Papers

In this paper, we establish the noncommutative maximal weak type $(1,1)$ and strong type $(p,p)$ estimates for the family of operators $(T_N)_N$, defined by $$T_Nf=\sum_{k=N_1}^{N_2}\nu_{k}(M_{k}-\mathsf{E}_k)f,$$ where $M_k$ denotes the dyadic Hardy–Littlewood average operator, $\mathsf{E}_{k}$ is the conditional expectation with respect to the dyadic cubes of side-length $2^{-k}$, $N=(N_1,N_2)$ with $N_1<N_2$ and $(\nu_{k})\in\ell_{\infty}$. The main novelty of our approach is the development of a noncommutative Cotlar-type inequality for non-smooth kernels, a result that is new even in classical harmonic analysis. As an application, we obtain the boundedness theory of the noncommutative maximal differential transforms for averaging operators.

We develop a one-dimensional theory of dispersive elasticity within Rational Extended Thermodynamics, taking local first-order balance laws rather than higher spatial gradients as the fundamental description. A supplementary mechanical-energy law and the Ruggeri–Strumia main-field principle determine the admissible stress, internal fluxes and production. A canonical two-field hierarchy is symmetric hyperbolic under explicit convexity conditions and contains nonlinear elasticity and a generalized-stress theory as principal subsystems. For the reversible linear singular two-field class, elimination of the fast stress mode yields the Love–Rosenau equation exactly, together with a necessary and sufficient realizability condition. Nonlinear elastic stresses and non-quadratic higher-field energies are compatible with the same RET architecture; for the exact nonlinear Love–Rosenau reduction we retain quadratic higher-field inertia while allowing nonlinear elastic stress. The reduced equation admits a travelling-wave first integral. For the leading cubic elastic correction we obtain an exact parametric smooth solitary pulse in a supersonic velocity window, while the truncated-cosine compacton is excluded. Direct simulations of the hyperbolic parent system show finite-time pulse persistence in the reversible regime and slow decay under weak dissipation. The local parent energy flux is kept distinct from interstitial working in the reduced theory.

We investigate the mixed Stirling numbers of the second kind, $\mathcal{S}(n; \mathbf{c})$, which count partitions of $n$ distinct elements into $m$ unlabeled and $k$ labeled non-empty blocks encoded by $\mathbf{c} = (m, 1^k)$. We establish their recurrence relations and exponential generating functions, and analyze their behavior modulo a prime $p$ and $p^2$. In particular, we extend the classical Touchard congruence to this mixed framework. Our results show that these configurations possess unique number-theoretic signatures distinct from classical set partitions.

In this work, for a network of differential equations with a prescribed graph structure, our goal is to show how to select the Laplacian of the network in order to obtain the most favorable outcome insofar as synchronizability. That is, we will want to: (i) guarantee asymptotic stability of a synchronous solution (as measured by a negative value of the master stability function), (ii) minimize the normalized spread of the Laplacian eigenvalues, and (iii) have a transient as short as possible. Within the class of tridiagonal Laplacians, we give both necessary and sufficient conditions for satisfying our three criteria above, and give extension to banded Laplacians as well. Finally, we give extensive numerical results to elucidate our theoretical results and to compare to existing works.

This paper investigates the asymptotic behavior of the out-of-sample prediction risk of the high-dimensional ridgeless least-squares estimator when the feature dimension $p$ and the sample size $n$ grow proportionally. We consider a generalized spiked population covariance model with multiple latent factors, where the number of spiked eigenvalues may remain finite or increase with $n$, and the spiked eigenvalues may be bounded or diverge at arbitrary rates. Beyond characterizing the impact of covariance spectra, we reveal a new mechanism underlying benign overfitting: the prediction behavior of ridgeless interpolation is fundamentally governed by the alignment between the regression coefficient $\boldsymbol\beta$ and the spiked eigenspaces of the population covariance matrix. In particular, we show that the signal energy distributed along latent spike directions determines whether interpolation leads to benign, tempered, or catastrophic overfitting. Our theoretical framework establishes sharp prediction risk limits under minimal moment conditions, requiring only finite fourth moments rather than Gaussianity. We characterize how the number, strength, and geometric structure of the spikes jointly influence the double-descent phenomenon. These results provide a unified understanding of when latent covariance structures facilitate or hinder generalization in overparameterized regression.

Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity. We prove that if $A\subset\mathbb{N}^3$ has positive upper Banach density, then there are $(x,y,z)\in\mathbb{N}^3$ and $d\in\mathbb{N}$ such that $$(x,y,z),(x+d,y,z),(x,y+d,z),(x,y,z+\Omega(d))\in A.$$ To establish the above result, we give an $L^2$-decoupling theorem for the triple ergodic averages $$ \frac1N\sum_{n=1}^N T_1^n f_1\,T_2^n f_2\,S^{\Omega(n)}g $$ associated with three commuting transformations by isotropy factors and nilpotent structures in $\mathbb{Z}^2$-actions.

Let $G$ be any given continuous positive function on $\mathbb{R}_+$. Let $V$ be radial with $|V(x)|\leq G(|x|)$. We prove a Landis-type theorem for any real-valued solution of $\Delta u=Vu$ on $\mathbb{R}^n$. We construct a decay threshold $e^{-g(r)}$, where $g$ is a strictly increasing function which can be computed explicitly in terms of $G$. Under suitable assumptions the exponent in the decay threshold is proportional to the Agmon distance associated with $G$.

A group is an E-group if every element commutes with each of its endomorphic images. Caranti asked whether a finite E-group can have nilpotency class three. We prove that the $3$-group of order $3^{84}$ introduced by Abdollahi, Faghihi, and Mohammadi Hassanabadi, and later shown by Abdollahi, Faghihi, Linton, and O'Brien to have the corresponding automorphism property, is an E-group. Let $P$ denote this group and put $V=P/\Phi(P)\cong \mathbb{F}_3^9$. The nine power relations of $P$ determine a linear map $q:V\longrightarrow\Lambda^2 V$. We prove that $q$ has no nonzero proper subspace $U$ satisfying $q(U)\subseteq\Lambda^2 U$. Since the image induced by any endomorphism of $P$ on $V$ has precisely this closure property, every endomorphism acts on $V$ either invertibly or trivially. The invertible case is the known A-group case. In the trivial case the image first lies in $\Phi(P)=P'$, and the power relations then force it into $\Omega_1(P')=Z(P)$. Thus every element commutes with every endomorphic image. The tensor rigidity is reduced to an exact finite calculation on the $9841$ points of $\mathrm{PG}(8,3)$.

Quiver-invariant dualities relate distinct 4d N=1 supersymmetric gauge theories that arise as worldvolume theories on a D3-brane probing the same toric Calabi-Yau 3-fold. These dual theories share an identical quiver and differ only in their superpotentials. We show that quiver-invariant dualities are realized by tilting mutations on the brane tilings that realize these theories. Among the 42 toric phases of the H1121 model, we identify three general families of tilting mutations, all characterized by the reversal of the orientations of zig-zag paths within the mutation region of the brane tiling. We present new examples of quiver-invariant dualities, given by five doublets and one triplet of toric phases with identical quivers, for which we verify that the brane tilings related by the tilting mutations correspond to the same toric Calabi-Yau 3-fold H1121.

We establish a finite-sample learning-to-control theory for geometrically supervised latent models of nonlinear deterministic systems. Geometric supervision is used only during training: simulator state, proprioception, or state estimates with independently validated metric and directional error bounds supply observable-state distances and tangent directions, while deployment remains observation- and action-conditioned. We introduce an encoder-only local–global metric hinge that enforces directional resolution and separated-state discrimination. Under regular observable-factor, coverage, finite-capacity approximation, and uniform $C^{1,1}$ hypotheses, a computable one-sided regularization regime has a strong selection property: with high probability, every approximate empirical minimizer is simultaneously pointwise co-Lipschitz and uniformly approximately semiconjugate to the controlled dynamics. Approximation, sampling, and optimization errors remain explicit and separate. Norm-constrained tensor-product B-spline classes constructively realize the approximation hypotheses, and the interpolation exponent converting mean residual control into a uniform bound is sharp. A modular deterministic corollary transfers the learned certificates to trajectory, finite-horizon cost, learned-cost-head, and optimizer guarantees, while a validated finite-net result enables sharper model-specific certification. Controlled experiments isolate collapse and folding, quantify the analytic certificate's reserve, and demonstrate the control benefit of restored metric resolution. The principal contribution is a complete finite-sample implication from approximate empirical optimization to metric faithfulness, uniform controlled dynamics, and reliable planning for the same learned model.

We construct coordinates on orthogonal conjugacy classes of traceless real symmetric matrices with arbitrary spectrum (the isospectral manifold) that diagonalize the non-periodic Toda vector field. The coordinates, defined on a neighborhood of any diagonal matrix decouple the Toda vector field into a sum of multiples of the Euler vector field in $\mathbb{R}$. The domain of each set of coordinates is dense and their union covers the isospectral manifold. The construction relies on a new matrix factorization which is of independent interest.

In this paper, we investigate the existence of primitive polynomials over $\mathbb{F}_{q^n}$ whose constant term is a primitive element of $\mathbb{F}_{q^n}$. We prove that such polynomials do not exist if $q$ is odd, $q^n\equiv3\pmod{4}$, and the degree $m$ of the polynomial is odd. In particular, the polynomials $f(x)=g(x)+\lambda$, where $g(x)\in\mathbb{F}_q[x]$ satisfies $g(0)=0$ and $\lambda\in\mathbb{F}_{q^n}$ is primitive, cannot be primitive under the same conditions. Further, for the cubic polynomial $x^3+x^2+x+\lambda$, we establish non-existence results in characteristics $2$ and $3$. These results, in particular, provide counterexamples to previously proposed existence conjectures. We also study the family $x^p+x+\lambda$ over $\mathbb{F}_{p^n}$. For an odd prime $p$, we prove that, provided $\sum_{i=0}^{n-1}(-1)^i\lambda^{p^i}\neq0$, the polynomial $x^p+x+\lambda$ is irreducible over $\mathbb{F}_{p^n}$ if and only if $n$ is even. Motivated by this result and supported by computational evidence, we formulate a conjecture concerning the existence of such primitive polynomials.

Maps $\Phi:D\to D$ of the unit disk preserve frame generator sets for single orbits of normal diagonal operators exactly when they preserve Carleson interpolating sequences bidirectionally and satisfy $1-|\Phi(z)|^2\asymp1-|z|^2$ uniformly. Denote this class by $\mathcal P$. Equality of frame generator sets forces the boundary estimate, since every frame generator $f=(f_j)$ satisfies $|f_j|^2\asymp 1-|\lambda_j|^2$. No continuity, measurability, or analyticity is assumed; Carleson preservation alone remains flexible and, to our knowledge, unclassified in general. Equivalently, normalized Szego kernel Riesz sequences are preserved bidirectionally, with uniformly comparable unnormalized kernel norms at $z$ and $\Phi(z)$. Equivalently, we require uniform two-sided Carleson norm bounds for ordinary pushforwards of finite positive discrete measures and bidirectional preservation of pairwise pseudohyperbolic separation. Every $\Phi\in\mathcal P$ has a canonical bilipschitz radial trace $h_\Phi(\zeta)=\lim_{r\to1^-}\Phi(r\zeta)$ on $T=\partial D$, with uniform convergence. We give an intrinsic geometric characterization of $\mathcal K=\{\Psi\in\mathcal P:h_\Psi=\mathrm{id}\}$ and prove the unique factorization $\Phi=\Psi\circ E_{h_\Phi}$, $\Psi\in\mathcal K$, where $E_h(r\zeta)=rh(\zeta)$ and $E_h(0)=0$; thus $\mathcal P\cong\mathcal K\rtimes\mathrm{BiLip}(T)$ as monoids. For $m$ orbits, $\mathcal P_m=\mathcal P$ for every finite $m\ge1$, whereas $\mathrm{Aut}(D)\subsetneq\mathcal P_\omega\subsetneq\mathcal P$. Countable-orbit preservers in $\mathcal P_\omega$ are uniform pseudohyperbolic homeomorphisms. If $\varphi\in\mathrm{Aut}(D)$ and $|\Phi(z)-\varphi(z)|=o(1-|z|)$ uniformly as $|z|\to1$, then $\Phi\in\mathcal P_\omega$ exactly when $\Phi$ is a disk homeomorphism. Two-sided tests characterize $\mathcal P_\omega$. Holomorphic members of these classes are exactly disk automorphisms.

The leading asymptotic growth of Lanczos coefficients is controlled by spectral tails and furnishes a coarse classification of Krylov dynamics. We show that the global topology of the spectral measure, specifically the number of connected components, the gap structure, and the behaviour at gap-closing transitions, encodes a finer hierarchy of dynamical invariants invisible to tail-based arguments. Using the Riemann-Hilbert formulation of orthogonal polynomials and Deift-Zhou steepest descent, we recover the Freud growth laws $b_n\sim n^{1/\beta}$ for single-cut measures and determine their sub-leading corrections from endpoint data. Gapped spectra produce quasiperiodic Lanczos oscillations at a frequency fixed by the filling fraction of the spectral bands alone, and hence predictable from the band edges. We verify this in the SSH chain and its next-nearest-neighbour deformation. At a gap-closing transition the oscillation amplitude is governed by the Hastings-McLeod solution of Painlevé II, decaying as $n^{-1/3}$ at criticality and interpolating between the gapped and merged phases, so that the topology change of the spectral curve is realised as a Krylov phase transition. We also demonstrate that, while in the conformal limit of SYK the operator scaling dimension is invisible in the leading rate $\alpha = \pi T$, it can be extracted from the subleading offset $b_0 = \pi T(\Delta - \frac{1}{2})$. These results establish a refined notion of universality in operator growth, classified by spectral topology rather than spectral tails alone.

This paper studies the inverse reinforcement learning (RL) problem for linear-quadratic mean-field (MF) social optimization. The considered system features multiplicative noise and indefinite cost weights, which violate standard convexity assumptions and pose analytical challenges. The goal is to recover unknown social cost weights from expert demonstrations and reproduce the optimal control policies. This requires solving coupled stochastic algebraic Riccati equations and Lyapunov equations with unknown system dynamics. To this end, we first propose a model-based inverse RL algorithm with two sequential loops that separately handle individual and MF dynamics, and we prove its convergence and closed-loop stabilizability. Moreover, we characterize the non-uniqueness of the recovered cost weights. To eliminate reliance on system dynamics, we develop a model-free inverse RL algorithm using integral RL and least-squares identification, which requires only measured trajectory data satisfying mild rank conditions. Finally, numerical simulations validate the effectiveness of the proposed approaches.

Sequence convergence to a boundary Karush–Kuhn–Tucker (KKT) point has long remained unclear for nonconvex mirror descent with Legendre kernels. The difficulty arises from the blow-up of the gradient of the Legendre kernel at the boundary. Recent work~\cite{dingtoh2026nonkkt} shows that mirror descent can accumulate at non-KKT boundary points despite decreasing objective values, precluding a convergence guarantee to KKT points in general. Despite this negative result, mirror descent remains effective in many real applications. Motivated by this contrast, we address the boundary difficulty directly and establish KKT convergence of mirror descent for a broad class of structured nonconvex problems. We analyze mirror descent in reparameterized variables, where the Hessian metric is flattened and remains nondegenerate as the boundary is approached. Under extension and definability conditions jointly coupling the objective, the Legendre kernel, and the feasible region, the reparameterized sequence has finite length and converges, thereby recovering convergence to a KKT point of the original sequence. Our general framework applies to some concrete instances: Shannon entropy, Fermi–Dirac entropy, and power kernels on polyhedron.

The quantum-mechanical two-sided Bogoliubov inequality provides upper and lower bounds for the free energy required to separate a system of interacting particles into independent subsystems. The bounds can be calculated straightforwardly from the ensemble average of the interface energy, bypassing the direct evaluation of the free energy. In this work, we generalize the two-sided Bogoliubov inequality to arbitrary von Neumann algebras by employing the Araki-Uhlmann relative entropy and the framework of unbounded perturbation theory of KMS states. Furthermore, we obtain variational expressions for the relative free energy that extend existing bounded-perturbation principles to the unbounded setting. Crucially, these mathematical developments yield a physically well-founded thermodynamic criterion for the quantification of entanglement in infinite-dimensional systems.

We develop a constrained Onsager variational framework for parametric finite element approximations of Willmore and Helfrich flows. The weak curvature relation is used as a PDE constraint to express the first variation of the bending energy in curvature-vector form. Onsager's principle then combines this variation with dissipation based on the normal velocity, while an independent relaxed minimal-deformation-rate (relaxed-MDR) constraint selects the tangential velocity. The resulting mixed formulation admits a semidiscretization by continuous piecewise linear elements that satisfies an exact energy-dissipation law and exactly preserves area and volume when the corresponding constraints are imposed. The framework accommodates spatially varying spontaneous curvature and applies to both closed surfaces and open surfaces with fixed boundaries under Navier or clamped conditions. We also propose a linearly implicit fully discrete scheme, prove its unique solvability, and present numerical experiments demonstrating convergence, energy decay, constraint preservation, and effective mesh redistribution.

For integers \(n,\ell\geq1\), set \(Q_\ell=2(n+\ell)+2\) and \(q_\ell=2Q_\ell/(Q_\ell-2)\). On the Siegel domain \(\Ucal=\Heis^n\times(0,\infty)\), we study \[ -\Delta_{\Heis^n}v -4\rho\bigl(v_{\rho\rho}+T^2v\bigr)-4\ell v_\rho =|v|^{q_\ell-2}v. \] For its Dirichlet form \(E_\ell\), we determine the sharp Sobolev constant and all extremals, prove \[ S_\ell\|v\|_{q_\ell}^2\leq E_\ell(v),\qquad E_\ell(v)-S_\ell\|v\|_{q_\ell}^2 \geq\kappa_{n,\ell} \operatorname{dist}_{\dot S^1_\ell} \bigl(v,\mathfrak M_\ell^{\R}\bigr)^2, \] where \(\mathfrak M_\ell^{\R}\) is the extremal cone. We classify nonnegative finite-energy solutions and establish the linearized kernel, profile decomposition, and attainment of the optimal stability quotient. Cayley conjugation yields ground-state symmetry and nondegeneracy, global Palais–Smale compactness, and perturbative existence for critical equations on \(\CHyp^{n+1}\). The mechanism is the radial lift \(v^\uparrow(z,w,t)=v(z,t,|w|^2)\), which reverses the interior-to-boundary construction by realizing the Siegel domain as a symmetry-reduced slice of a larger Heisenberg group. The completed-space reduction resolves the degenerate axis, hidden auxiliary concentration, and the mismatch of extremal cones. Together with the Cayley transform, this supplies the missing nonlinear layer—classification, stability, bubbling, and variational compactness—on complex hyperbolic space and provides a blueprint for other rank-one symmetric spaces.

We develop a unified framework for constructing combinatorial structures under local constraints. Our approach extends the configuration model for random graphs with a prescribed degree sequence, and covers many special cases, including bipartite graphs, directed graphs, oriented graphs, edge-colored (bipartite) graphs, and (directed) hypergraphs. By reformulating half-edge matching as an independent set problem in an auxiliary graph, we identify 2-uniformity, a property characterising when greedy sampling preserves asymptotic uniformity. We classify all 2-uniform graphs and show that only two classes, the configuration space and the bipartite configuration space, have unbounded independence number, enabling the asymptotic regime. Our main theorem then gives the asymptotic sampling distribution and enumeration formulae for configurations, with error terms of order $O(d_{\max}^4\log m/m+d_{\max}^2(\log m)^2/m)$ as the number of edges $m$ tends to infinity with maximum degree $d_{\max}=O(m^{1/4}/\log m)$. This settles the long-standing $O(m^{1/4-\tau})$ bound (for some fixed $\tau > 0$), making the critical exponent explicit. Furthermore, our theorem accommodates forbidden edges, provided that each vertex participates in at most $O(m^{1/4}/\log m)$ of them. In particular, this enables the sampling of edge-colored graphs with prescribed degree sequences for each color class by constructing the colored subgraphs one at a time.

One hallmark of exactly solvable KPZ random growth models is product-form invariant measures. In the setting of exponential last-passage percolation (LPP), this corresponds to the independence of Busemann increments along any down-right path. However, this independence breaks down when multiple asymptotic directions are considered simultaneously, owing to the fact that jointly invariant measures are not jointly product-form. This paper shows that the failure of independence is one-sided: Busemann increments across arbitrary directions are negatively associated. As an application, we derive an exponential concentration inequality for sums of Busemann increments on the diffusive scale, even when the increments are not independent. While our argument relies on a Burke property that is special to exponential weights, all other proof ingredients$\unicode{x2014}$including hidden LPP monotonicities and braid relations for queueing maps$\unicode{x2014}$hold for arbitrary weights.

It is well established that an appropriate modification of test functions may lead to pressure-robust mixed methods for Stokes problems. However, for immersed finite element approximations of Stokes interface problems on unfitted meshes, it remains unclear whether the velocity error is independent of the pressure, since the velocity and the pressure are coupled in one of the interface conditions. In this paper, we provide a positive answer through a novel decomposition of the discontinuous pressure into a continuous component and a velocity-dependent discontinuous component. We demonstrate that the immersed Crouzeix–Raviart/$P_0$ element method achieves pressure robustness via an $H(\operatorname{div})$-conforming reconstruction of the test functions on the right-hand side. The stability and optimal error estimates of the proposed method are established with constants independent of the interface position relative to the mesh. Numerical experiments are presented to validate the theoretical findings.

Bier spheres arise as deleted joins of simplicial complexes with their combinatorial Alexander duals and form one of the largest known families of simplicial spheres. We study centrally symmetric Bier spheres and give a simple criterion for when they cannot arise as boundaries of centrally symmetric polytopes. From this, we obtain a large new family of simplicial polytopes with combinatorial automorphisms that cannot be realized geometrically. Prior to our construction, the Bokowski–Ewald–Kleinschmidt polytope was the only known simplicial example exhibiting these properties. By Smith theory, these polytopes have noncontractible realization spaces. Finally, we establish that every Bier sphere with at most $12$ vertices is polytopal.

We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p\leqslant\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. This resolves, in the classical strictly spacelike setting, the nonexistence conjecture of Byeon, Ikoma, Malchiodi, and Mari, including the critical endpoint. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key ingredient is a universal bound, valid for every $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. Then a weighted trace-free tensor identity from the invariant-tensor approach, combined with a common cutoff estimate, a core-counting argument and Souplet-type feedback inequality, yields a unified proof in the subcritical and critical ranges. The upper endpoint is sharp for $n\geqslant3$, as supercritical radial solutions exist. The theorem also gives half-space rigidity for complete spacelike hypersurfaces, including at the critical exponent.

When a reinforcement learning agent cannot observe the full state, we usually blame its policies: it cannot see enough to represent a good one. We show that in a solvable case the bigger problem lies elsewhere. Even when a good policy is available and the agent's value function is expressive enough to describe it exactly, learning still ends up somewhere far worse. We study a partially observed linear-quadratic problem in which a standard actor-critic learner can be solved in closed form. At our default setting the best policy the agent can represent is already close to optimal, costing 10.4% more than the ideal controller that observes everything. Learning does not find it. The algorithm instead comes to rest at a policy that is 35% worse than the best one available to it, and we can say exactly where and why. The cause is a bias in what the critic learns rather than a limit on what the actor can express. Because the agent cannot attribute what it sees to the part of the state it cannot observe, the critic misreads that unexplained variation as sharp curvature in its own value estimates, and the actor follows that error away from the optimum. We derive closed-form expressions for the resulting policy, for its cost, and for the one design choice that removes the problem, which is how far the learner looks ahead before trusting its own value estimates. Deep reinforcement learning experiments follow these predictions closely. Notably, giving the agent memory of past observations does not help, while changing how far it looks ahead does.

We establish Quillen equivalences between successive truncations of multicomplexes for the model structures that detect weak equivalences simultaneously on prescribed pages of the two associated spectral sequences. In particular, the natural comparison from $4$-multicomplexes to bicomplexes, which is relevant to the comparison between almost complex and complex geometry, is a Quillen equivalence. This answers a question of Cirici, Livernet, and Whitehouse.

Topological defects in quantum field theories are believed to assemble into higher categories with extra structure. This has been made precise for defects in $2$- and $3$-dimensional oriented topological quantum field theories by Davydov, Kong, and Runkel and by Carqueville, Meusburger, and Schaumann, respectively. In this paper we study the extra structure present on these categories when the topological field theory is additionally reflection positive. We define reflection defect TQFTs as symmetric monoidal functors out of a defect bordism category that intertwine orientation reversal with complex conjugation; they are reflection positive if they satisfy an additional positivity condition. Our main result is that in two dimensions the bicategory of defects $\mathcal{T}_\mathcal{Z}$ associated to a reflection defect TQFT carries the natural structure of an $O(2)$-dagger bicategory, a structure we define explicitly. If the theory is reflection positive, $\mathcal{T}_\mathcal{Z}$ can be equipped with additional structure closely related to the definition of a 3-Hilbert space (the two agree up to some finiteness and completeness conditions).

Let $K \subset \mathbb{R}^n$ be an origin-symmetric convex body and assume that its uniform probability measure is isotropic in the probabilistic normalization, namely \[ \int_K x \otimes x \, d\mu_K(x) = \mathrm{Id}_n. \] We give deterministic geometric proofs of \[ M(K) \leq C \frac{\log(n)}{\sqrt{n}} \qquad \text{and} \qquad M^*(K) \leq C \sqrt{n} \, \log(n), \] where \[ M(K) = \int_{\mathbb{S}^{n-1}} \|\theta\|_K \, d\sigma(\theta), \qquad M^*(K) = \int_{\mathbb{S}^{n-1}} h_K(\theta) \, d\sigma(\theta). \] Combining both estimates yields \[ M(K) M^*(K) \leq C \log^2(n). \] The first proof uses a quadratic aggregate of dyadic centroid bodies. The second uses the analogous weighted aggregate of the Laplace bodies $p\{\Lambda_K \leq p\}^{\circ}$, which are equivalent to the centroid bodies by the work of Klartag and E. Milman. In both cases, curvature at each dyadic scale outside a subspace of codimension $O(p)$ leads, via the min–max principle, Legendre duality, and the spherical Laplacian, to the required estimate. The only high-dimensional input is the dimension-free small-ball consequence of the slicing theorem.

Let $C_m:=\mathrm{BCH}(3,m)$ be the binary primitive triple-error-correcting BCH code of length $2^m-1$. We determine its second generalized covering radius exactly: $R_2(C_m)=8$ for every $m\geq5$. Equivalently, every two-dimensional syndrome subspace is contained in the binary span of at most eight parity-check columns, and eight columns are necessary in the worst case. This matches the known lower bound.

We prove that every reduced real plane curve singularity admits a real morsification. This settles a question of A'Campo and Gusein-Zade, later stated as conjectures by Leviant–Shustin and by Fomin–Pylyavskyy–Shustin–Thurston. In particular we overcome the main obstruction that remained posed by conjugate pairs of nonreal branches. Our new main ingredient is a construction that produces the divide from a nodal smoothing of two normalization disks. This is what we call the trace map. For real branches, it recovers Gusein-Zade's construction using Chebyshev polynomials. For pairs of complex conjugate branches with distinct tangents, the construction gives an explicit formula for the divide in terms of the Puiseux data. The general method consists in a delicate combination of the trace map with A'Campo's translations and contractions to produce divides and real morsifications for all reduced real plane curve singularities.