Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma=SL(2,\mathbb{Z})$. Denote by $\lambda_{\text{sym}^{2}f}(n)$ the $n$th normalized coefficient of the Dirichlet series expansion of the symmetric square $L$-function $L(s,\text{sym}^{2}f)$. In this paper, we are interested in the shifted convolution sum

\begin{equation*}\sum_{h\leq H}\sum_{N \lt n\leq 2N}\lambda_{\textrm{sym}^{2}f}(n)\lambda_{\textrm{sym}^{2}f}(n+h).\end{equation*}

We establish a non-trivial bound for $H\gg N^{1/4}$.

We show that, for a finite spectrum $X$, Spanier–Whitehead duality induces an isomorphism between the cohomological and homological Atiyah–Hirzebruch spectral sequences. As an application, it follows that Poincaré duality for a Poincaré duality complex that is oriented over a ring spectrum $\mathcal{R}$ induces an isomorphism between the two spectral sequences.

An exoflop takes a gauged Landau-Ginzburg (LG) model, partially compactifies it, and then performs certain birational transformations on it. When certain criteria hold, this can provide a crepant categorical resolution or equivalence of derived categories associated to the gauged LG models. We provide sufficient criteria for when this provides categorical resolutions for (or equivalences between) certain complete intersections in toric stacks.

The integral identity conjecture of Kontsevich and Soibelman plays an important role in proving the existence of motivic Donaldson-Thomas invariants for three-dimensional noncommutative Calabi-Yau manifolds. There are a number of different formulations of this conjecture in different contexts, and accordingly, there are corresponding solutions to them. The methods devoted to solving this conjecture are diverse, ranging from $\ell $-adic cohomology of rigid analytic varieties to Hrushovski-Kazhdan motivic integration and motivic Fubini theorem for tropicalization maps,... In [Ivo24], Ivorra deduces a functorial version of the integral identity in the motivic stable homotopy categories of schemes, from the Braden hyperbolic localization theorem. This functorial version concerns Ayoub’s nearby cycles functor associated with a $\mathbb {G}_m$-equivariant function $f \colon \mathbb {V}(\mathcal {E}) \longrightarrow \mathbb {A}^1$ on a vector bundle $\mathbb {V}(\mathcal {E})$ over a field of characteristic zero. In the present work, we follow the functorial approach from [Ivo24] and extend the scope of the original conjecture by Kontsevich and Soibelman by studying more generally the case of $\mathbb {G}_m$-equivariant functions on algebraic S-spaces with a $\tau $-locally linearizable action of $\mathbb {G}_m$ over a noetherian base scheme S.

In this short note, we prove that every closed, oriented, connected 3-manifold arises as Dehn surgery along a braid positive link.

We show that two simple, separable, nuclear, and ${\mathcal{Z}_0}$-stable $\mathrm{C}^{*}$-algebras are isomorphic if they are trace-preservingly homotopy equivalent. This result does not assume the Universal Coefficient Theorem and can be viewed as a tracial stably projectionless analogue of the homotopy rigidity theorem for Kirchberg algebras.

We study the class of transitive skew products associated with iterated function systems of circle diffeomorphisms. We approximate any of those skew products by maps in this class with a robustly zero Lyapunov exponent. In particular, we prove the existence of non-hyperbolic ergodic measures for an open and dense subset of transitive skew products. Moreover, these measures have full support and are the weak$^*$ limit of periodicmeasures.

In this paper, we use a degenerate reaction-diffusion system with free boundaries to model the spatial spread of rabies among foxes, whose population is divided into three sub-populations: susceptible ($S$), infected ($E$), and rabid ($I$). Based on established biological observations, susceptible and infected foxes are assumed to be territorial (random diffusion is ignored for $E$ and $S$ in the model), whereas rabid foxes disperse randomly (random diffusion is assumed for $I$), causing the spread of the disease. While $S$ evolves over the entire real line $\mathbb{R}$, $E$ and $I$ are found only in the infected region represented by an interval $[g(t), h(t)]$, which expands with moving fronts $x = g(t)$ and $x = h(t)$ as time $t$ increases. We show that this system admits a unique global solution and then analyse its dynamics and establish a spreading-vanishing dichotomy in certain natural parameter regimes. Moreover, we supply some simple sufficient conditions for the vanishing and spreading of the rabies, respectively. For example, we show that if the corresponding ODE system has basic reproduction number $\mathcal{R}_0 \gt 1$, then a spreading-vanishing dichotomy holds, and the outcome depends on the initial size of the infected region, while if a certain quantity $\mathcal R_0^* \in (\mathcal R_0,\infty)$ is no bigger than 1, then the rabies will always vanish. The degenerate nature of the model, combined with the evolving infected region, causes considerable difficulties in the mathematical treatment, both in proving the well-posedness and in understanding the long-time dynamics. This paper appears to be the first to treat a free boundary model where one reaction-diffusion equation is coupled with two ordinary differential equations.

In this paper, we establish general results for the asymptotic behaviour of solutions of dynamical systems in Banach spaces. We show that if the initial datum possesses a certain decay, then the corresponding solution emanating from the Cauchy problem studied inherits the same behaviour at any further time for which it exists. Our results are applied to a wide class of linear and non-linear models. In particular, we use our main results to show persistence properties for classical linear and non-linear ordinary differential equations (ODEs), the Benjamin–Bona–Mahony (BBM) equation, and the generalized Boussinesq equation.

Let $G(\mathbb {R})$ be a real reductive group. Suppose $\pi $ is an irreducible representation of $G(\mathbb {R})$ having a Whittaker model, and consider three invariants of $\pi $ related to nilpotent elements of the Lie algebra of G (or its dual): the associated variety, the wave-front set, and the set of Whittaker data for which $\pi $ has a Whittaker model. If $\pi $ is a discrete series representation, these invariants are known to determine each other. We provide a self-contained account of this and related matters. Many of the results were known: we give simplified proofs for several of them, for instance a simple proof (for generic discrete series) that the associated variety and the wave-front set are related by the Kostant–Sekiguchi correspondence.

We give a new definition of a Frobenius structure on an algebra object in a monoidal category, generalising Frobenius algebras in the category of vector spaces. Our definition allows Frobenius forms valued in objects other than the unit object and can be seen as a categorical version of Frobenius extensions of the second kind. When the monoidal category is pivotal, we define a Nakayama morphism for the Frobenius structure and explain what it means for this morphism to have finite order. Our main example is a well-studied algebra object in the (additive and idempotent completion of the) Temperley–Lieb category at a root of unity. We show that this algebra has a Frobenius structure and that its Nakayama morphism has order 2. As a consequence, we obtain information about Nakayama morphisms of preprojective algebras of Dynkin type, considered as algebras over the semisimple algebras on their vertices.

The present paper is concerned with the optimal weight of vibrating string equations with the first two eigenvalues $\lambda_1$ and $\lambda_2$ being given. Applying the method of critical equations in $L^p[0,1]$ for $p \gt 1$ and the inverse spectral theory of Sturm–Liouville problems with measure coefficients, we find that the optimal weight can be uniquely determined if and only if $\lambda_2 \ge 2\lambda_1$ provided that the weight is non-negative and symmetrical. As an application, we provide an estimation of the extremum for partial trace of the first two eigenvalues on a sphere in $L^1[0,1]$.

Biochemical reaction networks (RNs) are widely applied across scientific disciplines to model complex dynamic systems. We investigate the diffusion approximation of RNs with mass-action kinetics, focusing on the identifiability of the stochastic differential equations associated to the reaction network. We derive conditions under which the law of the diffusion approximation is identifiable and provide theorems for verifying identifiability in practice. Notably, our results show that some RNs have non-identifiable reaction rates, even when the law of the corresponding stochastic process is completely known. Moreover, we show that RNs with distinct graphical structures can generate the same diffusion law under specific choices of reaction rates. Finally, we compare our framework with identifiability results in the deterministic ordinary differential equation setting and the discrete continuous-time Markov chain models for RNs.

We study the dynamics of a delayed predator–prey system with Holling type II functional response, focusing on the interplay between time delay and carrying capacity. Using local and global Hopf bifurcation theory, we establish the existence of sequences of bifurcations as the delay parameter varies and prove that the connected components of global Hopf branches are nested under suitable conditions. A novel contribution is the demonstration that the classical limit cycle of the non-delayed system belongs to a connected component of the global Hopf bifurcation in Fuller’s space. Our analysis combines rigorous functional differential equation theory with continuation methods to characterize the structure and boundedness of bifurcation branches. We further demonstrate that delays can induce oscillatory coexistence at lower carrying capacities than in the corresponding ordinary differential equation model, yielding counterintuitive biological insights. The results contribute to the broader theory of global bifurcations in delay differential equations while providing new perspectives on nonlinear population dynamics.

For a $G$-equivariant fibration $p \colon E\to B$, we introduce and study the invariant analogue of Cohen, Farber, and Weinberger’s parametrized topological complexity, called the invariant parametrized topological complexity. This notion generalizes the invariant topological complexity introduced by Lubawski and Marzantowicz. When $G$ is a compact Lie group acting freely on $E$, we show that the invariant parametrized topological complexity of the $G$-fibration $p \colon E\to B$ coincides with the parametrized topological complexity of the induced fibration $\overline{p} \colon \overline{E} \to \overline{B}$ between the orbit spaces. Furthermore, we compute the invariant parametrized topological complexity of equivariant Fadell–Neuwirth fibrations, which measures the complexity of motion planning in the presence of obstacles with unknown positions, where the order of their placement is irrelevant. In addition, we study the equivariant sectional category and the equivariant parametrized topological complexity, which serve as essential tools for obtaining several results in this paper.

We continue our study of Ulam’s measure problem. In contrast to our previous works, we shift our focus from measures stratified by their additivity, to measures stratified by their indecomposability. The breakthrough here is obtained by replacing the classical ‘least’ function associated with ideals by a two-dimensional ‘last’ function associated with walks on ordinals. Consequently, we obtain conditions under which a measure admits not just infinite pairwise disjoint families of positive sets, but in fact families of maximum possible size. As an application we solve a problem left open in Shelah’s Cardinal Arithmetic book, proving that for every weakly inaccessible cardinal $\kappa $, if there exists a stationary subset of $\kappa $ that does not reflect at regulars, then the strong Ramsey relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ holds.

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.

We compute log canonical thresholds of reduced plane curves of degree $d$ at points of multiplicity $d-1$. As a consequence, we describe all possible values of log canonical threshold that are less than $2/(d-1)$ for reduced plane curves of degree $d$. In addition, we compute log canonical thresholds for all reduced plane curves of degree less than 6.