We shall discuss how the algebra norm can be used to identify structure in groups. No prior familiarity with the area will be assumed.
This is joint work with Uri Onn. We use motivic integration to get the growth rate of the sequence consisting of the number of conjugacy classes in quotients of G(O) by congruence subgroups, where $G$ is suitable algebraic group over the rationals and $O$ the ring of integers of a number field.
The proof uses tools from the work of Nir Avni on representation growth of arithmetic groups and results of Cluckers and Loeser on motivic rationality and motivic specialization.
Automorphisms of right-angled Artin groups interpolate between $Out(F_n)$ and $GL_n(\mathbb{Z})$. An active area of current research is to extend properties that hold for both the above groups to $Out(A_\Gamma)$ for a general RAAG. After a short survey on the state of the art, we will describe our recent contribution to this program: a study of how higher-rank lattices can act on RAAGs that builds on the work of Margulis in the free abelian case, and of Bridson and the author in the free group case.
In just the last year it has been realized that one can define supersymmetric gauge theories on non-trivial compact curved manifolds, coupled to a background R-symmetry gauge field, and moreover that expectation values of certain BPS operators reduce to finite matrix integrals via a form of localization. I will argue that a general approach to this topic is provided by the gauge/gravity correspondence. In particular, I will present several examples of supersymmetric gauge theories on different 1-parameter deformations of the three-sphere, which have a large N limit, together with their gravity duals (which are solutions to Einstein-Maxwell theory). The Euclidean gravitational partition function precisely matches a large N matrix model evaluation of the field theory partition function, as an exact \emph{function} of the deformation parameter.
I will study the sequestering of blow-up fields through a CFT in a toroidal orbifold setting. In particular, I will examine the disk correlator between orbifold blow-up moduli and matter Yukawa couplings. Blow-up moduli appear as twist fields on the worldsheet which introduce a monodromy
condition for the coordinate field X. Thus I will focus on how the presence of twist field affects
the CFT calculation of disk correlators. Further, I will explain how the results are relevant to
suppressing soft terms to scales parametrically below the gravitino mass. Last, I want to explore the
relevance of our calculation for the case of smooth Calabi-Yaus.
We introduce the notion of a real cubing. Roughly speaking, real cubings are to CAT(0) cube complexes what real trees are to simplicial trees. We develop an analogue of the Rips’ machine and establish the structure of groups acting nicely on real cubings.