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Losev introduced the scheme X of almost commuting elements (i.e., elements commuting upto a rank one element) of g = sp (V) for a symplectic vector space V and discussed its algebro-geometric properties. We construct a Lagrangian subscheme X nil of X and show that it is a complete intersection of dimension dim (g) + 1 2 dim (V) and compute its irreducible onents. We also study the quantum Hamiltonian reduction of the algebra D (g) of differential operators on the Lie algebra g tensored with the Weyl algebra with respect to the action of the symplectic group, and show that it is isomorphic to the spherical subalgebra of a certain rational Cherednik algebra of Type C. We contruct a category C c of D -modules whose characteristic variety is contained in X nil and construct an exact functor from this category to the category O of the above rational Cherednik algebra. Simple objects of the category C c are mirabolic analogs of Lusztig's character sheaves. [ABSTRACT FROM AUTHOR]

The Plücker matrix B L (n , E) of the Lagrangian Grassmannian L (n , E) , is determined by the linear envelope 〈 L (n , E) 〉 of the Lagrangian Grassmannian. The linear envelope 〈 L (n , E) 〉 is the intersection of linear relations of Plücker of Lagrangian Grassmannian, defined here. The Plücker matrix B L (n , E) is a direct sum of the incidence matrix of the configuration of subsets. These matrices determine the isotropy index r n and r n -atlas which are invariants associated with the symplectic vector space E. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present an algebraic theoretical basis for the study of ω -Hamiltonian vector fields defined on a symplectic vector space (V , ω) with respect to coordinates that are not necessarily symplectic. We introduce the concepts of ω -symplectic and ω -semisymplectic groups, and describe some of their properties that may not coincide with the classical context. We show that the Lie algebra of such groups is a useful tool in the recognition and construction of ω -Hamiltonian vector fields. [ABSTRACT FROM AUTHOR]

Supercuspidal representations are usually infinite-dimensional, so the size of such a representation cannot be measured by its dimension; the formal degree is a better alternative. Hiraga, Ichino, and Ikeda conjectured a formula for the formal degree of a supercuspidal in terms of its L-parameter only. Our first main result is to compute the formal degrees of the supercuspidal representations constructed by Yu. Our second result, using the first, is to verify that Kaletha's regular supercuspidal L-packets satisfy the conjecture. [ABSTRACT FROM AUTHOR]

Let (G, H) be one of the equal rank reductive dual pairs M p 2 n , O 2 n + 1 or U n , U n over a nonarchimedean local field of characteristic zero. It is well-known that the theta correspondence establishes a bijection between certain subsets, say G ^ θ and H ^ θ , of the tempered duals of G and H. We prove that this bijection arises from an equivalence between the categories of representations of two C ∗ -algebras whose spectra are G ^ θ and H ^ θ . This equivalence is implemented by the induction functor associated to a Morita equivalence bimodule (in the sense of Rieffel) which we construct using the oscillator representation. As an immediate corollary, we deduce that the bijection is functorial and continuous with respect to weak inclusion. We derive further consequences regarding the transfer of characters and preservation of formal degrees. [ABSTRACT FROM AUTHOR]

In this work we make the connection between the Unruh–DeWitt (UDW) particle detector model applied to quantum field theory in curved spacetimes and the rigorous construction of the spin-boson (SB) model. With some modifications, we show that existing results about the existence of a SB ground state can be adapted to the UDW model. In the most relevant scenario involving massless scalar fields in (3+1)-dimensional globally hyperbolic spacetimes, where the UDW model describes a simplified model of light–matter interaction, we argue that common choices of the spacetime smearing functions regulate the ultraviolet behaviour of the model but can still exhibit infrared (IR) divergences. In particular, this implies the well-known expectation that the joint interacting Hilbert space of the model cannot be described by the tensor product of a two-dimensional complex Hilbert space and the Fock space of the vacuum representation. We discuss the conditions under which this problem does not arise and the relevance of the operator-algebraic approach for better understanding of particle detector models and their applications.Our work clarifies the connection between obstructions due to Haag's theorem and IR bosons in the SB models, and paves the way for more rigorous study of entanglement and communication in the UDW framework involving multiple detectors. [ABSTRACT FROM AUTHOR]

We look at a momentum map associated with the Heisenberg group. We show that the cocycle associated with its momentum mapping is the value of a modulus of an associated coadjoint orbit. [ABSTRACT FROM AUTHOR]

We study a family of (possibly non topological) deformations of BF theory for the Lie algebra obtained by quadratic extension of so (3 , 1) by an orthogonal module. The resulting theory, called quadratically extended General Relativity (qeGR), is shown to be classically equivalent to certain models of gravity with dynamical torsion. The classical equivalence is shown to promote to a stronger notion of equivalence within the Batalin–Vilkovisky formalism. In particular, both Palatini–Cartan gravity and a deformation thereof by a dynamical torsion term, called (quadratic) generalised Holst theory, are recovered from the standard Batalin–Vilkovisky formulation of qeGR by elimination of generalised auxiliary fields. [ABSTRACT FROM AUTHOR]

We introduce the wrapped Donaldson--Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi--Yau 3-fold.We compute the wrapped Floer cohomology of a Lagrangian section and find that it is the algebra of functions on the Hori--Vafa mirror. The latter result is the key step in proving homological mirror symmetry for this case. The techniques developed here allow the construction in general of the wrapped Fukaya category on an open Calabi--Yau manifold carrying an SYZ fibration with nice behavior at infinity. We discuss the relation of this to the algebraic vs analytic aspects of mirror symmetry. [ABSTRACT FROM AUTHOR]

Buchholz and Grundling (Commun Math Phys 272:699–750, 2007) introduced a C∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hbox {C}^*$$\end{document}-algebra called the resolvent algebra as a canonical quantisation of a symplectic vector space and demonstrated that this algebra has several desirable features. We define an analogue of their resolvent algebra on the cotangent bundle T∗Tn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T^*\mathbb {T}^n$$\end{document} of an n-torus by first generalising the classical analogue of the resolvent algebra defined by the first author of this paper in earlier work (van Nuland in J Funct Anal 277:2815–2838, 2019) and subsequently applying Weyl quantisation. We prove that this quantisation is almost strict in the sense of Rieffel and show that our resolvent algebra shares many features with the original resolvent algebra. We demonstrate that both our classical and quantised algebras are closed under the time evolutions corresponding to large classes of potentials. Finally, we discuss their relevance to lattice gauge theory. [ABSTRACT FROM AUTHOR]

The mixed equation, defined as a combination of the anti-self-duality equation in gauge theory and Cauchy–Riemann equation in symplectic geometry, is studied. In particular, regularity and Fredholm properties are established for the solutions of this equation, and it is shown that the moduli spaces of solutions to the mixed equation satisfy a compactness property which combines Uhlenbeck and Gormov compactness theorems. The results of this paper are used in a sequel to study the Atiyah–Floer conjecture. [ABSTRACT FROM AUTHOR]

This article is an interview with mathematician Elisabetta Strickland, who has been actively involved in promoting women in mathematics. The interview covers her family background, her passion for mathematics, and her career in the field. Strickland discusses her preference for abstract proofs and her work in algebraic geometry, particularly in studying special varieties with a lot of structure. She also mentions her interest in groups and her work in invariant theory. The conversation explores various topics including mathematics, career experiences, women in mathematics, and personal interests. It highlights the importance of collaboration, the challenges faced by women in mathematics, the role of travel and conferences in the field, and the value of face-to-face interactions. The interview also reveals Strickland's interests in literature, writing, sailing, photography, and interior decorating. [Extracted from the article]

We introduce a notion of strong closing property of contact forms, inspired by the C ∞ closing lemma for Reeb flows in dimension three. We then prove a sufficient criterion for strong closing property, which is formulated by considering a monoidal functor from a category of manifolds with contact forms to a category of filtered vector spaces. As a potential application of this criterion, we propose a conjecture which says that a standard contact form on the boundary of any symplectic ellipsoid satisfies strong closing property. [ABSTRACT FROM AUTHOR]

High-resolution simulations of particle-based kinetic plasma models typically require a high number of particles and thus often become computationally intractable. This is exacerbated in multi-query simulations, where the problem depends on a set of parameters. In this work, we derive reduced order models for the semi-discrete Hamiltonian system resulting from a geometric particle-in-cell approximation of the parametric Vlasov–Poisson equations. Since the problem's nondissipative and highly nonlinear nature makes it reducible only locally in time, we adopt a nonlinear reduced basis approach where the reduced phase space evolves in time. This strategy allows a significant reduction in the number of simulated particles, but the evaluation of the nonlinear operators associated with the Vlasov–Poisson coupling remains computationally expensive. We propose a novel reduction of the nonlinear terms that combines adaptive parameter sampling and hyper-reduction techniques to address this. The proposed approach allows decoupling the operations having a cost dependent on the number of particles from those that depend on the instances of the required parameters. In particular, in each time step, the electric potential is approximated via dynamic mode decomposition (DMD) and the particle-to-grid map via a discrete empirical interpolation method (DEIM). These approximations are constructed from data obtained from a past temporal window at a few selected values of the parameters to guarantee a computationally efficient adaptation. The resulting DMD-DEIM reduced dynamical system retains the Hamiltonian structure of the full model, provides good approximations of the solution, and can be solved at a reduced computational cost. [ABSTRACT FROM AUTHOR]

In this paper we study coisotropic reduction in different types of dynamics according to the geometry of the corresponding phase space. The relevance of coisotropic reduction is motivated by the fact that these dynamics can always be interpreted as Lagrangian or Legendrian submanifolds. Furthermore, Lagrangian or Legendrian submanifolds can be reduced by a coisotropic one. [ABSTRACT FROM AUTHOR]

Take a sequence of contactomorphisms of a contact three-manifold that C^0-converges to a homeomorphism. If the images of a Legendrian knot limit to a smooth knot under this sequence, we show that it is contactomorphic to the original knot. We prove this by establishing that, on one hand, non–Legendrian knots admit a type of contact-squashing (similar to squeezing) onto transverse knots while, on the other hand, Legendrian knots do not admit such a squashing. The non-trivial input from contact topology that is needed is (a local version of) the Thurston–Bennequin inequality. [ABSTRACT FROM AUTHOR]

We exhibit an infinite family of discrete subgroups of Sp 4 (R) which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations with parameters N - 3 2 N , N - 1 2 N , N + 1 2 N , N + 3 2 N at infinity and maximal unipotent monodromy at zero, for any integer N ≥ 4 . Additionally, we relate the cones used for ping-pong in R 4 with crooked surfaces, which we then use to exhibit domains of discontinuity for the monodromy groups in the Lagrangian Grassmannian. These domains of discontinuity lead to uniformizations of variations of Hodge structure with Hodge numbers (1, 1, 1, 1). [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the natural action of the real symplectic group, Sp (4 , R) , on the algebraic set of 4-dimensional Lie algebras admitting symplectic structures and to give a complete classification of orbit closures. We present some applications of such classification to the study of the Ricci curvature of left-invariant almost Kähler structures on four dimensional Lie groups. [ABSTRACT FROM AUTHOR]

We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a 'stop removal equals localization' result, and (4) that the Fukaya–Seidel category of a Lefschetz fibration with Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Künneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful. [ABSTRACT FROM AUTHOR]

We identify Whittaker vectors for W k (g) -modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of G-bundles over P 2 for G a complex simple Lie group, can be computed by a non-commutative version of the Chekhov–Eynard–Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure N = 2 four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods. [ABSTRACT FROM AUTHOR]

We present a novel background-independent framework for cosmic inflation, starting with a matrix model. In this framework, inflation is portrayed as a dynamic process responsible for the generation of both space and time. This stands in contrast to conventional inflation, which is characterized as a mere (exponential) expansion of an already existing spacetime, driven by the vacuum energy associated with an inflaton field. We observe that the cosmic inflation is triggered by the condensate of Planck energy into a vacuum and responsible for the dynamical emergence of spacetime. The emergent spacetime picture admits a background-independent formulation so that the inflation is described by a conformal Hamiltonian system which requires neither an inflaton field nor an ad hoc inflation potential. This implies that the emergent spacetime may incapacitate all the rationales to introduce the multiverse hypothesis. [ABSTRACT FROM AUTHOR]

We develop a geometric approach to Poisson electrodynamics, that is, the semi-classical limit of noncommutative U(1) gauge theory. Our framework is based on an integrating symplectic groupoid for the underlying Poisson brackets, which we interpret as the classical phase space of a point particle on noncommutative spacetime. In this picture gauge fields arise as bisections of the symplectic groupoid while gauge transformations are parameterized by Lagrangian bisections. We provide a geometric construction of a gauge invariant action functional which minimally couples a dynamical charged particle to a background electromagnetic field. Our constructions are elucidated by several explicit examples, demonstrating the appearances of curved and even compact momentum spaces, the interplay between gauge transformations and spacetime diffeomorphisms, as well as emergent gravity phenomena. [ABSTRACT FROM AUTHOR]

We prove a seesaw identity for theta functions with polynomials, and establish a formula for the corresponding theta contractions. We then use it to determine the restrictions of theta lifts to sub-Grassmannians, with some applications. [ABSTRACT FROM AUTHOR]

Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important properties of the system. For example, a time-evolution PDE may have an observable that satisfies a local conservation law, such as the multisymplectic conservation law for Hamiltonian PDEs. We introduce the concept of functional equivariance, a natural sense in which a numerical integrator may preserve the dynamics satisfied by certain classes of observables, whether or not they are invariant. After developing the general framework, we use it to obtain results on methods preserving local conservation laws in PDEs. In particular, integrators preserving quadratic invariants also preserve local conservation laws for quadratic observables, and symplectic integrators are multisymplectic. [ABSTRACT FROM AUTHOR]

Gauging a finite group 0-form symmetry G of a quantum field theory (QFT) results in a QFT with a Rep(G) symmetry implemented by Wilson lines. The group G determines the fusion of Wilson lines. However, in general, the fusion rules of Wilson lines do not determine G. In this paper, we study the properties of G that can be determined from the fusion rules of Wilson lines and surface operators obtained from higher-gauging Wilson lines. This is in the spirit of Richard Brauer who asked what information in addition to the character table of a finite group needs to be known to determine the group. We show that fusion rules of surface operators obtained from higher-gauging Wilson lines can be used to distinguish infinite pairs of groups which cannot be distinguished using the fusion of Wilson lines. We derive necessary conditions for two non-isomorphic groups to have the same surface operator fusion and find a pair of such groups. [ABSTRACT FROM AUTHOR]

Following Nag–Sullivan, we study the representation of the group Diff + (S 1) of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space. This lift provides quantization of conformal welding, gives a new representative of the Bott–Virasoso cocycle class, and leads to a surprising identity for the Takhtajan–Teo energy functional on Diff + (S 1) . [ABSTRACT FROM AUTHOR]

What is striking about de Broglie's foundational work on wave–particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein–Gordon equation, he was led to the revolutionary idea that a variable rest mass was essential. DeWitt later explained that in order to obtain a covariant quantum Hamiltonian, one must supplement the classical Hamiltonian with an additional energy ℏ 2 Q from which the quantum potential emerges, a potential that Berry has recently shown also arises in classical wave optics. In this paper, we show how these ideas emerge from an essentially geometric structure in which the information normally carried by the wave function is contained within the algebraic description of the geometry itself, within an element of a minimal left ideal. We establish the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, we have the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow. [ABSTRACT FROM AUTHOR]

The Plücker matrix BL(n,E) of the Lagrangian Grassmannian L(n,E), is determined by the linear envelope ⟨L(n,E)⟩ of the Lagrangian Grassmannian. The linear envelope ⟨L(n,E)⟩ is the intersection of linear relations of Plücker of Lagrangian Grassmannian, defined here. The Plücker matrix BL(n,E) is a direct sum of the incidence matrix of the configuration of subsets. These matrices determine the isotropy index rn and rn-atlas which are invariants associated with the symplectic vector space E.