Papers

The objective of this paper is to identify a finite group based on the orbit of a vector under its action. When a finite group acts unitarily on a Hilbert space, the associated Gram matrix lies in the group algebra and satisfies identities that encode the algebraic group structure. We ask about the converse: Given a Gram matrix obtained from the orbit of a vector under the group action, can one recover the underlying abstract group structure? Our main result identifies a short list of conditions on a family of orthogonal matrices that force the family to be the right regular representation of a finite group. Building on this characterization, we develop a staged optimization framework that enforces these conditions progressively. Numerical experiments on the dihedral group $D_4$ and the tetrahedral rotation group $A_4$ recover $D_4$ exactly and all twelve $A_4$ matrices up to modest precision.

In this paper, we investigate the singular Dirichlet problem \begin{equation}\label{abs1} \left\{ \begin{array}{rclr} -\Delta_1u &=&\dfrac{\lambda}{|x|}\hbox{Sgn}\,(u)+\dfrac{f}{u^\gamma}&\quad\mbox{in}\; \Omega,\\[1ex] u &=&0& \quad\mbox{on}\; \partial\Omega, \end{array} \right. \end{equation} where $\Delta_1u=\hbox{div}\left(\frac{Du}{|Du|}\right)$ denotes the $1$-Laplacian operator, the parameters are $\lambda\in {R}$ and $\gamma>0$, and $f$ is a non-negative function belonging to the Lorentz space $L^{N,\infty}(\Omega)$. Our main goal is to establish the existence of non-trivial solutions under the sharp restriction $\lambda<N-1$, regardless of the magnitude of the datum $f$ or the value of the singular exponent $\gamma>0$. We also show that these solutions are globally bounded and that, on the contrary, no solution exists as soon as $\lambda\geq N-1$ and $f$ is positive. These results are achieved by rigorously analyzing the asymptotic behavior, as $p\to 1^+$, of the solutions to the approximating $p$-Laplace problems \begin{equation} \left\{ \begin{array}{rclc} -\Delta_pu&=&\dfrac{\lambda}{|x|^p}|u|^{p-2}u+\dfrac{f}{u^\gamma}&\quad\mbox{ in }\; \Omega,\\[1ex] u&=&0&\quad\mbox{ on }\; \partial\Omega. \end{array} \right. \end{equation} Finally, we provide a family of explicit examples designed to illustrate the sharp optimality of our main assumptions.

We investigate $c$-interior pointwise maximal leakage (PML) as a tool for contraction analyses and disclosure control. Based on the strong adversarial threat models from maximal leakage, $c$-interior PML generalizes local differential privacy (LDP) to data-generating distributions with densities uniformly bounded away from zero by $c>0$. Viewing $c$-interior PML as an algebraic constraint on a kernel yields more flexible (and often tighter) contraction analyses than standard LDP. We provide tight bounds on the Dobrushin coefficient, and bound the contraction coefficient of the Hockeystick-divergence. We further derive strong data processing inequalities on $f$-divergences under $c$-interior PML constraints when the input distributions to the divergence are restricted to be in the $c$-interior. These results extend beyond the regime of pure LDP to cover a larger class of kernels, including, e.g., arbitrary stochastic matrices. We apply the results to minimax theory and provide asymptotically optimal strategies under $c$-interior PML constraints for binary hypothesis testing and mean estimation. The results show that disclosure control with PML allows analysts to reason about systems in a more differentiated manner: For example, it allows us to quantify the privacy leakage of deterministic systems, and can give precise adversarial guarantees with respect to arbitrary distributional assumptions. Interestingly, a recurring theme in the disclosure analyses is that if the privacy problem is relatively regular (if the density bound $c$ is large), private inference can be possible without incurring any additional cost in terms of sample complexity.

Let $H$ be a nontrivial finite-dimensional nonnegatively graded connected Hopf algebra, with $\ell=\max\{d:H_d\ne0\}$, and let $A\ge0$ be a locally finite $H$-module algebra with finite-dimensional split semisimple, $H$-trivial $A_0$. Under the gap $A_d=0$ for $0<d<\ell$, the shifted standard cells $\{q^{-r}Ae_x:0\le r<\ell\}$ form a simple-minded collection. A socle estimate for positive syzygies proves negative-Hom vanishing, while the gap makes the degree-zero endomorphism algebra semisimple. A bounded $t$-structure on the compact derived category with these cells is constructed, with standard cells as its simple heart objects. It is also shown that every compact object has a finite-cell representative and that compact $K_0$ is free on $[Ae_x]$ over $\mathbb{O}_H=K_0(H\mbox{-}\underline{\mathrm{gmod}})$. The same connected-Hopf hypotheses give a Keller–Nicolás weight structure on the large derived category, without a claim of boundedness or preservation of compact objects. Neither a silting generator nor a bounded weight structure on compacts is forced by the gap. These arguments require neither cocommutativity nor finite representation type of $H$. The $p$-DG case, with $\mathrm{deg}(\partial)=2$ and $\ell=2p-2$, is treated as a specialization. Separately, under the stronger gap condition $A_1=\cdots=A_\ell=0$, literal cellularity of every finitely generated graded-projective hopfological module holds. Explicit $p$-DG higher-cycle retracts and algebra-valued traces show why additional grading hypotheses are needed; in particular, dropping the gap can produce nonzero $p$-torsion in the Grothendieck group of compact derived categories.

We report an explicit construction of a 27 x 27 x 27 symmetric tensor with rational entries that has tensor rank 55 over the rational numbers and symmetric tensor rank 56 over the complex numbers, providing a small counterexample to Comon's conjecture over the rational, real, and complex numbers. The construction follows the framework of symmetric adjoins introduced by Shitov. The primary technical contribution of this work is to prove a special case of Conjecture 6 appearing in Shitov's seminal 2018 work.

We study supinf problems for perturbations of a given function of two variables by continuous bounded functions in the underlying spaces and the properties of the associated solution mapping. We provide conditions on the initial data such that most (in the Baire category sense) of the functions from the perturbation space secure existence and uniqueness of the solution of the supinf problem for the corresponding perturbed functions. In this way we obtain generic variational principles for such problems.

We study the Dirichlet problem for an infinite series of $p$-Laplacians, $$-\sum_{p=2}^{\infty}a_{p}\Delta_{p}u=f \ \text{ in } \Omega, \qquad u=g \ \text{ on } \partial\Omega,$$ where $\{a_{p}\}$ is a sequence of nonnegative numbers whose power series has radius of convergence $\sigma\in(0,\infty]$. The operator is formally $-\operatorname{div}( {A}(|\nabla u|)\nabla u)$ with $ {A}$ singular at $|\nabla u|=\sigma$; model cases are the mean curvature operator in Minkowski space and the Born-Infeld operator of nonlinear electrostatics. The radius of convergence forces the gradient constraint $\|\nabla u\|_\infty\le\sigma$, so the natural variational problem is constrained. A unique minimizer of the associated energy always exists (for boundary data compatible with the constraint) and always solves a variational inequality, and we identify the saturation flux $\Lambda:=\sum_{p\ge2}a_p\sigma^{p-1}$ as the quantity governing solvability of the equation: if $\Lambda<\infty$ a weak solution exists only if $|\int_E f|\le\Lambda P(E)$ for every set of finite perimeter $E\Subset\Omega$, so for $f\equiv\lambda$ no solution exists once $\lambda>\Lambda h(\Omega)$, $h(\Omega)$ the Cheeger constant of $\Omega$. The threshold is sharp on balls, where the minimizer has a saturation region $\{|\nabla u|=\sigma\}$ of positive measure. When $\Lambda=\infty$, as for Born-Infeld type operators, no such obstruction is present, and the minimizer solves the equation whenever a Lipschitz bound below $\sigma$ is available; for $f\equiv0$ we obtain such a bound, uniform in the truncations of the series, under a bounded slope condition on the boundary datum. As applications we obtain an isoperimetric bound on the charge densities supported by a nonlinear electrostatics with saturating displacement, and a quantitative convergence rate for the weak field expansion of the Born-Infeld model.

For every fixed $m>1$, we construct nonzero global forward self-similar mild solutions of the three-dimensional spatially inhomogeneous Boltzmann equation for Maxwell molecules. We assume that the angular kernel is bounded and satisfies a weighted integrability condition at the grazing and head-on angles. The initial data are homogeneous under the kinetic scaling, singular at the origin in phase space, and have infinite mass. Their size is small in a weighted norm invariant under this scaling. We first prove a bilinear estimate for the gain operator in this norm, using free-transport coordinates and the orthogonality of the collision displacements. This estimate gives a global solution of the equation without the loss term, which bounds the lower and upper approximations in the Kaniel–Shinbrot iteration. We prove that their difference tendsto zero to arbitrarily high order at the initial time on a fixed annulus in phase space. The argument uses a finite number of iterations of the collision estimate and stops characteristics when they leave a spatial annulus. Self-similarity then gives finite mass and a finite first absolute velocity moment for the difference. Comparing its mass balance with its scaling law shows that the two limits agree. The resulting solution attains its initial trace locally in $L^\infty$ away from the origin in phase space and has infinite mass at every positive time.

Let $F=(f_1(x_1,\ldots,x_k),\ldots, f_m(x_1,\ldots,x_k))$ be a system of nonconstant polynomials of $k$ variables with integer coefficients and let \[ {\gcd}_F(x_1,\ldots,x_k)= \gcd(f_1(x_1,\ldots,x_k),\ldots, f_m(x_1,\ldots,x_k)). \] We obtain an unconditional asymptotic formula for the sum \[ \sum_{1\le x_1,\ldots,x_k\le x} h({\gcd}_F(x_1,\ldots,x_k)), \] where $F$ is a system of $m\ge 2$ polynomials of $k\ge 2$ variables subject to certain general properties, and $h$ is a bounded strongly multiplicative function. In particular, we deduce an asymptotic formula with error term concerning the density of $k$-tuples of positive integers $(x_1,\ldots,x_k)$ such that ${\gcd}_F(x_1,\ldots,x_k)=1$, given by the Ekedahl-Poonen formula.

The precise and efficient evaluation of large-scale lattice sums involving power-law kernels is a fundamental computational problem in the simulation of classical and quantum systems with long-range interactions. While methods for spatially isotropic kernels, some based on Epstein zeta functions, have advanced considerably in recent years, the anisotropic case has lagged behind, despite its broad relevance to both fundamental and effective interactions such as the dipole interaction in magnetic materials. In this work, we solve this issue by defining and analyzing anisotropic Epstein zeta functions for which we derive stably computable representations obtained from wave vector derivatives of lattice sums over isotropic interaction kernels. These functions find direct application in the analytical and numerical study of anisotropically interacting lattice systems. Going further, they provide the correction term in an exact equivalence between discrete lattices and their continuous analogs in a recent generalization of the classical Euler-Maclaurin summation formula to lattices and summands involving power-law kernels. Their connection to high-order derivatives of zeta functions can be used to improve convergence rates of numerical algorithms, for instance in micromagnetics, or to provide rapidly convergent expansions suitable for precomputations of generalized zeta functions. We derive a stably computable representation of anisotropic Epstein zeta functions, including the possibility for analytically removing Rayleigh–Wood singularities, and we develop a numerical algorithm for their stable evaluation for any lattice, power-law decay exponent and anisotropy order. We benchmark the algorithm against closed-form identities, direct summation and multi-precision results, obtaining machine precision across various lattices, power-law exponents, and anisotropy orders.

The IIB system is a system of PDEs for an SU(3)–structure and a positive function on a 6-manifold. Its solutions describe conformally balanced pluriclosed Hemitian metrics on complex 3-folds with holomorphically trivial canonical line bundle. In particular, solutions of the IIB system are steady solitons for the generalized Ricci-flow and Bismut–Hermitian–Einstein metrics. In this paper we study cohomogeneity one solutions of the IIB system. We construct 1-parameter families (up to scaling symmetries) of complete non-compact non-Kähler solutions on the smoothing of the conifold with controlled geometry at infinity. The generic member of the family is asymptotically conical with tangent cone at infinity the Calabi–Yau cone metric on the conifold. As a limit of the family we recover an explicit solutions known in the physics literature as the Chamseddine–Volkov/Maldacena–Nuñez solution, which has an exotic asymptotic geometry. We show that there are no complete cohomogeneity one solutions on crepant resolutions of the conifold (or its quotient), but we also establish the existence of an analogous 1-parameter family of forward complete solutions defined on exterior domains of the conifold with prescribed incomplete behaviour along the interior boundary and similar asymptotic behaviour at infinity. As a byproduct of these existence results, we produce infinitely many complete non-compact non-Kähler Bismut–Hermitian–Einstein metrics in complex dimension 3 with full holonomy of the Bismut connection, in contrast to the holonomy reduction forced upon compact examples. We also find infinitely many such complete non-compact full-holonomy non-Kähler examples (with at least two ends) in complex dimension 2 by revisiting a construction due to Callan–Harvey–Strominger in the physics literature.

We prove that transcendental entire functions do not have bounded-orbit wandering domains. This answers a long-standing open question. In fact, we prove a more general theorem, establishing the absence of certain wandering domains even for locally defined analytic functions. The proof is based on a recent remarkable argument of Ye, who obtained an elementary proof of Sullivan's classical theorem that rational maps do not have wandering domains.

The axisymmetric profile \(\sin\psi(\rho)=c_0\rho\log(\rho/\rho_B)\) introduced by Naito, Okuda, and Ouyang is an exact solution of the unconstrained Helfrich shape equation for every \(\rho>0\), but its mean curvature diverges logarithmically at the dimple center. A covariant local analysis shows that in Cartesian graph coordinates the immersion is nondegenerate and belongs to \(W^{2,p}\) for every finite \(p\); it is \(C^{1,\alpha}\) for every \(\alpha<1\), but neither \(C^{1,1}\) nor \(C^2\). The conformal factor has a finite nonzero limit, so the point has branch multiplicity one and is not a branch point in the sense used in weak-immersion compactness theory. The Helfrich energy is finite, whereas the complete Euler–Lagrange operator carries the distributional residue \(\E_{\CH}=4\pi c_0\delta_p\) under the convention specified below. Hence the profile is a classical solution on the punctured surface and a finite-energy weak immersion, but it is not a free weak critical point of the unforced Canham–Helfrich functional. Stationarity is restored after adding the corresponding point-force potential or under a pinned-point constraint. Area and volume multipliers cannot cancel the Dirac residue. The apparent sign discrepancy between the shape equation used in the original papers and the modern convention is explained by the opposite definition of mean curvature. These results delimit precisely what can and cannot be inferred from modern existence and regularity theorems for Canham–Helfrich minimizers.

We study a class of Dirichlet problems for coupled quasilinear elliptic systems driven by the $p$–Laplacian in a bounded domain, where the nonlinear terms split into a critical homogeneous part and a convex subcritical perturbation. Using variational methods, a nonlinear eigenvalue theory based on the Fadell–Rabinowitz $\mathbb Z_2$–cohomological index, and a refined linking construction, we prove the existence of a nontrivial solution for every parameter value under suitable dimensional assumptions. We also treat a nonhomogeneous version with small forcing terms by applying a recent abstract perturbation theorem, yielding the existence of two distinct nontrivial solutions.

We realize the $\mathrm{SU}(3)$ Verlinde sums $V_n(m)$, up to an explicit factor, as the values at $n$ of the spectral zeta function of a higher-order Laplace operator on the triangular discrete torus on $m^2$ vertices. For fixed $m$, we express their generating function in terms of the logarithmic derivative of an associated even spectral polynomial and express this polynomial as an explicit iterated resultant. Exploiting permutation symmetry, we prove that this polynomial is a cube over $\mathbb{Q}$, apart from an explicit quadratic factor when $3\mid m$. This factorization yields shorter linear recurrences satisfied by $V_n(m)$ with constant coefficients. We also derive Binet-type formulas expressing $V_n(m)$ as finite linear combinations of powers of rescaled inverse squares of the roots of the spectral polynomial. These results provide an efficient algorithm for computing these Verlinde sums. Several fully developed examples demonstrating computational efficiency of the method are given, including expressions in terms of Fibonacci and Lucas numbers.

In [Be17], we introduced the notion of emergence to quantify the statistical complexity of a dynamical system. Roughly speaking, emergence measures the number of probability measures required to describe, up to a given precision $\epsilon$, the statistical behavior of most orbits. A system is said to have high emergence when this number grows super-polynomially as $\epsilon\to0$. We survey recent developments in a program aimed at understanding the prevalence of high emergence in differentiable dynamics. We review several notions of emergence and their connections with quantization, ergodic decompositions, and entropy, and discuss examples exhibiting high or maximal emergence in conservative, symplectic, analytic, and dissipative dynamics, as well as in constrained families such as unimodal, Hénon, and rational maps. We also present new results concerning variants of metric and topological emergence, convexity properties, local emergence, and variational principles, together with a collection of open problems on the typicality of high emergence.

The growth of large language model (LLM) inference and search services increases the scale of online linear programming problems, motivating computationally efficient algorithms. We develop resource-adaptive stochastic gradient descent (RASGD) for stochastic online linear programming. The algorithm uses one request and current inventory to update resource prices, requiring O(m) operations for m resources and memory per arrival and no LP or sample-average optimization. The central idea is to express the current-resource pricing logic of re-solving through a first-order SGD update: each arrival refreshes the remaining-inventory allowance in the dual objective, while the stepsize decreases for early learning and increases later to match the speed of inventory adjustment. Under standard non-degeneracy conditions, our algorithm is feasible on every sample path and achieves O(\log T) expected regret against the realized fractional hindsight optimum, which matches the lower bound, even for policies that know the distribution and have unrestricted computation. The analysis converts curvature around the fixed reference price into inventory stability without tracking optimal prices at changing resource levels. Numerical experiments show that RASGD achieves regret competitive with per-arrival LP re-solving and improves upon the tested first-order baselines, while retaining the computational efficiency of first-order methods. These results establish RASGD as a computationally efficient approach to achieving high allocation quality in large-scale OLP.

We investigate a class of mixed-order quasilinear elliptic equations in \(\mathbb R^N\) driven by the \(q\)-biharmonic and \(p\)-Laplacian operators, \[ \Delta_q^2u-\Delta_pu = \lambda\frac{|u|^{r_\sigma-2}u}{|x|^\sigma} +|u|^{p^*-2}u +|u|^{q^{**}-2}u, \] where \(1<q<N/2\), \(1<p<q^*\), \(p\ne q\), \(q\ne p^*\), and the weighted exponent is determined by the natural scaling of the equation. We establish compact weighted embeddings compatible with this mixed scaling and develop the corresponding variational framework. As an application, we prove existence and multiplicity of nontrivial solutions under suitable assumptions on the parameters. We also establish a local regularity result for the more general equation $\Delta_q^2u-\Delta_pu=f(x,u),$ where \(f\) is a Carathéodory function with local critical growth. By combining nonlinear potential estimates, a regularity-lifting argument, and a finite bootstrap for an associated second-order system, we obtain higher local regularity of both \(u\) and the nonlinear flux \(|\Delta u|^{q-2}\Delta u\). This regularity result, which is also of independent interest, enables us to derive a Pohozaev-type identity for the equations under consideration.

For finite groups $G$, we discuss the enrichment of equivariant $KK$- and $E$-theory in genuine $G$-spectra.

We give an elementary exposition of the tilted sieve introduced by GPT-5.6~Sol \cite{GPT2026}, showing that one residue class modulo each prime $p \le x$ can cover an interval of length \begin{equation*} \gg \frac{x\log x}{(\log_{2} x)\log_{3} x}. \end{equation*} This improves the classical Erdős–Rankin bound by a factor of $(\log_{2} x)/(\log_{3} x)^{2}$. Although weaker than the strongest known bounds, it shows what the tilt and its associated covering of composite survivors achieve without Maynard sieve weights or a hypergraph covering theorem. The proof uses the prime number theorem, Mertens' reciprocal-prime formula, and elementary probability. The appendices provide a historical survey and self-contained proofs of the classical Erdős–Rankin bound.

We prove nonlinear Landau damping for the two-dimensional Vlasov–Poisson system in the unconfined Euclidean setting near the homogeneous Poisson equilibrium \[ \mu(v)=\frac{1}{2\pi}(1+|v|^2)^{-3/2}. \] For small, smooth, localized, and neutral perturbations we establish global existence, quantitative pointwise decay estimates for the electric field, and scattering to free transport. The result we prove here appears to be the first nonlinear Landau damping result for solution of the unconfined 2D Vlasov–Poisson system. The uniform Penrose stability bound degenerates at low spatial frequencies, giving rise to weakly damped plasma oscillations. We separate a faster-decaying regular field from oscillatory components with temporal factors $\cos(t)$ and $\sin(t)$. The proof combines a Volterra representation of the density, weighted estimates for the characteristic flow and nonlinear moments, and integration by parts in time. Neutrality supplies the spatial cancellation needed in the critical two-dimensional estimates.

For smooth convex–concave minimax optimization, the higher-order lower bound of Chen et al. (2026) applies to a restricted tensor-algorithm class with prescribed regularized Taylor-model updates. We establish the same bound for arbitrary adaptive deterministic and randomized algorithms, matching, up to logarithmic factors, the upper bound of Zhang et al. (2026). Fix an integer $p\ge 2$ and let $L_p>0$ bound the Lipschitz constant of the objective's $p$-th derivative on a compact convex product domain of diameter at most $D_Z>0$. Each feasible query returns the objective value and all derivatives through order $p$. For accuracy $\epsilon>0$, set $Q_{\mathrm{tan}}=L_pD_Z^p/\epsilon$ for tangent residual and $Q_{\mathrm{gap}}=L_pD_Z^{p+1}/\epsilon$ for saddle gap. Let $T_E^{\mathrm{det}}(\epsilon)$ and $T_E^{\mathrm{rand}}(\epsilon)$ denote the high-dimensional minimax query complexities for criterion $E\in\{\mathrm{tan},\mathrm{gap}\}$, with randomized success probability at least $2/3$ on every instance. Our lower bounds and the existing upper bound give $c_pQ_E^{2/(3p-1)}\le T_E^{\mathrm{rand}}(\epsilon)\le T_E^{\mathrm{det}}(\epsilon)\le C_pQ_E^{2/(3p-1)}[1+\log(3+Q_E)]^{6(p-1)}$ for sufficiently large $Q_E$, where $c_p,C_p>0$ depend only on $p$. Thus the same accuracy exponent holds beyond tensor update rules, even for randomized queries and arbitrary feasible outputs. The proof constructs a scalar convex–concave chain with exactly flat gates that hide complete derivative information. Direct product-domain error witnesses and adaptive transcript arguments establish the lower bounds for both criteria.

We study the nonparametric statistical inverse problem of recovering the space-dependent permittivity in a coupled Fokker-Planck-Darcy system from discrete, noisy observations of the Fokker-Planck solution. We consider the coupled parabolic-elliptic system on a bounded domain with the physically natural no-flux boundary condition for the Fokker-Planck equation and homogeneous Dirichlet boundary conditions for the Darcy equation. The inverse problem is indirect, since the unknown coefficient enters only through the elliptic equation and is observed only through its effect on the density in the parabolic equation. We first develop the analytical theory of the forward problem required for the statistical analysis, establishing well-posedness and a priori bounds uniform over the admissible set of permittivities. We then derive two stability estimates for the inverse problem, namely a Lipschitz-type forward estimate and a generalized backward estimate. Placing a rescaled Gaussian process prior on the log-shifted permittivity, we show that the posterior contracts around the truth at an explicit polynomial rate in the number of observations, with the posterior mean converging at the same rate. Numerical experiments in one and two dimensions, using preconditioned Crank-Nicolson and ensemble Kalman filter algorithms, complement our theoretical results.

We establish tight polynomial tolerance bounds for stochastic second-order stationarity when each fresh oracle response is a derivative of one common random scalar loss. For a population objective $F$ with Lipschitz gradient and Hessian, the target is $\|\nabla F(x)\|\le \epsilon$ and $\lambda_{\min}(\nabla^2 F(x))\ge -\gamma$, with independent tolerances $\epsilon,\gamma>0$. Under bounded gradient variance and almost-surely bounded Hessian error, the minimax number of fresh gradient or Hessian-vector-product calls is $\widetilde{\Theta}\!\left(\epsilon^{-3}+\gamma^{-5}\right)$. The characterization fixes positive gap, smoothness, and noise parameters, suppresses logarithmic factors, and allows dimension to grow within an explicit polynomial envelope. The upper bound removes the mixed term $\epsilon^{-2}\gamma^{-2}$ from the earlier fresh-HVP guarantee. Direct random-line Hessian estimates and a dyadic gradient tracker separate gradient drift from randomly signed curvature motion. The lower bound realizes the endpoint costs through globally defined smooth random losses: a smooth partition localizes scalar noise without a chain-length penalty, while exact cancellation limits the information in the entire response. Consequently, the same tolerance exponents hold even for joint value, gradient, and full-Hessian responses with bounded value variance. For a population Hessian with Hölder exponent $\nu\in(0,1]$, fresh gradient/HVP complexity becomes $\widetilde{\Theta}\!\left(\epsilon^{-3}+\gamma^{-(3+2/\nu)}\right)$ under the corresponding dimension envelope.

We prove results about stability under mixtures for transport–entropy inequalities and restricted functional inequalities of Poincaré and log-Sobolev type. We treat Talagrand inequalities of type $\mathbf{T}_2$ and $\mathbf{T}_1$ together with their generalizations, as well as weak transportation inequalities introduced by Gozlan et al. Our main results provide mild sufficient conditions under which a given inequality may be transferred from mixture components to the mixture itself.

In this paper, we verify a bifurcation phenomenon regarding the multiplicity of weak solutions, subject to the Dirichlet boundary condition, of a regularized two-phase free boundary problem associated with the $p$-Laplacian. In fact, we prove the existence of a mountain pass solution when the boundary data is small. In this case, three weak solutions of the regularized problem exist: the $p$-harmonic function, a minimizer of the corresponding functional, and the mountain pass solution. Finally, we consider the special case of radially symmetric solutions. By solving the associated nonlinear ordinary differential equation in the unregularized case, we show explicitly how the Bernoulli condition at the free boundary gives rise to the bifurcation of solutions.

We study homotopy theories of higher groupoids in specified geometries, distinguishing geometric representability from realization and descent. Ordinary smooth and Banach-open sheaves admit full Brown categories of fibrant objects. For ordinary geometric groupoids, a crossing-axes obstruction motivates an empty-compatible incomplete Brown structure. Its split-Banach version requires a fixed chart-compatible plot site with local kernel-product closure. These results include all finite unique-horn bounds and their union. For represented derived models, finite matching and an outer-prism filtration give the structural Brown calculus. Nuiten's finite-geometric realization theorem yields the finite derived-smooth structure and its localization. Enriched realization base change on the original small Kan-enriched site gives the outer-unbounded hypercomplete structure, with positive-degree geometric horn conditions and Reedy fibrancy in every degree. For Banach domains, structured spectra, strict-open gluing and split-open hyperdescent establish geometric finite limits and hypersheaf representability. They give the represented Brown structure, using the same hypercompleted weak-equivalence class at every outer bound. A square-zero test detects nonordinary self-intersections of the chosen Banach charts, including infinite-dimensional charts. For connective associative affines, homotopy split horns, Pridham's finite effectivity theorem and a pointwise diagonal-fibration argument give categories of fibrant objects for his Artin and Deligne–Mumford classes. The associative theory uses quasi-isomorphisms of algebras and realization in presheaves.

We prove a sharp quantitative stability result for the affine \(L^p\)-Sobolev inequality, for \(1<p<2\), introduced by Lutwak–Yang–Zhang (J. Differential Geom., 62 (2002),17–38). Moreover, the stability exponent is shown to be optimal, and equal to \(2\).

Let $\F_q$ be the field of order $q = p^f$ for prime $p$. If $G(\chi)$ is the Gauss sum attached to a multiplicative character $\chi$ on $\F_q^\times$, then $G(\chi) = G(\chi^p)$. We investigate the converse question: when does the equality of Gauss sums $G(\chi_2) = G(\chi_1)$ imply that $\chi_2 = \chi_1^{p^j}$ for some integer $j$. If $\chi_2 = \chi_1^{p^j}$, we say that $\chi_1$ and $\chi_2$ are Frobenius-conjugate. We use the Stickelberger factorization of ideals in cyclotomic fields to give an easily testable criterion for equality of Gauss sums, based on $p$-adic digit expansion. As an application, we develop several conditions under which Gauss sum equalities between characters on $\F_q^\times$ are explained by Frobenius-conjugacy. For example, if $\chi_1$ has order $q-1$ or $\frac{1}{2}(q-1)$ and $G(\chi_2) = G(\chi_1)$, then $\chi_1$ and $\chi_2$ are Frobenius-conjugate. We also include several examples, based on the explicit evaluation of certain pure Gauss sums by R.J. Evans.

This is an English translation of a book originally published in Japanese by Saiensu-sha in 2015. The book is an introduction to the 3d-3d correspondence, the relation between 3d $\mathcal{N}=2$ supersymmetric gauge theories and the geometry of 3-manifolds, which arises from the compactification of the 6d $\mathcal{N}=(2,0)$ theory. The presentation is bottom-up rather than top-down: instead of starting from the 6d theory, we begin by asking what a quantum field theory is, and let the geometry emerge on its own. Along the way we discuss renormalization and low-energy effective theories, gauging as an operation which glues field theories together, the resulting $Sp(2n, \mathbb{Z})$-action on 3d theories, 3d $\mathcal{N}=2$ supersymmetric theories and their dualities, the squashed three-sphere partition function and supersymmetric localization, duality domain walls of 4d $\mathcal{N}=2$ theories, quantum Teichmuller theory, complex Chern-Simons theory and the quantization of the moduli space of flat $SL(2, \mathbb{C})$-connections on 3-manifolds, with quantum dilogarithm functions as a recurring thread. The emphasis throughout is on geometric and algebraic structures which are invisible in a single field theory, and appear only in the theory space of quantum field theories. Appendices summarize supersymmetry in various dimensions, classical and quantum dilogarithm functions, and cluster algebras. Exercises with difficulty ratings are included in each chapter.