L3

Free groups, free abelian groups and fundamental groups of

closed orientable surfaces are the most basic and well-understood

examples of infinite discrete groups. The automorphism groups of

these groups, in contrast, are some of the most complex and intriguing

groups in all of mathematics. In these lectures I will concentrate

on groups of automorphisms of free groups, while drawing analogies

with the general linear group over the integers and surface mapping

class groups. I will explain modern techniques for studying

automorphism groups of free groups, which include a mixture of

topological, algebraic and geometric methods.

I will discuss quasi-isometries of the free group that preserve an

equivariant pattern of lines.

There is a type of boundary at infinity whose topology determines how

flexible such a line pattern is.

For sufficiently complicated patterns I use this boundary to define a new

metric on the free group with the property that the only pattern preserving

quasi-isometries are actually isometries.

Chataur and Menichi showed that the homology of the free loop space of the classifying space of a compact Lie group admits a rich algebraic structure: It is part of a homological field theory, and so admits operations parametrised by the homology of mapping class groups.  I will present a new construction of this field theory that improves on the original in several ways: It enlarges the family of admissible Lie groups.  It extends the field theory to an open-closed one.  And most importantly, it allows for the construction of co-units in the theory.  This is joint work with Anssi Lahtinen.

The partition function of 3d N=2 superconformal theories on the

3-sphere can be computed exactly by localization methods. I will explain

some sublteties associated to that important result. As a by-product, this

analysis establishes the so-called F-maximization principle for N=2 SCFTs in

3d: the exact superconformal R-charge maximizes the 3-sphere free energy

F=-log Z.

We study pure Seiberg-Witten theories with $SU(r+1)$ and $Sp(2r)$ gauge groups with no flavors. We study singularity loci of moduli space of the Seiberg-Witten curve. Using exterior derivative and discriminant operators, we can find Argyres-Douglas loci of the SW theory. We also compute BPS charges of the massless dyons of $SU$ and $Sp$ SW theory. In a detailed example of $C_2=Sp(4)$, we find 6 points in the moduli space where we have 2 massless BPS dyons, and 3 of them give Argyres-Douglas loci. We show that BPS charges of the massless dyons jump as we go across Argyres-Douglas loci, giving an explicit example of Argyres-Douglas loci living inside the wall of marginal stability. (Based on work in progress with Keshav Dasgupta)

The Eilenberg-Ganea conjecture is the statement that every group of cohomological dimension two admits a two-dimensional classifying space.  This problem is unsolved after 50 years.  I shall discuss the background to this question and negative answers to some other related questions.  This includes recent joint work with Martin Fluch.

The idea of three-tier conformal field theory (CFT) was first proposed by Greame Segal. It is an extension of the functorial approach to CFT, where one replaces the bordism category of Riemann surfaces by a suitable bordism 2-category, whose objects are points, whose morphism are 1-manifolds, and whose 2-morphisms are pieces of Riemann surface. The Baez-Dolan cobordism hypothesis is a meta-mathematical principle. It claims that functorial quantum field theory (i.e. quantum field theory expressed as a functor from some bordism category) becomes simper once "you go all the way down to points", i.e., once you replace the bordism category by a higher category. Three-tier CFT is an example of "going all the way down to points". We will apply the cobordism hypothesis to the case of three-tier CFT, and show how the modular invariance of the partition function can be derived as a consequence of the formalism, even if one only starts with genus-zero data.

Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the first talk.

I shall describe the construction of the four-brane giant graviton on $\mathrm{AdS}_4\times \mathbb{CP}^3$ (extended and moving in the complex projective space), which is dual to a subdeterminant operator in the ABJM model. This dynamically stable, BPS configuration factorizes at maximum size into two topologically stable four-branes (each wrapped on a different $\mathbb{CP}^2 \subset \mathbb{CP}^3$ cycle) dual to ABJM dibaryons. Our study of the spectrum of small fluctuations around this four-brane giant provides good evidence for a dependence in the spectrum on the size, $\alpha_0$, which is a direct result of the changing shape of the giant’s worldvolume as it grows in size. I shall finally comment upon the implications for operators in the non-BPS, holomorphic sector of the ABJM model.