Khintchine's Theorem is one of the cornerstones in metric Diophantine approximation. The question of removing the monotonicity condition on the approximation function in Khintchine's Theorem led to the recently proved Duffin-Schaeffer conjecture. Gallagher showed an analogue of Khintchine's Theorem for multiplicative Diophantine approximation, again assuming monotonicity. In this talk, I will discuss my joint work with L. Frühwirth about a Duffin-Schaeffer version for Gallagher's Theorem. Furthermore, I will give a broader overview on various questions in metric Diophantine approximation and demonstrate the deep connection to both analytic and combinatorial number theory that is hidden inside the proof of these statements.

Functoriality is a key feature in Langlands’ conjectured relationship between automorphic representations and Galois representations; it predicts that certain Galois representations are automorphic, i.e. should come from automorphic representations. We discuss the idea of $p$-adic propagation of automorphy, which seeks to establish the automorphy of everything in a “neighborhood” given the automorphy of something in that neighborhood. The “neighborhoods” that we consider will be the irreducible components of a $p$-adic analytic space called the eigenvariety, which parameterizes $p$-adic automorphic representations. This technique was introduced by Newton and Thorne in their proof of symmetric power functoriality, and can be adapted to investigate similar problems.

An interesting twist on classical subgraph containment problems in graph theory is the following: given a graph $H$ and a collection $\{G_1, \dots , G_m\}$ of graphs on a common vertex set $[n]$, what conditions on $G_i$ guarantee a copy of $H$ using at most one edge from each $G_i$? Such a subgraph is called transversal, and the above problem is closely related to the study of temporal graphs in Network Theory. In 2020 Joos and Kim showed that if $\delta(G_i)\geq n/2$, the collection contains a transversal Hamilton cycle. We improve on their result by showing that it actually contains every transversal Hamilton cycle if $\delta(G_i)\geq (1/2+o(1))n$. That is, for every function $\chi:[n]\to[m]$, there is a Hamilton cycle whose $i$-th edge belongs to $G_{\chi(i)}$.

This is joint work with Candida Bowtell, Patrick Morris and Katherine Staden.

Let $G$ be a $d$-regular graph of growing degree on $n$ vertices. Form a random subgraph $G_p$ of $G$ by retaining edge of $G$ independently with probability $p=p(d)$. Which conditions on $G$ suffice to observe a phase transition at $p=1/d$, similar to that in the binomial random graph $G(n,p)$, or, say, in a random subgraph of the binary hypercube $Q^d$?

We argue that in the supercritical regime $p=(1+\epsilon)/d$, $\epsilon>0$ a small constant, postulating that every vertex subset $S$ of $G$ of at most $n/2$ vertices has its edge boundary at least $C|S|$, for some large enough constant $C=C(\epsilon)>0$, suffices to guarantee likely appearance of the giant component in $G_p$. Moreover, its asymptotic order is equal to that in the random graph $G(n,(1+\epsilon)/n)$, and all other components are typically much smaller.

We also give examples demonstrating tightness of our main result in several key senses.

A joint work with Sahar Diskin, Joshua Erde and Mihyun Kang.