An important conjecture in Iwasawa theory predicts that the fine Selmer group of any p-adic Galois representation is cotorsion over the relevant Iwasawa algebra with vanishing $\mu $-invariant, generalizing Iwasawa’s original conjecture for class groups. In this article, we recast this conjecture in terms of higher Galois cohomological operations called Massey products, stated purely in terms of the residual representation. This characterization implies that if the $\mu $-invariant vanishes for a given $\mathbb {Z}_p$-extension, then it also vanishes for all $\mathbb {Z}_p$-extensions the Greenberg neighborhood of radius $1/p$. Furthermore, we establish that the unobstructedness of deformation rings over $\mathbb {Z}_p$-extensions is equivalent to the vanishing of the $\mu $-invariant of the fine Selmer group attached to the adjoint representation. For ordinary deformation rings attached to two-dimensional automorphic Galois representations, we establish a close connection between the Noetherian property and the vanishing of the $\mu $-invariant.

In this article, we consider the non-vanishing problem for the family of special Hecke–Maass L-values $ L (1/2+it_f, f) $ with $f (z)$ in an orthonormal basis of (even or odd) Hecke–Maass cusp forms of Laplace eigenvalue $1/4 + t_f^2$ ($t_f> 0$). We prove that 20% of $L (1/2+it_f, f)$ for $ t_f \leqslant T$ do not vanish as $T \rightarrow \infty $. For comparison, it is known that the non-vanishing proportion is at least 25% for the central L-values $L (1/2, f)$. Further, 20% may be raised to 50% conditionally on the generalized Riemann hypothesis. Moreover, we prove non-vanishing results on the short interval $|t_f-T| \leqslant T^{\mu }$ for any $0 < \mu < 1$.

Let $\mathbb {X}$ bea Riemannian symmetric space of non-compact type. For a real-valued $f \in L^1(\mathbb {X})$ with $\int _{\mathbb {X}} f \ge 0$, we prove that if f satisfies the iterated convolution inequality:

$$ \begin{align*} f \ge \sum_{n=2}^N a_n \left(*^n f\right),\:\:\text{ a.e. on } \mathbb{X}, \end{align*} $$

$N \ge 2$

$2 \le n \le N$

$a_n$

$\int _{\mathbb {X}} f \le t_{\mathcal {Q}}\:$

$\mathcal {Q}(t):=t-\sum _{n=2}^N a_n\:t^n$

$t_{\mathcal {Q}}$

$\mathcal {Q}'$

$(0,\infty )$

$\mathbb {R}^d$

$\mathbb R^d$

In this article, we study Iwasawa theory for the Selmer group of an elliptic curve over a certain non-commutative p-adic Lie extension whose Galois group is isomorphic to an open subgroup of $\mathrm {PGL}_2(\mathbb {Z}_p)$. Unlike the classes of extension usually considered in non-commutative Iwasawa theory, this extension does not contain the cyclotomic $\mathbb {Z}_p$-extension. We give a formula for its Euler characteristic in this case. We also show that the dual Selmer group contains no nonzero pseudonull submodules, a result analogous to a theorem of Ochi and Venjakob for $\mathrm {GL}_2$-extensions. Finally, we provide an explicit formula computing the change in Iwasawa $\mu $-invariants of dual Selmer groups when two elliptic curves are isogenous; this in the same spirit as that of Howson’s result for the $\mathrm {GL}_2$-extension which extends results in the commutative case by Schneider and Perrin-Riou.

The Moore–Tachikawa conjecture is that each connected complex semisimple group $G$ determines a two-dimensional topological quantum field theory (TQFT) in a category of Hamiltonian symplectic varieties. We view the Moore–Tachikawa conjecture as a first step in systematically assigning new TQFTs to purely Lie-theoretic data. At the same time, one should expect these new TQFTs to bear a close relation to those conjectured by Moore and Tachikawa. Our manuscript aims to integrate these two points of view. In more detail, let $\mathfrak {g}$ be the Lie algebra of $G$. Consider a conjugacy class $\mathcal {C}$ of parabolic subalgebras of $\mathfrak {g}$. This class determines partial Grothendieck–Springer resolutions $\mu _{\mathcal {C}}:\mathfrak {g}_{\mathcal {C}}\longrightarrow \mathfrak {g}^*=\mathfrak {g}$ and $\nu _{\mathcal {C}}:G_{\mathcal {C}}\longrightarrow G$. We construct a canonical symplectic groupoid $(T^*G)_{\mathcal {C}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;\mathfrak {g}_{\mathcal {C}}$ and quasi-symplectic groupoid $\mathrm {D}(G)_{\mathcal {C}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\; G_{\mathcal {C}}$. By considering a Kostant slice $\mathrm {Kos}\subseteq \mathfrak {g}$ and Steinberg slice $\mathrm {Ste}\subseteq G$, we prove that the pairs $(((T^*G)_{\mathcal {C}})_{\text {reg}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;(\mathfrak {g}_{\mathcal {C}})_{\text {reg}},\mu _{\mathcal {C}}^{-1}(\mathrm {Kos}))$ and $((\mathrm {D}(G)_{\mathcal {C}})_{\text {reg}}\;\substack {\longrightarrow \\[-9pt] \longrightarrow }\;(G_{\mathcal {C}})_{\text {reg}},\nu _{\mathcal {C}}^{-1}(\mathrm {Ste}))$ determine new and explicit TQFTs in a $1$-shifted Weinstein symplectic category. We then show that certain symplectic varieties arising from our new TQFTs have canonical Lagrangian relations to the open Moore–Tachikawa varieties.

This article investigates how classical homological invariants extend from module categories to a triangulated category $\mathcal {T}$ with a compact silting generator M. We construct sup-projective, inf-injective, and sup-flat resolutions for objects in $\mathcal {T}$ and use them to establish basic properties of the corresponding homological dimensions. We then introduce the global dimension $\mathrm {gldim}\mathcal {T}$ and prove that it coincides with the global dimension of the heart $\mathcal {H}_M=M^{\bot _{\neq 0}}$ and with the $\mathrm {Add}(M)$-dimension of $\mathcal {T}^{\mathrm {b}}$ when M is tilting. These results provide a unified framework for homological dimension theory in triangulated categories and reveal how silting and tilting objects encode essential homological information. As applications, we study homological properties, such as Gorensteinness and regularity for $\mathcal {T}$ with a focus on the derived category over rings and dg rings.

We consider a hybrid system of delay-algebraic equations motivated by an epidemic model involving both disease transmission and behavioral adaptation, and we show how the $S^1$-degree theoretic approach toward global bifurcations of nonlinear systems with two parameters can be adopted to develop a global Hopf bifurcation theory for such a hybrid system. We use the epidemic models, previously developed to explore mechanisms of multiple waves during the acute phase and post-pandemic phase of an emerging disease, to illustrate the effectiveness and challenges of applying the global Hopf bifurcation theory to examine the global continuation of local Hopf bifurcations when the critical parameter (implementation delay) moves away from the critical value for onset of the oscillation. We also show how a three-dimensional ordinary differential competitive system with built-in cyclic symmetry and embedded negative loops arises naturally from our global Hopf bifurcation study.

This article investigates coefficient randomizations of holomorphic functions and the resulting random analytic functions within the framework of mixed norm spaces. Adopting the perspective of the random symbol space $(\mathcal {X})_{\star }=\{f\in H(\mathbb {B}_n): \mathbb {P}(\mathcal {R} f\in \mathcal {X})=1\}$, we first focus on the unit disk $\mathbb {D}$. Specifically, for $q\geq 4$, we identify the conditions under which the random symbol space of the mixed-norm space $H^{p,q,u}(\mathbb {D})$ coincides with $H^{2,q,u}(\mathbb {D})$. Furthermore, we extend our analysis to the higher-dimensional unit ball $\mathbb {B}_n$. We establish sufficient conditions and necessary constraints linking the deterministic membership of a function to the almost sure membership of its randomization in mixed norm spaces. A key finding is the identification of a sharp, dimension-dependent weight loss phenomenon that arises for $n>1$.

We study torsors for groups defined by algebraic difference equations. Our main result provides necessary and sufficient conditions on the base difference field for all such torsors to be trivial. We also present an application to the uniqueness of difference Picard–Vessiot extensions for linear differential equations.

This article is devoted to the study of the mosquito growth incorporating metamorphic stages (including aquatic stage and adult stage), mating behavior, and the spatial movement of adult mosquitoes in a seasonal environment. Our model is a reaction–diffusion system that describes the dynamics of aquatic mosquitoes, mature male mosquitoes, and mature female mosquitoes. In the case of a bounded habitat, we establish the threshold dynamics of the model system in terms of the basic reproduction ratio, which is defined by a time-periodic homogeneous evolution system without compactness. In the case of a spatially periodic habitat, we show that the system admits the leftward and rightward spreading speeds that coincide with the minimal wave speeds of leftward and rightward time–space periodic traveling waves, respectively. Further, we derive a formula to compute the spreading speed via the homogeneous systems approach and quantitatively investigate the effects of parameters on it in the autonomous case.

Let K be a number field with ring of integers $\mathcal {O}_K$, and let $f(x)\in \mathcal {O}_K[x]$ be a monic, irreducible polynomial. We establish necessary and sufficient conditions in terms of the critical points of $f(x)$ for the iterates of $f(x)$ to be monogenic polynomials. More generally, we give necessary and sufficient conditions for the backward orbits of elements of $\mathcal {O}_K$ under $f(x)$ to be monogenerators. We apply our criteria to construct novel examples of dynamically monogenic polynomials, yielding infinite towers of monogenic number fields with the backward orbit of one monogenerator giving a monogenerator at the next level.

Let $k\ge 2$ and let X be a subset of the natural numbers that is k-automatic and not eventually periodic. We show that the following dichotomy holds: either all k-automatic subsets are definable in the expansion of Presburger arithmetic in which we adjoin the predicate X, or $(\mathbb {N},+,X)$ has the same definable sets as $(\mathbb {N},+,k^{\mathbb {N}})$.

We introduce the relative Matsui spectrum, a new invariant associated with a stable infinity category equipped with an action. This construction generalizes both Balmer’s tensor triangular spectrum and Matsui’s triangular spectrum and provides a unified framework for classifying thick submodules. We establish its fundamental properties, including universality, comparisons with existing spectra and Stevenson’s support, fiberwise decomposition, and descent. Applications show that the relative Matsui spectrum recovers the underlying classical geometric spaces from categorical data in various settings: categories of perfect complexes of schemes, twisted derived categories, categories of singularities, and derived matrix factorization categories. Thus, the relative Matsui spectrum extends tensor triangular geometry beyond settings with a global tensor structure, while preserving geometric intuition.

We investigate order isomorphisms, which are not assumed to be linear, between complete order-unit spaces. We show that two such spaces are order isomorphic if and only if they are linearly order isomorphic. We then introduce a condition that determines whether all order isomorphisms on a complete order-unit space are automatically affine. This characterization is in terms of the geometry of the state space. We consider how this condition applies to several examples, including the space of bounded self-adjoint operators on a Hilbert space. Our techniques also allow us to show that in a unital $C^*$-algebra there is an order isomorphism between the space of self-adjoint elements and the cone of positive invertible elements if and only if the algebra is commutative.

Let R be a commutative Noetherian ring. We establish a close relationship between the strong generation of the singularity category of R and the nonvanishing of the annihilator of the singularity category of R. As an application, we prove that the singularity category of R has a strong generator if and only if the annihilator of the singularity category of R is nonzero when R is a Noetherian domain with Krull dimension at most one. We introduce the notion of the co-cohomological annihilator of modules. If the category of finitely generated R-modules has a strong generator, we show that the infinite injective dimension locus of a finitely generated R-module M is closed, with the defining ideal given by the co-cohomological annihilator of M. Finally, we provide a connection between the existence of an extension generator of the category of finitely generated R-modules and the finiteness of the Krull dimension of R.

Assuming the Generalized Riemann Hypothesis, it is known that the least quadratic non-residue modulo a prime p is less than or equal to $(\log p)^2$. In the present article, we establish unconditional results on the distribution of partitions associated with quadratic non-residues in even smaller intervals of size $(\log p)^A$ with $A> 0$, for almost all primes p.

Let X be a rationally elliptic space. Utilizing the Gorenstein algebra structure of X, we present three algorithms that together induce a generating class of $\mathrm{Ext}^N_{(\Lambda V,d)}(\mathbb {Q},(\Lambda V,d))$ with N being the formal dimension of X. From these algorithms, we develop an algorithm to compute the rational Lusternik–Schnirelmann category $\mathrm{cat}_0(X)$. Furthermore, by applying a spectral sequence argument based on the Eilenberg–Moore spectral sequence, we compute the rational Ginsburg invariant $l_0(X)$ introduced by M. Ginsburg in (1963, Ann. Math. 77, 89–96).

An $S_k$-set in a group $\Gamma $ is a set $A\subseteq \Gamma $ such that $\alpha _1\dots \alpha _k=\beta _1\dots \beta _k$ with $\alpha _i,\beta _i\in A$ implies $(\alpha _1,\ldots ,\alpha _k)=(\beta _1,\ldots ,\beta _k)$. An $S_k'$-set is a set such that $\alpha _1\beta _1^{-1}\dots \alpha _k\beta _k^{-1}=1$ implies that there exists i such that $\alpha _i=\beta _i\text { or }\beta _i=\alpha _{i+1}$. We give explicit constructions of large $S_k$-sets in the groups $\mathrm {Sym}(n)$ and $\mathrm {Alt}(n)$ and $S_2$-sets in $\mathrm {Sym}(n)\times \mathrm {Sym}(n)$ and $\mathrm {Alt}(n)\times \mathrm {Alt}(n)$. We give probabilistic constructions which yield large $S_2'$-sets in $\mathrm {Sym}(n)$. We also give upper bounds on the size of $S_k$-sets in certain groups, improving the trivial bound by a constant multiplicative factor. We describe some connections between $S_k$-sets and extremal graph theory. In particular, we determine up to a constant factor the minimum outdegree of a digraph which guarantees even cycles with certain orientations. As applications, we improve the upper bound on Hamilton paths which pairwise create a two-part cycle of given length, and we show that a directed version of the Erdős–Simonovits compactness conjecture is false.

In 1967, Grünbaum conjectured that the function

$$ \begin{align*}\phi_k(d+s,d):=\binom{d+1}{k+1}+\binom{d}{k+1}-\binom{d+1-s}{k+1},\; \text{for } 2\leqslant s\leqslant d \end{align*} $$

provides the minimum number of $k$-faces for a $d$-dimensional polytope (abbreviated as a $d$-polytope) with $d+s$ vertices. In 2021, Xue proved this conjecture for each $k\in [1, d-2]$ and characterized the unique minimizers, each having $d+2$ facets. In this article, we refine Xue’s theorem by considering $d$-polytopes with $d+s$ vertices ($2\leqslant s\leqslant d$) and at least $d+3$ facets. If $s=2$, then there is precisely one minimizer for many values of $k$. For other values of $s$, the number of $k$-faces is at least $\phi _k(d+s,d)+\binom {d-1}{k}-\binom {d+1-s}{k}$, which is met by precisely two polytopes in many cases, and up to five polytopes for certain values of $s$ and $k$. We also characterize the minimizing polytopes.

We study the subsymmetric basic sequence structure of variable exponent Lebesgue spaces $L_{\boldsymbol {P}}$ built from index functions $\boldsymbol {P}\colon \Omega \to (0,\infty ]$ on $\sigma $-finite measure spaces $(\Omega ,\Sigma ,\mu )$. Specifically, we prove that if $\boldsymbol {P}$ is bounded away from infinity, then any complemented subsymmetric basic sequence of $L_{\boldsymbol {P}}$ is equivalent to the canonical basis of $\ell _r$ for some $r\ge 1$ in the essential range of $\boldsymbol {P}$.