University of Oxford

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

Consider the following "controlled" random graph process: edges of the complete graph are revealed one by one in random order to an online algorithm, which immediately decides whether to retain each observed edge. The algorithm's objective is to construct a graph property within specified constraints on the total number of observed edges ("time") and the total number of retained edges ("budget").

During this talk, I will present results in this model for natural graph properties, such as connectivity, Hamiltonicity, and containment of fixed-size subgraphs. Specifically, I will describe a strategy to construct a Hamilton cycle at the hitting time for minimum degree 2 by retaining a linear number of edges. This extends the classical hitting time result for Hamiltonicity originally established by Ajtai–Komlós–Szemerédi and Bollobás.

The talk is based on joint work with Alan Frieze and Michael Krivelevich.

Abstract: Shifted moments of the Riemann zeta function, introduced by Chandee, are natural generalizations of the moments of zeta. While the moments of zeta capture large values of zeta, the shifted moments also capture how the values of zeta are correlated along the half line. I will describe recent work giving sharp bounds for shifted moments assuming the Riemann hypothesis, improving previous work of Chandee and Ng, Shen, and Wong. I will also discuss some unconditional results about shifted moments with small exponents.

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.

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.

Kaplansky's Zerodivisor Conjecture predicts that the group algebra kG is a domain, where k is a field and G is a torsion-free group. Though the general sentiment is that the conjecture is false, it still remains wide open after more than 70 years. In this talk we will survey known positive results surrounding the Zerodivisor Conjecture, with a focus on the technique of embedding group algebras into division rings. We will also present some new results in this direction, which are joint with Pablo Sánchez Peralta.

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.