We introduce the singular cohomology ring of a matroid which extends the Chow ring of a matroid. This is defined as the singular cohomology ring of a certain quasi-projective toric variety associated to the matroid. Using the matroidal flips of Adiprasito, Huh, and Katz, we prove sharp vanishing results for the cohomology ring and compute the dimension of the top-weight cohomology in terms of the Möbius invariant of the matroid. In the case of uniform matroids, these techniques give a recursive formula for the Hodge numbers. Finally, we generalize the singular cohomology ring to arbitrary building sets on the lattice of flats, and we show how the cohomology depends on the building set.

We present a general proof that non-Hermitian dynamics and Lindblad dynamics with only decay terms are equivalent in the highest particle subspace. We then propose an unbiased method to determine if a system's dynamics in the highest-particle subspace is non-Hermitian. We exemplify this for a simple two-site decay system connected to two baths, and find that the exact solution is well approximated by non-Hermitian dynamics only in the weak-coupling and in the singular-coupling limits, where a Lindbladian description was already known to be accurate. The fact that an accurate non-Hermitian description is so limited, even for such a simple system, raises doubts about how valid such descriptions are for more complicated systems away from these asymptotic limits. Finally, we prove that for models with a nondegenerate system Hamiltonian, exceptional points cannot occur in the weak-coupling limit. This result is relevant for the design of experiments that aim to identify such exceptional points.

The accurate computation of low-energy spectra of strongly correlated quantum many-body systems, typically accessed via Green's-functions, is a long-standing problem posing enormous challenges to numerical methods. When the spectral decomposition is obtained from Fourier transforming a time series, the Nyquist-Shannon theorem limits the frequency resolution $Δω$ according to the numerically accessible time domain size $T$ via $Δω= 2π/T$. In tensor network methods, increasing the domain size is exponentially hard due to the ubiquitous spread of correlations, limiting the frequency resolution and thereby restricting this ansatz class mostly to one-dimensional systems with small quasi\hyp particle velocities. Here, we show how this limitation can be overcome by augmenting the time series with complex-time Krylov states. At the example of the critical $S-1/2$ Heisenberg model and light bipolarons in the two-dimensional Su-Schrieffer-Heeger model, we demonstrate the enormous improvements in accuracy, which can be achieved using this method.

This work explores the implications of the exclusivity principle (EP) in the context of quantum and postquantum correlations. We first establish a key technical result demonstrating that given the set of correlations for a complementary experiment, the EP restricts the maximum set of correlations for the original experiment to the antiblocking set. Based on it, we can prove our central result: if all quantum behaviors are accessible in Nature, the EP guarantees that no postquantum behaviors can be realized. This can be seen as a generalization of the result of B. Amaral et al. [Phys. Rev. A 89, 030101(R) (2014)], to a wider range of scenarios. It also provides novel insights into the structure of quantum correlations and their limitations.

Recent literature proposes combining short-term experimental and long-term observational data to provide alternatives to conventional observational studies for the identification of long-term average treatment effects (LTEs). This paper re-examines the identification problem and uncovers that assumptions restricting temporal link functions -- relationships between short-term and mean long-term potential outcomes -- are central in this context. The experimental data serve to amplify the identifying power of such assumptions; absent them, the combined data are no more informative than the observational data alone. Plausible inference thus hinges on justifiable restrictions in this class. Motivated by this, I introduce two treatment response assumptions that may be defensible based on economic theory or intuition. To utilize them and facilitate future developments, I develop a novel unifying identification framework that computationally produces sharp bounds on the LTE for a general class of temporal link function restrictions and accommodates imperfect experimental compliance -- thereby also extending existing approaches. I illustrate the method by estimating the long-term effects of Head Start participation. The findings indicate that the effects on educational attainment, employment, and criminal involvement are lasting but smaller in magnitude than those established by sibling comparisons.

In this paper, we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to exhibit a presentation for each of them.

In this paper, we establish a conformal scattering theory for defocusing semilinear wave equations on Schwarzschild spacetime. We combine the energy and pointwise decay results for solutions obtained in \cite{Yang} with a Sobolev embedding on spacelike hypersurfaces to derive two-sided energy estimates between the energy flux of solutions through the Cauchy initial hypersurface $Σ_0 = \{ t = 0 \}$ and that through the null conformal boundaries $\mathfrak{H}^+ \cup \scri^+$ (respectively, $\mathfrak{H}^- \cup \scri^-$). By combining these estimates with the well-posedness of the Cauchy and Goursat problems for nonlinear wave equations, we construct a bounded linear and locally Lipschitz scattering operator that maps past scattering data to future scattering data.

We investigate the Hurwitz action of the $m$-braid group on the $m$-fold Cartesian product of the universal dihedral quandle. We introduce three computable invariants and prove that they give a complete classification of the orbits under this action. As a consequence, we describe an explicit complete system of orbit representatives. We further obtain analogous classifications for the corresponding Hurwitz actions of the pure $m$-braid group, the virtual $m$-braid group, and the virtual pure $m$-braid group.

Let G be a locally compact Hausdorff group in which every element is of finite order, and let P(G) denote the class of all regular probability measures on G. In this note, it is observed that a characterization of algebraically regular elements in certain subsemigroups of P(G) (Theorem 4.1 [11]) for compact G remains true for locally compact G. In addition, a complete description of algebraically regular elements in P(G) has been established when G is countable or uncountable where every proper subgroup is countable. In this case the standing assumption that every element is of finite order is not required. For compact Lie groups, Fourier transform techniques are also used to get more information on P(G). Several concrete examples are provided to illustrate the observations.

We construct bypass attachments in higher dimensional contact manifolds that, when attached to a neighborhood of a Weinstein hypersurface, yield a neighborhood of a new Weinstein hypersurface, obtained via local modifications to the Weinstein handle decomposition of the first. For context, we give $3$-dimensional analogues of these bypass attachments and discuss their appearance in nature. We then show that our bypass attachments give a necessary and sufficient set of moves relating any two Weinstein domains which become almost symplectomorphic after one stabilization. Finally, we use our construction to produce several examples of interesting convex hypersurfaces and recover an existence $h$-principle for Weinstein hypersurfaces.

This is an expanded version of the notes by the second author of the lectures on Hitchin systems and their quantization given by the first author at the Beijing Summer Workshop in Mathematics and Mathematical Physics ``Integrable Systems and Algebraic Geometry" (BIMSA-2024).

In this paper we investigate a construction of scattering for wave-type equations with singular potentials on the whole space $\mathbb{R}^n$ in a framework of weak-$L^p$ spaces. First, we use an Yamazaki-type estimate for wave groups on Lorentz spaces and fixed point arguments to prove the global well-posedness for wave-type equations on weak-$L^p$ spaces. Then, we provide a corresponding scattering results in such singular framework. Finally, we use also the dispersive estimates to establish the polynomial stability and improve the decay of scattering in weak-$L^p$ spaces.

Two central problems in extremal combinatorics are concerned with estimating the number $ex(n,H)$, the size of the largest $H$-free hypergraph on $n$ vertices, and the number $forb(n,H)$ of $H$-free hypergraph on $n$ vertices. While it is known that $forb(n,H)=2^{(1+o(1))ex(n,H)}$ for $k$-uniform hypergraphs that are not $k$-partite, estimates for hypergraphs that are $k$-partite (or degenerate) are not nearly as tight. In a recent breakthrough, Ferber, McKinley, and Samotij proved that for many degenerate hypergraphs $H$, $forb(n, H) = 2^{O(ex(n,H))}$. However, there are few known instances of degenerate hypergraphs $H$ for which $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds. In this paper, we show that $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds for a wide class of degenerate hypergraphs known as $2$-contractible hypertrees. This is the first known infinite family of degenerate hypergraphs $H$ for which $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds. As a corollary of our main results, we obtain a surprisingly sharp estimate of $forb(n,C^{(k)}_\ell)=2^{(\lfloor\frac{\ell-1}{2}\rfloor+o(1))\binom{n}{k-1}}$ for the $k$-uniform linear $\ell$-cycle, for all pairs $k\geq 5, \ell\geq 3$, thus settling a question of Balogh, Narayanan, and Skokan affirmatively for all $k\geq 5, \ell\geq 3$. Our methods also lead to some related sharp results on the corresponding random Turan problem. As a key ingredient of our proofs, we develop a novel supersaturation variant of the delta systems method for set systems, which may be of independent interest.

Superconductivity emerging near charge-density-wave (CDW) quantum critical points often defies a conventional BCS description, particularly in multi-orbital systems with small and orbitally distinct Fermi surfaces. In TiSe$_2$, superconductivity appears under pressure near the suppression of a chiral CDW, yet its microscopic origin has remained unresolved. Here we show that the chiral CDW quantum criticality in TiSe$_2$ originates from a fluctuation-induced intertwining of charge-order and phonon modes that are symmetry incompatible at the Brillouin-zone center but become mixable at the CDW ordering wave vector. This resolution of symmetry frustration enables a single continuous chiral CDW transition and strongly enhances collective fluctuations near criticality. We demonstrate that these critical chiral CDW fluctuations drive a finite-center-of-mass-momentum inter-orbital pairing instability fundamentally different from BCS superconductivity. Because electrons near the $Γ$ and $L$ points occupy small $p$- and $d$-orbital Fermi pockets connected only by the CDW ordering vector, the inter-orbital pair susceptibility does not develop a Cooper logarithm. As a result, superconductivity is governed by an interaction-driven pairing mechanism rather than by the density of states. Using a symmetry-constrained low-energy theory and a random-phase-approximation analysis, we show that the fluctuation-enhanced pairing interaction is maximized near the chiral CDW quantum critical point, giving rise to a dome-shaped superconducting phase. A group-theoretical analysis further identifies an orbital-selective $s$-wave pairing symmetry as the most likely superconducting state.

The aim of this paper is to provide a construction of stationary discrete solitons in an extended one-dimensional Discrete NLS model with non-nearest neighbour interactions. These models, models of the type with long-range interactions were studied in various other contexts. In particular, it was shown that, if the interaction strength decays sufficiently slowly as a function of distance, it gives rise to bistability of solitons, which may find applications in their controllable switching. Dynamical lattices with long-range interactions also serve as models for energy and charge transport in biological molecules. Using a dynamical systems method we are able to construct, with great accuracy, stationary discrete solitons for our model, for a large region of the parameter space.

We consider the initial value problem (IVP) for the 2D generalized Zakharov-Kuznetsov (ZK) equation \begin{equation} \begin{cases} \partial_{t}u+\partial_{x}Δu+μ\partial_{x}u^{k+1}=0, \,\;\; (x, y) \in \mathbb{R}^2, \, t \in \mathbb{R},\\ u(x,y,0)=u_0(x,y), \end{cases} \end{equation} where $Δ=\partial_x^2+\partial_y^2$, $μ=\pm 1$, $k=1,2$ and the initial data $u_0$ is real analytic in a strip around the $x$-axis of the complex plane and have radius of spatial analyticity $σ_0$. For both $k=1$ and $k=2$ we prove that there exists $T_0>0$ such that the radius of spatial analyticity of the solution remains the same in the time interval $[-T_0, T_0]$. We also consider the evolution of the radius of spatial analyticity when the local solution extends globally in time. For the Zakharov-Kuznetsov equation ($k=1$), we prove that, in both focusing ($μ=1$) and defocusing ($μ=-1$) cases, and for any $T> T_0$, the radius of analyticity cannot decay faster than $cT^{-4+ε}$, $ε>0$, $c>0$. For the modified Zakharov-Kuznetsov equation ($k=2)$ in the defocusing case ($μ=-1$), we prove that the radius of spatial analyticity cannot decay faster than $cT^{-\frac{4}{3}}$, $c>0$, for any $T>T_0$. These results on the algebraic lower bounds for the evolution of the radius of analyticity improve the ones obtained by Shan and Zhang in [J. Math. Anal. Appl., 501 (2021) 125218] and by Quian and Shan in [Nonlinear Analysis, 235 (2023) 113344] where the authors have obtained lower bounds involving exponential decay.

The Bernstein degree ($\operatorname{Deg}$) is a fundamental invariant of admissible representations of a real reductive Lie group $G_{\mathbb{R}}$. Our main result concerns the classical dual pairs $(G_{\mathbb{R}}, H_{\mathbb{R}}(k))$, namely $(\operatorname{U}(p,q), \: \operatorname{U}(k))$, $(\operatorname{Mp}(2n, \mathbb{R}), \: \operatorname{O}(k))$, and $(\operatorname{O}^*(2n), \: \operatorname{Sp}(k))$, where $k$ is any positive integer. In this setting, via Howe duality, each irreducible representation $σ$ of $H_{\mathbb{R}}(k)$ corresponds to a unitary highest weight module $L_{λ(σ)}$ for $G_{\mathbb{R}}$. A landmark result of Nishiyama-Ochiai-Taniguchi (2001) expressed $\operatorname{Deg} L_{λ(σ)}$ as a product of two quantities: the dimension of $σ$ and the degree of the associated variety. However, this result was limited to a specific range of the parameter $k$ (namely $k \leq r$, the real rank of $G_{\mathbb{R}}$). The present paper resolves this limitation by introducing, for all $k$, the combinatorial interpretation $\operatorname{Deg} L_{λ(σ)} = \#( \mathcal{Q}_k(σ) \times \mathcal{P}_k)$, where $\mathcal{Q}_k(σ)$ is a certain set of semistandard tableaux and $\mathcal{P}_k$ is a set of plane partitions. (The result remains partly conjectural in the $\operatorname{Mp}(2n, \mathbb{R})$ case.) Beyond the dual pair setting, we generalize the set $\mathcal{P}_k$ to all groups $G_{\mathbb{R}}$ of Hermitian type, and we exhibit analogues of the Nishiyama-Ochiai-Taniguchi result for certain families of unitary highest weight modules of $\operatorname{E}_6$ and $\operatorname{E}_7$.

In federated learning (FL), although the original intention of available but not visible data is to allay data privacy concerns, it potentially brings new security threats, particularly poisoning attacks that target such not visible local data. Intuitively, such data poisoning attacks have great potential in stealthily degrading global FL outcomes, and are expected to be even stealthier if being enhanced by generative models like generative adversarial networks (GANs). However, existing defense methods have not been thoroughly challenged in this regard and generally fail to be aware of a local generation of seemingly legitimate poisoned data. With a growing concern on potentially stealthier attacks, in this paper, a cost-effective defense mechanism named Model Consistency-Based Defense (MCD) is proposed, which offers a comprehensive examination of available local models across multiple feature dimensions, providing an indirect yet effective means of identifying hidden data poisoning attackers. To push the limit of MCD against stealthier attacks, we propose a new GAN-based data poisoning attack model named VagueGAN and an unsupervised variant of it, which can be flexibly deployed to generate seemingly legitimate but noisy poisoned data. The consistency of GAN outputs revealed by VagueGAN helps strengthen MCD to work against stealthier GAN-based attacks as well as other mainstream ones. Extensive experiments on multiple open datasets (MNIST, Fashion-MNIST, CIFAR-10, CIFAR-100, and Mini-Imagenet) indicate that our attack method better balances the trade-off between attack effectiveness and stealthiness with low complexity. More importantly, our defense mechanism is shown to be more competent in identifying a variety of poisoned data, particularly stealthier GAN-poisoned ones.

We propose that generalized symmetries in some string-constructed QFTs are given by K-theory. We thus have \textit{even-form} and \textit{odd-form} symmetries determined by $K_N(\partial X)$, the twisted K-theory as D-brane charges on the asymptotic boundary $\partial X$ of internal geometry $X$ with twist class $N$. For these QFTs, ``\textit{$p$-form symmetries}" are no longer separately well-defined for individual $p$, but are instead mixed together. We discuss 6D ADE-type (2,0) SCFTs and some 6d (1,0) LSTs as examples and demonstrate their twisted K-theoretic symmetries, and we checked them to be compatible with T-duality. We further point out, through explicit examples, that K-theory leads to symmetry extensions that cannot be detected by cohomology for Type II string theory on certain orbifolds of $\mathbb{C}^3$ and $\mathbb{C}^4$. We also discuss the implications of these results in the dual brane descriptions.

The simulation hypothesis has recently excited renewed interest in the physics and philosophy communities. However, the hypothesis specifically concerns {\textit{computers}} that simulate physical universes. So to formally investigate the hypothesis, we need to understand it in terms of computer science (CS) theory. In addition we need a formal way to couple CS theory with physics. Here I couple those fields by using the physical Church-Turing thesis. This allow me to exploit Kleene's second recursion, to prove that not only is it possible for {us} to be a simulation being run on a computer, but that we might be in a simulation being run a computer \emph{by us}. In such a ``self-simulation'', there would be two identical instances of us, both equally ``real''. I then use Rice's theorem to derive impossibility results concerning simulation and self-simulation; derive implications for (self-)simulation if we are being simulated in a program using fully homomorphic encryption; and briefly investigate the graphical structure of universes simulating other universes which contain computers running their own simulations. I end by describing some of the possible avenues for future research. While motivated in terms of the simulation hypothesis, the results in this paper are direct consequences of the Church-Turing thesis. So they apply far more broadly than the simulation hypothesis.

On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. Moreover, this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary. The non-generic case (also called non-generic domains) corresponds to static manifolds with boundary. We discuss their geometric properties, which also work as the necessary conditions of non-generic metrics. In space forms and the Schwarzschild manifold, we classify simple non-generic domains (with only one boundary component) and show their connection with rigidity theorems and the Schwarzschild photon sphere.

The Arellano-Bond estimator is a fundamental method for dynamic panel data models, widely used in practice. It can be severely biased when the time series dimension of the data, $T$, is long. The source of the bias is the large degree of overidentification. We propose a simple two-step approach to deal with this problem. The first step applies LASSO to the cross-section data at each time period to select the most informative moment conditions, exploiting the approximately sparse structure of these conditions. The second step applies a linear instrumental variable estimator using the instruments constructed from the moment conditions selected in the first step. Using asymptotic sequences where the two dimensions of the panel grow with the sample size, we show that the new estimator is consistent and asymptotically normal under much weaker conditions on $T$ than the Arellano-Bond estimator. Our theory covers models with high-dimensional covariates including multiple lags of the dependent variable and strictly exogenous covariates, which are becoming common in modern applications. We illustrate our approach by applying it to weekly county-level panel data from the United States to study opening K-12 schools and other mitigation policies' short and long-term effects on COVID-19's spread.

It is well known that an element of the algebra of noncommutative *-polynomials is positive in all *-representations if and only if it is a sum of squares. This provides an effective way to determine if a given *-polynomial is positive, by searching through sums of squares decompositions. We show that no such procedure exists for the tensor product of two noncommutative *-polynomial algebras: determining whether a *-polynomial of such an algebra is positive is coRE-hard. We also show that it is coRE-hard to determine whether a noncommutative *-polynomial is trace-positive. Our results hold if noncommutative *-polynomial algebras are replaced by other sufficiently free algebras such as group algebras of free groups or free products of cyclic groups.

We say that a convergence law holds for a sequence of random combinatorial objects if, for any first-order sentence $φ$, the proportion of objects satisfying $φ$ converges to a limiting value as the size of the objects tends to infinity. In this paper, we show that the convergence law holds for random $321$-avoiding permutations, settling an open problem posed in Albert, Bouvel, Féray, and Noy (2024). Our proof relies on an infinite-dimensional version of the Perron-Frobenius theorem.

Monotone polytopes, also known as smooth reflexive polytopes, are the polytopes associated to monotone symplectic toric manifolds and Gorenstein Fano toric varieties. We first show that the only monotone polytopes admitting blow-ups at vertices are the simplex and the result of a codimension-two blow-up in it (this is the polyhedral version of a result of Bonavero from 2002). Then we show that the $n$-simplex admits disjoint blow-ups at faces if and only if the faces are disjoint and have dimensions adding up to $n-1$ or $n-2$. These results answer a question posed by Dusa McDuff in 2011.

Debiased machine learning estimators for smooth functionals in nonparametric models can exhibit substantial variability and instability, often leading practitioners to instead rely on parametric or semiparametric working models. Such models, however, may be misspecified and can therefore introduce bias. We study how data-driven model selection can be combined with debiased machine learning to construct estimators that adapt to structure in the data-generating distribution. To this end, we propose Adaptive Debiased Machine Learning (ADML), a nonparametric framework for constructing superefficient estimators of pathwise differentiable parameters. The framework unifies a broad class of previously proposed adaptive estimators, including methods based on variable selection, learned feature representations, and collaborative targeted learning. It requires only high-level conditions and approximate validity of the selection procedure, which are implied by lower-level conditions already assumed in important settings, including sieve-based selection, sparsity-based methods such as the Lasso, and data-adaptive feature representations. We show that ADML estimators yield regular and efficient root-\(n\) inference for an oracle projection parameter induced by a data-adaptive oracle submodel. This oracle parameter coincides with the target parameter at the true distribution but typically has a smaller efficiency bound, thereby yielding superefficiency for the target parameter. As a practical illustration, we introduce a broad class of automatic ADML estimators for continuous linear functionals of the outcome regression, in which model selection is performed directly on the regression itself. Motivated by overlap challenges in causal inference, we develop new superefficient plug-in estimators for the average treatment effect based on calibration in semiparametric regression models.

Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On the other hand, for a compact convex surface with boundary smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we can estimate its intrinsic diameter in terms of its mean curvature field integral and the length of its boundary. These results are supplements of previous work of Topping, Wu-Zheng and Paeng.

The no-broadcasting theorem is a fundamental result in quantum information theory. It guarantees that a class of attacks on quantum protocols, based on eavesdropping and indiscriminate copying of quantum information, are impossible. Due to its fundamental importance, it is natural to ask whether it is an intrinsic quantum property or whether it also holds for a broader class of non-classical theories. To address this question, one could use the framework of correlation scenarios. Under this standpoint, Joshi, Grudka, and the Horodeckis conjectured that one cannot locally broadcast nonlocal behaviours. In this paper, we prove their conjecture based on the monotonicity of the relative entropy for behaviours. Additionally, following a similar reasoning, we obtain an analogous no-go theorem for steerable assemblages.

In this article, I put forward the idea that the neoplastic process (NP) has deep evolutionary roots and make specific predictions about the connection between cancer and the formation of the first embryo, which allowed for the evolutionary radiation of metazoans. My main hypothesis is that the NP is at the heart of cellular mechanisms responsible for animal morphogenesis and, given its embryological basis, also at the center of animal evolution. It is thus understood that NP-associated mechanisms are deeply rooted in evolutionary history and tied to the formation of the first animal embryo. In my consideration of these arguments, I expound on how cancer biology is perfectly intertwined with evolutionary biology. I describe essential cellular components of unicellular holozoans that served as a basis for the formation of the neoplastic functional module (NFM) and its subsequent exaptation, which brought forth two great biophysical revolutions within the first embryo. Finally, I examine the role of Physics in the modeling of the NFM and its contribution to morphogenesis to reveal the totipotency of the zygote.

The decisive role of Embryology in understanding the evolution of animal forms is founded and deeply rooted in the history of science. It is recognized that the emergence of multicellularity would not have been possible without the formation of the first embryo. We speculate that biophysical phenomena and the surrounding environment of the Ediacaran ocean were instrumental in co-opting a neoplastic functional module (NFM) within the nucleus of the first zygote. Thus, the neoplastic process, understood here as a biological phenomenon with profound embryologic implications, served as the evolutionary engine that favored the formation of the first embryo and cancerous diseases and allowed to coherently create and recreate body shapes in different animal groups during evolution. In this article, we provide a deep reflection on the Physics of the first embryogenesis and its contribution to the exaptation of additional NFM components, such as the extracellular matrix. Knowledge of NFM components, structure, dynamics, and origin advances our understanding of the numerous possibilities and different innovations that embryos have undergone to create animal forms via Neoplasia during evolutionary radiation. The developmental pathways of Neoplasia have their origins in ctenophores and were consolidated in mammals and other apical groups.