Tue, 09 Feb 2016
14:30
L6

The Chromatic Number of Dense Random Graphs

Annika Heckel
(Oxford University)
Abstract

The chromatic number of the Erdős–Rényi random graph G(n,p) has been an intensely studied subject since at least the 1970s. A celebrated breakthrough by Bollobás in 1987 first established the asymptotic value of the chromatic number of G(n,1/2), and a considerable amount of effort has since been spent on refining Bollobás' approach, resulting in increasingly accurate bounds. Despite this, up until now there has been a gap of size O(1) in the denominator between the best known upper and lower bounds for the chromatic number of dense random graphs G(n,p) where p is constant. In contrast, much more is known in the sparse case.

In this talk, new upper and lower bounds for the chromatic number of G(n,p) where p is constant will be presented which match each other up to a term of size o(1) in the denominator. In particular, they narrow down the optimal colouring rate, defined as the average colour class size in a colouring with the minimum number of colours, to an interval of length o(1). These bounds were obtained through a careful application of the second moment method rather than a variant of Bollobás' method. Somewhat surprisingly, the behaviour of the chromatic number changes around p=1-1/e^2, with a different limiting effect being dominant below and above this value.

Tue, 16 Feb 2016
14:30
L6

Product-Free Subsets of the Alternating Group

Sean Eberhard
(Oxford University)
Abstract

There is an obvious product-free subset of the symmetric group of density 1/2, but what about the alternating group? An argument of Gowers shows that a product-free subset of the alternating group can have density at most n^(-1/3), but we only know examples of density n^(-1/2 + o(1)). We'll talk about why in fact n^(-1/2 + o(1)) is the right answer, why
Gowers's argument can't prove that, and how this all fits in with a more general 'product mixing' phenomenon. Our tools include some nonabelian Fourier analysis, a version of entropy subadditivity adapted to the symmetric group, and a concentration-of-measure result for rearrangements of inner products.

Tue, 02 Feb 2016
14:30
L6

Monochromatic Sums and Products

Ben Green
(Oxford University)
Abstract

Fix some positive integer r. A famous theorem of Schur states that if you partition Z/pZ into r colour classes then, provided p > p_0(r) is sufficiently large, there is a monochromatic triple {x, y, x + y}. By essentially the same argument there is also a monochromatic triple {x', y', x'y'}. Recently, Tom Sanders and I showed that in fact there is a
monochromatic quadruple {x, y, x+y, xy}. I will discuss some aspects of the proof.

Wed, 03 Feb 2016
16:00
C3

Quasi-isometry invariants of groups

Giles Gardam
(Oxford University)
Abstract

We will discuss various familiar properties of groups studied in geometric group theory, whether or not they are invariant under quasi-isometry, and why.

Wed, 27 Jan 2016
16:00
C3

Profinite rigidity of Seifert fibre spaces

Gareth Wilkes
(Oxford University)
Abstract

One can ask whether the fundamental groups of 3-manifolds are distinguished by their sets of finite quotients. I will discuss the recent solution of this question for Seifert fibre spaces.

Wed, 20 Jan 2016
16:00
C3

Expanders and Warped Cones

Federico Vigolo
(Oxford University)
Abstract

I will illustrate how to build families of expanders out of 'very mixing' actions on measure spaces. I will then define the warped cones and show how these metric spaces are strictly related with those expanders.

Thu, 12 May 2016

16:30 - 18:00
L1

Marcus du Sautoy - What We Cannot Know

Marcus du Sautoy
(Oxford University)
Abstract

Science is giving us unprecedented insight into the big questions that have challenged humanity. Where did we come from? What is the ultimate destiny of the universe? What are the building blocks of the physical world? What is consciousness?

‘What We Cannot Know’ asks us to rein in this unbridled enthusiasm for the power of science. Are there limits to what we can discover about our physical universe? Are some regions of the future beyond the predictive powers of science and mathematics? Are there ideas so complex that they are beyond the conception of our finite human brains? Can brains even investigate themselves or does the analysis enter an infinite loop from which it is impossible to rescue itself? 

To coincide with the launch of his new book of the same title, Marcus du Sautoy will be answering (or not answering) those questions. He will also be signing copies of the book before and after the lecture.

To book please email @email

Mon, 30 Nov 2015

17:00 - 18:00
L1

Slightly Rubbish Modular Ax-Lindemann

Haden Spence
(Oxford University)
Abstract

In quite an elementary, hands-on talk, I will discuss some Ax-Lindemann type results in the setting of modular functions.  There are some very powerful results in this area due to Pila, but in nonclassical variants we have only quite weak results, for a rather silly reason to be discussed in the talk.

Mon, 23 Nov 2015

17:00 - 18:00
L3

Functors of points and moduli problems

Alexander Betts
(Oxford University)
Abstract

In algebraic and arithmetic geometry, there is the ubiquitous notion of a moduli space, which informally is a variety (or scheme) parametrising a class of objects of interest. My aim in this talk is to explain concretely what we mean by a moduli space, going through the functor-of-points formalism of Grothendieck. Time permitting, I may also discuss (informally!) a natural obstruction to the existence of moduli schemes, and how one can get around this problem by taking a 2-categorical point of view.

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