Papers

This is my written Laudatio for Graeme Segal on receiving the Chern Medal at the 2026 International Congress of Mathematicians. I make some general remarks about Segal's qualities as a mathematician. Then I discuss the origins of his work on classifying spaces of categories and Gamma-spaces. In the last section I describe his axiom system for Wick-rotated quantum field theory and some of his subsequent work in the area. Of course, this is necessarily only a small slice of Segal's total research output.

Symmetries of a quantum field theory or a quantum lattice model are expected to be described not by a group but by a homotopy type, whose homotopy groups are the higher symmetry groups and whose Postnikov invariants encode 't Hooft anomalies. We propose a homotopy-theoretic definition of 't Hooft anomalies based on the structure of localized symmetries in the operator-algebraic approach. The input is a crossed precosheaf: a collection of groups attached to regions of space, equipped with pairings recording that the commutator of two transformations localized in two regions is localized in their intersection. To a crossed precosheaf and a finite family of regions we attach a pointed connected homotopy type, the Cech-Loday classifying space. As the family of regions ranges over finite covers of space, the classifying spaces form a pro-object in the homotopy category. There is a canonical map from the ordinary classifying space of the symmetry group to this pro-type. Composing this map with a group action gives an anomaly whose nontriviality obstructs gauging. In favorable cases, the Cech-Loday space is essentially independent of the cover. In such cases, the homotopy fiber of the canonical map is a candidate for the classifying space of the higher group of symmetries. We give examples of crossed precosheaves arising from gauge transformations in continuum field theory and from quantum cellular automata and finite-depth circuits. For quantum spin systems in one and two dimensions, our construction recovers the Else-Nayak index and its generalizations to two dimensions.

We construct giant graviton expansions of the thermal and $(-1)^F$-graded partition functions of $\mathcal{N}=4$ and $\mathcal{N}=1$ super Yang-Mills theories at zero 't Hooft coupling. These observables are generically non-BPS and do not admit a direct bulk interpretation in a supergravity regime. Instead, we exploit the corresponding self-dual theories, which arise as chiral zero-coupling limits and admit twistor string holographic duals. We apply a quiver generalization of Murthy's expansion motivated by twistor strings. Notably, we obtain expansions at both finite and infinite $N$. We reproduce the terms in these expansions from bulk closed string and D-brane sectors, identifying the nonperturbative corrections with D-instanton/anti-D-instanton pairs. At infinite $N$, resumming the D-instanton sectors reproduces the Hagedorn temperatures of the free gauge theories, providing a microscopic bulk origin for their Hagedorn singularities.

Chiral topological order is incompatible with strictly two-dimensional local commuting-projector Hamiltonians on finite-dimensional tensor-product Hilbert spaces. Invertible subalgebras (ISAs) provide an alternative framework. We construct an ISA supporting chiral semions, together with its anti-semion commutant, on a square lattice with one qubit per edge. We give explicit non-Pauli generators for both algebras and prove that together they generate all local operators on the Hilbert space. Starting from the $\mathbb{Z}_4$ toric code, we condense the boson $e^2m^2$ and identify the semion and anti-semion algebras. We then decompose each $\mathbb{Z}_4$ qudit into two qubits and use a finite-depth circuit to disentangle one qubit per edge fixed by the condensation constraints. Pumping the semion ISA along the $z$ direction defines an explicit three-dimensional non-Clifford quantum cellular automaton (QCA). We show that pumping eight times is equivalent to a translation followed by a finite-depth quantum circuit with ancillary qubits. We also give an explicit finite-depth circuit relating this construction to the semion QCA previously obtained from topological quantum field theory. A similar condensation construction yields a $U(1)_4$ ISA on a square lattice. Together with the semion and earlier odd-prime ISAs, finite stacking realizes every pointed Witt class of bosonic Abelian anyon theories. Finally, we extend both constructions to all spatial dimensions $d=4k{-}2$, with one qubit or $\mathbb{Z}_4$ qudit on each $(2k{-}1)$-cell for the semion or $U(1)_4$ ISA, respectively.

Motivated by the 3d-3d correspondence, we introduce an index $\mathcal{I}_g^\phi$ for a mutation loop $\phi$ of a quiver and an integer $g \geq 0$. We prove its integrality $\mathcal{I}_g^\phi \in \mathbb{Z}$ under certain conditions, which extends the integrality conjecture of Gang–Kim–Yoon for $3$-manifolds to the broader framework of mutation loops in cluster algebras. Our index also has an interpretation as the genus-$g$ partition function of a $(1+1)$-dimensional TQFT. We provide explicit computations for mutation loops of period one. In particular, in the case of the generalized Kronecker quivers, we give an explicit formula expressing the generating function of the indices for all genera $g$ as a ratio of polynomials. Altogether, we provide a new insight to the interplay between the geometry of $3$-manifolds and cluster algebra.

We consider the Asymmetric Simple Exclusion Process (ASEP) at its critical dimension $d=2$, started from the Bernoulli product measure with density $\rho \in (0,1)$. We first derive the sharp $\log(t)^{2/3}$ divergence of the bulk diffusivity, predicted via mode-coupling theory in [H. van Beijeren, R. Kutner, H. Spohn, Phys. Rev. Lett., 1985], including the exact prefactor, thereby removing the subleading diverging multiplicative corrections in the Tauberian bounds of [H.-T. Yau, Ann. of Math., 2004]. More importantly, we establish an invariance principle for the density fluctuation field on the corresponding superdiffusive scales. On the time scale $\tau$, we rescale space by $\sqrt{\tau} \log(\tau)^{1/3}$ in the direction of the drift and by $\sqrt{\tau}$ in the transverse direction, and prove functional convergence of the rescaled fluctuation process to the solution of a conservative stochastic heat equation with additive noise. This is the same explicit Gaussian Fixed Point obtained in [G. Cannizzaro, Q. Moulard, F. Toninelli, arXiv:2501.00344, 2025] for the two-dimensional Stochastic Burgers Equation (SBE). We also deduce a superdiffusive central limit theorem for the second-class particle under the same anisotropic rescaling. The result holds for any finite-range jump kernel with nonzero drift and uniaxial asymmetry. This is the first space-time scaling limit result for a nonequilibrium interacting particle system at the critical dimension, where Gaussian fluctuations emerge despite logarithmic corrections to the scaling. The proof uses an approximate fluctuation-dissipation relation, which is obtained by combining a sharp quantitative comparison between the ASEP and SBE generators with refined estimates on the SBE Ansatz. An additional key novelty is a tightness argument that overcomes the limitations of the Itô trick on superdiffusive scales.

It is proved that every Borel ensemble of pure states on a separable bipartite Hilbert space with finite average entanglement admits a countable ensemble with the same barycenter and arbitrarily small additional average entanglement. Consequently, the discrete and continuous versions of the Entanglement of Formation coincide on the whole space of states of infinite-dimensional bipartite quantum system (with infinite values allowed). The main idea of the proof (realized with the AI-assistance) is the identification of a Borel ensemble of pure bipartite states with a Borel probability measure on the unit sphere of the space of Hilbert–Schmidt operators followed by dividing the support of this measure into appropriate cells and replacing each cell by its conditional barycenter. The resulting countable ensemble is obtained by a recursive construction.

We introduce a diagram algebra $\mathcal{K}_L$ based on Kuperberg's $A_2$-webs and prove a Schur–Weyl duality $\mathcal{K}_L \cong \operatorname{End}_{U_q(\mathfrak{sl}_3)}((V^+ \oplus V^-)^{\otimes L})$, where $V^\pm$ are the fundamental representations of $U_q(\mathfrak{sl}_3)$. Thus, $\mathcal{K}_L$ is a natural $A_2$ analogue of the Temperley–Lieb algebra. We construct an explicit cellular structure for $\mathcal{K}_L$, with the web basis as its cellular basis and cell modules indexed by dominant $U_q(\mathfrak{sl}_3)$-weights. The bases of these modules are parametrized by dominant lattice paths via a recursive growth algorithm. Using this structure, we determine local generators of $\mathcal{K}_L$ and establish web formulations of the first and second fundamental theorems of invariant theory. In particular, we show that the braid-group image, augmented by local idempotents, does not generate $\mathcal{K}_L$, and identify an additional local generator. Finally, we derive a bimodule decomposition of $(V^+ \oplus V^-)^{\otimes L}$, yielding a complete set of irreducible $\mathcal{K}_L$-modules for formal or generic $q$.

We study the invariant rank-four deformed Hermitian–Yang–Mills equation associated with the chain $\mathcal{O}(1)\to\mathcal{O}(2)^{\oplus2}\to\mathcal{O}(3)$ over $\mathbb P^1$, with vertical weights $(4,2,0)$ and a fixed product Kähler class and phase. For arbitrary linear arrows $\psi=Bz$ and $\phi=(Az)^T$, smooth strictly calibrated reduced solutions exist exactly when $A$ is invertible and $z^TA^TBz$ is not identically zero. The same condition characterizes stability, semistability and polystability in the specified equivariant coherent category. The classification includes all eleven nonempty coefficient strata and both coincident and distinct divisor points on the rank-one lower boundary. The analytic proof retains the ordered, noncommuting middle equation. After constructing positive-parameter solutions, it rules out unbounded metric values by combining the source budgets with moving spectral tests; on the rank-one lower boundary, the divisor mass must also be retained. Global value, gradient and symbol estimates then give a smooth strictly calibrated endpoint on the original chain. No equivalence with full coherent or derived-category stability, uniqueness, or estimate uniform across changing chains is asserted.

We study the long-time, large-distance asymptotics of finite-temperature dynamical Fredholm determinants using effective form factors. The method expresses these asymptotics as a functional of an effective phase shift. We determine its two branches on either side of a quadratic stationary point and the transition profile joining them. The profile is expressed through parabolic-cylinder functions. Its connection to the high-temperature Gaussian $\beta$ ensemble fixes the finite normalization, in agreement with an independent Riemann–Hilbert analysis. We also show that the leading exponential asymptotics suffice to recover the Euler-scale prediction of generalized hydrodynamics for the dynamical structure factor of hard rods.

In the present article the self-contained derivation of main formulas of the inverse eigenvalue problem for the Sturm-Liouville operator is given on the base of its discrete approximation and the subsequent transition to the limit of the continuum coordinate. In doing so, we adhere to Tikhonov concept, where the discrete approximation is performed as a regularization of the ill-posed operator problem. While the continuum limit for the three-diagonal matrix Hamiltonian is impossible as such, following the Tikhonov procedure leads not to a limit for the potential matrix itself, but rather for a certain combination of the matrix elements, and this limit is precisely the desired potential in the continuum case.

We prove the Labastida–Mariño–Ooguri–Vafa conjecture for oriented links with arbitrary integer framings. Consequently, the framed reformulated invariants have finite integral LMOV expansions for every integer framing vector.

The Klein-Gordon equation is a second order differential equation that appears in a variety of applications in mathematics, physics and engineering. This equation is often solved using numerical methods, since a general analytical solution is unattainable. However, one can find very useful analytical solutions for some special cases. In cosmology, the Klein-Gordon equation of a field can explain the physics of inflation and has important consequences on understanding the origin and evolution of the universe. An analytical solution to the Klein-Gordon equation in cosmology has been obtained in the small-field excursion approximation, and it turns out that the solution can be written in terms of the confluent hypergeometric limit function. In this work, we derive this result again while showing the details of the calculations and then generalize to the case of two coupled Klein-Gordon equations. We show that the solution of this coupled system can be written in terms of Kummer's hypergeometric limit function while considering minimal assumptions. We carry out a full numerical calculation of the coupled system and compare it to the results from the analytical expressions.

The modular data of the doubles of the Haagerup–Izumi categories is a grafting of two finite metric groups (conjectured by Evans–Gannon, proved for untwisted categories by Gannon–Schopieray–Yadav). Gang and Kim found non-unitary modular data of this shape in the topological twist of rank-$0$ $S$-fold theories, from the Bethe vacua of an abelianised gauge theory. The aim is to understand rigorously the significance of the handle-gluing and fibering operators, and to construct the Bethe vacua. For every hyperbolic once-punctured-torus monodromy $\varphi$, a word in $L$ and $R$, the two groups are $A_\mp=\coker(\varphi\mp1)$, with the torsion linking forms of the mapping tori of $\varphi$ and $-\varphi$. Over all words of trace $m^2+2$ these forms reproduce the Evans–Gannon table of Q-systems, one printed entry being corrected, and the distinct modular data in the list of Gannon–Schopieray–Yadav. The Bethe data are given in closed form by the word: the level matrix, the positive vacuum, $c_{\eff}=\#L$, and the handle-gluing values $2^{\#L}\lambda^{\pm1}|A_-||A_+|$ at the identity classes, $\lambda$ the stretch factor. Each kernel vector of $\varphi\mp1$ modulo $|A_\mp|$ at which an explicit trigonometric formula does not vanish gives a solution of the Bethe equations in closed form. Under the half-index dictionary, with the unit in the class of $\sigma V^*$, $c\equiv\#L-6\#R\pmod{24}$.

We consider the linear operator for metamaterial modeling $$L_{c,d }=(\chi_{\RE^{3}\backslash\overline \Omega}+c)\nabla\!\cdot (\chi_{\RE^{3}\backslash\overline \Omega}+d)\nabla\,,$$ where the material parameters $c>0$ and $d$ are constant on each connected component of the bounded domain $\Omega$, $d$ is negative on at least one component of $\Omega$ and transmission boundary conditions hold at the interfaces. It turns out that $L_{c,d }$ has, as in the sign-definite case, an absolutely continuous spectrum equal to $[0,+\infty)$; however, contrary to the latter, it has a non-empty point spectrum; in particular, we find conditions on the parameters so that zero is a threshold eigenvalue. In the case $\Omega$ is connected, we study how such a threshold eigenvalue behaves under $\varepsilon$-small analytic perturbations of the parameters. We show that, depending on the position of the number $\frac12\,\frac{1+d(\ve)}{1-d(\ve)}$ relative to the nearest eigenvalue of the Neumann-Poincaré operator of the interface, the zero eigenvalue splits into two branches (analytic in $\varepsilon^{1/2}$), consisting either of two branches of resonances, or of one branch of negative eigenvalues and one of resonances.

The Sachdev–Ye–Kitaev (SYK) model is a quantum many-body model of Majorana fermions with all-to-all random interactions. It has attracted wide interest in condensed matter and high-energy physics as a solvable model of non-Fermi liquids and holography. Fix an even integer $q\geq4$, and let $Z_N(\beta)$ be the partition function of the Gaussian $q$-body SYK model. We prove that for all sufficiently small inverse temperatures $\beta > 0$, there exists $\rho=\rho(\beta)>0$ such that \[ N^{q/2 -1}(\log Z_N(\beta)-\mathbb{E}[\log Z_N(\beta)]) \Longrightarrow \mathcal{N}(0,\rho^2). \] In addition to the mathematical results, we discuss how the proofs were developed and refined through interaction between human researchers and AI, and we describe the subsequent formalization of the main arguments in Lean 4. This work thus serves as a case study of a research workflow that integrates human insight, AI-assisted exploration, iterative proof refinement, and formal verification.

We prove that, for every $c>0$, the image of the generalized Markov-Krein transform on probability measures with finite logarithmic moment is closed under classical additive convolution and under the Mellin convolution. This resolves the positivity conjecture for the additive high-temperature convolution, a one-parameter interpolation between classical and free convolution arising in random matrix theory and representation theory in the regime $N\to\infty, N\beta_N/2\to c$. Our proof relies on a convolution theorem for universally prestarlike functions.

We study a one-dimensional single-excitation model of a scalar photon field coupled to a continuum of two-level atoms with quadratic density. The resulting matrix Hamiltonian has an unbounded spatially varying coupling and is not semibounded. We prove that it is essentially self-adjoint and determine its spectrum explicitly. The continuous spectrum is the half-line $(-\infty,\Omega]$, where $\Omega$ is the atomic resonance frequency, while the spectrum above $\Omega$ consists of a strictly increasing sequence of simple eigenvalues. These eigenvalues are characterized by the interlaced zeros of the Airy function and its derivative and satisfy \begin{align*} \omega_n = \left(\frac{3\pi g n}{4}\right)^{1/2} + \frac{\Omega}{4} + O(n^{-1/2}). \end{align*} The analysis is based on a nonlinear spectral reduction that yields a confining massless relativistic oscillator above $\Omega$ and an inverted oscillator below $\Omega$.

Polynomial excitations consisting of creation and annihilation operators are studied. Subsystem entanglement is proven produced and preserved by such manipulations. NPT entanglement already present between any two subsystems of a general pure or mixed multipartite Gaussian state is preserved under all polynomial excitations. Nevertheless, these excitations can create NPT entanglement between initially separable subsystems, implying that the number of NPT entangled pairs cannot decrease. Polynomial excitations of a wide class of initially separable pure states are considered. For these states, the method reduces the infinite-dimensional positivity problem to a finite-dimensional one: after partial transposition, NPT entanglement is equivalent to a corresponding bilinear representation matrix not being positive semi-definite. Furthermore, using symplectic diagonalization and spectral decomposition, an explicit transformation is constructed between the Fock-space and position representations. It is applied to quadratic excitations of a three-mode mixed Gaussian state. The behavior of the NPT entanglement is tracked between subsystems as a function of a parameter $\kappa$, which tunes the non-locality of the excitation, and quantum negativity is evaluated. Under a chosen quadratic Hamiltonian, the resulting unitary time evolution maintains NPT entanglement for long intervals, interrupted only by short periods of PPT states. Finally, a closed analytical expression is derived for the covariance matrix of a general quadratic operator for an arbitrary number of degrees of freedom in phase space. Our framework directly connects mathematical foundations, theoretical modeling, and experimentally feasible implementations.

We study spectral measures of one-dimensional quasiperiodic Schrödinger operators. For subcritical Schrödinger operator with Diophantine frequency, we prove sharp measurable-set bounds, answering a question of Avila and Jitomirskaya \cite{AJ}; we prove optimal Fourier decay and the sharp decay order of the time-averaged quantum return probability, confirming a conjecture of Aloisio, Carvalho, and de Oliveira \cite{ACO} and a prediction of Zhong and Mosseri \cite{ZhongMosseri}. For the subcritical almost Mathieu operator, we establish summable square-root envelopes for the spectral density, with an optimal density-one formulation of the lower bound, thereby answering a question of Avila \cite{Aqi}. These results follow from a general theorem for reflectionless measures under a summable square-root Craig-type condition. We establish an exact residue expansion of the squared Borel transform, which connects local density blow-up with long-time averaged return probabilities: the endpoint residues determine both the square-root edge profiles and the leading coefficient of the time-averaged return probability.

This work is a step towards a general understanding of fluctuations for systems of particles described at large scale by fractional diffusion. First, we derive formally a Large Deviation Principle for the empirical measure of mean-field interacting particles with a Lévy-type noise. In the case of independent particles, this unveils a generalized gradient flow structure at large scales. Second, we derive and solve numerically a stochastic Partial Differential Equation describing the fluctuating hydrodynamics of this system. The results compare well with direct numerical integration of the particles' dynamics.

We study the contributions of half-BPS surface defects to supersymmetric Casimir energy and the universal logarithmic term of supersymmetric Rényi entropy in four- and six-dimensional SCFTs. Using equivariant integration of defect anomaly polynomials and its extension to supersymmetric Rényi backgrounds, we obtain closed expressions for these observables in terms of defect 't Hooft anomalies. For Gukov–Witten defects preserving transverse rotations in four-dimensional $\mathcal{N} = 4$ super Yang–Mills theories, both observables depend on a single R-symmetry anomaly. For $\mathcal{N} = (0, 4)$ defects in six-dimensional $\mathcal{N} = (1, 0)$ theories, additional terms arise from the gravitational anomaly. We provide complementary checks using chiral algebras, five-dimensional localization, and holographic probe branes at large $N$, while localized chiral fermions reproduce the gravitational-anomaly contributions. The framework also recovers Wilson-surface results in six-dimensional $\mathcal{N} = (2, 0)$ theories.

We study the optimization inverse spectral problem (OISP) for the first $m$ eigenvalues of the Schrödinger operator on a compact metric graph. We prove the existence of a solution for every strictly increasing finite set of prescribed eigenvalues. On a graph without cycles, we obtain a representation of the optimal correction as a linear combination of squared eigenfunctions, provided that the selected eigenvalues are simple and separated from the next eigenvalue. On a graph with cycles, under the same simplicity and separation assumptions, we obtain a strong limiting representation of the optimal potential in terms of eigenfunctions of approximating operators. For a graph with a cycle, we give a counterexample with simple eigenvalues and linearly dependent squared eigenfunctions. We also prove that linear dependence of the squares of nonzero restrictions to an edge of eigenfunctions corresponding to distinct eigenvalues forces the potential on that edge to have an analytic representative

We study the eigenvalues of the negative Dirichlet Laplacian $-\Delta$ in domain $D_{\epsilon,N}$ obtained by removing $N$ Euclidean balls of radius $\epsilon$ from a bounded open set $D\subseteq\mathbb R^d$ ($d\ge2$). It is known that, for the balls centered at points drawn independently from a distribution with probability density $\rho$ and for $N\to\infty$ and $\epsilon\downarrow0$ so that the aggregate Newtonian capacity of the removed balls tends to some $\kappa>0$, the eigenvalues of $-\Delta$ in $D_{\epsilon,N}$ homogenize to those of the Schrödinger operator $-\Delta+\kappa\rho$ in $D$. Here we prove that, for the eigenvalues that are simple in the limit, a multivariate Central Limit Theorem holds in all $d\ge2$ under centering by expected values. Unlike earlier work in $d=3$ by Figari, Orlandi and Teta and, independently, Ozawa from 1980s that was based on resolvent expansions, we rely on martingale representations and one-ball perturbation asymptotic controlled by methods of harmonic analysis.

We use Loewner-Kufarev evolution to decompose the Brownian loop soup into a Poisson point process on Brownian bubbles which we call the Brownian bubble tea driven by $\rho$, where $\rho$ is the driving measure for the Loewner chain. The only assumption on $\rho$ is that the evolving boundaries are locally connected. This generalizes a similar disintegration result of Lawler and Werner to essentially optimal regularity, and includes foliations with finite Loewner-Kufarev energy as well as radial and whole-plane $\mathrm{SLE}_\kappa$ for all $\kappa \geqslant 0$. As applications, we give Brownian loop interpretations of the conformal distortion formula for Loewner-Kufarev energy and the relationships between the Loewner energy, Loewner-Kufarev energy, and the Schwarzian action.

We study the one-dimensional nonlinear Schrödinger system \[ -u_i''-2\Big(\sum_{k=1}^N u_k^2\Big)^{p-1}u_i=\mu_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \tag{S} \] where $u=(u_1,\cdots,u_N)\in H^1(\mathbb{R})^N$, $\mu_1\leq\mu_2\leq\cdots\leq\mu_N<0$, $1<p<3$ and $N\geq 2$ are arbitrary. For the cubic case where $p=2$ and $N\ge 2$, we establish the following structural results of the system $(S)$: (i) all nontrivial solutions can be classified completely; (ii) the linearized operator at any nontrivial solution is non-degenerate; (iii) exact $L^2$-mass formulas of all nontrivial solutions are derived in terms of distinct Lagrange multipliers $\mu_i$. As the applications of the above structural results, we first show that the system $(S)$ with $p=2$ does not have any orthonormal solution for any $N\ge2$. We then investigate a fermionic variational problem $J_N(p)$ with $N\ge2$, whose minimizers correspond to ground states of the system $(S)$ under orthonormal constraints. More precisely, we prove that there exists a sufficiently small $\delta_N>0$ such that $J_N(p)$ is attained for any $p\in(2-\delta_N,2)$, whereas $J_N(p)$ is not attained for any $p\in[2,2+\delta_N)$. These settle some conjectures of [Gontier–Lewin–Nazar, ARMA, 2021] and [Frank–Gontier–Lewin, CMP, 2021].

The dynamical formation of a complete Bose-Einstein condensate and the asymptotic purification of trapped bosons coupled to a quantized radiation field are rigorously established. By tracking the macroscopic one-particle density matrix $\kappa_T$ through a nonlinear resonance cascade, it is demonstrated that radiative relaxation irreversibly depletes all excited trap modes, even when the initial macroscopic state $\kappa_0$ is genuinely mixed. Here $\kappa_T$ is an average, with respect to a de Finetti measure $\mu_0$, over mean-field particle-field orbits, so that the mixedness of $\kappa_0$ reflects a statistical ensemble of orbits. Under the assumptions of a non-vanishing initial ground state component and strictly positive downward Fermi-Golden-Rule rates, strong trace-norm convergence to a pure condensate is proven, $$ \lim_{T\to\infty} \| \kappa_T-|\chi_0\rangle\langle\chi_0| \|_{\mathfrak{S}_1(\mathfrak{h})} = 0 \, . $$ Furthermore, the von Neumann entropy of the macroscopic one-particle density matrix is shown to vanish asymptotically, $$ \lim_{T\to\infty}S[\kappa_T]=0 \, , $$ without any claim of monotonicity in $T$. Thus, any initially mixed, positive-entropy state ($0<S[\kappa_0]<\infty$) is purified to the ground state $|\chi_0\rangle\langle\chi_0|$. For the proof, we derive the Hartree hierarchy of the coupled system of bosons and photons from the finite-$N$ BBGKY hierarchy of the microscopic quantum many-body system, establishing trace-norm convergence in the mean-field limit as $N\to\infty$. The limits are taken successively; first $N\to\infty$, then the weak-coupling limit, and finally $T\to\infty$. Although the global particle-field evolution remains unitary, the macroscopic one-particle state converges to a pure state as energy and correlations are continuously radiated into the dispersive field.

We show that every symplectic manifold admits a symplectic form whose metaplectic-c prequantizations give quantizability conditions on all its Lagrangian submanifolds. Metaplectic-c prequantizations descend to prequantizations over the Lagrangian Grassmannian bundle that are fibrewise flat with holonomies primitive fourth roots of unity on the generators of the fundamental groups. This fibrewise holonomy is consistently interpreted as that of both Leray two-orientations and normalized Blattner–Kostant–Sternberg half-forms. The prequantizations pull back to flat circle bundles on each Lagrangian submanifold whose holonomy character defines a quantizability condition. The first and second Stiefel–Whitney classes are identified as splitting obstructions to the Lagrangianwise holonomies. These holonomies split into a product of holonomies on any orientable Lagrangian submanifold, and split consistently for all Lagrangian submanifolds of a manifold with metaplectic structure. When the first Chern class vanishes, the holonomy splits into the Bohr–Sommerfeld factor and a quarter phase Maslov correction, recovering the classical Einstein–Brillouin–Keller condition. We develop the underlying obstruction theory for circle extensions associated to general Lie group double coverings with first hypercohomology groupoids.

Spin wave theory plays a central role in the analysis of low-energy properties of the quantum ferromagnetic Heisenberg model. In this work, we present a first step towards a rigorous justification of this approximation in the context of quantum dynamics in one dimension. According to spin wave theory, the evolution of a spin wave state should be well approximated by its free evolution with the appropriate energy of free quasiparticles called magnons. Our main result establishes this idea rigorously for a class of spin wave packets. We focus on spin-$\frac{1}{2}$ systems, and consider wave packets composed of two magnons. We argue our result is optimal with respect to the timescales.

Any pair of sufficiently generic Hamiltonians on $\mathbb{C}^d$ generates the full unitary group, i.e., any $W\in U(d)$ can be synthesized to arbitrary precision by alternating between the two evolutions for suitably chosen durations. For any given pair of Hamiltonians, the minimal number of switches required to cover the entire group is at least $d$ and unbounded in general. Despite this, we show that it scales linearly in $d$ in expectation. Assuming that the two spectra are sufficiently generic, we derive constructive upper bounds on the minimal switching number as a function of the change-of-basis matrix between the two eigenbases. In particular, if this matrix has a column whose entries all have equal moduli, as for a complex Hadamard matrix, then $9d$ switches suffice to reach every unitary. To obtain these results, we construct a short exact representation of any unitary as a product of unitaries that are alternately diagonal in two given bases. All bounds are extended to non-generic spectra and bounded evolution times.