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In this study, we aim to prove the generalized Hyers-Ulam-Rassias (GHUR) stability for the following Euler-Lagrange (EL) type cubic functional equation f(bx+y)+f(x+by)=(b+1)(b-1)²[f(x)+f(y)]+b(b+1)f(x+y) in non-Archimedean (n.A) quasi-Banach spaces and n.A (n,β) Banach spaces. In recent decades, the stability of functional equations has emerged as one of the most intriguing and engaging topics, as it leads to the applications of functional equations in various domains such as algebraic geometry, Group theory, Mechanics etc., This study is to investigate the GHUR stability for the above equation using Hyers direct method. Furthermore, we obtain the stability results for the aforementioned equation with an illustrative example for the n.A case. With the study of the example one may easily understand how the stability result of functional equations in n.A case differs from the setting of classical Banach spaces. [ABSTRACT FROM AUTHOR]

Algebraic geometry over completely simple semigroups is considered. We prove theorems that define coordinate semigroups and irreducible coordinate semigroups in several classes of completely simple semigroups. [ABSTRACT FROM AUTHOR]

Algebraic simple groups, being non-abelian and lacking proper normal subgroups, represent a profound and captivating domain within group theory and algebraic geometry. In the pursuit of understanding these enigmatic entities, the study of conjugacy classes and products emerges as a key to unlocking their underlying symmetries and structures. Group members can be partitioned into different sets with the use of conjugacy classes, which group together equivalent elements under inner automorphisms to show similar algebraic features. The analysis of conjugacy classes provides deep insights into the dynamics and character of algebraic simple groups. On the other hand, group products explore the interactions between distinct elements, generating new elements within the group and unraveling its internal symmetries. [ABSTRACT FROM AUTHOR]

Abstract: Let be a numerical semigroup. For each , let denote the number of pairs : it is well-known that there exists an integer such that the sequence is non-decreasing for . The problem of finding is solved only in special cases. By way of a suitable parameter , we improve the known bounds for and in several cases we determine m explicitly. In particular we give the value of m when the Cohen–Macaulay type of the semigroup is three or when the multiplicity is less than or equal to six. When is the Weierstrass semigroup of a family of one-point algebraic geometry codes, these results give better estimates for the order bound on the minimum distance of the codes . [Copyright &y& Elsevier]

This paper is of an overview nature, accumulating results on algebraic geometry over Abelian groups and close to them model-theoretic results related to the description of principal universal classes and quasi-varieties. [ABSTRACT FROM AUTHOR]

We define dilatations of general schemes and study their basic properties. Dilatations of group schemes are in favorable cases-again group schemes, called Néron blowups. We give two applications to their cohomology in degree 0 (integral points) and degree 1 (torsors): we prove a canonical Moy-Prasad isomorphism that identifies the graded pieces in the congruent filtration of G with the graded pieces in its Lie algebra g, and we show that many level structures on moduli stacks of G-bundles are encoded in torsors under Néron blowups of G. [ABSTRACT FROM AUTHOR]

Let S be a KLT threefold singularity over an algebraically closed field of positive characteristic p > 5. We prove that its localétale fundamental group is tame and finite. Further, we show that every finite unipotent torsor over a big open subset of S is realized as the restriction of a finite unipotent torsor over S. [ABSTRACT FROM AUTHOR]

A Fano–Enrique's threefold is a 3-dimensional non-Goren stein Fano variety of index 1 with at most canonical singularities. We study the birational geometry of Fano– Enrique's threefold with terminal cyclic quotient singularities. We investigate their rationality and also provide an example of a Fano–Enrique's threefold whose pliability is 9, that is, a Fano–Enrique's threefold birationally equivalent to exactly nine Mori fibre spaces in the Sarkisov category. [ABSTRACT FROM AUTHOR]

In this work, we give an algebro-geometric proof of Hrushovski’s generalization of the Lang-Weil estimates on the number of points in the intersection of a correspondence with the graph of the Frobenius map. This result has numerous applications to various areas of mathematics, including model theory, algebraic dynamics, group theory and arithmetic algebraic geometry. [ABSTRACT FROM AUTHOR]

The Connes embedding problem (CEP) is a problem in the theory of tracial von Neumann algebras and asks whether or not every tracial von Neumann algebra embeds into an ultrapower of the hyperfinite II_1 factor. The CEP has had interactions with a wide variety of areas of mathematics, including \mathrm {C}^*-algebra theory, geometric group theory, free probability, and noncommutative real algebraic geometry, to name a few. After remaining open for over 40 years, a negative solution was recently obtained as a corollary of a landmark result in quantum complexity theory known as \operatorname {MIP}^*=\operatorname {RE}. In these notes, we introduce all of the background material necessary to understand the proof of the negative solution of the CEP from \operatorname {MIP}^*=\operatorname {RE}. In fact, we outline two such proofs, one following the "traditional" route that goes via Kirchberg's QWEP problem in \mathrm {C}^*-algebra theory and Tsirelson's problem in quantum information theory and a second that uses basic ideas from logic. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of residually finite rationally p (RFR p) groups, where p is a prime. We first prove a series of results about the structure of finitely generated RFR p groups (either for a single prime p , or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR p groups, which we use to study the boundary manifolds of algebraic curves ℂ ℙ 2 and in ℂ 2 . We show that boundary manifolds of a large class of curves in ℂ 2 (which includes all line arrangements) have RFR p fundamental groups, whereas boundary manifolds of curves in ℂ ℙ 2 may fail to do so. [ABSTRACT FROM AUTHOR]

The leading theme of the meeting was to understand nonpositively curved complexes and groups acting on them. Motivations, questions, results, and techniques being presented and discussed come from various areas of mathematics, including algebraic geometry, Lie groups, metric graph theory, geometric topology, algebraic topology, coarse geometry, K-theory, and, in general, geometric and analytic group theory. The subject discussed focused around participant's latest achievements, and important open questions in the area. [ABSTRACT FROM AUTHOR]

Algebraic error correcting codes (ECC) are widely used to implement reliability features in modern servers and systems and pose a formidable verification challenge. We present a novel methodology and techniques for provably correct design of ECC logics. The methodology is comprised of a design specification method that directly exposes the ECC algorithm's underlying math to a verification layer, encapsulated in a tool 'BLUEVERI', which establishes the correctness of the design conclusively by using an apparatus of computational algebraic geometry (Buchberger's algorithm for Gröbner basis construction). We present results from its application to example circuits to demonstrate the effectiveness of the approach. The methodology has been successfully applied to prove correctness of large error correcting circuits on IBM's POWER systems to protect memory storage and processor to memory communication, as well as a host of smaller error correcting circuits. [ABSTRACT FROM AUTHOR]

We introduce a new construction of towers of algebraic curves over finite fields and provide a simple example of an optimal tower. [ABSTRACT FROM AUTHOR]

Mechanical design and assembly planning inherently involve geometric constraint satisfaction or scene feasibility (GCS/SF) problems. Such problems imply the satisfaction of proposed relations placed between undefined geometric entities in a given scenario. If the degrees of freedom remaining in the scene are compatible with the proposed relations or constraints, a set of entities is produced that populate the scenario satisfying the relations. Otherwise, a diagnostic of inconsistency of the problem is emitted. This problem appears in various forms in assembly planning (assembly model generation), process planning, constraint driven design, computer vision, etc. Previous attempts at solution using separate numerical, symbolic or procedural approaches suffer serious shortcomings in characterizing the solution space, in dealing simultaneously with geometric (dimensional) and topological (relational) inconsistencies, and in completely covering the possible physical variations of the problem. This investigation starts by formulating the problem as one of characterizing the solution space of a set of polynomials. By using theories developed in the area of algebraic geometry, properties of Grobner Bases are used to assess the consistency and ambiguity of the given problem and the dimension of its solution space. This method allows for the integration of geometric and topological reasoning, The high computational cost of Grobner Basis construction and the need for a compact and physically meaningful set of variables lead to the integration of known results on group theory. These results allow the characterization of geometric constraints in terms of the subgroups of the Special Group of Euclidean displacements in E³, SE(3). Several examples are developed which were solved with computer algebra systems (MAPLE and Mathematical. They are presented to illustrate the use of the Euclidean group-based variables, and to demonstrate the theoretical completeness of the... [ABSTRACT FROM AUTHOR]

We study the BPS spectrum of four-dimensional N = 2 supersymmetric Yang-Mills theory with gauge group SU(2) and a massive adjoint hypermultiplet, which has an extremely intricate structure with infinite spectrum in all chambers of its Coulomb moduli space, and is not well understood. We build on previous results by employing the BPS quiver description of the spectrum, and explore the qualitative features in detail using numerical techniques. We find novel and unexpected behaviour in the form of wall-crossings involving interactions between BPS particles with negative electric-magnetic pairings, which we interpret in terms of the reverse orderings of the usual wall-crossing formulas for rank one N = 2 field theories. This identifies new a priori unrelated states in the spectrum. [ABSTRACT FROM AUTHOR]

We compute explicitly the Chow motive of any generalized Kummer variety associated to any abelian surface. In fact, it lies in the rigid tensor subcategory of the category of Chow motives generated by the Chow motive of the underlying abelian surface. One application of this calculation is to show that the Hodge conjecture holds for arbitrary products of generalized Kummer varieties. As another application, all numerically trivial 1-cycles on arbitrary products of generalized Kummer varieties are smash-nilpotent. [ABSTRACT FROM AUTHOR]

We study the geometry and arithmetic of so-called primary Burniat surfaces, a family of surfaces of general type arising as smooth bidouble covers of a del Pezzo surface of degree 6 and at the same time as étale quotients of certain hypersurfaces in a product of three elliptic curves. We give a new explicit description of their moduli space and determine their possible automorphism groups. We also give an explicit description of the set of curves of geometric genus 1 on each primary Burniat surface. We then describe how one can try to obtain a description of the set of rational points on a given primary Burniat surface S defined over [ABSTRACT FROM AUTHOR]

Understanding bounds for the effective differential Nullstellensatz is a central problem in differential algebraic geometry. Recently, several bounds have been obtained using Dicksonian and antichains sequences (with a given growth rate). In the present paper, we make these bounds more explicit and, therefore, more applicable to understanding the computational complexity of the problem, which is essential to designing more efficient algorithms. [ABSTRACT FROM AUTHOR]

In this paper, considering the geometric equivalence for algebras of a variety $_{S}\mathcal{A}$ of S-acts over a monoid S, we obtain representation theorems describing all types of the equivalence classes of geometrically equivalent S-acts of varieties $_{S}\mathcal{A}$ over groups S. [ABSTRACT FROM AUTHOR]

We classify 2n-order groups and 2n-order square commutative groups with n-order elements. [ABSTRACT FROM AUTHOR]

The category of strict polynomial functors inherits an internal tensor product from the category of divided powers. To investigate this monoidal structure, we consider the category of representations of the symmetric group S d which admits a tensor product coming from its Hopf algebra structure. It is classical that there exists a functor F from the category of strict polynomial functors to the category of representations of the symmetric group. Our main result is that this functor F is monoidal. In addition we study the relations under F between projective strict polynomial functors and permutation modules and the link to symmetric functions. [ABSTRACT FROM AUTHOR]

Let p be a prime number. In this article we present a theorem, suggested by Peter Scholze, which states that Gal(Qp/Qp) is the étale fundamental group of a certain object Z which is defined over an algebraically closed field. This object is the quotient of the "punctured perfectoid open disk" by an action of the group Qxp . The proof of this fact combines two themes: the tilting equivalence for perfectoid spaces, and the Fargues-Fontaine curve. [ABSTRACT FROM AUTHOR]

Let A be a principally polarized CM abelian variety of dimension d defined over a number field F containing the CM-field K . Let ℓ be a prime number unramified in K / Q . The Galois group G ℓ of the ℓ -division field of A lies in a maximal torus of the general symplectic group of dimension 2 d over F ℓ . Relying on a method of Weng, we explicitly write down this maximal torus as a matrix group. We restrict ourselves to the case that G ℓ equals the maximal torus. If p is a prime ideal in F with p | p , let A p be the reduction of A modulo p . By counting matrices with eigenvalue 1 in G ℓ we obtain a formula for the density of primes p such that the ℓ -rank of A p ( F p ) is at least 1. Thereby we generalize results of Koblitz and Weng who computed this density for d = 1 and 2. Both authors gave conjectural formulae for the number of primes p with norm less than n such that A p ( F p ) has prime order. We describe the involved heuristics, generalize these conjectures to arbitrary d and provide examples with d = 3 . [ABSTRACT FROM AUTHOR]

We impose a rather unknown algebraic structure called a ‘hyperstructure’ to the underlying space of an affine algebraic group scheme. This algebraic structure generalizes the classical group structure and is canonically defined by the structure of a Hopf algebra of global sections. This paper partially generalizes the result of A. Connes and C. Consani in [1] . [ABSTRACT FROM AUTHOR]

Let K be a field, and let f : X → Y be a finite étale cover between reduced and geometrically irreducible K -schemes of finite type such that f K s is Galois. Assuming f admits a Galois K -form f ¯ : X ¯ → Y , we use it to analyze fields of definition over K for the Galois property of f and the presence of K -points in general K -forms f ′ : X ′ → Y over Y ( K ) . Additionally, we show that if K is Hilbertian, the group G is non-abelian, and the base variety is rational, then there are finite separable extensions L / K such that some L -form of f L does not descend to a cover of Y . [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld. [ABSTRACT FROM AUTHOR]

Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to G. M. Kelly in the 1960s that there is a very close connection between these phenomena. In this first part of a two-part series on this subject, we show that the assignment to each symmetric monoidal closed category V its associated V -enriched category V extends to a 2-functor valued in an op-2-fibred 2-category of symmetric monoidal closed categories enriched over various bases. For a fixed V , we show that this induces a 2-functorial passage from symmetric monoidal closed categories over V (i.e., equipped with a morphism to V ) to symmetric monoidal closed V -categories over V . As a consequence, we find that the enriched adjunction determined a symmetric monoidal closed adjunction can be obtained by applying a 2-functor and, consequently, is an adjunction in the 2-category of symmetric monoidal closed V -categories. [ABSTRACT FROM AUTHOR]

We give a lower bound for the Waring rank and cactus rank of forms that are invariant under an action of a connected algebraic group. We use this to improve the Ranestad–Schreyer–Shafiei lower bounds for the Waring ranks and cactus ranks of determinants of generic matrices, Pfaffians of generic skew-symmetric matrices, and determinants of generic symmetric matrices. [ABSTRACT FROM AUTHOR]

Abstract: Text: We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space of a global field . After promoting to a hyperfield K, we prove that a hyperring of the form (where R is a ring and is a subgroup of its multiplicative group) is a hyperring extension of K if and only if is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field of positive characteristic, the groupoid of the prime elements of the hyperring is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3LSKD_PfJyc. [ABSTRACT FROM AUTHOR]

MR2415383 AN APPRECIATION OF JEAN BOURGAIN'S WORK 153 [BG08b] J. Bourgain and A. Gamburd, On the spectral gap for finitely-generated subgroups of SU(2), Invent. The four articles in this issue give very clear and insightful accounts of some highlights of Jean's work in functional analysis (Ball), harmonic analysis (Demeter), dispersive partial differential equations (Kenig), and a deconstruction of some of the techniques that Jean invented (Tao). BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 58, Number 2, April 2021, Pages 151-153 https://doi.org/10.1090/bull/1732 Article electronically published on February 12, 2021 AN APPRECIATION OF JEAN BOURGAIN'S WORK PETER SARNAK Jean Bourgain viewed himself as an "analyst", and as the record shows he was uniquely gifted as such, and much more. [Extracted from the article]

We prove a dynamical version of the Mordell-Lang conjecture for étale endomorphisms of quasiprojective varieties. We use p-adic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. As special cases of our result we obtain a new proof of the classical Mordell-Lang conjecture for cyclic subgroups of a semiabelian variety, and we also answer positively a question of Keeler/Rogalski/Stafford for critically dense sequences of closed points of a Noetherian integral scheme. [ABSTRACT FROM AUTHOR]