Papers

This paper presents a high-order rank-adaptive implicit integrator for the tensor solution of high-dimensional diffusion equations. We extend the 3D version of this method from the Tucker decomposition to higher dimensions using the hierarchical Tucker (HT) decomposition, since the storage complexity for the Tucker decomposition increases exponentially with the number of dimensions $d>3$. The HT format avoids this issue by decomposing the solution according to a binary tree consisting of bases for each dimension and core tensors which connect the bases. Spectral methods are considered for spatial discretization, and diagonally implicit Runge-Kutta methods are considered for time discretization. At each stage of the Runge-Kutta method, the bases computed at the previous stages are augmented to predict the upcoming basis and construct projection subspaces. By projecting onto these enriched subspaces, the bases and cores can be updated in a sequential manner going up the tree from leaf-to-root. Unlike the 3D Tucker method which has a single core tensor, the HT method also updates the intermediate core tensors. Numerical experiments demonstrate that the method observes high-order accuracy, and test how well the integrator captures the solution rank for various sets of time-dependent diffusion coefficients.

A finite-dimensional two-field system for divergence-free edge cochains is introduced. Its MHD-type designation refers only to a quadratic exchange pattern and exact total-energy cancellation; it is not a physical MHD discretization. A general cancellation class is separated from a corrected explicit realization: the anticommutator $D(a)J+JD(a)$ is skew-symmetric for diagonal $D(a)$ and skew-symmetric $J$, and its projected bilinear map has the required trilinear antisymmetry. The central result is a Hodge coercivity criterion: the full divergence-free space admits the Poincaré-type estimate needed for dissipativity if and only if its harmonic $1$-cochain space is trivial. Under this condition, global existence, an exact energy identity, an absorbing ball, and a compact global attractor follow. When harmonic modes are present, a harmonic-decoupled interaction class yields invariant harmonic affine fibres and fibre-wise attractors. Deterministic disk, annular, and two-hole examples illustrate the spectral criterion, energy law, and distinction between general harmonic exchange and harmonic-fibre invariance.

We establish an optimal Poincaré inequality for real-valued symmetric observables of the complex Ginibre log-gas. Equality is attained by the real and imaginary parts of the center-of-mass observable. Equivalently, we determine the exact spectral gap of the associated overdamped Langevin dynamics, for real symmetric observables. The Hessian of the energy of this log-gas is unbounded below, so standard convexity arguments do not directly apply. The proof instead combines a Vandermonde transform, a holomorphic projection, and a complex Gaussian d-bar spectral-gap estimate, corresponding to the constant-curvature case of the Hörmander-Berndtsson estimate. It is short and self-contained. It remains valid, beyond the quadratic potential, for rotationally invariant additive plurisubharmonic potential perturbations. Additionally, we provide four alternative proofs, two of which yield sum-of-squares formulas for the deficit, based respectively on a Hermite expansion and on an integrated Bochner-Kodaira formula, the other two use the spectral analysis of a two-sided number-operator factorization and a Hermite-Slater polynomial expansion. We furthermore present new results related to linear statistics, Gaussian factorization, polynomial eigenfunctions, log-Sobolev inequalities, matrix lift and eigenvector overlaps, and non-quadratic potentials.

Let \(\Omega\Subset\mathbb H^N\), \(N\in\{3,4,5\}\), be a bounded connected \(C^2\) domain. We prove that the pure critical Dirichlet problem \[ -\Delta_{\mathbb H}u=u^{\frac{N+2}{N-2}} \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega \] admits a positive solution whenever \(H_d(\Omega;\mathbb F_2)\neq0\) for some \(1\le d\le N-1\). This gives a hyperbolic Bahri-Coron theorem for \(3\le N\le5\) under \(C^2\) boundary regularity. Under conformal reduction, the hyperbolic geometry produces a positive potential and leads to dimension-dependent bubbling mechanisms. In dimension three, the required energy drop follows from the balance between diagonal corrections and pair interactions at fixed large multiplicity. In dimensions four and five, it is obtained at the matched scale through a normalized defect estimate and an all-pairs source-transfer bound. A unified Thom-barycenter construction converts these analytic estimates into the topological contradiction. Thus nontrivial domain topology forces existence despite critical loss of compactness and the additional geometric potential.

Characterizing isolated points on the modular curve $X_1(N)$ is a key obstruction to classifying all points of a fixed degree. These points do not lie in infinite parameterized families, making them difficult to obtain through geometric constructions. In this paper, we focus on the collection of "isolated $j$-invariants" for $X_1(N)$, which are the values obtained by mapping isolated points to the $j$-line. Prior work of the author in collaboration with Ejder, Liu, Odumodu, and Viray asks whether there are only finitely many isolated $j$-invariants lying in extensions of bounded degree. Here, we explore how this question relates to other uniformity problems in the field and give new finiteness results for isolated $j$-invariants in $\mathbb{Q}$. As an application, we show similar methods give sharpened polynomial bounds on torsion for non-CM elliptic curves having rational $j$-invariant.

The connective constant $\mu(G)$ of a graph $G$ is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We prove upper and lower bounds for the connective constants of Cayley graphs $G_\omega$ of a general Grigorchuk group encoded by a sequence $\omega\in\{0,1,2\}^{\Bbb N}$. In particular, $\mu(G_\omega) > \phi$ for any such Cayley graph (subject to a simple condition on $\omega$), where $\phi:= \frac12(1+\sqrt 5)$ is the golden mean. This extends earlier work of the author and Zhongyang Li in "Cubic graphs and the golden mean”, Discrete Math. 343 (2020), article 111638, where it was conjectured that $\mu(G)\ge\phi$ for all infinite, vertex-transitive, cubic graphs. The current work includes an analysis of the proportions of appearances of given label-sequences in the orbital Schreier graphs of general Grigorchuk groups.

In a $k$-uniform hypergraph, the minimum $d$-degree for some $0\le d\le k-1$ is the minimum number of edges containing any given $d$-set of vertices. An extension of the classical Dirac theorem guarantees that whenever the minimum $d$-degree of a $k$-uniform $n$-vertex hypergraph, $k\mid n$, is larger than a certain Dirac threshold, it contains at least one perfect matching. Moreover, it has been known for some time, due to Kwan, Safavi, and Wang, that for $d\ge k/2$ such hypergraphs contain not only one, but “many” perfect matchings, that is, at least as many as are expected in a random hypergraph with the same edge density. However, it has also been known that such a result could not be hoped for in general, as it already fails for $(d,k)=(1,3)$. In this paper we introduce new notions of the counting thresholds and approximate counting thresholds, above which a hypergraph is guaranteed to have at least this many perfect matchings. We show that these thresholds are well-defined and nontrivial for all $d,k,n$, that they are asymptotically related, and finally, we derive improved upper bounds by reducing to cases with smaller $d$ and $k$.

We study a long-range perturbation of the perimeter functional by a nonlocal repulsive term defined through the Yukawa kernel, minimized under a volume constraint. Because the kernel decays exponentially, the competition between surface tension and repulsion is governed by two independent quantities, the screening rate and the volume. Specifically we prove that (i) above an explicit critical screening rate, minimizers exist at every volume, whereas earlier work required the volume to be large; (ii) below an explicit volume threshold minimizers exist for every screening rate; (iii) at small volume, and uniformly in the screening rate, the ball is the unique minimizer up to translation; and (iv) there exists a sharp volume threshold, depending on the screening rate, for the ball to be a stable volume-constrained critical point. The threshold is given in closed form for every screening rate and reduces to the known unscreened value when $\alpha=0$. Our proofs overcome new technical changes due to the lack of homogeneity in the Yukawa kernel.

We define a map from solutions of the Bethe Ansatz equations (BAEs) of four-dimensional $\mathcal{N}=4$ super-Yang–Mills with arbitrary semisimple gauge algebra $\mathfrak{g}$ to extrema and poles of the potential of the untwisted elliptic Calogero–Moser system of type $\mathfrak{g}$. We conjecture the map to be a bijection on the preimage of the Calogero–Moser extrema, and show that it intertwines the symmetries of the two systems, both the gauge ones (torus, Weyl and center invariance) and a $\mathrm{PSL}(2,\mathbb{Z})$ action, so that solutions on both sides organize into orbits, each BAE orbit mapping onto a single orbit of Calogero–Moser extrema or poles. That the system produced is the untwisted one has a consequence: the conjectured correspondence between BAE solutions and vacua of the $\mathcal{N}=1^\ast$ deformation of $\mathcal{N}=4$ on $\mathbb{R}^{3,1}$, which are extrema of the twisted system, cannot extend to non-simply-laced $\mathfrak{g}$. It also fails within the simply-laced cases, though not for $\mathfrak{su}(N)$: we exhibit an $\mathfrak{so}(8)$ solution that flows to a pole of the Calogero–Moser potential rather than to an extremum, and so has no $\mathcal{N}=1^\ast$ counterpart. We illustrate the map in detail for every rank-two $\mathfrak{g}$, classical and exceptional alike, and use these cases as evidence for the conjecture.

Quantum state transfer on qubit lattices is a crucial step in a panoply of quantum information processing tasks. Understanding its fundamental limits in the presence of practical imperfections remains a pressing open question. In particular, it is unclear how thermal noise in the intermediate ancilla sites impedes transfer. In this work, we derive tight lower bounds on the necessary growth of commutator norms to achieve approximate state transfer. Specializing to 1D power-law systems with thermal ancilla states, we demonstrate that state transfer runtimes depend sensitively on the scaling of temperature with system size, improving logarithmic bounds to algebraic ones. Our work extends previous results on exact state transfer to the qualitatively different and more practically relevant regime of approximate state transfer.

Gaussian boson sampling (GBS) is a sampling task proposed to demonstrate quantum advantage. We consider Gaussian boson sampling on $M$ optical modes, with $K$ equally squeezed input modes and $N$ observed photon counts. We complete the proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezers $K$, which is a part of the argument for classical hardness of GBS. In particular, we show that for any $K$ and $N=o(\sqrt{K})$, the symmetric product $MK^{-1/2}U_{NK}U_{NK}^T$, for $U_{NK}$ the top left $N\times K$ submatrix of an $M\times M$ Haar random unitary $U$, is close in total variation distance to both an $N\times N$ symmetric complex Gaussian matrix $\mathbf G$ with independent entries, and the symmetric product $GG^T/\sqrt{K}$ for $G$ an $N\times K$ matrix of iid standard complex Gaussians. We show however that the density-based instance generating method of [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013), Lemma 5.8] used to efficiently implement a hiding procedure fails for Gaussian boson sampling with $K=cM$ if $c<1/2$. Instead we use approximate instance generating to implement the hiding for the usual classical hardness reduction.

In view of embedding 3D gravity into topological string theory, we show that the dimensional reduction of 6D Kodaira-Spencer gravity on $\text{AdS}_3\times S^3$, i.e. the low-energy description of twisted holography, contains a subsector that coincides with the action of chiral 3D gravity on $\text{AdS}_3$. Furthermore, we show that the full 3D Einstein action is obtained as the reduction of two independent complex-conjugate copies of 6D Kodaira-Spencer gravity. We perform the dimensional reduction explicitly at the level of the classical actions, extending our previous work that related the equations of motion of the two theories. Our reduction procedure is based on a novel rewriting of the non-local 6D Kodaira-Spencer action as a local 6D holomorphic Chern-Simons action for the gauge group $\mathrm{SL}(2,\mathbb{C})$, valid within a subsector of complex structure deformations that we identify. Our results provide a necessary step toward embedding the Euclidean path integral of 3D gravity into topological string theory.

We construct a polarized smooth projective fivefold and prove that it is K-polystable but does not admit a constant scalar curvature Kähler metric. This disproves the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics. The main result of this paper was obtained using generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system. A detailed report on the use of generative AI in this paper is enclosed in the appendix, joint with Bin Dong and Guoxiong Gao.

In this paper we study generalised total colourings of graphs where the colour classes formed by vertices and edges, respectively, induce forests, while incident edges/vertices receive distinct colours. In [M. Borowiecki and I. Broere, Hamiltonicity and Generalised Total Colourings of Planar Graphs, Discussiones Mathematicae Graph Theory 36 (2016) 243–257] it was conjectured that for planar graphs, four colours suffice for this type of colouring. We confirm this conjecture for two infinite families of planar graphs.

We prove the existence of periodic solutions for Hamiltonian differential inclusions under growth conditions involving a G-function.

In this article, we introduce and study the class $\mathcal{M}_r(H)$ of minimum attaining operators on a separable Hilbert space $H$ whose restrictions to every nonzero reducing subspace are minimum attaining. This class lies between the classes of absolutely minimum attaining operators and minimum attaining operators. We establish a spectral characterization of positive operators in $\mathcal{M}_r(H)$: a positive operator $T$ belongs to $\mathcal{M}_r(H)$ if and only if $T=\sum_{\lambda\in\Lambda}\lambda P_\lambda$, where $\Lambda\subseteq[0,\|T\|]$ is a countable well-ordered set, $(P_\lambda)_{\lambda\in\Lambda}$ is a family of mutually orthogonal nonzero projections, and $\sum_{\lambda\in\Lambda}P_\lambda=I$ in the strong operator topology. We also establish a connection between $\mathcal{M}_r(H)$ and the class of operators attaining their norm on every nonzero reducing subspace. Further, we obtain stability results under suitable finite-rank and compact perturbations, and prove that $\mathcal{M}_r(H)$ is norm dense in $\mathcal{B}(H)$. We also establish the norm density of operators in this class having a nontrivial invariant half-space. Finally, we characterize normal operators in $\mathcal{M}_r(H)$, obtain a corresponding characterization for multiplication operators on $L^2(\mathbb{T})$, and characterize the condition $\mathcal{R}_T=\mathcal{R}_{|T|}$ in terms of the von Neumann algebra generated by $|T|$, showing that this condition forces $T$ to be normal.

Wasserstein gradient flows are intimately connected with evolution partial differential equations and diffusion processes. We take the first step in developing such connections for inner product Gromov–Wasserstein (IGW) gradient flows by studying the IGW gradient flow of the relative entropy $\mathsf{H}(\cdot\|\gamma)$ with respect to the standard Gaussian measure $\gamma$. We first show that $\mathsf{H}(\cdot\|\gamma)$ fails to be $\lambda$-convex along generalized or modified generalized IGW geodesics for any $\lambda \in \mathbb{R}$, and therefore falls outside the scope of the existing IGW gradient flow theory from Zhang et al. (2026). We bridge this gap by establishing a suitable local convexity estimate that enables the construction of the gradient flow and its extension to the infinite time horizon. We then obtain increasingly explicit representations of the resulting dynamics. Starting from a partial integro-differential equation, we derive a nonlinear Fokker–Planck equation and show that its second-moment dynamics decouple from the law as they satisfy an autonomous matrix ODE. This reduces the IGW dynamics to a linear, time-inhomogeneous Fokker–Planck equation, yielding a probabilistic representation as the time-marginal flow of a linear stochastic differential equation resembling the Ornstein–Uhlenbeck process. Finally, we study its asymptotic behavior by establishing exponential convergence of the flow to $\gamma$ in relative entropy.

We study volume comparison for closed Riemannian manifolds satisfying a positive Ricci curvature lower bound together with an improved scalar curvature lower bound. We prove that if a closed Riemannian manifold $(N^n, g)$ satisfies $\operatorname{Ric}_g\ge (n-1)g$ and the scalar curvature $R_g\ge n(n-1)(1+\varepsilon)$, then its volume satisfies $$\lvert N\rvert_g \le \frac{1}{\sqrt{1+n\varepsilon}}\lvert\mathbb S^n\rvert. $$ In fact, assuming only $\mathrm{Ric}_g\ge(n-1)g$, we can prove that $$\frac{|N|_g}{\left|\mathbb{S}^n\right|} \le \frac{1}{|N|_g} \int_N\left(\frac{R_g}{n-1}-(n-1)\right)^{-\frac{1}{2}} d \mathrm{vol}_g. $$ The equality holds if and only if $N$ is isometric to the unit sphere. The proof combines a coefficient-adapted Jacobian comparison with a new integral shuffling comparison for scalar Jacobi solutions, inspired by Brown and Freedman \cite{BrownFreedman2022}. This yields an estimate that retains the full Ricci spectrum. The resulting volume bound agrees to first order in $\varepsilon$ with the factor appearing in Bray's conjecture. A further consequence of the argument is an averaged volume comparison for metric balls involving the scalar curvature. We also obtain a volume comparison theorem under a weighted integral lower bound on the Branson $Q$-curvature.

We reconstruct a piecewise-smooth source in a Poisson equation on a semi-infinite strip from noisy solution values measured along an interior line. Applying the one-dimensional Dirichlet Laplacian to the observation reduces the inverse problem to regularized second-order differentiation followed by a boundedly invertible correction. The differentiated trace and the source differ by an analytic smoothing term and therefore have the same interior singular support and jump data. This structure motivates a localized Fourier extension method that combines two staggered detection partitions, GTSVD-regularized local derivative coefficients, mollified conjugate Fourier sums, and structure-aligned numerical differentiation. The source is then recovered by an exponentially decaying spectral correction. For an exact partition, the reconstruction inherits the piecewise differentiation rate $\mathcal O(\delta^{(\bar s-2)/\bar s})$; local peak and partition perturbation estimates describe the additional effect of breakpoint errors. Comparisons with full-grid total-variation regularization and truncated Fourier inversion show competitive high-noise performance and a pronounced low-noise advantage of the localized method. Repeated Gaussian-noise tests confirm robustness in the moderate- and low-noise regimes.

Gaussian elimination is one of the oldest algorithms in mathematics, and the most popular method for solving an unstructured linear system. Its stability in finite precision is controlled by its growth factor, which measures how large the entries produced during elimination can become. Understanding the worst-case behavior of this quantity has been a central problem in numerical analysis since the 1940s. Here we make a significant leap in that understanding, settling several open problems. In particular, we determine the asymptotic behavior of the maximum growth factor under complete and rook pivoting, proving that both are quasi-polynomial in dimension. We also show that the exponential growth under partial pivoting persists for sparse matrices and that randomized partial pivoting suffers the same instability. In contrast, we show that every non-singular matrix has a row permutation with polynomial growth, though finding the optimal row permutation is NP-hard.

A set of points $C\subset \mathbb{R}^n$ is canonically Ramsey if there is some larger set of points $S\subset \mathbb{R}^{n'}$ such that any colouring of $S$ contains either a monochromatic copy of $C$ or a rainbow copy of $C$. Mao, Ozeki, and Wang introduced this notion, showing that the 30-60-90 triangle is canonically Ramsey. Since then, many other configurations have been shown to be canonically Ramsey. The author showed that cuboids are canonically Ramsey. Ge, Shu, Xu, and Yu later showed that all simplices are canonically Ramsey, after which the author showed that all products of simplices are canonically Ramsey, a class which, together with its closure under taking subsets, includes all previously known canonically Ramsey sets. We prove that regular polygons with a prime number of sides are canonically Ramsey—the first known sets outside this class.

The kernel of a random symmetric p-adic matrix is a random abelian group, equipped with a symmetric pairing. If we consider not only the matrix but also its top-left corners, we get a process valued in isomorphism classes of abelian groups, equipped with such a pairing. We show that when the matrix is Haar random, this process is a Markov chain, generated by an operator that we explicitly describe. We will also prove that this operator is reversible with respect to a Cohen-Lenstra type measure.

The unfair $0$–$1$ polynomial conjecture asks whether a factorization \[C(x)=A(x)B(x),\] with $A$ and $B$ monic and having nonnegative real coefficients, and $C$ a polynomial with all the coefficients 0 and 1, must already be a factorization into $0$–$1$ polynomials. Let $k$ be odd and $0<a<1$. We study the possibility that \[1+a x^2+x^k\] divides a $0$–$1$ polynomial with a nonzero cofactor having nonnegative real coefficients. Ghidelli settled the first nontrivial case $k=5$. We prove that no such factorization exists for any odd $k\ge 341$. The intermediate cases (5 < k < 341) are treated in the companion paper.

This article introduces a numerical optimal control framework for minmax constrained density control for a class of noisy linear parabolic partial differential equations (PDEs), in particular the noisy heat equation. The goal is to transport an initial density to a target density while minimizing a specified cost with respect to control actions and maximizing it with respect to disturbances, all within a fixed time horizon while satisfying given convex path constraints. To address this, the spatial derivatives in the PDE are discretized using finite-difference approximations, transforming the problem into a system of ordinary differential equations in time. The admissible space of control and disturbance trajectories is then finitely parametrized, and the resulting optimal control problem is formulated as a convex semi-infinite program (SIP) under mild assumptions. By leveraging new numerical tools from convex SIP theory, we establish guarantees for exact solutions that account for constraint satisfaction under an infinite family of disturbance realizations, and we establish an optimization-based computationally efficient algorithm to recover these solutions. Comprehensive numerical examples to demonstrate and validate our findings are included.

In this work we analyze the eigendecomposition of the Hessian matrix of the Robin function $\mathcal{R}(x)$ for the spectral fractional Laplacian in orthogonally invariant domains. We prove that, if $\Omega$ a smooth bounded convex domain invariant under the action of an orthogonal transformation $\mathcal{O}$ then, for $\overline{t}\in\{a\in\Omega:\mathcal{O}(a)=a\}$, the gradient vector $\nabla\mathcal{R}(\overline{t})$ is an eigenvector of the Jacobian $D\mathcal{O}$ associated to the eigenvalue $1$. Moreover, if $\Omega$ is a domain invariant under the reflection about a hyperplane $\pi_{v}=\{x\in\mathbb{R}^N:x\cdot v=0\}$, there exists $\eta>0$ such that $\mathbb{H}(\overline{t})v=\eta v$ where $\mathbb{H}$ denotes the Hessian matrix of $\mathcal{R}(x)$. Consequently, if $\Omega$ is invariant under the reflection about the hyperplanes $\pi_{v_i}$ for a linearly independent set $\{v_1,\ldots,v_N\}$, then the origin is a non degenerate critical point of $\mathcal{R}(x)$. A short proof of the Brezis-Peletier-like formulas for $\nabla \mathcal{R}$ is also provided which allows us to prove the former results for any $0<s<1$.

Let $\Phi$ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann $\Xi$-function, set $s(t)=\Phi(\sqrt t)$, and define the first Laguerre expression $f(t)=s'(t)^2-s(t)s''(t)$. Csordas and Dimitrov (2000) conjectured that $\log f$ is strictly concave on $(0,\infty)$; Csordas (2015) later restated the assertion as Open Problem~4.14. We prove the conjecture by two complementary methods, sharing only a local certificate near the modular fixed point. The first proof is a direct theta-series argument: a directed-rounding Taylor certificate near $t=0$ is joined to a dominant-first-summand estimate with rigorous theta-tail bounds. The second proof uses Jacobi's nonlinear third-order differential equation for $\theta_3$ to obtain a three-dimensional autonomous phase space, a sharp elliptic monotonicity theorem, and an exact quartic reduction of the target inequality. The quartic boundary is a polynomial shear of the quadratic cone $XY=Z^2$. A directed interval certificate proves that every possible cone contact on the only remaining compact interval points strictly into the desired region; beyond that interval a pointwise monotonicity theorem closes the argument. The finite certificates and exact symbolic checks are supplied as reproducible scripts. The result implies the associated double Turán inequalities through the theorem of Csordas–Dimitrov, but no assertion of the Riemann Hypothesis is made.

We propose a local Legendre frame method for the accurate computation of Caputo fractional derivatives of order \(0<\alpha<1\). On each local subinterval, the function is represented by a restricted Legendre frame obtained from scaled Legendre polynomials on an extended interval. The local coefficients are computed from equispaced samples by an exponentially weighted GTSVD regularization. The Caputo derivative is then evaluated by applying the weakly singular fractional integral to the derivatives of the local frame basis functions. Since these derivatives are polynomials, the corresponding Caputo weights can be written in terms of finite weighted moments, so that the singular kernel is treated analytically rather than by a low-order quadrature rule. For uniform partitions, the history weights have a block-dependent structure and can be reused efficiently. The error analysis separates the local frame reconstruction from the Caputo integration. In particular, the Caputo error is bounded by the derivative reconstruction error, while the latter is obtained from the \(L^2\) reconstruction error and a weighted smoothness bound of the GTSVD approximation through an interpolation argument. For analytic local functions with exponential coefficient decay, this leads to exponential-type convergence of the derivative and hence of the Caputo approximation. Numerical experiments confirm the accuracy of the exact moment weights, the effectiveness of the local weighted reconstruction, and the efficiency of the block implementation.

Let $A$ be a commutative simple algebra in a braided finite tensor category $\mathcal{B}$. We identify the largest transparent subalgebra of $A$ as the algebra induced by a central lift of the free-module functor. This identification gives formulas for the Frobenius-Perron dimension and the Müger center of the category of local $A$-modules. These formulas give criteria for nondegeneracy, symmetry, and modularity, together with sharp bounds on $\mathrm{FPdim}_{\mathcal{B}}(A)$. We also realize the Müger center of $\mathcal{B}$ as a category of local modules over an adjoint algebra. Finally, we prove a relative-center factorization and deduce that taking the category of local modules preserves the relative Witt class.

We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of Kähler potentials. Consequently, every noncollapsed Kähler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.

We study stochastic control of multivariate Hawkes-driven stochastic differential equations with machine learning algorithms in a non-Markovian setting. Due to the path dependence of the memory of the Hawkes intensity, this problem does not fall within classical stochastic control theory outside particular Markovian kernels. We first develop a finite-dimensional Markovianization procedure and algorithm to approximate multivariate Hawkes processes with mixtures of exponential kernels. We prove the convergence of the Markovianized approximation of the Hawkes process, its intensity, and the value of the problem to the original non-Markovian processes and the value of the primal problem. We then formulate continuous-time deterministic policy gradient learning on the Markovianized approximation of the problem, called Hawkes-CT DDPG. We propose a model-free algorithm to solve the non-Markovian Hawkes-driven optimization by observing only the event times of the process, the realization of the solution to the SDE, and a chosen set of decay filters, while the Hawkes kernel coefficients remain unknown. We compare our continuous time reinforcement learning Hawkes-CT DDPG method with discrete time reinforcement learning techniques under three different types of kernels: simple exponential, Erlang, and power-law kernels.