Let Λ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of Λ-mod with $n \geq 2$. From the viewpoint of higher homological algebra, a question that naturally arose in Ebrahimi and Nasr-Isfahani (The completion of d-abelian categories. J. Algebra 645 (2024), 143–163) is when $\mathcal{M}$ induces an n-cluster tilting subcategory of Λ-Mod. In this article, we answer this question and explore its connection to Iyama’s question on the finiteness of n-cluster tilting subcategories of Λ-mod. In fact, our theorem reformulates Iyama’s question in terms of the vanishing of Ext and highlights its relation with the rigidity of filtered colimits of $\mathcal{M}$. Also, we show that ${\rm Add}(\mathcal{M})$ is an n-cluster tilting subcategory of Λ-Mod if and only if ${\rm Add}(\mathcal{M})$ is a maximal n-rigid subcategory of Λ-Mod if and only if $\lbrace X\in \Lambda-{\rm Mod}~|~ {\rm Ext}^i_{\Lambda}(\mathcal{M},X)=0 ~~~ {\rm for ~all}~ 0 \lt i \lt n \rbrace \subseteq {\rm Add}(\mathcal{M})$ if and only if $\mathcal{M}$ is of finite type if and only if ${\rm Ext}_{\Lambda}^1({\underrightarrow{\lim}}\mathcal{M}, {\underrightarrow{\lim}}\mathcal{M})=0$. Moreover, we present several equivalent conditions for Iyama’s question which shows the relation of Iyama’s question with different subjects in representation theory such as purity and covering theory.

We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over $\mathbb{F}_q$. Among other results, this allows us to prove that the $\mathbb{Q}$-vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields.

We prove structural results for measure-preserving systems, called Furstenberg systems, naturally associated with bounded multiplicative functions. We show that for all pretentious multiplicative functions, these systems always have rational discrete spectrum and, as a consequence, zero entropy. We obtain several other refined structural and spectral results, one consequence of which is that the Archimedean characters are the only pretentious multiplicative functions that have Furstenberg systems with trivial rational spectrum, another is that a pretentious multiplicative function has ergodic Furstenberg systems if and only if it pretends to be a Dirichlet character, and a last one is that for any fixed pretentious multiplicative function, all its Furstenberg systems are isomorphic. We also study structural properties of Furstenberg systems of a class of multiplicative functions, introduced by Matomäki, Radziwiłł, and Tao, which lie in the intermediate zone between pretentiousness and strong aperiodicity. In a work of the last two authors and Gomilko, several examples of this class with exotic ergodic behavior were identified, and here we complement this study and discover some new unexpected phenomena. Lastly, we prove that Furstenberg systems of general bounded multiplicative functions have divisible spectrum. When these systems are obtained using logarithmic averages, we show that a trivial rational spectrum implies a strong dilation invariance property, called strong stationarity, but, quite surprisingly, this property fails when the systems are obtained using Cesàro averages.

The delta invariant interprets the criterion for the K-(poly)stability of log terminal Fano varieties. In this paper, we determine local delta invariants for all weak del Pezzo surfaces with the anti-canonical degree $\geq 5$.

We prove a large deviation principle for the slow-fast rough differential equations (RDEs) under the controlled rough path (RP) framework. The driver RPs are lifted from the mixed fractional Brownian motion (FBM) with Hurst parameter $H\in (1/3,1/2)$. Our approach is based on the continuity of the solution mapping and the variational framework for mixed FBM. By utilizing the variational representation, our problem is transformed into a qualitative property of the controlled system. In particular, the fast RDE coincides with Itô stochastic differential equation (SDE) almost surely, which possesses a unique invariant probability measure with frozen slow component. We then demonstrate the weak convergence of the controlled slow component by averaging with respect to the invariant measure of the fast equation and exploiting the continuity of the solution mapping.

This work is concerned with the exponential turnpike property for optimal control problems of particle systems and their mean-field limit. Under the assumption of the strict dissipativity of the cost function, exponential estimates for both optimal states and optimal control are proven. Moreover, we show that all the results for particle systems can be preserved under the limit in the case of infinitely many particles.

We investigate and compare applications of the Zilber–Pink conjecture and dynamical methods to rigidity problems for arithmetic real and complex hyperbolic lattices. Along the way, we obtain new general results about reconstructing a variation of Hodge structure from its typical Hodge locus that may be of independent interest. Applications to Siu’s immersion problem are also discussed, the most general of which only requires the hypothesis that infinitely many closed geodesics map to proper totally geodesic subvarieties under the immersion.

Recently it has been shown that the unique local perimeter minimizing partitioning of the plane into three regions, where one region has finite area and the other two have infinite measure, is given by the so-called standard lens partition. Here we prove a sharp stability inequality for the standard lens, hence strengthening the local minimality of the lens partition in a quantitative form. As an application of this stability result we consider a nonlocal perturbation of an isoperimetric problem.

Early warning for epilepsy patients is crucial for their safety and well being, in particular, to prevent or minimize the severity of seizures. Through the patients’ electroencephalography (EEG) data, we propose a meta learning framework to improve the prediction of early ictal signals. The proposed bilevel optimization framework can help automatically label noisy data at the early ictal stage, as well as optimize the training accuracy of the backbone model. To validate our approach, we conduct a series of experiments to predict seizure onset in various long-term windows, with long short-term memory (LSTM) and ResNet implemented as the baseline models. Our study demonstrates that not only is the ictal prediction accuracy obtained by meta learning significantly improved, but also the resulting model captures some intrinsic patterns of the noisy data that a single backbone model could not learn. As a result, the predicted probability generated by the meta network serves as a highly effective early warning indicator.

Motivated by the study of algebraic classes in mixed characteristic, we define a countable subalgebra of ${\overline {\mathbb {Q}}}_p$ which we call the algebra of André’s p-adic periods. The classical Tannakian formalism cannot be used to study these new periods. Instead, inspired by ideas of Drinfel’d on the Plücker embedding and further developed by Haines, we produce an adapted Tannakian setting which allows us to bound the transcendence degree of André’s p-adic periods and to formulate the p-adic analog of the Grothendieck period conjecture. We exhibit several examples where special values of classical p-adic functions appear as André’s p-adic periods, and we relate these new conjectures to some classical problems on algebraic classes.

This paper analyzes the initial value problem for the Toda lattice with almost periodic initial data: let $J(t; J_{0})$ denote the family of Jacobi matrices which are solutions of the Toda flow equation with initial condition $J(0; J_{0})=J_{0},$ then, the given almost periodic datum $J_{0}$ is a discrete linear Schrödinger operator with almost periodic potential, which plays a fundamental role in our considerations. We show that, under some given hypotheses, the spectrum of the Schrödinger operator is pure absolute continuous and homogeneous (measure-theoretically) by establishing exponential asymptotics on the size of spectral gaps. These two conclusions enable us to show the boundedness and almost periodicity in the time of solutions for Toda lattice equation with almost periodic initial data. As a consequence, our result presents a positive answer to the discrete Deift’s conjecture [Some open problems in random matrix theory and the theory of integrable systems. Integrable Systems and Random Matrices (Contemporary Mathematics, 458). American Mathematical Society, Providence, RI, 2008, pp. 419–430; Some open problems in random matrix theory and the theory of integrable systems. II. SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), Paper no. 016].

We propose and unify classes of different models for information propagation over graphs. In a first class, propagation is modelled as a wave, which emanates from a set of known nodes at an initial time, to all other unknown nodes at later times with an ordering determined by the arrival time of the information wave front. A second class of models is based on the notion of a travel time along paths between nodes. The time of information propagation from an initial known set of nodes to a node is defined as the minimum of a generalised travel time over subsets of all admissible paths. A final class is given by imposing a local equation of an eikonal form at each unknown node, with boundary conditions at the known nodes. The solution value of the local equation at a node is coupled to those of neighbouring nodes with lower values. We provide precise formulations of the model classes and prove equivalences between them. Finally, we apply the front propagation models on graphs to semi-supervised learning via label propagation and information propagation on trust networks.

Let $G$ be a group. The notion of linear sofic approximations of $G$ over an arbitrary field $F$ was introduced and systematically studied by Arzhantseva and Păunescu [3]. Inspired by one of the results of [3], we introduce and study the invariant $\kappa _F(G)$ that captures the quality of linear sofic approximations of $G$ over $F$. In this work, we show that when $F$ has characteristic zero and $G$ is linear sofic over $F$, then $\kappa _F(G)$ takes values in the interval $[1/2,1]$ and $1/2$ cannot be replaced by any larger value. Further, we show that under the same conditions, $\kappa _F(G)=1$ when $G$ is torsion-free. These results answer a question posed by Arzhantseva and Păunescu [3] for fields of characteristic zero. One of the new ingredients of our proofs is an effective non-concentration estimates for random walks on finitely generated abelian groups, which may be of independent interest.

The rank of a tiling’s return module depends on the geometry of its tiles and is not a topological invariant. However, the rank of the first Čech cohomology $\check H^1(\Omega )$ gives upper and lower bounds for the rank of the return module. For all sufficiently large patches, the rank of the return module is at most the rank of $\check H^1(\Omega )$. For a generic choice of tile shapes and an arbitrary reference patch, the rank of the return module is at least the rank of $\check H^1(\Omega )$. Therefore, for generic tile shapes and all sufficiently large patches, the rank of the return module is equal to the rank of $\check H^1(\Omega )$.

The notion of effective topological complexity, introduced by Błaszczyk and Kaluba, deals with using group actions in the configuration space in order to reduce the complexity of the motion planning algorithm. In this article, we focus on studying several properties of this notion of topological complexity. We introduce a notion of effective LS category which mimics the behaviour the usual LS category has in the non-effective setting. We use it to investigate the relationship between these effective invariants and the orbit map with respect to the group action, and we give numerous examples. Additionally, we investigate non-vanishing criteria based on a cohomological dimension bound of the saturated diagonal.

In this paper, we study the twisted Ruelle zeta function associated with the geodesic flow of a compact, hyperbolic, odd-dimensional manifold X. The twisted Ruelle zeta function is associated with an acyclic representation $\chi \colon \pi _{1}(X) \rightarrow \operatorname {\mathrm {GL}}_{n}(\mathbb {C})$, which is close enough to an acyclic, unitary representation. In this case, the twisted Ruelle zeta function is regular at zero and equals the square of the refined analytic torsion, as it is introduced by Braverman and Kappeler in [6], multiplied by an exponential, which involves the eta invariant of the even part of the odd-signature operator, associated with $\chi $.