L6

You may be surprised to see the generalised Riemann hypothesis appear in algorithmic topology. For example, knottedness was originally shown to be in NP under the assumption of GRH.
Where does this condition come from? We will discuss this in the context of 3-sphere recognition, and examine why the approach fails for higher dimensions.

It is a truth universally acknowledged, that an infinite group in possession of a good algebraic structure, must be in want of a hyperbolic space to act on. For the mapping class group of a surface, one of the most popular choices is the curve graph. This is a combinatorial object, built from curves on the surface and intersection patterns between them.
Hyperbolicity of the curve graph was proved by Masur and Minsky in a celebrated paper in 1999. In the same article, they showed how the geometry of the action on this graph reflects dynamical/topological properties of the mapping class group; in particular, loxodromic elements are precisely the pseudo-Anosov mapping classes.
In light of this, one would like to better understand distances in the curve graph. The graph is locally infinite, and finding a shortest path between two vertices is highly non-trivial. In this talk, we will see how to use the machinery of train tracks to overcome this issue and compute (approximate) distances in the curve graph. If time permits -- which, somehow, it never does -- we will also analyse this construction from an algorithmic perspective.

An old and challenging conjecture proposed by R.H. Fox in 1962 states that the absolute values of the coefficients of the Alexander polynomial of an alternating knot are trapezoidal i.e. strictly increase, possibly plateau, then strictly decrease. We give a survey of the known results and use them to motivate the study of branched double covers. The second part of the talk focuses on the properties of the branched double covers of alternating knots.

The space of subgroups of a countable group is a compact topological space which encodes many of the properties of its non-free actions. We will discuss some approaches to studying the Cantor-Bendixson decomposition of this space in the context of hyperbolic groups and groups which act (nicely) on trees. We will also give some conditions under which the conjugation action on the perfect kernel is highly topologically transitive and see how this can be applied to find new examples of groups (including all virtually compact special groups) which admit faithful transitive amenable actions. This is joint work with Damien Gaboriau.

Algebraic fibring is the group-theoretic analogue of fibration over the circle for manifolds. Generalising the work of Agol on hyperbolic 3-manifolds, Kielak showed that many groups virtually fibre. In this talk we will discuss the geometry of groups which fibre, with some fun applications to Poincare duality groups - groups whose homology and cohomology invariants satisfy a Poincare-Lefschetz type duality, like those of manifolds - as well as to exotic subgroups of Gromov hyperbolic groups. No prior knowledge of these topics will be assumed.

Disclaimer: This talk will contain many manifolds.

We study the distribution of eigenvalues of kernel random matrices where each element is the empirical covariance between the feature map evaluations of a random fully-connected neural network. We show that, under mild assumptions on the non-linear activation function, namely Lipschitz continuity and measurability, the limiting spectral distribution can be written as successive free multiplicative convolutions between the Marchenko-Pastur law and a nonrandom measure specific to the neural network. The latter has no known analytical expression but can be simulated empirically, separately from the random matrices of interest.

If some structure, mathematical or otherwise, is giving you grief, then often the first thing to do is to attempt to break the offending object down into (finitely many) simpler pieces.

In group theory, when we speak of questions of *accessibility* we are referring to the ability to achieve precisely this. The idea of an 'accessible group' was first coined by Terry Wall in the 70s, and since then has left quite a mark on our field (and others). In this talk I will introduce the toolbox required to study accessibility, and walk you and your groups through some reasons to be accessible.

One of the fundamental problems in representation theory is determining the representation type of algebras. In this talk, we will introduce the representation type of cyclotomic quiver Hecke algebras, also known as cyclotomic Khovanov-Lauda-Rouquier algebras, especially in affine type A and affine type C. Our main result relies on novel constructions of the maximal dominant weights of integrable highest weight modules over quantum groups. This talk is based on collaborations with Susumu Ariki, Berta Hudak, and Linliang Song.

There are several functional encodings of random trees which are commonly used to prove (among other things) scaling limit results.  We consider two of these, the height process and Lukasiewicz path, in the classical setting of a branching process tree with critical offspring distribution of finite variance, conditioned to have n vertices.  These processes converge jointly in distribution after rescaling by n^{-1/2} to constant multiples of the same standard Brownian excursion, as n goes to infinity.  Their difference (taken with the appropriate constants), however, is a nice example of a discrete snake whose displacements are deterministic given the vertex degrees; to quote Marckert, it may be thought of as a “measure of internal complexity of the tree”.  We prove that this discrete snake converges on rescaling by n^{-1/4} to the Brownian snake driven by a Brownian excursion.  We believe that our methods should also extend to prove convergence of a broad family of other “globally centred” discrete snakes which seem not to be susceptible to the methods of proof employed in earlier works of Marckert and Janson.

This is joint work in progress with Louigi Addario-Berry, Serte Donderwinkel and Rivka Mitchell.