16:00
Generalising Mirzakhani’s curve counting result
Abstract
On any hyperbolic surface, the number of curves of length at most L is finite. However, it is not immediately clear how quickly this number grows with L. We will discuss Mirzakhani’s breakthrough result regarding the asymptotic behaviour of this number, along with recent efforts to generalise her result using currents.
16:00
Automorphisms of free groups and train tracks
Abstract
Let phi be an outer automorphism of a free group. A topological representative of phi is a marked graph G along with a homotopy equivalence f: G → G which induces the outer automorphism phi on the fundamental group of G. For any given outer automorphism, the choice of topological representative is far from unique. Handel and Bestvina showed that sufficiently nice automorphisms admit a special type of topological representative called a train track map, whose dynamics can be well understood.
In this talk I will outline the definition and motivation for train tracks, and give a sketch of Handel and Bestvina’s algorithm for finding them.
Growth in soluble linear groups over finite fields
Abstract
In joint work with James Wheeler, we show that if a subset $A$ of $GL_n(\mathbb{F}_q)$ is a $K$-approximate group and the group $G$ it generates is soluble, then there are subgroups $U$ and $S$ of $G$ and a constant $k$ depending only on $n$ such that:
$A$ quickly generates $U$: $U\subseteq A^k$,
$S$ contains a large proportion of $A$: $|A^k\cap S| \gg K^{-k}|A|, and
$S/U$ is nilpotent.
Briefly: approximate soluble linear groups over any finite field are (almost) finite by nilpotent.
The proof uses a sum-product theorem and exponential sum estimates, as well as some representation theory, but the presentation will be mostly self-contained.
Efficient congruence and discrete restriction for (x,x^3)
Abstract
We will outline the main features of Wooley's efficient congruencing method for the parabola. Then we will go on to prove new bounds for discrete restriction to the curve (x,x^3). The latter is joint work with Trevor Wooley (Purdue).
Oxford Mathematics Public Lecture. Alan Champneys: Why pedestrian bridges wobble - synchronisation and the wisdom of the crowd
There is a beautiful mathematical theory of how independent agents tend to synchronise their behaviour when weakly coupled. Examples include how audiences spontaneously rhythmically applause and how nearby pendulum clocks tend to move in sync. Another famous example is that of the London Millennium Bridge. On the day it opened, the bridge underwent unwanted lateral vibrations that are widely believed to be due to pedestrians synchronising their footsteps.
In this talk Alan will explain how this theory is in fact naive and there is a simpler mathematical theory that is more consistent with the facts and which explains how other bridges have behaved including Bristol's Clifton Suspension Bridge. He will also reflect on the nature of mathematical modelling and the interplay between mathematics, engineering and the real world.
Alan Champneys is a Professor of Applied Non-linear Mathematics at the University of Bristol.
Please email @email to register.
Watch live:
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https://livestream.com/oxuni/Champneys
The Oxford Mathematics Public Lectures are generously supported by XTX Markets.
Steklov eigenvalue problem on orbisurfaces
Abstract
The Steklov eigenvalue problem is an eigenvalue problem whose spectral parameters appear in the boundary condition. On a Riemannian surface with smooth boundary, Steklov eigenvalues have a very sharp asymptotic expansion. Also, a number of interesting sharp bounds for the $k$th Steklov eigenvalues have been known. We extend these results on orbisurfaces and discuss how the structure of orbifold singularities comes into play. This is joint work with Arias-Marco, Dryden, Gordon, Ray and Stanhope.
16:00
Surfaces via subsurfaces: an introduction to Masur-Minsky
Abstract
The mapping class group of a surface is a group of homeomorphisms of that surface, and these groups have been very well studied in the last 50 years. The talk will be focused on a way to understand such a group by looking at the subsurfaces of the corresponding surface; this is the so-called "Masur-Minsky hierarchy machinery". We'll finish with a non-technical discussion of hierarchically hyperbolic groups, which are a popular area of current research, and of which mapping class groups are important motivating examples. No prior knowledge of the objects involved will be assumed.
15:45
Random triangular Burnside groups
Abstract
In this talk I will discuss recent joint work with Dominik Gruber where
we find a reasonable model for random (infinite) Burnside groups,
building on earlier tools developed by Coulon and Coulon-Gruber.
The free Burnside group with rank r and exponent n is defined to be the
quotient of a free group of rank r by the normal subgroup generated by
all elements of the form g^n; quotients of such groups are called
Burnside groups. In 1902, Burnside asked whether any such groups could
be infinite, but it wasn't until the 1960s that Novikov and Adian showed
that indeed this was the case for all large enough odd n, with later
important developments by Ol'shanski, Ivanov, Lysenok and others.
In a different direction, when Gromov developed the theory of hyperbolic
groups in the 1980s and 90s, he observed that random quotients of free
groups have interesting properties: depending on exactly how one chooses
the number and length of relations one can typically gets hyperbolic
groups, and these groups are infinite as long as not too many relations
are chosen, and exhibit other interesting behaviour. But one could
equally well consider what happens if one takes random quotients of
other free objects, such as free Burnside groups, and that is what we
will discuss.
Confined Rayleigh Taylor instabilities and other mushy magma problems
Abstract
The magma chamber - an underground vat of fluid magma that is tapped during volcanic eruptions - has been the foundation of models of volcanic eruptions for many decades and successfully explains many geological observations. However, geophysics has failed to image the postulated large magma chambers, and the chemistry and ages of crystals in erupted magmas indicate a more complicated history. New conceptual models depict subsurface magmatic systems as dominantly uneruptible crystalline networks with interstitial melt (mushes) extending deep into the Earth's crust to the mantle, containing lenses of potentially eruptible (low-crystallinity) magma. These lenses would commonly be less dense than the overlying mush and so Rayleigh Taylor instabilities should develop leading to ascent of blobs of magma unless the growth rate is sufficiently slow that other processes (e.g. solidification) dominate. The viscosity contrast between a buoyant layer and mush is typically extremely large; a consequence is that the horizontal dimension of a magma reservoir is commonly much less than the theoretical fastest growing wavelength assuming an infinite horizontal layer.
I will present laboratory experiments and linear stability analysis for low Reynolds number, laterally confined Rayleigh Taylor instabilities involving one layer that is much thinner and much less viscous than the other. I will then apply the results to magmatic systems, comparing timescales for development of the instability and the volumes of packets of rising melt generated, with the frequencies and sizes of volcanic eruptions. I will then discuss limitations of this work and outstanding fluid dynamical problems in this new paradigm of trans-crustal magma mush systems.