Article
Mathematics
Eva Elduque, Moises Herradon Cueto, Laurentiu Maxim, Botong Wang
Summary: In this article, we introduce a method that associates a complex algebraic variety with a morphism to a complex affine torus and defines a natural mixed Hodge structure on the corresponding multivariable cohomological Alexander modules. By applying this method, we prove the quasi-unipotence of monodromy, obtain upper bounds on the sizes of the Jordan blocks of monodromy, and explore the change in the Alexander modules after removing fibers of the map. We also provide an example of a variety whose Alexander module has non-semisimple torsion.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics
Bernt Tore Jensen, Alastair King, Xiuping Su
Summary: This paper presents a method of constructing Grassmannian cluster algebras using categorification, and introduces a new module invariant Kappa(M, N) with an intriguing link to mirror symmetry.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics
Nicolas Libedinsky, Geordie Williamson
Summary: This study focuses on the diagrammatic categorification (the anti-spherical category) of the anti-spherical module for any Coxeter group. The results show that Deodhar's (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients and validate Brenti's monotonicity conjecture. The key technical observation is a localization procedure for the anti-spherical category, which enables the construction of a light leaves basis of morphisms. These techniques can be applied to calculate numerous new elements of the p-canonical basis in the anti-spherical module.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics
Jun Maillard
Summary: We prove a categorification of the stable elements formula of Cartan and Eilenberg. Our formula expresses the derived category and the stable module category of a group as a bilimit of the corresponding categories for the p-subgroups.
ADVANCES IN MATHEMATICS
(2022)
Article
Cell Biology
Fabienne Mauxion, Jerome Basquin, Sevim Ozgur, Marion Rame, Jana Albrecht, Ingmar Schaefer, Bertrand Seraphin, Elena Conti
Summary: The CCR4-NOT complex is an important mRNA deadenylase involved in constitutive and regulated mRNA decay pathways in the cytoplasm. While the structure and function of most modules in this complex are known, the N-terminal module remains unclear. However, recent studies using structural approaches have revealed the high-resolution architecture of the human N-terminal module, composed of CNOT1, CNOT10, and CNOT11. Further analysis showed that GGNBP2, a protein involved in tumor suppression and spermatogenesis, interacts with the conserved antenna domain of CNOT11. These findings highlight the N-terminal module as a crucial protein-protein interaction platform.
Article
Mathematics
Farid Aliniaeifard, Nathaniel Thiem
Summary: This paper categorifies combinatorial Hopf algebras by using the representation theory of a tower of p-groups, providing a natural way to prove positivity results. The application of supercharacter theory has made the construction of a Hopf structure on the representation theory of a tower of groups more feasible. Specifically, functors on the representation theory are designed to realize the Hopf structure of the Malvenuto-Reutenauer algebra, where the fundamental basis corresponds to a supercharacter basis.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Manuel Arrayas, Alfredo Tiemblo, Jose L. Trueba
Summary: In this study, new exact solutions of the Maxwell equations in vacuum for null electromagnetic fields are discovered, expanding the concept of the hopfion. The hopfion represents a specific solution of the Maxwell equations in vacuum, where all field lines, both electric and magnetic, are topologically equivalent to closed and linked circles, forming a mathematical structure known as Hopf fibration. A generalization is presented to include other field line topologies, such as the Seifert fibration, where the field lines form linked torus knots. This generalization also encompasses fields that ergodically fill torus surfaces.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Mathematics
Spela Spenko, Michel Van Den Bergh
Summary: This article investigates the application of perverse schobers in certain mathematical theories, and provides a categorization theory for GKZ hypergeometric systems with non-resonant parameters.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Haimiao Chen
Summary: In this paper, we apply the Dijkgraaf-Witten invariants over semidirect products of abelian groups to investigate the constraints on the parameters of a knot surgery that result in a small Seifert 3-manifold. We show that these constraints can be determined from the Alexander polynomial of the knot.
TOPOLOGY AND ITS APPLICATIONS
(2022)
Article
Physics, Particles & Fields
E. Lanina, A. Morozov
Summary: This passage discusses the depth of factorization of terms in the differential (cyclotomic) expansions of knot polynomials in the Chern-Simons theory, i.e. the non-perturbative Wilson averages. It proves the conjecture that the defect can be described as the degree in q (+/- 2) of the fundamental Alexander polynomial, which corresponds to the case of no colors. It also raises the question of whether these Alexander polynomials can be arbitrary integer polynomials of a given degree. Preliminary analysis of specific torus knots provides evidence for positive answers in the case of defect zero knots.
EUROPEAN PHYSICAL JOURNAL C
(2022)
Review
Cell Biology
Maria A. Pajares, Elena Hernandez-Gerez, Milos Pekny, Dolores Perez-Sala
Summary: Alexander disease is a rare neurodegenerative disorder caused by mutations in the glial fibrillary acidic protein, leading to characteristic aggregates in astrocytes. The impact of mutations on protein expression, posttranslational modifications, and protein-protein interactions of the mutant protein may impair astrocyte function and affect neurons. Experimental models are being developed to study the disease and potential therapeutic strategies.
NEURAL REGENERATION RESEARCH
(2023)
Article
Mathematics
Mee Seong Im, Can Ozan Oguz
Summary: This article discusses the structural properties of the group algebra of an n-step iterated wreath product, denoted as CA(n). It explores the centers, centralizers, and right and double cosets of A(n). The results are applied to derive the Mackey theorem and partially describe the natural transformations between induction and restriction functors on the representations of the iterated wreath product tower.
Review
Engineering, Chemical
Xiaotian Lu, Jiachen Huang, Manuel Pinelo, Guoqiang Chen, Yinhua Wan, Jianquan Luo
Summary: This paper summarizes the application of pervaporation (PV) technology in various fields and provides an overview of recent progress and challenges in the optimization of PV modules. The presented rules and tools can offer valuable guidance and suggestions for module modeling and optimization, not only for PV modules but also for other membrane modules.
JOURNAL OF MEMBRANE SCIENCE
(2022)
Article
Mathematics
Iva Halacheva, Anthony Licata, Ivan Losev, Oded Yacobi
Summary: This article studies a finite type semisimple simply-laced Lie algebra g and the action of Artin braid group B of g on a categorical representation of the quantum group Uq(g). It proves that the triangulated equivalence provided by the action is t-exact up to shift when the representation is isotypic, and proves that it is a perverse equivalence with respect to a Jordan-Holder filtration of the representation for general cases. Furthermore, it constructs an action of the cactus group on the crystal of the representation, connecting categorical representation theory and crystal bases. It also provides new proofs for the theorems of Berenstein-Zelevinsky, Rhoades, and Stembridge regarding the action of symmetric group on the Kazhdan-Lusztig basis of its Specht modules.
ADVANCES IN MATHEMATICS
(2023)
Article
Mathematics, Applied
Qingyuan Jiang, Naichung Conan Leung, Ying Xie
Summary: In this paper, the fundamental theorem of Kuznetsov's homological projective duality (HPD) theory is generalized beyond linear sections, showing the existence of semiorthogonal decompositions and equivalence of primitive parts for any two pairs of HP-dual varieties intersecting properly.
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2021)
