Martin Bridson will give a "repeat" performance of his Abel Lecture which he delivered a few weeks ago in Oslo as part of the scientific programme in honour of Abel Prize laureate Mikhail Gromov.
Abstract:
Gromov has illuminated great swathes of mathematics with the bright light of geometry. By means of example, I hope to convey the sense of wonder that his work engenders and something of the profound influence he has had on the way my generation thinks about mathematics.
I shall focus particularly on Geometric Group Theory. Gromov's ideas turned the study of discrete groups on its head, infusing it with an array of revolutionary ideas and unveiling deep connections to many other branches of mathematics.
We generalize rings, Banach algebras and C*-algebras to ringoids, Banach algebroids and C*-algebroids. We construct algebraic and topological K-theory of these objects. As an application we can formulate Farrell-Jones Conjecture in algebraic K-theory, Bost- and Baum-Connes-Conjecture in topological K-theory
I shall discuss joint work with Angus Macintyre on the model theory of the ring of adeles of a number field
I will explain what a perfect obstruction theory is, and how it gives rise to a "virtual" fundamental class of the right expected dimension, even when the dimension of the moduli space is wrong. These virtual fundamental classes are one of the main preoccupations of "modern" moduli theory, being the central object of study in Gromov-Witten and Donaldson-Thomas theory. The purpose of the talk is to remove the black-box status of these objects. If there is time I will do some cheer-leading for dg-schemes, and try to convince the audience that virtual fundamental classes are most happily defined to live in the dg-world.
String theory suggests the simultaneous presence of many ultralight axions possibly populating each decade of mass down to the Hubble scale 10^-33eV. Conversely the presence of such a plenitude of axions (an "axiverse") would be evidence for string theory, since it arises due to the topological complexity of the extra-dimensional manifold and is ad hoc in a theory with just the four familiar dimensions. We investigate how upcoming astrophysical experiments will explore the existence of such axions over a vast mass range from 10^-33eV to 10^-10eV. Axions with masses between 10^-33eV to 10^-28eV cause a rotation of the CMB polarization that is constant throughout the sky. The predicted rotation angle is of order \alpha~1/137. Axions in the mass range 10^-28eV to 10^-18eV give rise to multiple steps in the matter power spectrum, that will be probed by upcoming galaxy surveys and 21 cm line tomography. Axions in the mass range 10^-22eV to 10^-10eV affect the dynamics and gravitational wave emission of rapidly rotating astrophysical black holes through the Penrose superradiance process. When the axion Compton wavelength is of order of the black hole size, the axions develop "superradiant" atomic bound states around the black hole "nucleus". Their occupation number grows exponentially by extracting rotational energy from the ergosphere, culminating in a rotating Bose-Einstein axion condensate emitting gravitational waves. This mechanism creates mass gaps in the spectrum of rapidly rotating black holes that diagnose the presence of axions. The rapidly rotating black hole in the X-ray binary LMC X-1 implies an upper limit on the decay constant of the QCD axion f_a
Let $A$ be a finite set in some ambient group. We say that $A$ is a $K$-approximate group if $A$ is symmetric and if the set $A.A$ (the set of all $xy$, where $x$, $y$ lie in $A$) is covered by $K$ translates of $A$. I will illustrate this notion by example, and will go on to discuss progress on the "rough classification" of approximate groups in various settings: abelian groups, nilpotent groups and matrix groups of fixed dimension. Joint work with E. Breuillard.
The Gilbert model of a random geometric graph is the following: place points at random in a (two-dimensional) square box and join two if they are within distance $r$ of each other. For any standard graph property (e.g. connectedness) we can ask whether the graph is likely to have this property. If the property is monotone we can view the model as a process where we place our points and then increase $r$ until the property appears. In this talk we consider the property that the graph has a Hamilton cycle. It is obvious that a necessary condition for the existence of a Hamilton cycle is that the graph be 2-connected. We prove that, for asymptotically almost all collections of points, this is a sufficient condition: that is, the smallest $r$ for which the graph has a Hamilton cycle is exactly the smallest $r$ for which the graph is 2-connected. This work is joint work with Jozsef Balogh and B\'ela Bollob\'as