Research

We classify the Hom–Lie structures on finite-dimensional simple Lie algebras over an algebraically closed field of characteristic different from $2$ and $3$. Except for the three-dimensional simple Lie algebra and the Zassenhaus algebras, every such structure is a scalar multiple of the identity. In the two exceptional cases, we determine the full Hom–Lie space: it has dimension six for the three-dimensional algebra, while for a Zassenhaus algebra it consists precisely of the multiplication operators arising from the corresponding divided-power algebra. This proves a conjecture of Makhlouf and Zusmanovich.

We prove that the multiplicities of every symmetric $Q$-polynomial association scheme form a log-concave sequence. This resolves the 1984 unimodality conjecture of Bannai and Ito and establishes both related inequalities conjectured by Stanton. Our proof is a novel application of strong subadditivity and weak monotonicity inequalities for quantum entropy.

We prove a positivity theorem for the universal Plücker coordinates of Karp and Purbhoo when exactly one parameter is negative, with a sharp uniform threshold governed by the largest Plancherel up-transition probability. At the critical specialisation, positivity of the normalised $(2,2)$-coordinate is equivalent to the Octopus inequality, the main technical tool in the proof of Aldous's spectral gap conjecture. By connecting the inequality with Schubert calculus, this answers questions raised by Caputo and Aldous. For a Wronskian with distinct real zeros, we determine the maximal common half-line on which all branching-balanced coordinates are positive semidefinite. We also give Plücker-theoretic proofs of two hypergraph inequalities of Alon, Kozma, and Puder.

We introduce group-valued coloured interchange processes, a class of continuous-time random walks on wreath products generated by transpositions and arbitrary symmetric base-group transitions. We give a complete representation-theoretic characterisation of their spectral gaps by decomposing the associated quasi-regular representation and determining precisely which irreducible representations are required. For abelian base groups, we realise these representations on multislices and identify their Laplacians with discrete Schrödinger operators. We then classify the minimal families of irreducible representations that determine the spectral gap for every choice of transition rates.

We prove that the $\mathrm{iSWAP}$ gate maximises the spectral gap of the Hermitian second-moment operator on every connected graph with at least three vertices, among all two-local unitary circuit ensembles. We further prove that the polyhedral cone defined by asymmetric four-point inequalities is invariant under the transpose of the $\mathrm{iSWAP}$ semigroup, yielding a componentwise comparison certificate for its positive Perron--Frobenius eigenvector. These results resolve a conjecture of Kong, Li, and Liu.

We prove that every ironed two-qubit gadget whose KAK-derived parameter satisfies $a=5/9$ has, on the complete graph $K_n$ with $n\geqslant 5$, the same second-moment spectral gap as the iSWAP gadget. The central step is a representation-theoretic localisation theorem: the largest strictly negative eigenvalue of the associated $\mathfrak S_n$-invariant operator always occurs in the highest-spin $\mathrm{SU}(2)$ summand. A local positive-semidefinite decomposition separates every spin sector except the two highest. This settles a conjecture of Kong, Li, and Liu.

We prove that, over an arbitrary field and whenever $q$ is not a root of unity, the Baseilhac--Kolb root vectors form a PBW basis of the $q$-Onsager algebra for every total order on the positive roots of $\widehat{\mathfrak{sl}}_2$. This removes the previous transcendence hypothesis. We establish twelve PBW bases in the alternating generators and show that they persist under arbitrary scalar central specialisation of the alternating central extension. We determine the centraliser of the negative alternating subalgebra and that of the first imaginary alternating generator, and deduce that the four single-family alternating polynomial subalgebras are maximal commutative. Together, these results settle four conjectures of Terwilliger and, in characteristic different from $2$, a conjecture of Baseilhac and Belliard.

We show that the Hurwitz problem for sums of squares can depend on the base field. More precisely, we construct an explicit formula of type $[12,12,18]$ over every field of characteristic different from $2$ in which $-1$ is a square, whereas no such formula exists over any formally real field. In particular, a formula of this type exists over $\mathbb Q(i)$ and over $\mathbb C$, but not over $\mathbb Q$ or over $\mathbb R$. This settles, in the negative, a longstanding conjecture of Shapiro from 1984, a conjecture of Adem from 1975, and answers a signed-formula problem raised by Shapiro in 2000.

We prove that if $\Gamma$ is a connected graph with minimum degree $\delta$ and Laplacian eigenvalues $0=\mu_1\lt\mu_2\leqslant \cdots \leqslant \mu_n$, then the toughness of $\Gamma$ is bounded below by $\mu_2/(\mu_n-\delta)$.

We prove an exact finite-volume symmetry formula for two-point functions in the periodic $N$-state superintegrable chiral Potts spin chain. We show that, for every chain length $L$ and every simultaneous eigenvector of the Hamiltonian and the one-site translation operator, the correlations satisfy $\langle Z_0^r Z_R^{\dagger r}\rangle^*=\langle Z_0^r Z_{L-R}^{\dagger r}\rangle$ for $1\leqslant r\leqslant N-1$. Hence, whenever $L$ is even, the midpoint correlation $\langle Z_0^r Z_{L/2}^{\dagger r}\rangle$ is real. Then we generalise the three-state chain case to arbitrary $N$ and to every translation eigensector. This resolves a conjecture of Fabricius and McCoy.

We establish a sharp lower bound on the spectral gap of the biased adjacent-transposition Markov chain on the symmetric group. As a consequence, we resolve a longstanding conjecture of Fill, proving that among all regular probability vectors, the minimum spectral gap of the transition matrix is attained by the uniform probability vector. We also characterise the regular probability vectors attaining the minimum spectral gap and determine the exact multiplicity of the corresponding second-largest eigenvalue. Our proof relies on a novel algebraic decomposition of the transition matrix into elementary orthogonal projections.

We show that whenever \[ [\,\cdot,\cdot]_t = [\,\cdot,\cdot]_0 + t[\,\cdot,\cdot]_1,\qquad \alpha_t = \mathrm{id} + t\alpha_1 \] define an infinitesimal Hom--Lie deformation of $\mathfrak{sl}_2(\mathbb K)$ over $\mathbb K[t]/(t^2)$ and $(\mathfrak{sl}_2(\mathbb K),[\,\cdot,\cdot]_0,\alpha_1)$ is a Hom--Lie algebra, then the deformed bracket $[\,\cdot,\cdot]_t$ satisfies the ordinary Jacobi identity over $\mathbb K[t]$. This solves a conjecture of Makhlouf and Silvestrov from 2010.

We show that the full-flag Johnson graph has spectral gap equal to that of its Schreier quotient arising from the point-stabiliser equitable partition. Our results confirm two conjectures posed by Huang, Huang, and Cioabă, which imply an Aldous-type spectral-gap phenomenon for full-flag Johnson graphs.

We study the $t$-Schur measure on partitions, defined by $\mathbb{P}(\lambda)=Z^{-1}S_\lambda(x;t)s_\lambda(y)$, where $S_\lambda(x;t)$ denotes the $t$-Schur symmetric functions and $s_\lambda(y)$ the ordinary Schur functions, and $Z$ is the normalising constant. Using vertex operator calculus, we realise $S_\lambda(x;t)$ in the charged free-fermion Fock space, yielding a $t$-deformation of the classical boson-fermion correspondence. These realisations give vertex-algebraic proofs of the $t$-Cauchy identities and $t$-Gessel identity. Building on this framework, we compute the correlation functions of the $t$-Schur measure and show that the associated point process is determinantal, with an explicit correlation kernel. The Poissonised $t$-Plancherel measure appears as a specialisation of our construction, so its correlation functions follow as a corollary. As an application, we derive the limiting distribution for the length of the longest ascent pair in a random permutation. Our results interpolate the Schur case at $t=0$, connect to the Schur-$Q$ theory at $t=-1$, and provide a probabilistic interpretation of a natural $t$-refinement of increasing subsequences via a generalised RSK correspondence.

We determine the cutoff profile for Cartesian powers of every fixed finite abelian Cayley graph. We give a construction of simple random walks on abelian Cayley graphs that preserves cutoff and its limiting profile, with applications to deterministic and random generating sets. For an explicit family of Cayley graphs on elementary abelian groups, we determine the complete spectrum, mixing-time asymptotics and limiting profiles. The product condition and cutoff have distinct thresholds in this family, giving walks that satisfy the product condition without exhibiting cutoff. This answers a longstanding question of Peres and a problem of Salez, and settles a conjecture of Hermon and Olesker-Taylor.

We establish sharp quadratic bounds on the relaxation time of simple random walk on connected regular graphs, with leading constants $3/(2\pi^2)$ and $1/\pi^2$ for even and odd orders, respectively. This resolves a longstanding conjecture known as Aldous--Fill spectral gap conjecture (2002). We also prove uniqueness and stability theorems for the corresponding cubic and quartic chains, and settle the quartic uniqueness conjecture posed by Abdi, Ghorbani, and Imrich (2021) and by Abdi and Ghorbani (2023). We establish sharp bounds on algebraic connectivity under minimum-degree and regularity constraints and prove the conjectured chain structure of the minimisers, identifying their repeating blocks. This resolves a longstanding conjecture of Guiduli and Mohar (1996) and a structural conjecture of Abdi and Ghorbani (2024). Our quantitative stability theorem shows that nearly minimum algebraic connectivity forces nearly maximum diameter, proving their diameter conjecture. We further obtain sharp relaxation-time bounds in terms of edge-connectivity. For nonregular graphs, we establish sharp bounds for the gap between maximum degree and adjacency spectral radius, confirming a conjecture of Liu (2024). As applications, we prove the sharp bounds on hitting and commute times conjectured by Aldous and Fill (2002) for regular graphs.

We prove that the second-moment Hamiltonian of the open qubit eSWAP chain has a diffusive spectral gap, resolving a conjecture of Li, Zheng, Liu, Jiang, and Liu. We also characterise the ground space on every connected interaction graph with at least five vertices and prove a uniform lower bound for the unnormalised complete-graph gap. We apply these results to sequential and brickwork circuits and prove that the periodic chain also has a diffusive spectral gap.

We study the random walk on a Cayley graph of a finite group whose generators are chosen independently and uniformly at random. We show that, for every finite group, the walk exhibits cutoff with high probability whenever the number of generators is at least a constant multiple of the logarithm of the order of the group and small compared with every positive power of it. This holds for both undirected and directed Cayley graphs, in discrete and continuous time, and the lower bound on the number of generators is best possible, up to the value of the constant. The cutoff time is the time at which the expected entropy of the walk comes close to its maximum, and in general it depends on the structure of the group. This settles a conjecture of Aldous and Diaconis.

We prove that the total-variation $\varepsilon$-mixing time of the $k$-particle symmetric exclusion process is at most $34$ times the single-particle mixing time at error $\varepsilon/k$. The bound holds for every $\varepsilon\in(0,1)$ on every finite connected graph with symmetric edge rates, and $k$ may be replaced by the number of vacancies. This settles a conjecture of Oliveira. We also obtain bounds for the $L^2$ and $L^\infty$ mixing times in terms of independent walks, and a Dirichlet-form inequality with the complete graph whose coefficient is optimal up to a universal constant. Further consequences include logarithmic Sobolev inequalities and energy bounds for the spin-$1/2$ Heisenberg ferromagnet.