |
Richard Williamson (University of Oxford) |
Algebra Kinderseminar |
ChCh, Tom Gate, Room 2 |
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From a categorical point of view, the standard Zermelo-Frankel set theoretic approach to the foundations of mathematics is fundamentally deficient: it is based on the notion of equality of objects in a set. Equalities between objects are not preserved by equivalences of categories, and thus the notion of equality is 'incorrect' in category theory. It should be replaced by the notion of 'isomorphism'. Moving higher up the categorical ladder, the notion of isomorphism between objects is 'incorrect' from the point of view of 2-category, and should be replaced by the notion of 'equivalence'... Recently, people have started to take seriously the idea that one should be less dogmatic about working with set-theoretic axiomatisiations of mathematics, and adopt the more fluid point of view that different foundations of mathematics might be better suited to different areas of mathematics. In particular, there are currently serious attempts to develop foundations for mathematics built on homotopy types, or, in another language, ∞-groupoids. An (∞,1)-topos should admit an internal 'homotopical logic', just as an ordinary (1-)topos admits an internal logic modelling set theory. It turns out that formalising such a logic is rather closely related to the problem of finding good foundations for 'intensional dependent type theory' in theoretical computer science/logic. This is sometimes referred to as the attempt to construct a 'homotopy lambda calculus'. It is expected that a homotopy theoretic formalisation of the foundations of mathematics would be of genuine practical significance to the average mathematician! In this talk we will give an introduction to these ideas, and to the recent work of Vladimir Voevodsky and others in this area. |
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Wed, 03/11/2010 11:30 |
Ric Wade (University of Oxford) |
Algebra Kinderseminar |
ChCh, Tom Gate, Room 2 |
| This talk will be an introduction to property (T). It was originally introduced by Kazhdan as a method of showing that certain discrete subgroups of Lie groups are finitely generated, but has expanded to become a widely used tool in group theory. We will take a short tour of some of its uses. | |||
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Wed, 10/11/2010 11:30 |
Owen Cotton-Barratt (University of Oxford) |
Algebra Kinderseminar |
ChCh, Tom Gate, Room 2 |
| We will investigate what one can detect about a discrete group from its profinite completion, with an emphasis on considering geometric properties. | |||
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Wed, 17/11/2010 11:30 |
Elisabeth Fink (University of Oxford) |
Algebra Kinderseminar |
ChCh, Tom Gate, Room 2 |
| I am going to introduce Thompson's groups F, T and V. They can be seen in two ways: as functions on [0,1] or as isomorphisms acting on trees. | |||
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Wed, 24/11/2010 11:30 |
Algebra Kinderseminar |
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Wed, 01/12/2010 11:30 |
Jason Semeraro (University of Oxford) |
Algebra Kinderseminar |
ChCh, Tom Gate, Room 2 |
| Given a block, b, of a finite group, Alperin's weight conjecture predicts a miraculous equality between the number of isomorphism classes of simple b-modules and the number of G-orbits of b-weights. Radha Kessar showed that the latter can be written in terms of the fusion system of the block and Markus Linckelmann has computed it as an Euler characteristic of a certain space (provided certain conditions hold). We discuss these reformulations and give some examples. | |||
