In their celebrated paper [CLR10], Caputo, Liggett and Richthammer proved Aldous’ conjecture and showed that for an arbitrary finite graph, the spectral gap of the interchange process is equal to the spectral gap of the underlying random walk. A crucial ingredient in the proof was the Octopus Inequality — a certain inequality of operators in the group ring $\mathbb{R}\left[{\mathrm{Sym}}_{n}\right]$ of the symmetric group. Here we generalise the Octopus Inequality and apply it to generalising the Caputo–Liggett–Richthammer Theorem to certain hypergraphs, proving some cases of a conjecture of Caputo.

Let M be a closed oriented 3-manifold equipped with an Euler structure e and an acyclic representation of its fundamental group. We define a twisted self-linking homology class of the diagonal of the two-point configuration space of M with respect to e. This twisted self-linking homology class appears as an obstruction in the Chern–Simons perturbation theory. When the representation is the maximal free abelian representation $\rho_0$, we prove that our self-linking class is a properly defined “logarithmic derivative” of the Reidemeister–Turaev torsion of $(M,\rho_0,e)$ equipped with the given Euler structure.

We study a family of Thompson-like groups built as rearrangement groups of fractals introduced by Belk and Forrest in 2019, each acting on a Ważewski dendrite. Each of these is a finitely generated group that is dense in the full group of homeomorphisms of the dendrite (studied by Monod and Duchesne in 2019) and has infinite-index finitely generated simple commutator subgroup, with a single possible exception. More properties are discussed, including finite subgroups, the conjugacy problem, invariable generation and existence of free subgroups. We discuss many possible generalisations, among which we find the Airplane rearrangement group $T_A$. Despite close connections with Thompson’s group F, dendrite rearrangement groups seem to share many features with Thompson’s group V.

Let $k{\geqslant} 2$ be an integer and let $\lambda$ be the Liouville function. Given k non-negative distinct integers $h_1,\ldots,h_k$, the Chowla conjecture claims that $\sum_{n{\leqslant} x}\lambda(n+h_1)\cdots \lambda(n+h_k)=o(x)$. An unconditional answer to this conjecture is yet to be found, and in this paper, we take a conditional approach. More precisely, we establish a non-trivial bound for the sums $\sum_{n{\leqslant} x}\lambda(n+h_1)\cdots \lambda(n+h_k)$ under the existence of a Landau–Siegel zero for x in an interval that depends on the modulus of the character whose Dirichlet series corresponds to the Landau–Siegel zero. Our work constitutes an improvement over the previous related results of Germán and Kátai, Chinis and Tao and Teräväinen.

We derive new cases of conjectures of Rubin and of Burns–Kurihara–Sano concerning derivatives of Dirichlet L-series at $s = 0$ in p-elementary abelian extensions of number fields for arbitrary prime numbers p. In naturally arising examples of such extensions one therefore obtains annihilators of class groups from S-truncated Dirichlet L-series for ‘large-enough’ sets of places S.

We prove the following restricted projection theorem. Let $n\ge 3$ and $\Sigma \subset S^{n}$ be an $(n-1)$-dimensional $C^2$ manifold such that $\Sigma$ has sectional curvature $\gt1$. Let $Z \subset \mathbb{R}^{n+1}$ be analytic and let $0 \lt s \lt \min\{\dim Z, 1\}$. Then

\begin{equation*} \dim \{z \in \Sigma\; :\; \dim (Z \cdot z) \lt s\} \le (n-2)+s = (n-1) + (s-1) \lt n-1.\end{equation*}

$z \in \Sigma$

$\dim (Z \cdot z) = \min\{\dim Z, 1\}$

The core idea, originated from Käenmäki–Orponen–Venieri, is to transfer the restricted projection problem to the study of the dimension lower bound of Furstenberg sets of cinematic family contained in $C^2([0,1]^{n-1})$. This cinematic family of functions with multivariables are extensions of those of one variable by Pramanik–Yang–Zahl and Sogge. Since the Furstenberg sets of cinematic family contain the affine Furstenberg sets as a special case, the dimension lower bound of Furstenberg sets improves the one by Héra, Héra–Keleti–Máthé and Dąbrowski–Orponen–Villa.

Moreover, our method to show the restricted projection theorem can also give a new proof for the Mattila projection theorem in $\mathbb{R}^n$ with $n \ge 3$.

Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and 4-term arithmetic progression (4-AP) density at most $\alpha^C$, for arbitrarily large C. Gowers constructed Fourier uniform sets with density $\alpha$ and 4-AP density at most $\alpha^{4+c}$ for some small constant $c \gt 0$. We show that an affirmative answer to Ruzsa’s question would follow from the existence of an $N^{o(1)}$-colouring of [N] without symmetrically coloured 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$, we show that Ruzsa’s question is equivalent to our arithmetic Ramsey question.

We prove analogous results for all even-length APs. For each odd $k\geq 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and k-AP density at most $\alpha^{c_k \log(1/\alpha)}$. We also prove generalisations to arbitrary one-dimensional patterns.

Given a number field K, we show that certain K-integral representations of closed surface groups can be deformed to being Zariski dense while preserving many useful properties of the original representation. This generalises a method due to Long and Thistlethwaite who used it to show that thin surface groups in $\textrm{SL}(2k+1,\mathbf{Z})$ exist for all k.

The aim of this work is to prove a new sure upper bound in a setting that can be thought of as a simplified function field analogue. This result is comparable to a recent result of the author concerning an almost sure upper bound of random multiplicative functions. Having a simpler quantity allows us to make the proof more accessible.

We study the notion of inhomogeneous Poissonian pair correlations, proving several properties that show similarities and differences to its homogeneous counterpart. In particular, we show that sequences with inhomogeneous Poissonian pair correlations need not be uniformly distributed, contrary to what was till recently believed.

We calculate the orbifold Euler characteristics of all the degree d fine universal compactified Jacobians over the moduli space of stable curves of genus g with n marked points, as defined by Pagani and Tommasi. We show that this orbifold Euler characteristic agrees with the Euler characteristic of $\overline{\mathcal{M}}_{0, 2g+n}$ up to a combinatorial factor, and in particular, is independent of the degree d and the choice of degree d fine compactified universal Jacobian.

A simple probabilistic argument shows that every r-uniform hypergraph with m edges contains an r-partite subhypergraph with at least $({r!}/{r^r})m$ edges. The celebrated result of Edwards states that in the case of graphs, that is $r=2$, the resulting bound $m/2$ can be improved to $m/2+\Omega(m^{1/2})$, and this is sharp. We prove that if $r\geq 3$, then there is an r-partite subhypergraph with at least $({r!}/{r^r}) m+m^{3/5-o(1)}$ edges. Moreover, if the hypergraph is linear, this can be improved to $({r!}/{r^r}) m+m^{3/4-o(1)}$, which is tight up to the o(1) term. These improve results of Conlon, Fox, Kwan and Sudakov. Our proof is based on a combination of probabilistic, combinatorial, and linear algebraic techniques, and semidefinite programming.

A key part of our argument is relating the energy $\mathcal{E}(G)$ of a graph G (i.e. the sum of absolute values of eigenvalues of the adjacency matrix) to its maximum cut. We prove that every m edge multigraph G has a cut of size at least $m/2+\Omega({\mathcal{E}(G)}/{\log m})$, which might be of independent interest.