Papers

We construct representations of the symmetry algebra and derive the exact spectrum of a three-dimensional oscillator with two spin-$\tfrac12$ degrees of freedom. The Hamiltonian and its first-order scalar and pseudo-scalar integrals were obtained in earlier classifications. Here we determine their action on the complete spectral basis. At fixed total angular momentum, a quadratic algebra acts on the three triplet orbital channels and admits a three-dimensional deformed-oscillator representation, while an overlapping two-state exchange algebra joins the middle triplet to the singlet. These finite modules determine angular–spin multiplicities but do not quantize the radial energy. A separate positive-discrete-series $su(1,1)$ representation of the common radial Hamiltonian generates the energy sequence. We give the finite matrices, Casimir and structure functions, normalized eigenfunctions, radial ladder actions, the exceptional $j=0$ sector, complete energy-shell multiplicities, and the effect of particle interchange.

We study classical local conserved charges obtained after finite-defect reduction of doubled four-dimensional Chern–Simons theory. At a finite simple zero, the inhomogeneous derivative term in the affine gauge transformation induced by the twist function vanishes, so the residue evaluations of the two Gaudin currents transform by the ordinary adjoint action. For each simple ideal, we use the Evans-compatible invariant tensors of the principal Drinfeld–Sokolov hierarchy as external algebraic input, construct local circle integrals, and treat the same-zero, distinct-zero, and mixed sectors separately, proving pairwise Poisson commutativity. Under the explicit assumptions of Stedman's simple-pole/simple-zero boundary Hamiltonian, the periodic trivial-monodromy sector, and boundary weights independent of phase-space variables, the reduced Hamiltonian decomposes term by term into finite-pole, infinity, finite-zero, and residual first-class-constraint contributions, which yields conservation of the constructed charges. We distinguish the finite-defect reduced space, the extended space carrying the residual constraint, and the physical quotient. The result is a conditional existence statement for a classical local Abelian Poisson subalgebra; we make no claim of functional independence, maximality, or Liouville completeness. Extensions to higher-order zeros and nontrivial monodromy sectors are outside the scope of this work.

In this PhD thesis, we take an axiomatic approach to studying multivariate divergences, which are measures of the information content of a tuple of classical or quantum states. These divergences are studied in the context of large-sample and catalytic convertibility of tuples of states. This type of convertibility is characterized by the ordering of an axiomatically defined family of divergences. In this thesis, we identify this family of divergences explicitly for two scenarios, one classical and one quantum. In the classical setting, we show that the relevant divergences are certain generalizations of the bivariate Rényi divergences. Unlike previous studies, we do not require the distributions to have the same support, yielding interesting new divergences. In the quantum setting, we consider pairs of classical-quantum states with pure component states, and find that majorization is characterized by all $\alpha$-$z$ relative entropies for $\alpha<1$. This demonstrates the first operational interpretation of these relative entropies.

We present a nonabelian complex self-dual antisymmetric tensor model, with gauged internal symmetry, comprising a complex multiplet of massless self-dual antisymmetric tensor fields and its conjugate (interacting with massless Yang-Mills gauge fields). We build the model at the classical level, via a first order formulation, assigning local fields to a component expansion over the Dirac algebra and a basis of internal matrices, with Lorentz and gauge invariance ensured by tracing, and applying an appropriate bilinear form. The full expansion requires auxiliary place-holder fields, which are not potentials, but become curvatures (field strengths) after elimination via their algebraic equations of motion. The component expansion over the Dirac algebra is closed under multiplication by additional internal matrices, which are not gauge group generators, but mix internal components, and link auxiliary curvatures with potentials. The model is shown to be invariant under such transformations, and thus possesses a type of hidden internal symmetry. We analyze the specific case of gauge group $SU(2)\! \times \! U(1)$, and as higher symmetry candidates, we consider both the Lie group $SU(3)$ and the Lie superalgebra $SU(2/1)$ (with the additional matrices acting via Lie brackets, or graded Lie brackets, respectively).

Let $P$ be a short-range projector on $\ell^2(\mathbb{Z}^{2n},\mathbb{C}^m)$ and $U=(U_1,\dots,U_{2n})$ be a transverse family of open subsets of $\mathbb{R}^{2n}$. We give an elementary proof of a recent result of Martinian-Stoiber, which extends a two-dimensional result of Drouot-Zhu to all even dimensions. Specifically, using only Stokes' theorem and the Poincaré lemma, we show that the geometric Kubo formula $\operatorname{Tr}\big(P[P,\mathbf{1}_{U_1}]\wedge\cdots\wedge[P,\mathbf{1}_{U_{2n}}]\big)$ is the product of a topological integer $\mathcal{X}_U$, which depends only on the relative positions of $U_1, \dots, U_{2n}$, and of the bulk conductance of $P$. Our proof writes the geometric bulk conductance as the pairing of $P$ with a cochain on $\mathbb{Z}^{2n}$ - an idea originally due to Kitaev - and deforms the cochain rather than the sets $(U_1, \dots, U_{2n})$. The proof presented in this paper is due to Claude Opus 5.5.

The spectral form factor (SFF), which is the Fourier transform of the two-point spectral correlation, is a fundamental diagnostics of quantum chaos. Nevertheless, its experimental detection in quantum many-body systems was less unexplored until recently. Here, we establish a relationship between the SFF with the quantum Fisher information (QFI) by means of quantum sensing protocols. Furthermore, when the eigenvalues and eigenstates of the disordered Hamiltonian are statistically independent, we prove that QFI is typical both in short and long timescales while numerics suggest that it is also typical at intermediate timescales, which ensures self-averaging of QFI. The self-averaging property renders the experimental measurement of QFI with our protocol less noisy in contrast with the direct measurement of SFF which is known to be non-self-averaging. Our results reveal an intriguing connection between quantum noise and the signatures of quantum chaos.

We determine the Poisson algebras generated by partial Casimirs of undeformed and non-standard $q$-deformed $\mathfrak{sl}(2)$ coalgebras, respectively. The derivation is performed on the abstract algebraic level and does not depend on any particular symplectic realization. For the 3- and 4-site cases the constructions associated with the undeformed $\mathfrak{sl}(2)$ produce the rank-one and rank-two Racah Poisson algebras, respectively, while those with the non-standard deformed $\mathfrak{sl}$ yield finite quadratic Poisson algebras generated by connected partial Casimirs. At both the three and four sites, the final closure coefficients do not explicitly depend on the deformation parameter. The results hold at the Poisson coalgebra level and hence apply to any realizations which preserve the defining brackets and coproducts. Canonical symplectic realizations give rise to Hamiltonians in spaces of non-constant curvature. For the Stäckel $q$-deformed Smorodinsky-Winternitz model we show explicitly how the universal coalgebra sector is completed to maximal superintegrability. An exact two-block identity, proved directly in the localized Poisson algebra without appealing to a symplectic realization, controls the additional Stäckel integrals. In three dimensions, it generates a quadratic Poisson block with its own Casimir relation; in four dimensions, its coupling to the full left/right coalgebra family is organized into three nested quadratic blocks, and the same identity extends this nested coupling to arbitrary $N$. Explicit higher-site partial Casimirs and factorized bracket generators are collected in the appendices.

We prove that for every choice of five positive masses the planar five-body problem has only finitely many central configurations up to rotation, translation and scaling. This answers Smale's sixth problem for five bodies. Albouy and Kaloshin proved the statement for all masses outside a closed algebraic subset of codimension two of the mass space. We keep their classification of singular sequences and replace the construction of the exceptional set by a direct exclusion: we use finite boundary charts for seven of the eight diagrams that can carry an unbounded four-product, and an exhaustive order census for Diagram 5. We derive the necessary mass conditions where needed and show that no germ with an unbounded four-product lies on a curve of normalized central configurations through a positive configuration. Five diagrams are excluded for all complex germs, two on curves through real configurations, and one on curves through positive configurations. Parts of the argument are computer-assisted: exact algebra, interval arithmetic and exact linear programming.

We study nematic liquid crystal phases confined to bounded, two-dimensional (2D) domains, in the reduced Landau-de Gennes (rLdG) theoretical framework. The rLdG framework is equivalent to the celebrated Ginzburg-Landau (GL) theoretical setting, under certain assumptions. In the asymptotic limit of large domains (or low temperatures), we recall the concept of the finite-dimensional renormalized energy in the GL theory [Bethuel et al., GL Vortices, Prog. Nonlinear Differ. Equ. Appl. 13, Birkhäuser, 1994.] and generalize it to polygons with sharp corners. The renormalized energy is fully defined by the triplet, $(\mathbf{a},\mathbf{c}, \mathbf{d}$), of topological degrees of the interior defects ($\mathbf{d}$) and the winding number contribution of polygon vertices ($\mathbf{c}$) and interior defect locations ($\mathbf{a}$), subject to sum rules dictated by the imposed Dirichlet boundary condition. We use the critical points of the renormalized energy to construct initial conditions for the rLdG solver, recover known energy-minimizing and non-energy minimizing saddle points of the rLdG energy on a disc, regular hexagon and rectangle along with the discovery of previously unreported rLdG solution branches. The renormalized energy approach is computationally substantially more efficient than a direct PDE-solver in the rLdG setting but misses solution branches with line defects and finite-size effects.

We consider the three-dimensional Fröhlich polaron at fixed coupling strength $\alpha>0$ and total momentum $P \in \bbR^3$. We show that the bottom of the spectrum reaches the essential spectrum at a finite value of $|P|$, implying that the fiber Hamiltonian has no ground state at sufficiently large total momentum. This extends the known weak-coupling result to arbitrary coupling. Our approach describes states near the spectral threshold by a fully interacting polaron with small residual momentum and one additional phonon carrying almost all of the total momentum.

An exactly solvable lattice with periodic inverse-square singularities exhibits two physically distinct mechanisms of spectral-gap closure. The model arises from the generalized trigonometric Pöschl-Teller potential in the coupling range where both local solutions at each singularity are square-integrable. A transmitting, current-preserving self-adjoint matching condition connects neighboring cells and generates a nontrivial Bloch spectrum. For a generic lattice with two inequivalent singular sublattices, the spectrum contains two alternating families of gaps, and the band edges reproduce the scale-invariant spectra of the associated finite singular system. One gap family closes when the two sublattices become equivalent. In this symmetric limit, however, the primitive period halves, so the apparent band crossing results from Brillouin-zone folding. A different family closes at a critical but genuinely asymmetric configuration. Because the primitive period is preserved, adjacent bands then meet at the center of the original Brillouin zone in a genuine massless Dirac-like cone. Near either critical limit, a small finite gap acts as an effective Dirac mass: the dispersion remains quadratic near a band edge but crosses over to a massive Dirac-like form beyond that narrow region. The band curvatures define effective electron and hole masses. Exact Bloch eigenfunctions further reveal that isospectral lattices can display different single-particle probability densities. This model therefore provides an analytically controlled one-dimensional setting in which conventional quadratic carriers, massive Dirac-like band dispersions, and a tunable massless contact coexist within the same singular lattice.

We present two and three dimensional direct numerical simulations of planar intrusions injected from a continuously supplied, narrow source into a linearly stratified ambient at the level of neutral buoyancy. The evolution passes through three phases. A laminar leading vortex dipole first advances at constant speed with a varicose wave train behind it; the trailing jet then undergoes a sinuous instability whose initial growth rate and wavelength agree with inviscid unstratified planar-jet theory evaluated on the local jet; the flow finally relaxes into a buoyancy-controlled wedge whose area grows linearly in time. Three-dimensional simulations show that the transition remains organized by the planar instability, while three-dimensionality reshapes the near-source region into a slowly evolving, actively mixing zone that entrains an additional twenty to one hundred percent of the source flux depending on source width. Rescaled with this measured effective flux, the downstream intrusion collapses reasonably well onto the similarity solution for a wide-source intrusion.

We consider singular Schrödinger operators with an attractive $\delta'$ interaction of strength $\beta$ supported by a star graph and prove that the ground-state eigenvalue is uniquely maximized by stars of the maximum symmetry. Two different proofs are presented, one for an even number of edges using a comparison to the $\delta$ interaction star, the other general based on Birman-Schwinger technique.

Stationary scattering admits a dynamical formulation in which the scattering data are extracted from the evolution operator for an effective non-stationary and non-unitary quantum system. This provides a mathematical method of solving scattering problems. We consider the possibility of realizing such a quantum system and relating the scattering information to outcomes of quantum measurements performed on it. To this end, we examine stationary scattering problems in one dimension where the scattering data consist of reflection and transmission amplitudes and the corresponding effective quantum system is a non-unitary two-level system. The states of this system identify points in a Bloch sphere $S^2$ and their evolution yields trajectories in $S^2$. Quantum measurements turn out to be capable of providing information about the reflection amplitudes. We identify configurations displaying reflectionlessness with situations where one or both of the poles of $S^2$ undergo cyclic evolutions. Spectral singularities and coherent perfect absorption correspond to dynamical spin flips which map one of the poles of $S^2$ to the other.

This paper explores topological phase transitions in the momentum space of the $\kappa$-deformed $\phi^4$ lattice model within the mean-field approximation. We show that the double-well structure of the momentum space gives rise to degenerate momentum states at high energies and that this change is marked by a phase transition. We investigate this transition using both numerical and analytical methods, with the analytical approach based primarily on singularity theory. Our numerical analysis shows that the energy at which the topological phase transition occurs is related to the noncommutative parameter by the simple power law $E \sim 1/\alpha^{4}$. By mapping the constant-energy-level surface to the $A_{3}$ singularity, we obtain the same result analytically. We also present results unique to noncommutative systems, showing that the fraction of particles in degenerate momentum states has a nonzero commutative limit. As a nonperturbative effect, this result provides a testable prediction for the existence of noncommutative spacetime.

Let $\mathcal B=(B_g)_{g\in G}$ be a saturated Fell bundle over a countable discrete group, equipped with a spectral triple. We give explicit sufficient conditions for lifting this triple to the reduced cross-sectional C\Star-algebra $C_r^*(\mathcal B)$. Using finite projective realizations of the homogeneous Morita equivalence $B_e$-bimodules, we transport the base Dirac operator to the individual fibres and assemble the resulting operators into a fibrewise Dirac operator. We introduce a spectral-regularity condition ensuring bounded commutators and give a sufficient criterion in terms of uniform derivative bounds on smooth factor-system data. Combining the fibrewise operator with a length operator associated with a proper translation-bounded matrix-valued length function yields the lifted spectral triple. When $B_e$ is separable, the lifted triple, with a suitable grading, represents the Kasparov product of the vertical class with the class of the unit-fibre triple. We also establish a summability criterion and sufficient conditions for stability under central cocycle twisting. A range of examples illustrates the scope of the construction, including twisted crossed products, free compact Abelian actions, and Fell bundles with nonfree homogeneous fibres.

For the first time, we study the effects of mass transfer on alloy droplet impact, solidification, and solute non-homogeneity or MacroSegregation (MS). For this purpose, a multi-physics model, including fully-coupled hydrodynamics of multiphase flow, wetting, conjugate heat transfer, phase change, and mass transfer, is established and validated carefully across several benchmarks. In addition, a concentration-flux-correction algorithm is proposed to conserve the solute mass within the droplet at the numerical discretization level. The validated model is subsequently used to explore the impact and solidification of an alloy droplet. The results show that accounting for mass transfer and solute transport improves the prediction of the dynamics of a solidifying splat, in a consistent manner with experimental observations, while this effect has been neglected in all previous studies. The analysis demonstrates that this improvement results from the flow sensitivity to the mushy region thickness, which is regulated by local solute concentration through the solidus and liquidus temperatures. Furthermore, analyses of the MS map identify several critical segregation regions most susceptible to defect initiation, such as cracking, including a long negatively segregated vein within the lamella, formed by the severe advective flow through the mush in the middle stages of the impact, a positively segregated region near the rim of the splat, appearing as a result of hook formation, and a major positively segregated zone at the top of the rim, which develops following the accumulation of the solute-rich melt sustained by washout from the lamella.

We introduce and study trigonometric weight functions of type $B$ and type $C$ arising from integrable lattice models. These functions give $q$-deformations of the factorial $K$-theoretic Schur $P$- and $Q$-functions of Ikeda and Naruse, with the $P$-functions specialized to the type $B$ case. We realize them as specializations of partition functions of a six-vertex model with triangular boundary conditions and derive their algebraic properties using the Yang-Baxter equation and properties of the $R$- and $K$-matrices. We introduce normalized weight functions and establish diagonal evaluation formulas, divisibility properties, and bounds on Newton polytopes analogous to those in type $A$, together with recursive relations and stability. We further derive branching formulas relating these functions to type $A$ weight functions, and prove equal-cardinality orthogonality and all-cardinality biorthogonality relations.

Non-intrusive reduced-order modeling is essential for systems simulated with black- or gray-box solvers, yet computing accurate data-driven models remains challenging when the dynamics exhibit large-amplitude transient energy growth. This is because state-of-the-art non-intrusive methods typically rely on orthogonal projection onto high-energy linear or nonlinear manifolds identified by minimizing static reconstruction losses, which truncates low-energy, yet dynamically important, states. Intrusive methods such as Balanced Truncation and its extensions address this issue by using adjoint information to define dynamics-aware oblique projections. Here, we show that balancing-based oblique projections for nonlinear systems can instead be computed non-intrusively from forward simulation data. Specifically, we develop a framework to compute the nonlinear empirical reachability Gramian and balance it against its observability counterpart, projected onto a suitable restriction subspace. This procedure provably recovers the oblique projection of Balanced Truncation when the dynamics are linear and the range of either Gramian is contained in the restriction subspace. Moreover, in the linear case and in the absence of adjoint information, we show that the minimax-optimal restriction subspace spans the dominant eigenspaces of the reachability Gramian, which informs an efficient balancing algorithm with cost comparable to a standard SVD. Once the oblique projection is computed, we use Operator Inference to learn the latent-space dynamics on its range. The method is demonstrated on several examples of increasing complexity, including the three-dimensional incompressible flow past a wing with $8\times 10^7$ degrees of freedom. In all examples, the Balanced Operator Inference models significantly outperform their POD-based counterparts, reducing forecasting error by up to two orders of magnitude.

Area laws express the expectation that entanglement in gapped quantum ground states scales with a boundary rather than a volume. In one dimension this holds for every Rényi entropy, and the Rényi area-law conjecture asserts the same in higher dimensions for each fixed $0<\alpha<1$. Here we disprove this conjecture in every spatial dimension two and higher. We construct finite-range Hamiltonians with fixed local dimension, bounded interaction strength, unique ground states and a uniform spectral gap, whose ground states exhibit volume-law Rényi entanglement while obeying a von Neumann area law and remaining connected to product states through uniformly gapped local paths. The mechanism is error-space complexity: locally identifiable components of arbitrarily small weight can encode states with high entanglement complexity. We also establish the worst-case rank-error trade-off for approximate ground-state projectors, a key tool in one-dimensional area-law proofs. At fixed gap, the optimal Schmidt rank at projection error $\epsilon$ across a planar cut of cross-section $A$ scales as $\exp[\tilde{\Theta}((A\log\epsilon^{-1})^{2/3})]$ when $\log\epsilon^{-1}\gtrsim A$. Thus $\alpha=1$ is the sharp threshold for a universal Rényi area law within this adiabatically connected class.

Motivated by three-dimensional wave diffraction by conical shapes, we consider the two-dimensional problem of wave scattering on the real sphere $\mathcal{S}^2$ resulting from a point-source incidence on a scatterer $\mathbb{S}$. Using the recently introduced notion of plane waves on $\mathcal{S}^2$ as well as recent results on the Laplace–Beltrami Green's function $G$ on $\mathcal{S}^2$, we derive three plane-wave representations for the scattered field $u$. We then explain how $u$ can be analytically continued to the complexified sphere $\mathcal{S}^2_{\mathbb{c}}$. This allows us to formulate a complexified version of Green's third identity, and to provide a characterisation of complex sources. All our results are then used to draw strong analogies between planar and spherical scattering problems, notably through the introduction of directivities.

We present DisCo algebra—a compositional framework for joint probability built from a symmetric cooperad (disintegration) and a symmetric operad (combination) on the bare family $\mathcal{P}_n=\mathrm{Prob}([0,1]^n)$. No probability integral transform, no external summary functions, no enrichment of objects with conditional kernels. The disintegration side uses block-wise joint CDFs as intrinsic, measure-determined 1D summaries, yielding a lossy compression. The combination side reads the same summaries and reconstructs via regular conditional measures, yielding a lossless embedding. We prove that combination followed by disintegration is the identity on admissible inputs, but disintegration followed by combination is not—the information loss is exactly the conditional KL divergence between the true conditional distribution and the conditionally independent product. This asymmetry reveals a hard limit on how much cross-block dependence can be captured by scalar block summaries.

We study the Gibbs measure of $H(h) = \Re\sum_{p^{j} \leq X}U_{p}^{j}/(j p^{j(\sigma + i h)})$, a random field modeling large values of the log-modulus of the Riemann zeta function on and near the critical line, where the $U_{p}$ are independent and uniformly distributed on the complex unit circle. Writing $t = \log \log T$, the cutoff $X$ and real part $\sigma$ may vary with $T$ subject to $T^{1/t^{\Phi}} \leq X \leq T^{1/t^{\phi}}$ and $0 \leq \sigma - 1/2 \leq C_{\sigma}t^{\Phi}/\log T$, for fixed $\Phi \geq \phi > 1/2$ and $C_{\sigma} > 0$. On $I_{\theta} = [ - (\log T)^{\theta},(\log T)^{\theta}]$, define the Gibbs measure $G_{\beta,\sigma,T}^{\mathrm{H}}(\mathrm{d}h) := e^{\beta H(h)}\mathrm{d}h/\int_{I_{\theta}}e^{\beta H(h')}\mathrm{d}h'$. For every fixed $\theta > - 1$ and inverse temperature $\beta > 0$, the limiting joint distributions of correlations (overlaps) among points sampled from this measure are determined after averaging over the Haar phases. At and below $\beta_{c}(\theta) = 2\sqrt{1 + (\theta \vee 0)}$, the limiting two-overlap distribution is concentrated at $q_{0} = |\theta \wedge 0|$. Above this threshold, it assigns masses $\beta_{c}(\theta)/\beta$ and $1 - \beta_{c}(\theta)/\beta$ to $q_{0}$ and $1$, respectively, and the limiting joint overlap distribution is described by a Poisson-Dirichlet process of parameter $\beta_{c}(\theta)/\beta$. Approximate Ghirlanda-Guerra identities are shown to hold uniformly for bounded deterministic functions of the sampled positions. This paper is the first in a series devoted to extending these results to $\log|\zeta|$ near the critical line.

Gauging fermion parity in a $(2+1)$-dimensional fermionic topological order requires lifting its symmetries to the resulting bosonic theory. We give a direct tensor-categorical proof that, for a slightly degenerate braided fusion category $\mathcal{C}$ and a fixed minimal nondegenerate extension ${\cal E}$, the restriction homomorphism $r\colon\operatorname{Aut}_{\mathrm{LR}}(\mathcal{E})\to \operatorname{Aut}_{\mathrm{LR}}(\mathcal{C})$ on classes of complex-linear locality-respecting (LR) braided autoequivalences is surjective with central kernel $\mathbb{Z}/2$, generated by fermion attachment. Thus each symmetry class has exactly two LR lift classes, related by attaching the fermion to every vortex. We apply graded Morita theory to reduce the kernel to two module types on the vortex sector, corresponding to tensoring with the vacuum or the fermion. We then prove relative rigidity to show that each type determines a unique LR class, which are distinguished by fermionic self-braiding even when their actions on simple-object labels coincide. We prove surjectivity by extending graph algebras in $\mathcal{C}^{\mathrm{rev}}\boxtimes\mathcal{C}$ to Lagrangian algebras in $\mathcal{E}^{\mathrm{rev}}\boxtimes\mathcal{E}$. Our results require neither unitarity nor a ribbon structure. We illustrate the lifting ambiguity in the Laughlin state, the $SO(3)_3$ theory and the Ising topological order.

We determine the best constants in estimates of the $L^\infty$ norm of functions on the sphere with bounded Laplacian whose first spherical harmonic components vanish. In every dimension $d\geq2$, these constants are expressed through weighted $L^1$ extremal polynomials. On $S^2$ the constant is an explicit positive series, and the best coefficient of $n^{-2}$ is $7\zeta(3)/4$. On $S^3$ the best coefficient of $n^{-2}$ is a positive integral involving the Hörmander–Bernhardsson extremal function. In higher dimensions the rescaled extremal polynomials converge to weighted entire extremals, which give asymptotic formulas for the Bohr constants on $S^4$ and $S^6$. The most unexpected and, in our view, interesting observation is that, for every fixed $n\geq2$ on $S^2$ and $S^3$, the constants in best approximation estimates are strictly smaller than the corresponding Bohr–Favard constants.

We study a class of periodic NLS-Dysthe (NLS-Dy) equations, also referred to as higher-order nonlinear Schrödinger equations (HNLS), of the form $i\partial_t u+a\partial_x^2u+b|u|^2u+i\varepsilon(c\partial_x^3u+d|u|^2\partial_xu+eu^2\partial_x\overline{u})=0$ with $ x\in\mathbb T$, $a,b,c,d,e\in\mathbb R$, $abc\neq0$, and $\varepsilon>0$. For $0<\varepsilon\ll1$, equations of this type arise as higher-order modulation models for some nonlinear dispersive PDEs, including the Klein-Gordon equation and the Serre-Green-Naghdi equation modeling the evolution of nonlinear shallow water waves. We prove local well-posedness in $H^s(\mathbb T)$ for $s>3/2$ for arbitrary admissible coefficients, and establish an $H^1$ continuation criterion for $s\ge2$. In the parameter regime $e\neq0$, a lower-order conservation law provides uniform $H^1$ control and, together with the continuation criterion, yields global well-posedness for $s\ge2$. This result does not require the non-resonance condition appearing in Takaoka's periodic theory, while the condition $e\neq0$ naturally arises in the modulation setting considered here. Finally, motivated by the long-time regime relevant for modulation, we establish uniform long-time control without imposing a smallness assumption on the initial datum. Using the conservation laws of cubic NLS and their invariance under the commuting complex mKdV flow, we obtain uniform $H^s$ bounds on time intervals of order $\varepsilon^{-1}$ for integer $s\ge2$. For non-integer real $s\ge2$, the maximal $H^s$ solution exists on the same time scale, while uniform control is obtained at a lower regularity, with a loss of less than one derivative.

Free classical communication can yield an unbounded single-shot saving in quantum compression, even though it leaves the asymptotic rate of an i.i.d. source unchanged. At every fixed error in $(0,1)$, we exhibit sources whose unassisted compression cost is arbitrarily large, yet a single qubit suffices with classical communication and no preshared entanglement. We establish this separation by determining the exact optimal source fidelity for compression through an $m$-dimensional noiseless quantum channel, with and without free classical communication, while preserving correlations with an inaccessible reference. Without classical assistance, the optimal fidelity is exactly the Ky Fan norm $\|\rho\|_{(m)}$, the sum of the $m$ largest eigenvalues of the source state. With classical assistance, we obtain an explicit spectral threshold formula and a protocol attaining it. Classical communication strictly improves fidelity whenever $m$ is smaller than the source rank. The corresponding compression costs are two smoothed versions of the log-rank entropy, also called max-entropy: trace-distance smoothing under local operations and quadratic smoothing under LOCC. For i.i.d. copies of a fixed source with positive entropy variance, however, the two costs have identical first-, second-, and third-order asymptotic terms. For nonlattice spectra, we determine the constant-order advantage: classical communication saves $\log_2(e)-1$ bits per block asymptotically, and the ratio of assisted to unassisted minimum communication dimensions converges to $2/e$.

In these lecture notes we review some constructions and results on the relation between topology of the moduli space of stable curves and integrable systems of evolutionary PDEs.

We consider spin-coupled Grover walks on hypercubes within the framework of quantum Bernoulli noises (QBN). We identify parity-dependent harmonic subspaces generated by the central Hamming layers: in even dimension they give a fixed-point subspace, whereas in odd dimension the walk reduces on the harmonic subspace to an explicit eight-dimensional reduced dynamics. Using symmetric-group representation theory, we characterize the corresponding isotypic structure and determine the reduced spectrum. The harmonic reduction also yields explicit long-time probability behavior and shows that the resulting states may occupy an asymptotically vanishing fraction of the hypercube while still spanning its full Hamming diameter.

We construct a chiral algebra $U$ of central charge eight inside the lattice vertex operator algebra $V_{E_6\oplus A_2}$. It is generated by two commuting affine $\mathfrak{sl}_2$ current algebras at levels $1$ and $39$, the stress tensor of the lattice theory, and a single weight-two field built from three lattice exponentials. We announce that $U$ is strongly rational and completely unitary, and that $Rep(U)$ is equivalent, as a unitary ribbon category, to the Drinfeld center $Z(\mathcal{H}_3)$ of the Haagerup fusion category $\mathcal{H}_3$. Its characters are the $\gamma=0$ Evans–Gannon vector. From $U$ we obtain a diagonal full CFT, and also a holomorphic $E_8$ extension on which $\mathcal{H}_3$ acts by topological defects.