The Riemann hierarchy is the simplest example of rank one, ($1$+$1$)-dimensional integrable system of nonlinear evolutionary PDEs. It corresponds to the dispersionless limit of the Korteweg--de Vries hierarchy. In the language of formal variational calculus, we address the classification problem for deformations of the Riemann hierarchy satisfiying different extra requirements (general deformations, defomations as systems of conservation laws, Hamiltonian deformations, and tau-symmetric deformations), under the natural group of coordinate transformations preserving each of those requirements. We present several results linking previous conjectures of Dubrovin--Liu--Yang--Zhang (for the tau-symmetric case) and of Arsie--Lorenzoni--Moro (for systems of conservation laws) to the double ramification hierarchy construction of integrable hierarchies from partial CohFTs and F-CohFTs. We prove that, if the conjectures are true, DR hierarchies of rank one are universal objects in the space of deformations of the Riemann hierarchy. We also prove a weaker version of the DLYZ conjecture and that the ALM conjecture implies (the main part of) the DLYZ conjecture. Finally we characterize those rank one F-CohFTs which give rise to Hamiltonian deformations of the Riemann hierarchy.

In this paper, we investigate solvable structures associated to Hamiltonian equations. For a completely integrable Hamiltonian system with $n$ degrees of freedom, we construct a canonical solvable structure consisting of $2n$ Hamiltonian vector fields. We derive explicit expressions for the corresponding Pfaffian forms, whose integration provides solutions to the Hamiltonian equations. We show that the upper $n$ forms give the action varibles, while the lower $n$ forms yield the angle variables of the system. This offers a novel interpretation of the Arnold--Liouville theorem in terms of solvable structures. We ilustrate the theory by deriving explicit solutions and action--angle variables for $n$ harmonic oscillators and the Calogero--Moser system.

A new approach to the study of spectral asymmetry for systems of partial differential equations (PDEs) on closed manifolds was proposed in a recent series of papers by the first author and collaborator. They showed that information on spectral asymmetry can be encoded within and recovered from a negative order pseudodifferential operator -- the asymmetry operator -- constructed from appropriately defined pseudodifferential (spectral) projections. In this manuscript we apply these techniques to the study of the massless Dirac operator; in particular, we compute the principal symbol of the asymmetry operator, accounting for the underlying gauge invariance.

In this paper we present a geometrical framework to study the uniformity of a composite material by means of double groupoid theory. The notions of vertical and horizontal uniformity are introduced, as well as other weaker ones that allows us to study other possible notions of more general uniformity.

We study from a mathematical point of view the nanoparticle model of a magnetic colloid, presented by G. Klughertz. Our objective is to obtain properties of stable stationary structures that arise in the long-time limit for the magnetic nanoparticles dynamics following this model. In this article, we present a detailed study of two specific structures using techniques from the calculus of variations. The first, called the spear, consists of a chain of aligned particles interacting via a Lennard-Jones potential. We establish existence and uniqueness results, derive bounds on the distances between neighboring particles, and provide a sharp asymptotic description as the number of particles tends to infinity. The second structure, the ring, features particles uniformly distributed along a circle. We prove its existence and uniqueness and derive an explicit formula for its radius.

In the present paper we revisit the Helmholtz equation on the Euclidean plane and make some remarks on normalization constants and completeness of wave function sets. The coefficients of interbasis expansions are also reconsidered.

General description of eigenfunctions of integrable Hamiltonians associated with the integer rays of Ding-Iohara-Miki (DIM) algebra, is provided by the theory of Chalykh Baker-Akhiezer functions (BAF) defined as solutions to a simply looking linear system. Solutions themselves are somewhat complicated, but much simpler than they could. It is because of simultaneous partial factorization of all the determinants, entering Cramer's rule. This is a conspiracy responsible for a relative simplicity of the Macdonald polynomials and of the Noumi-Shirashi functions, and it is further continued to all integer DIM rays. Still, factorization is only partial, moreover, there are different branches and abrupt jumps between them. We explain this feature of Cramer's rule in an example of a matrix that defines BAF and exhibits a non-analytical dependence on parameters. Moreover, the matrix is such that there is no natural expansion around non-degenerate approximations, which causes an unexpected complexity of formulas.

I point out that a possible minimal setting to realize Kitaev's proposal of a Topological Quantum Computation which offers topological protection from decoherence could in principle be realized by braided Majorana qubits. Majorana qubits and their braiding were introduced in Nucl. Phys. B 980, 115834 (2022) and further analyzed in J. Phys. A: Math. Theor. 57, 435203 (2024). Braided Majorana qubits implement a Gentile-type parastatistics with at most $s-1$ excited states accommodated in a multiparticle sector (the integer $s=2,3,4,\ldots$ labels quantum group reps at roots of unity). It is argued that braided Majorana qubits could play, for topological quantum computers, the same role as standard bits for ordinary computers and as qubits for "ordinary" quantum computers.

In this article we consider the linear inelastic Boltzmann equation in presence of a uniform and fixed gravity field, in the case of Maxwell molecules. We first obtain a well-posedness result in the space of finite, non-negative Radon measures. In addition, we rigorously prove the existence of a stationary solution under the non-equilibrium condition which is induced by the presence of the external field. We further show that this stationary solution is unique in the class of the finite, non-negative Radon measures with finite first order moment, and that all the solutions in this class converge towards the stationary solution in the weak topology of the measures.

The Markov property entails the conditional independence structure inherent in Gibbs distributions for general classical Hamiltonians, a feature that plays a crucial role in inference, mixing time analysis, and algorithm design. However, much less is known about quantum Gibbs states. In this work, we show that for any Hamiltonian with a bounded interaction degree, the quantum Gibbs state is locally Markov at arbitrary temperature, meaning there exists a quasi-local recovery map for every local region. Notably, this recovery map is obtained by applying a detailed-balanced Lindbladian with jumps acting on the region. Consequently, we prove that (i) the conditional mutual information (CMI) for a shielded small region decays exponentially with the shielding distance, and (ii) under the assumption of uniform clustering of correlations, Gibbs states of general non-commuting Hamiltonians on $D$-dimensional lattices can be prepared by a quantum circuit of depth $e^{O(\log^D(n/\epsilon))}$, which can be further reduced assuming certain local gap condition. Our proofs introduce a regularization scheme for imaginary-time-evolved operators at arbitrarily low temperatures and reveal a connection between the Dirichlet form, a dynamic quantity, and the commutator in the KMS inner product, a static quantity. We believe these tools pave the way for tackling further challenges in quantum thermodynamics and mixing times, particularly in low-temperature regimes.

Conditional mutual information (CMI) has recently attracted significant attention as a key quantity for characterizing quantum correlations in many-body systems. While it is conjectured that CMI decays rapidly in finite-temperature Gibbs states, a complete and general proof remains elusive. Previous work addressed this problem in the high-temperature regime using cluster expansion techniques [T. Kuwahara, K. Kato, F.G.S.L. Brand\~ao, Phys. Rev. Lett. 124, 220601 (2020)]; however, flaws in the proof have been pointed out, and the method does not provide a uniformly convergent expansion at arbitrarily high temperatures. In this work, we demonstrate that the cluster expansion approach indeed fails to converge absolutely, even at any high-temperatures. To overcome this limitation, we propose a new approach to proving the spatial decay of CMI. Our method leverages the connection between CMI and quantum recovery maps, specifically utilizing the Fawzi-Renner theorem. We show that such recovery maps can be realized through dissipative dynamics, and by analyzing the locality properties of these dynamics, we establish the exponential decay of CMI in high-temperature regimes. As a technical contribution, we also present a new result on the perturbative stability of quasi-local Liouvillian dynamics. Our results indicate that, contrary to common intuition, high-temperature Gibbs states can exhibit nontrivial mathematical structure, particularly when multipartite correlations such as CMI are considered.

We consider the Knizhnik-Zamolodchikov equations in Deligne Categories in the context of $(\mathfrak{gl}_m,\mathfrak{gl}_{n})$ and $(\mathfrak{so}_m,\mathfrak{so}_{2n})$ dualities. We derive integral formulas for the solutions in the first case and compute monodromy in both cases.

Liouville field theory has long been a cornerstone of two-dimensional quantum field theory and quantum gravity, which has attracted much recent attention in the mathematics literature. Timelike Liouville field theory is a version of Liouville field theory where the kinetic term in the action appears with a negative sign, which makes it closer to a theory of quantum gravity than ordinary (spacelike) Liouville field theory. Making sense of this "wrong sign" requires a theory of Gaussian random variables with negative variance. Such a theory is developed in this paper, and is used to prove the timelike DOZZ formula for the $3$-point correlation function when the parameters satisfy the so-called "charge neutrality condition". Expressions are derived also for the $k$-point correlation functions for all $k\ge 3$, and it is shown that these functions approach the correct semiclassical limits as the coupling constant is sent to zero.

The Landau-Zener formula describes the diabatic transition probability of a two-level system under linear driving. Its rigorous derivation typically relies on sophisticated mathematical tools, such as special functions, Laplace transforms, or contour integrals. In this work, we present a derivation of the Landau-Zener transition probability using a fundamentally different approach via functional equations. By leveraging integrability, we prove that this transition probability satisfies a functional equation, whose solutions establish the exponential form of the formula. The coefficient in the exponent is then determined through a lowest-order perturbation calculation. This derivation is rigorous and mathematically simple. Our work provides new insight into the origin of the exponential form of the Landau-Zener transition probability.

We present a novel approach to quantizing Markov chains. The approach is based on the Markov chain coupling method, which is frequently used to prove fast mixing. Given a particular coupling, e.g., a grand coupling, we construct a completely positive and trace preserving map. This quantum map has a unique fixed point, which corresponds to the quantum sample (qsample) of the classical Markov chain's stationary distribution. We show that the convergence time of the quantum map is directly related to the coupling time of the Markov chain coupling.

We present a variational framework in which Centroidal Voronoi Tessellations (CVTs) arise as local minimizers of a generalized electrostatic energy functional. By modeling interior point distributions in a convex domain as repelling charges balanced against a continuous boundary charge, we show that the resulting equilibrium configurations converge to CVT structures. We prove this by showing that CVTs minimize both the classical centroidal energy and the electrostatic potential, establishing a connection between geometric quantization and potential theory. Finally, we introduce a thermodynamic annealing scheme for global CVT optimization, rooted in Boltzmann statistics and random walk dynamics. By introducing a scheme for varying time steps (faster or slower cooling) we show that the set of minima of the centroid energy functional (and therefore the electrostatic potential) can be recovered. By recovering a set of generator locations corresponding to each minimum we can create a lattice continuation that allows for a customizable framework for individual minimum seeking.

In this paper, we study weakly interacting diffusion processes on random graphs. Our main focus is on the properties of the mean-field limit and, in particular, on the nonuniqueness of stationary states. By extending classical bifurcation analysis to include multichromatic interaction potentials and random graph structures, we explicitly identify bifurcation points and relate them to the eigenvalues of the graphon integral operator. Furthermore, we characterize the resulting McKean-Vlasov PDE as a gradient flow with respect to a suitable metric. We combine these theoretical results with the spectral analysis of the linearized McKean-Vlasov operator and extensive numerical simulations to gain insight into the stability and long-term behaviour of stationary solutions. In addition, we provide strong evidence that (minus) the interaction energy of the interacting particle system serves as a natural order parameter. In particular, beyond the transition point and for multichromatic interactions, we observe an energy cascade that is strongly linked to the dynamical metastability of the system.

We consider percolation of the vacant set of random interlacements at intensity $u$ in dimensions three and higher, and derive lower bounds on the truncated two-point function for all values of $u>0$. These bounds are sharp up to principal exponential order for all $u$ in dimension three and all $u \neq u_\ast$ in higher dimensions, where $u_*$ refers to the critical parameter of the model, and they match the upper bounds derived in the article arXiv:2503.14497. In dimension three, our results further imply that the truncated two-point function grows at large distances $x$ at a rate that depends on $x$ only through its Euclidean norm, which offers a glimpse of the expected (Euclidean) invariance of the scaling limit at criticality. The rate function is atypical, it incurs a logarithmic correction and comes with an explicit pre-factor that converges to $0$ as the parameter $u$ approaches the critical point $u_*$ from either side. A particular challenge stems from the combined effects of lack of monotonicity due to the truncation in the super-critical phase, and the precise (rotationally invariant) controls we seek, that measure the effects of a certain "harmonic humpback" function. Among others, their derivation relies on rather fine estimates for hitting probabilities of the random walk in arbitrary direction $e$, which witness this invariance at the discrete level, and preclude straightforward applications of projection arguments.