A previous work introduced pair space, which is spanned by the center of mass of a system and the relative positions (pair positions) of its constituent bodies. Here, I show that in the $N$-body Newtonian problem, a configuration that does not remain on a fixed line in space is a central configuration if and only if it conserves all pair angular momenta. For collinear systems, I obtain a set of equations for the ratios of the relative distances of the bodies, from which I derive some bounds on the minimal length of the line. For the non-collinear case I derive some geometrical relations, independent of the masses of the bodies. These are necessary conditions for a non-collinear configuration to be central. They generalize, to arbitrary $N$, a consequence of the Dziobek relation, which holds for $N=4$.

I introduce an extended configuration space for classical mechanical systems, called pair-space, which is spanned by the relative positions of all the pairs of bodies. To overcome the non-independence of this basis, one adds to the Lagrangian a term containing auxiliary variables. As a proof of concept, I apply this representation to the three-body problem with a generalized potential that depends on the distance $r$ between the bodies as $r^{-n}$. I obtain the equilateral and collinear solutions (corresponding to the Lagrange and Euler solutions if $n=1$) in a particularly simple way. In the collinear solution, this representation leads to several new bounds on the relative distances of the bodies.

We study dynamical models on a self-similar space-time lattice as toy models for multiscale motion in hydrodynamic turbulence. Here an ill-posed ideal system is regularized at small scales and the vanishing regularization (inviscid) limit is considered. By relating the inviscid limit to the dynamics of the RG operator acting on the flow maps, we explain the existence and universality (regularization independence) of the limiting solutions as a consequence of the fixed-point RG attractor. Considering the local linearized dynamics, we show that the approach to the inviscid limit is governed by the universal RG eigenmode. We also demonstrate that the RG attractor undergoes a period-doubling bifurcation with parameter variation, thereby changing the nature of the inviscid limit. In the case of chaotic RG dynamics, we introduce the stochastic RG operator acting on Markov kernels. Then the RG attractor becomes stochastic, which explains the existence and universality of spontaneously stochastic solutions in the limit of vanishing noise. We study a linearized structure (RG eigenmode) of the stochastic RG attractor and its period-doubling bifurcation. Viewed as prototypes of Eulerian spontaneous stochasticity, our models explain its mechanism, universality and potential diversity.

We consider extended Cartan homotopy formula (ECHF) for higher gauge theory. Firstly, we construct an oriented simplex based on 2-connections and present differential and integral forms of the higher ECHF. Then, we study the higher Chern-Simons-Antoniadis-Savvidy (ChSAS) theory and prove that the higher ECHF can reproduce the higher Chern-Weil theorem and give higher triangle equation. We finally conclude from the higher ECHF that a higher transgression form can be written as the difference of two higher ChSAS forms minus an exact form.

We prove a quenched version of the large deviation principle for Birkhoff-like sums along a sequence of random quantum measurements driven by an ergodic process. We apply the result to the study of entropy production in the two-time measurement framework.

In this paper we provide a complete characterisation of coisotropic embeddings of precosymplectic manifolds into cosymplectic manifolds. This result extends a theorem of Gotay about coisotropic embeddings of presymplectic manifolds. We also extend to the cosymplectic case some results of A. Weinstein which generalise the Darboux theorem. While symplectic geometry is the natural framework for developing Hamiltonian mechanics, cosymplectic geometry is the corresponding framework for time-dependent Hamiltonian mechanics. The motivation behind proving this theorem is to generalise known results for symplectic geometry to cosymplectic geometry, so that they can be used to study time-dependent systems, for instance for the regularization problem of singular Lagrangian systems.

It is well known that the chain map between the de Rham and Poisson complexes on a Poisson manifold also maps the Koszul bracket of differential forms into the Schouten bracket of multivector fields. In the generalized case of a $P_\infty$-structure, where a Poisson bivector $P$ is replaced by an arbitrary even multivector obeying $[[P,P]]=0$, an analog of the chain map and an $L_\infty$-morphism from the higher Koszul brackets into the Schouten bracket are also known; however, they differ significantly in nature. In the present paper, we address the problem of quantizing this picture. In particular, we show that the $L_\infty$-morphism is quantized into a single linear operator, which is a formal Fourier integral operator. This paper employs Voronov's thick morphism technique and quantum Mackenzie-Xu transformations in the framework of $L_\infty$-algebroids.

The purpose of this paper is to clarify the relations between various constructions of solutions of the Yang-Baxter equation from Leibniz algebras, racks, 3-Leibniz algebras, 3-racks, linear racks, trilinear racks, and give new constructions of solutions of the Yang-Baxter equation. First we show that a 3-Leibniz algebra naturally gives rise to a 3-rack on the underlying vector space, which generalizes Kinyon's construction of racks from Leibniz algebras. Then we show that a trilinear rack naturally gives rise to a linear rack. Combined with Lebed's construction of solutions of the Yang-Baxter equation from linear racks, our results give an intrinsic explanation of Abramov and Zappala's construction of solutions of the Yang-Baxter equation from trilinear racks. Next we show that a 3-Leibniz algebra gives rise to a trilinear rack, which generalizes Abramov and Zappala's construction from 3-Lie algebras. Finally, we construct solutions of the Yang-Baxter equation using central extensions of 3-Leibniz algebras and Leibniz algebras. In particular, given a 3-Leibniz algebra, there are two different approaches to construct solutions of the Yang-Baxter equation, namely either consider the central extension of the Leibniz algebra on the fundamental objects, or consider the Leibniz algebra on the fundamental objects of the central extension of the 3-Leibniz algebra. We also show that there is a homomorphism between the corresponding solutions.

We consider a two-dimensional system in which a charged particle is exposed to a homogeneous magnetic field perpendicular to the plane and a potential that is translationally invariant in one dimension. We derive several conditions on such a perturbation under which the Landau levels change into an absolutely continuous spectrum.

The notions of the Novikov deformation of a commutative associative algebra and the corresponding classical limit are introduced. We show such a classical limit belongs to a subclass of transposed Poisson algebras, and hence the Novikov deformation is defined to be the quantization of the corresponding transposed Poisson algebra. As a direct consequence, we revisit the relationship between transposed Poisson algebras and Novikov-Poisson algebras due to the fact that there is a natural Novikov deformation of the commutative associative algebra in a Novikov-Poisson algebra. Hence all transposed Poisson algebras of Novikov-Poisson type, including unital transposed Poisson algebras, can be quantized. Finally, we classify the quantizations of $2$-dimensional complex transposed Poisson algebras in which the Lie brackets are non-abelian up to equivalence.

To implement the dynamics of a projected Hamiltonian or Lindbladian, the quantum Zeno effect is a fundamental quantum phenomenon that approximates the effective dynamic by intersecting the Hamiltonian or Lindblad evolution by any quantum operation that converges to the desired projected subspace. Unlike the related Trotter product formula, the best-known convergence rate of the quantum Zeno effect is limited to the order $1/n$. In this work, we improve the convergence rate using a multi-product formula to achieve any power of $1/n^{K+1}$, employing a modified Chernoff Lemma, a modified Dunford-Segal approximation, and the holomorphic functional calculus. We then briefly illustrate this scheme using the bosonic cat code, as well as a broad class of examples governed by the `Bang-Bang' method used to decouple systems from their environment.

Cosmological correlators encode statistical properties of the initial conditions of our universe. Mathematically, they can often be written as Mellin integrals of a certain rational function associated to graphs, namely the flat space wavefunction. The singularities of these cosmological integrals are parameterized by binary hyperplane arrangements. Using different algebraic tools, we shed light on the differential and difference equations satisfied by these integrals. Moreover, we study a multivariate version of partial fractioning of the flat space wavefunction, and propose a graph-based algorithm to compute this decomposition.

The goal of this note is to study integrable properties of a generating function of the HOMFLY-PT invariants of the Hopf link colored with different representations. We demonstrate that such a generating function is a $\tau$-function of the KP hierarchy. Furthermore, this Hopf generating function in the case of composite representations, which is a generating function of the 4-point functions in topological string (corresponding to the resolved conifold with branes on the four external legs), is a $\tau$-function of the universal character(UC) hierarchy put on the topological locus. We also briefly discuss a simple matrix model associated with the UC hierarchy.

We consider hard-potential cutoff multi-species Boltzmann operators modeling microscopic binary elastic collisions and bimolecular reversible chemical reactions inside a gaseous mixture. We prove that the spectral gap estimate derived for the linearized elastic collision operator can be exploited to deduce an explicit negative upper bound for the Dirichlet form of the linearized chemical Boltzmann operator. Such estimate may be used to quantify explicitly the rate of convergence of close-to-equilibrium solutions to the reactive Boltzmann equation toward the global chemical equilibrium of the mixture.

This work extends the study of mean field equations arising in two-dimensional (2D) turbulence by introducing generalized weighted Sobolev operators. Employing variational methods, particularly the mountain pass theorem and a refined blow-up analysis, we establish the existence of nontrivial solutions under broader boundary conditions than those considered in previous studies. In contrast to the unweighted approach developed by Ricciardi (2006), the incorporation of variable weights \(\rho(x)\) enables a more accurate representation of complex geometric domains. This generalization addresses cases where the geometry of the manifold or external factors induce local fluctuations, thereby enhancing the applicability of mean field models to real-world turbulence phenomena. The weighted Sobolev framework also strengthens control over the nonlinearity in the problem, reducing the risk of blow-up and ensuring the convergence of solutions in the \(H^1(M)\) space. The analysis reveals new conditions for the existence of solutions, including situations where classical methods may fail due to a lack of compactness. The significant advancement lies in the flexibility of the model; the inclusion of weights allows for a more precise description of physical systems in varied settings, representing a key advantage over the original unweighted problem. These results contribute to a deeper understanding of the role of geometric and physical constraints in turbulent behavior and pave new pathways for future research on nonlinear partial differential equations in complex domains.

The Clifford algebra of the endomorphisms of the exterior algebra of a countably dimensional vector space induces natural bosonic shadows, i.e. families of linear maps between the cohomologies of complex grassmannians. The main result of this paper is to provide a determinantal formula expressing generating functions of such endomorphisms unifying several classical special cases. For example the action over a point recovers the Jacobi-Trudy formula in the theory of symmetric functions or the Giambelli's one in classical Schubert calculus, whereas the action of degree preserving endomorphisms take into account the finite type version of the Date-Jimbo-Kashiwara-Miwa bosonic vertex operator representation of the Lie algebra $gl(\infty)$. The fermionic actions on (finite type) bosonic spaces is described in terms of the classical theory of symmetric functions. The main guiding principle is the fact that the exterior algebra is a (non irreducible) representation of the ring of symmetric functions, which is the way we use to spell the ``finite type'' Boson-Fermion correspondence.

We analyze a model of the evolution of a (solid) magnetoelastic material. More specifically, the model we consider describes the evolution of a compressible magnetoelastic material with a non-convex energy and coupled to a gradient flow equation for the magnetization in the quasi-static setting. The viscous dissipation considered in this model induces an extended material derivative in the magnetic force balance. We prove existence of weak solutions based on De Giorgi's minimizing movements scheme, which allows us to deal with the non-convex energy as well as the non-convex state space for the deformation. In the application of this method we rely on the fact that the magnetic force balance in the model can be expressed in terms of the same energy and dissipation potentials as the equation of motion, allowing us to model the functional for the discrete minimization problem based on these potentials.

We consider learning the dynamics of a typical nonholonomic system -- the rolling penny. A nonholonomic system is a system subject to nonholonomic constraints. Unlike holonomic constraints, a nonholonomic constraint does not define a submanifold on the configuration space. Therefore, the inverse problem of finding the constraints has to involve the tangent space. This paper discuss how to learn the dynamics, as well as the constraints for such a system given the data set of discrete trajectories on the tangent bundle $TQ$.

In this paper we construct a non-perturbative action of the higher spin symmetry algebra on the gravitational phase space. We introduce a symmetry algebroid $\mathcal{T}$ which allows us to include radiation in an algebraic framework. We show that $\mathcal{T}$ admits a non-linear realization on the asymptotic phase space generated by a Noether charge defined non-perturbatively for all spins. Besides, this Noether charge is conserved in the absence of radiation. Moreover, at non radiative cuts, the algebroid can be restricted to the wedge symmetry algebra studied in ArXiv:2409.12178. The key ingredient for our construction is to consider field and time dependent symmetry parameters constrained to evolve according to equations of motion dual to (a truncation of) the asymptotic Einstein's equations. This result then guarantees that the underlying symmetry algebra is also represented canonically.

The Stefan problem with surface tension is well known to exhibit discontinuities in the associated moving aggregate (i.e., in the domain occupied by the solid), whose structure has only been understood under translational or radial symmetry so far. In this paper, we derive an auxiliary partial differential equation of second-order hyperbolic type, referred to as the cascade equation, that captures said discontinuities in the absence of any symmetry assumptions. Specializing to the one-phase setting, we introduce a novel (global) notion of weak solution to the cascade equation, which is defined as a limit of mean-field game equilibria. For the spatial dimension two, we show the existence of such a weak solution and prove a natural perimeter estimate on the associated moving aggregate.

Synchronization is a fundamental dynamical state of interacting oscillators, observed in natural biological rhythms and in the brain. Global synchronization which occurs when non-linear or chaotic oscillators placed on the nodes of a network display the same dynamics as received great attention in network theory. Here we propose and investigate Global Topological Dirac Synchronization on higher-order networks such as cell and simplicial complexes. This is a state where oscillators associated to simplices and cells of arbitrary dimension, coupled by the Topological Dirac operator, operate at unison. By combining algebraic topology with non-linear dynamics and machine learning, we derive the topological conditions under which this state exists and the dynamical conditions under which it is stable. We provide evidence of 1-dimensional simplicial complexes (networks) and 2-dimensional simplicial and cell complexes where Global Topological Dirac Synchronization can be observed. Our results point out that Global Topological Dirac Synchronization is a possible dynamical state of simplicial and cell complexes that occur only in some specific network topologies and geometries, the latter ones being determined by the weights of the higher-order networks

This paper demonstrates how to explicitly construct a bidiagonal factorization of the banded recurrence matrix that appears in mixed multiple orthogonality on the step-line in terms of the coeffcients of the mixed multiple orthogonal polynomials. The construction is based on the \(LU\) factorization of the moment matrix and Christoffel transformations applied to the matrix of measures and the associated mixed multiple orthogonal polynomials.

We propose a simple definition of a Born geometry in the framework of K\"unneth geometry. While superficially different, this new definition is equivalent to the known definitions in terms of para-quaternionic or generalized geometries. We discuss integrability of Born structures and their associated connections. In particular we find that for integrable Born geometries the Born connection is obtained by a simple averaging under a conjugation from the K\"unneth connection. We also give examples of integrable Born geometries on nilmanifolds.

We develop a hybrid classical-quantum method for solving the Lorenz system. We use the forward Euler method to discretize the system in time, transforming it into a system of equations. This set of equations is solved using the Variational Quantum Linear Solver (VQLS) algorithm. We present numerical results comparing the hybrid method with the classical approach for solving the Lorenz system. The simulation results demonstrate that the VQLS method can effectively compute solutions comparable to classical methods. The method is easily extended to solving similar nonlinear differential equations.

Using an approach developed by Melrose to study the geometry at infinity of the Nakajima metric on the reduced Hilbert scheme of points on $\mathbb{C}^2$, we show that the Nakajima metric on a quiver variety is quasi-asymptotically conical (QAC) whenever its defining parameters satisfy an appropriate genericity assumption. As such, it is of bounded geometry and of maximal volume growth. Being QAC is one of two main ingredients allowing us to use the work of Kottke and the second author to compute its reduced $L^2$-cohomology and prove the Vafa-Witten conjecture. The other is a spectral gap for the Hodge-deRham operator associated with an exact $3$-Sasakian wedge metric that we obtain using the Weitzenb\"ock formula of Semmelmann and Weingart on quaternionic-K\"ahler manifolds. This yields in particular a generalization and a different proof of the vanishing theorem in cohomology originally obtained by Galicki and Salamon for closed $3$-Sasakian manifolds.

We prove the global well-posedness of the dynamical sine-Gordon model up to the third threshold, i.e., for parameters $\beta^2 < 6\pi$. The key novelty in our approach is the introduction of the so-called resonant equation, whose solution is entirely deterministic and completely captures the size of the solution to the dynamical sine-Gordon model. The probabilistic fluctuations in the dynamical sine-Gordon model are then controlled using uniform estimates for modified stochastic objects.

We consider the variational principle for the Lagrangian 1-form structure for long-range models of Calogero-Moser (CM) type. The multiform variational principle involves variations with respect to both the field variables as well as the independent variables corresponding to deformations of the time-curves in a multi-time space. The ensuing generalised Euler-Lagrange (gEL) equations comprise a system of multi-time EL equations, as well as constraints from so-called `alien derivatives' and `corner equations' arising from how variations on different coordinate curves match up. The closure relation, i.e. closedness of the Lagrange 1-form on solutions of the EL system, guarantees the stationarity of the action functional under deformation of the time-curves, and hence the multidimensional consistency of the corresponding gEL system. Using this as an integrability criterion on the Lagrangian level, we apply the system to some ans\"atze on the kinetic form of the Lagrangian components, associated with models of CM type without specifying the potentials. We show that from this integrability criterion the general elliptic form of the three systems, Calogero-Moser, Ruijsenaars-Schneider, and Goldfish systems, can be derived. We extend the analysis to an associated Hamiltonian formalism, via Noether's theorem and by applying Legendre transformations. Thus, the multiform variational principle leads to a system of generalised Hamilton equations describing Hamiltonian commuting flows for the mentioned elliptic models.

We study a system of stochastic differential equations with singular drift which describes the dynamics of signed particles in two dimensions interacting by the Coulomb potential. In contrast to the well-studied cases of identical particles that either all repel each other or all attract each other, this system contains both `positive' and `negative' particles. Equal signs repel and opposite signs attract each other; apart from the sign, the potential is the same. We derive results on well-posedness of the system, on the type of collisions that can occur, and on the mean-field limit as the number of particles tends to infinity. Our results demonstrate that the signed system shares features of both the fully repulsive and the fully attractive cases. Our proof method is inspired by the work of Fournier and Jourdain (The Annals of Applied Probability, 27, pp. 2807-2861, 2017) on the fully attractive case; we construct an approximate system of equations, establish uniform estimates, and use tightness to pass to limits.

We modify the R\'enyi (1961) axioms for entropy to apply to negative (``signed") measures as arise, for example, in phase-space representations of quantum mechanics. We obtain two new measures of (lack of) information about a system -- which we propose as signed analogs to classical Shannon entropy and classical R\'enyi entropy, respectively. We show that signed R\'enyi entropy witnesses non-classicality of a system. Specifically, a measure has at least one negative component if and only if signed R\'enyi $\alpha$-entropy is negative for some $\alpha > 1$. The corresponding non-classicality test does not work with signed Shannon entropy. We next show that signed R\'enyi $2k$-entropy, when $k$ is a positive integer, is Schur-concave. (An example shows that signed Shannon entropy is not Schur-concave.) We then establish an abstract quantum H-theorem for signed measures. We prove that signed R\'enyi $2k$-entropy is non-decreasing under classical (``decohering") evolution of a signed measure, where the latter could be a Wigner function or other phase-space representation of a quantum system. (An example shows that signed Shannon entropy may be non-monotonic.) We also provide a characterization of the Second Law for signed R\'enyi $2$-entropy in terms of what we call eventual classicalization of evolution of a system. We conclude with an argument that signed R\'enyi $2$-entropy of the Wigner function is constant under Moyal bracket evolution.

We consider the two-dimensional nonlinear Schr\"odinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We provide a proof by employing a Kato's method along with Hardy inequalities with logarithmic correction. Moreover, we establish finite time blow-up for solutions with positive energy and infinite variance.

It has been known for some time that generalised geometry provides a particularly elegant rewriting of the action and symmetries of 10-dimensional supergravity theories, up to the lowest nontrivial order in fermions. By exhibiting the full symmetry calculations in the second-order formalism, we show in the $\mathcal N=1$ case that this analysis can be upgraded to all orders in fermions and we obtain a strikingly simple form of the action as well as of the supersymmetry transformations, featuring overall only five higher-fermionic terms. Surprisingly, even after expressing the action in terms of classical (non-generalised geometric) variables one obtains a simplification of the usual formulae. This in particular confirms that generalised geometry provides the natural set of variables for studying (the massless level of) string theory. We also show how this new reformulation implies the compatibility of the Poisson-Lie T-duality with the equations of motion of the full supergravity theory.

We justify Prandtl equations and higher order Prandtl expansion from the hydrodynamic limit of the Boltzmann equations. Our fluid data is of the form $\text{shear flow}$, plus $\sqrt{\kappa}$ order term in analytic spaces in $(x_1, x_2)\in\mathbb T^2$ and Sobolev in $x_3\in\mathbb{R}_+$. This work is the first to rigorously justify the Prandtl equations from the hydrodynamic limits of the Boltzmann equations. The novelty lies in obtaining estimates for the linearized Boltzmann equation with a diffusive boundary condition around a Prandtl layer shear flow in analytic spaces. The key techniques involve delicate commutator estimates and the use of local conservation law.

We propose a scheme for the construction of deformed matrix geometries using Landau models. The level projection method cannot be applied straightforwardly to extract matrix geometries to the Landau models on deformed manifolds, as they do not generally exhibit degenerate energy levels (Landau levels). We overcome this problem by exploiting the idea of spectral flow. Taking a symmetric matrix geometry as a reference point of the spectral flow, we evolve the matrix geometry by deforming the Landau model. In this process, unitarity is automatically preserved. The explicit matrix realization of the coordinates is derived straightforwardly even for a non-perturbative deformation. We clarify the basic properties of the matrix geometries through the analysis of concrete models. The matrix geometries of an expanding two-sphere and ellipsoids are investigated using the non-relativistic and relativistic Landau models. The obtained ellipsoidal matrix geometries show behavior quantitatively different in each Landau level, but qualitatively similar to their classical counterpart. The difference between the ellipsoidal matrix geometry and the fuzzy ellipsoid is investigated numerically.

In this paper, we prove the codimension-one nonlinear asymptotic stability of the extremal Reissner-Nordstr\"om family of black holes in the spherically symmetric Einstein-Maxwell-neutral scalar field model, up to and including the event horizon. More precisely, we show that there exists a teleologically defined, codimension-one "submanifold" $\mathfrak M_\mathrm{stab}$ of the moduli space of spherically symmetric characteristic data for the Einstein-Maxwell-scalar field system lying close to the extremal Reissner-Nordstr\"om family, such that any data in $\mathfrak M_\mathrm{stab}$ evolve into a solution with the following properties as time goes to infinity: (i) the metric decays to a member of the extremal Reissner-Nordstr\"om family uniformly up to the event horizon, (ii) the scalar field decays to zero pointwise and in an appropriate energy norm, (iii) the first translation-invariant ingoing null derivative of the scalar field is approximately constant on the event horizon $\mathcal H^+$, (iv) for "generic" data, the second translation-invariant ingoing null derivative of the scalar field grows linearly along the event horizon. Due to the coupling of the scalar field to the geometry via the Einstein equations, suitable components of the Ricci tensor exhibit non-decay and growth phenomena along the event horizon. Points (i) and (ii) above reflect the "stability" of the extremal Reissner-Nordstr\"om family and points (iii) and (iv) verify the presence of the celebrated "Aretakis instability" for the linear wave equation on extremal Reissner-Nordstr\"om black holes in the full nonlinear Einstein-Maxwell-scalar field model.

We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes. Viewing CSS codes as cochain complexes, we observe that cohomology invariants naturally give rise to diagonal logical gates. We show that such invariants exist if the quantum code has a structure that relaxes certain properties of a differential graded algebra. We show how to equip quantum codes with such a structure by defining cup products on CSS codes. The logical gates obtained from this approach can be implemented by a constant-depth unitary circuit. In particular, we construct a $\Lambda$-fold cup product that can produce a logical operator in the $\Lambda$-th level of the Clifford hierarchy on $\Lambda$ copies of the same quantum code, which we call the copy-cup gate. For any desired $\Lambda$, we can construct several families of quantum codes that support gates in the $\Lambda$-th level with various asymptotic code parameters.