L6

Let X be an integral projective variety of degree at least 2 defined over Q, and let B>0 an integer. The dimension growth conjecture, now proven in almost all cases following works of Browning, Heath-Brown, and Salberger, provides a certain uniform upper bound on the number of rational points of height at most B lying on X. 

Shifting to the geometric setting (where X may be defined over C(t)), the collection of C(t)-rational points lying on X of degree at most B naturally has the structure of an algebraic variety, which we denote by X(B). In ongoing work with Tijs Buggenhout and Floris Vermeulen, we uniformly bound the dimension and, when the degree of X is at least 6, the number of irreducible components  of X(B) of largest possible dimension​ analogously to dimension growth bounds. We do this by developing a geometric determinant method, and by using results on rational points on curves over function fields. 

Joint with Tijs Buggenhout and Floris Vermeulen.

A classical approach to studying the topology of a manifold is through the analysis of its submanifolds. The realm of 3-manifolds is particularly rich and diverse, and we aim to explore the complexity of surfaces within a given 3-manifold. After reviewing the fundamental definitions of the Thurston norm, we will present a constructive method for computing it on Seifert fibered manifolds and extend this approach to graph manifolds. Finally, we will outline which norms can be realized as the Thurston norm of some graph manifold and examine their key properties.

The hyperbolic plane and its higher-dimensional analogues are well-known
objects. They belong to a larger class of spaces, called rank-one
symmetric spaces, which include not only the hyperbolic spaces but also
their complex and quaternionic counterparts, and the octonionic
hyperbolic plane. By a result of Pansu, two of these families exhibit
strong rigidity properties with respect to their self-quasiisometries:
any self-quasiisometry of a quaternionic hyperbolic space or the
octonionic hyperbolic plane is at uniformly bounded distance from an
isometry. The goal of this talk is to give an overview of the rank-one
symmetric spaces and the tools used to prove Pansu's rigidity theorem,
such as the subRiemannian structure of their visual boundaries and the
analysis of quasiconformal maps.

Originally posed in the 1950s, the Hadwiger-Nelson problem interrogates the ‘chromatic number of the plane’ via an infinite unit-distance graph. This question remains open today, known only to be 5,6, or 7. We may ask the same question of the hyperbolic plane; there the lack of homogeneous dilations leads to unique behaviour for each length scale d. This variance leads to other questions: is the d-chromatic number finite for all d>0? How does the d-chromatic number behave as d increases/decreases? In this talk, I will provide a summary of existing methods and results, before discussing improved bounds through the consideration of semi-regular tilings of the hyperbolic plane.

Congruence Subgroup Property is a characterisation of finite-index subgroups of automorphism groups. It first arose from the study of subgroups of linear groups. In this talk, I will show a few examples where it holds and where it fails, and give an overview of what is known about the family $SL_n\mathbb{Z}$, $Out(F_n)$, $MCG(\Sigma)$. Then I will describe some related results in the case of Mapping Class Groups, and explain their relation to profinite rigidity of 3-manifolds.

Donaldson proved that there are pairs of 4-manifolds that are homeomorphic but not diffeomorphic, a phenomenon that does not appear for any lower dimensional manifolds. Until recently, proving this for compact manifolds has required smooth 4-manifold invariants coming from gauge theory. In this talk, we will give an introduction to an exciting new smooth 4-manifold invariant of Morrison Walker and Wedich, called a skein lasagna module that does not rely on gauge theory. Further, this talk will not assume any knowledge of 4-manifold topology.

A group is free-by-cyclic if it is an extension of a free group by a cyclic group. Knowing that a group is virtually free-by-cyclic is often quite useful; it implies that the group is coherent and that it is cohomologically good in the sense of Serre. In this talk we will give a homological characterisation of when a finitely generated RFRS group is virtually free-by-cylic and discuss some generalisations.

In this talk I will give an introduction to analogues to the classical finiteness conditions FP_n for totally disconnected locally compact groups. I will present a construction of non-discrete tdlc groups of arbitrary finiteness length. As a bi-product we also obtain a new collection of (discrete) Thompson-like groups which contains, for all positive integers n, groups of type FP_n but not of type FP_{n+1}. This is joint work with I. Castellano, B. Marchionna, and Y. Santos-Rego.

Let G be a free group of rank N, let f be an automorphism of G and let Fix(f) be the corresponding subgroup of fixed points. Bestvina and Handel showed that the rank of Fix(f) is at most N, for which they developed the theory of train track maps on free groups. Different arguments were provided later on by Sela, Paulin and Gaboriau-Levitt-Lustig. In this talk, we present a new proof which involves the Linnell division ring of G. We also discuss how our approach relates to previous ones and how it gives new insight into variations of the problem.