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Let H (n , q 2) be a non-degenerate Hermitian variety of P G (n , q 2) , n ≥ 2 . Let N U (n + 1 , q 2) be the graph whose vertices are the points of P G (n , q 2) \ H (n , q 2) and two vertices u, v are adjacent if the line joining u and v is tangent to H (n , q 2) . Then N U (n + 1 , q 2) is a strongly regular graph. In this paper we show that the automorphism group of the graph N U (3 , q 2) is isomorphic either to P Γ U (3 , q) , the automorphism group of the projective unitary group PGU(3, q), or to S 3 ≀ S 4 , according as q ≠ 2 , or q = 2 . [ABSTRACT FROM AUTHOR]

Let f : M → M be a self-map of a compact manifold and n ∈ N . In general, the least number of n-periodic points in the smooth homotopy class of f may be much bigger than in the continuous homotopy class. For a class of spaces, including compact Lie groups, a necessary condition for the equality of the above two numbers, for each iteration f n , appears. Here we give the explicit form of the graph of orbits of Reidemeister classes GOR (f ∗) for self-maps of projective unitary group PSU(2) and of P S U (2) × P S U (2) satisfying the necessary condition. The structure of the graphs implies that for self-maps of the above spaces the necessary condition is also sufficient for the smooth minimal realization of n-periodic points for all iterations. [ABSTRACT FROM AUTHOR]

The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that { π Λ (m , n) g 1 ⊕ ⋯ ⊕ π Λ (m , n) g k } m , n ∈ Z d is a frame for L 2 (R d) ⊕ ⋯ ⊕ L 2 (R d) if and only if ∪ i = 1 k { π Λ o (m , n) g i } m , n ∈ Z d is a Riesz sequence, and ∪ i = 1 k { π Λ (m , n) g i } m , n ∈ Z d is a frame for L 2 (R d) if and only if { π Λ o (m , n) g 1 ⊕ ⋯ ⊕ π Λ o (m , n) g k } m , n ∈ Z d is a Riesz sequence, where π Λ and π Λ o is a pair of Gabor representations restricted to a time–frequency lattice Λ and its adjoint lattice Λ o in R d × R d . [ABSTRACT FROM AUTHOR]

The vast majority of quantum states and unitaries have circuit complexity exponential in the number of qubits. In a similar vein, most of them also have exponential minimum description length, which makes it difficult to pinpoint examples of exponential complexity. In this work, we construct examples of constant description length but exponential circuit complexity. We provide infinite families such that each element requires an exponential number of two-qubit gates to be generated exactly from a product and where the same is true for the approximate generation of the vast majority of elements in the family. The results are based on sets of large transcendence degree and discussed for tensor networks, diagonal unitaries and maximally coherent states. [ABSTRACT FROM AUTHOR]

In a companion paper (Arone et al. in Noncommutative CW-spectra as enriched presheaves on matrix algebras, arXiv:2101.09775, 2021) we introduced the stable ∞ -category of noncommutative CW-spectra, which we denoted NSp . Let M denote the full spectrally enriched subcategory of NSp whose objects are the non-commutative suspension spectra of matrix algebras. In Arone et al. (2021) we proved that NSp is equivalent to the ∞ -category of spectral presheaves on M . In this paper we investigate the structure of M , and derive some consequences regarding the structure of NSp . To begin with, we introduce a rank filtration of M . We show that the mapping spectra of M map naturally to the connective K-theory spectrum ku, and that the rank filtration of M is a lift of the classical rank filtration of ku. We describe the subquotients of the rank filtration in terms of spaces of direct-sum decompositions which also arose in the study of K-theory and of Weiss's orthogonal calculus. We prove that the rank filtration stabilizes rationally after the first stage. Using this we give an explicit model of the rationalization of NSp as presheaves of rational spectra on the category of finite-dimensional Hilbert spaces and unitary transformations up to scaling. Our results also have consequences for the p-localization and the chromatic localization of M . [ABSTRACT FROM AUTHOR]

The link between modular functions and algebraic functions was a driving force behind the 19th century study of both. Examples include the solutions by Hermite and Klein of the quintic via elliptic modular functions and the general sextic via level 2 hyperelliptic functions. This paper aims to apply modern arithmetic techniques to the circle of "resolvent problems" formulated and pursued by Klein, Hilbert and others. As one example, we prove that the essential dimension at p = 2 for the symmetric groups S n is equal to the essential dimension at 2 of certain S n -coverings defined using moduli spaces of principally polarized abelian varieties. Our proofs use the deformation theory of abelian varieties in characteristic p, specifically Serre-Tate theory, as well as a family of remarkable mod 2 symplectic S n -representations constructed by Jordan. As shown in an appendix by Nate Harman, the properties we need for such representations exist only in the p = 2 case. In the second half of this paper we introduce the notion of E -versality as a kind of generalization of Kummer theory, and we prove that many congruence covers are E -versal. We use these E -versality result to deduce the equivalence of Hilbert's 13th Problem (and related conjectures) with problems about congruence covers. [ABSTRACT FROM AUTHOR]

In this paper we consider a group generated by two unipotent parabolic elements of SU (2 , 1) with distinct fixed points. We give several conditions that guarantee the group is discrete and free. We also give a result on the diameter of a finite R -circle in the Heisenberg group. [ABSTRACT FROM AUTHOR]

Constructing hemisystems of the Hermitian surface is a well known, apparently difficult, problem in Finite geometry. So far, a few infinite families and some sporadic examples have been constructed. One of the different approaches relies on the Fuhrmann-Torres maximal curve and provides a hemisystem in P G (3 , p 2) for every prime p of the form p = 1 + 4 a 2 , a even. Here we show that this approach also works in P G (3 , p 2) for every prime p = 1 + 4 a 2 , a odd. The resulting hemisystem gives rise to two weight linear codes and strongly regular graphs whose properties are also investigated. [ABSTRACT FROM AUTHOR]

In the present work, we study the Hausdorff dimension of the limit set of Schottky groups on the boundary of the complex hyperbolic group via the Eigenvalue algorithm as in [17]. The visual geometry on H C 2 ¯ allows a direct application of the Eigenvalue algorithm, but at the same time, it hardens the computations. Using the Heisenberg structure on ∂ H C 2 \ { ∞ } and the Cygan metric, we propose a Markov partition associated with the group that simplifies the application of the Eigenvalue algorithm. [ABSTRACT FROM AUTHOR]

This note provides new criteria on a unimodular group G and a discrete series representation (π , H π) of formal degree d π > 0 under which any lattice Γ ≤ G with vol (G / Γ) d π ≤ 1 (resp. vol (G / Γ) d π ≥ 1 ) admits g ∈ H π such that π (Γ) g is a frame (resp. Riesz sequence). The results apply to all projective discrete series of exponential Lie groups. [ABSTRACT FROM AUTHOR]

In pursuit of efficient time-frequency representation of Clifford-valued signals, we propose a novel integral transform coined as the Clifford-valued wave-packet transform. In the first instance, all the fundamental properties of the Clifford-valued wave-packet transform including the inner product relation, inversion formula and range theorem are studied. This is followed by the formulation of several uncertainty inequalities, such as the Heisenberg–Pauli–Weyl inequality, Pitt's inequality and logarithmic inequality for the proposed transform. We culminate our study by demonstrating the performance and efficiency of the Clifford-valued wave-packet transform for detection and orientation of pointwise singularities in the context of three-dimensional signal analysis. [ABSTRACT FROM AUTHOR]

We classify Coble surfaces with finite automorphism group in characteristic p ≥ 0 , p ≠ 2 . There are exactly 9 isomorphism classes of such surfaces. [ABSTRACT FROM AUTHOR]

The quantum transition probability assignment is an equiareal transformation from the annulus of symplectic spinorial amplitudes to the disk of complex state vectors, which makes it equivalent to the equiareal projection of Archimedes. The latter corresponds to a symplectic synchronization method, which applies to the quantum phase space in view of Weyl’s quantization approach involving an Abelian group of unitary ray rotations. We show that Archimedes’ method of synchronization, in terms of a measure-preserving transformation to an equiareal disk, imposes the integrality of the quantum of action, and requires the extension of the classical moment map from the real line to the circle. Additionally, the same synchronization method is encoded in the structure of the Heisenberg group, viewed as a principal bundle with a connection, whose curvature and anholonomy is expressed in terms of area bounding loops in relation to the underlying Abelian shadow on a symplectic plane. In this manner, we show that the geometric phase pertains to the minimal synchronized area π ħ of the 2-d symplectic Abelian shadow of the symplectic ball, modulo Z . The integrality condition naturally leads to the consideration of modular commutative observables pertaining to the role of the discrete Heisenberg group. We prove that the structural transition from non-commutativity to modular commutativity in accordance to Weyl’s group-theoretic commutation relations takes place via universal factorization through the discrete Heisenberg group. In this way, we derive a homology-theoretic formulation of the synchronization method in terms of the area-bounding cells of the modular lattice R 2 Z 2 in relation to any Abelian symplectic shadow. Thus, we finally obtain the physical interpretation of the analytic representation of quantum states as theta functions corresponding to the sections of a complex line bundle with an integral symplectic structure. [ABSTRACT FROM AUTHOR]

In a seminal series of papers from the 80's, Lubotzky, Phillips and Sarnak applied the Ramanujan–Petersson Conjecture for G L 2 (Deligne's theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat–Tits trees associated with S L 2 (Q p) . As a result, they obtained explicit Ramanujan Cayley graphs from P S L 2 F p , as well as optimal topological generators ("Golden Gates") for the compact Lie group PU(2). In higher dimension, the naive generalization of the Ramanujan Conjecture fails, due to the phenomenon of endoscopic lifts. In this paper we overcome this problem for P U 3 by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat–Tits buildings associated with S L 3 (Q p) and S U 3 (Q p) , while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from P S L 3 (F p) and P S U 3 (F p) , as well as golden gates for PU(3). [ABSTRACT FROM AUTHOR]

We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174, 960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds. [ABSTRACT FROM AUTHOR]

We study geometric structures on the complement of a toric mirror arrangement associated with a root system. Inspired by those root system hypergeometric functions found by Heckman–Opdam, and in view of the work of Couwenberg–Heckman–Looijenga on the geometric structures on projective arrangement complements, we consider a family of connections on a total space, namely, a C × -bundle on the complement of a toric mirror arrangement (=finite union of hypertori, determined by a root system). We prove that these connections are torsion free and flat, and hence define a family of affine structures on the total space, which is equivalent to a family of projective structures on the toric arrangement complement. We then determine a parameter region for which the projective structure admits a locally complex hyperbolic metric. In the end, we show that the space in question can be biholomorphically mapped onto a divisor complement of a ball quotient if the Schwarz conditions are invoked. [ABSTRACT FROM AUTHOR]

We develop a general operator algebraic method which focuses on projective representations of symmetry group for proving Lieb–Schultz–Mattis type theorems, i.e., no-go theorems that rule out the existence of a unique gapped ground state (or, more generally, a pure split state), for quantum spin chains with on-site symmetry. We first prove a theorem for translation invariant spin chains that unifies and extends two theorems proved by two of the authors (Ogata and Tasaki, Commun. Math. Phys. 372 951–962, (2019) https://doi.org/10.1007/s00220-019-03343-5). We then prove a Lieb–Schultz–Mattis type theorem for spin chains that are invariant under the reflection about the origin and not necessarily translation invariant. [ABSTRACT FROM AUTHOR]

A Correction to this paper has been published: https://doi.org/10.1007/s00605-020-01421-8 [ABSTRACT FROM AUTHOR]

For a manifold M with an integral closed 3-form ω, we construct a PU(H)-bundle and a Lie groupoid over its total space, together with a curving in the sense of gerbes. If the form is non-degenerate, we furthermore give a natural Lie 2-algebra quasi-isomorphism from the observables of (M, ω) to the weak symmetries of the above geometric structure, generalising the prequantisation map of Kostant and Souriau. [ABSTRACT FROM AUTHOR]

We study relations between special elliptic isometries in the complex hyperbolic plane. Relations of lengths 2, 3, and 4 are fully classified. Some relative SU (2 , 1) -character varieties of the quadruply punctured sphere are described and applied to the study of length 5 relations. [ABSTRACT FROM AUTHOR]

Mark and Paupert devised a general method for obtaining presentations for arithmetic non-cocompact lattices, Γ , in isometry groups of negatively curved symmetric spaces. The method involves a classical theorem of Macbeath applied to a Γ -invariant covering by horoballs of the negatively curved symmetric space upon which Γ acts. In this paper, we will discuss the application of their method to the Picard modular groups, PU (2 , 1 ; O d) , when d = 2 , 11 , and obtain presentations for these groups, which completes the list of presentations for Picard modular groups whose entries lie in Euclidean domains, namely those with d = 1 , 2 , 3 , 7 , 11 . [ABSTRACT FROM AUTHOR]

We study equivariant linear maps between finite-dimensional matrix algebras, as introduced in Collins et al. (Linear Algebra Appl 555:398–411, 2018). These maps satisfy an algebraic property which makes it easy to study their positivity or k-positivity. They are therefore particularly suitable for applications to entanglement detection in quantum information theory. We characterize their Choi matrices. In particular, we focus on a subfamily that we call (a, b)-unitarily equivariant. They can be seen as both a generalization of maps invariant under unitary conjugation as studied by Bhat (Banach J Math Anal 5(2):1–5, 2011) and as a generalization of the equivariant maps studied in Collins et al. (2018). Using representation theory, we fully compute them and study their graphical representation and show that they are basically enough to study all equivariant maps. We finally apply them to the problem of entanglement detection and prove that they form a sufficient (infinite) family of positive maps to detect all k-entangled density matrices. [ABSTRACT FROM AUTHOR]

We describe the nonabelian simple finite groups whose every nonsolvable local maximal subgroup is a Hall subgroup, and the nonsolvable finite groups whose all nonsolvable superlocals are Hall subgroups. [ABSTRACT FROM AUTHOR]

We describe classes of local coordinates on the Carnot–Carathéodory spaces of lower smoothness which permit the homogeneous approximation of quasimetrics and basis vector fields. We establish the minimal smoothness that is required for these classes to coincide with the class of the already-described privileged coordinates in the infinite smoothness case. Moreover, we apply these results to prove the analogs of the available theorems in the case of the canonical coordinates of the second kind. Also, we prove some convergence theorems in quasimetric spaces. [ABSTRACT FROM AUTHOR]

We construct a natural prequantization space over a monotone product of a toric manifold and an arbitrary number of complex Grassmannians of 2-planes in even-dimensional complex spaces, and prove that the universal cover of the identity component of the contactomorphism group of its total space carries a nonzero homogeneous quasi-morphism. The construction uses Givental's nonlinear Maslov index and a reduction theorem for quasi-morphisms on contactomorphism groups previously established together with M. S. Borman. We explore applications to metrics on this group and to symplectic and contact rigidity. In particular we obtain a new proof that the quaternionic projective space H P n - 1 , naturally embedded in the Grassmannian G 2 (C 2 n) as a Lagrangian, cannot be displaced from the real part G 2 (R 2 n) by a Hamiltonian isotopy. [ABSTRACT FROM AUTHOR]

For a fixed polynomial Δ , we study the number of polynomials f of degree n over F q such that f and f + Δ are both irreducible, an F q T -analogue of the twin primes problem. In the large-q limit, we obtain a lower-order term for this count if we consider non-monic polynomials, which depends on Δ in a manner which is consistent with the Hardy–Littlewood Conjecture. We obtain a saving of q if we consider monic polynomials only and Δ is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in Δ . This allows us to obtain additional saving from equidistribution results for L-functions. We do all this in a combinatorial framework that applies to more general arithmetic functions than the indicator function of irreducibles, including the Möbius function and divisor functions. [ABSTRACT FROM AUTHOR]

The concept of full points of abstract unitals has been introduced by Korchmáros, Siciliano and Szőnyi as a tool for the study of projective embeddings of abstract unitals. In this paper we give a detailed description of the combinatorial and geometric structure of the sets of full points in abstract unitals of finite order. [ABSTRACT FROM AUTHOR]

Every embedding of a hermitian unital with at least four points on a block into any pappian projective plane is standard, i.e. it originates from an inclusion of the pertinent fields. This result about embeddings also allows us to determine the full automorphism groups of (generalized) hermitian unitals. [ABSTRACT FROM AUTHOR]

By use of H. C. Wang's bound on the radius of a ball embedded in the fundamental domain of a lattice of a semisimple Lie group, we construct an explicit lower bound for the volume of a quaternionic hyperbolic orbifold that depends only on dimension. [ABSTRACT FROM AUTHOR]

This note presents an analysis of a class of operator algebras constructed as cross-sectional algebras of flat holomorphic matrix bundles over a finitely bordered Riemann surface. These algebras are partly inspired by the bundle shifts of Abrahamse and Douglas. The first objective is to understand the boundary representations of the containing C ∗ -algebra, i.e. Arveson's noncommutative Choquet boundary for each of our operator algebras. The boundary representations of our operator algebras for their containing C ∗ -algebras are calculated, and it is shown that they correspond to evaluations on the boundary of the Riemann surface. Secondly, we show that our algebras are Azumaya algebras, the algebraic analogues of n-homogeneous C ∗ -algebras. [ABSTRACT FROM AUTHOR]

Let Sp (2 , 1) be the isometry group of the quaternionic hyperbolic plane H H 2 . An element g in Sp (2 , 1) is hyperbolic if it fixes exactly two points on the boundary of H H 2 . We classify pairs of hyperbolic elements in Sp (2 , 1) up to conjugation. A hyperbolic element of Sp (2 , 1) is called loxodromic if it has no real eigenvalue. We show that the set of Sp (2 , 1) conjugation orbits of irreducible loxodromic pairs is a (C P 1) 4 bundle over a topological space that is locally a semi-analytic subspace of R 13 . We use the above classification to show that conjugation orbits of 'geometric' representations of a closed surface group (of genus g ≥ 2 ) into Sp (2 , 1) can be determined by a system of 42 g - 42 real parameters. Further, we consider the groups Sp (1 , 1) and GL (2 , H) . These groups also act by the orientation-preserving isometries of the four and five dimensional real hyperbolic spaces respectively. We classify conjugation orbits of pairs of hyperbolic elements in these groups. These classifications determine conjugation orbits of 'geometric' surface group representations into these groups. [ABSTRACT FROM AUTHOR]

Let π:P→M be a principal bundle and p an invariant polynomial of degree r on the Lie algebra of the structure group. The theory of Chern-Simons differential characters is exploited to define a homology map χk:H2r-k-1(M)×Hk(F/G)→R/Z, for k

For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of Kostant's symplectic spinor bundle. Defining a Hilbert C*-structure on this bundle for a suitable C*-algebra, we obtain an elliptic C*-complex in the sense of Mishchenko-Fomenko. Its cohomology groups appear to be finitely generated projective Hilbert C*-modules. The paper can serve as a guide for handling differential complexes and PDEs on Hilbert bundles. [ABSTRACT FROM AUTHOR]

We compute degree 3 unramified cohomology of adjoint semisimple group. New classes of non-rational adjoint semisimple groups are found. [ABSTRACT FROM AUTHOR]

Using quantum representations of mapping class groups, we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups. [ABSTRACT FROM AUTHOR]

We study the mean square of sums of the kth divisor function dk(n) over short intervals and arithmetic progressions for the rational function field over a finite field of q elements. In the limit as q→∞ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of dk(n) in terms of a lattice point count. This lattice point count can in turn be calculated in terms of a certain piecewise polynomial function, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known. [ABSTRACT FROM AUTHOR]

We consider separately radial (with corresponding group T) and radial (with corresponding group U( n)) symbols on the projective space P(C), as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the C*-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the C*-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between T and U( n). [ABSTRACT FROM AUTHOR]

We exhibit a class of Schottky subgroups of $$\mathbf {PU}(1,n)$$ ( $$n \ge 2$$ ) which we call well-positioned and show that the Hausdorff dimension of the limit set $$\Lambda _\Gamma $$ associated with such a subgroup $$\Gamma $$ , with respect to the spherical metric on the boundary of complex hyperbolic n-space, is equal to the growth exponent $$\delta _\Gamma $$ . For general $$\Gamma $$ we establish (under rather mild hypotheses) a lower bound involving the dimension of the Patterson-Sullivan measure along boundaries of complex geodesics. Our main tool is a version of the celebrated Ledrappier-Young theorem. [ABSTRACT FROM AUTHOR]

In this article, we provide a detailed construction and analysis of the mathematical conformal field theory of the free fermion, defined in the sense of Graeme Segal. We verify directly that the operators assigned to disks with two disks removed correspond to vertex operators, and use this to deduce analytic properties of the vertex operators. One of the main tools used in the construction is the Cauchy transform for Riemann surfaces, for which we establish several properties analogous to those of the classical Cauchy transform in the complex plane. [ABSTRACT FROM AUTHOR]

Let c: SU( n) → PSU( n) = SU( n)/ℤ be the quotient map of the special unitary group SU( n) by its center subgroup ℤ. We determine the induced homomorphism c : H (PSU( n)) → H (SU( n)) on cohomologies by computing with the prime orders of binomial coefficients. [ABSTRACT FROM AUTHOR]

We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, 'Unperformed experiments have no results.' The tools of quantum information theory, and in particular the symmetric informationally complete (SIC) measurements, provide a concise expression of how exactly Peres's dictum holds true. That expression is a constraint on how the probability distributions for outcomes of different, hypothetical and mutually exclusive experiments ought to mesh together, a type of constraint not foreseen in classical thinking. Taking this as our foundational principle, we show how to reconstruct the formalism of quantum theory in finite-dimensional Hilbert spaces. The central variety of mathematical entity in our reconstruction is the qplex, a very particular type of subset of a probability simplex. Along the way, by closely studying the symmetry properties of qplexes, we derive a condition for the existence of a d-dimensional SIC. Graphical abstract: [ABSTRACT FROM AUTHOR]

The sister of Eisenstein-Picard modular group was described in Parker (Duke Math J 94(3):433-464, 1998). In fact, one can define the sister groups of all Picard modular groups. In particular, we find the generators of the sister of Euclidean Picard modular groups by using a geometric method. [ABSTRACT FROM AUTHOR]

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Let M be a real hypersurface in a complex space form M ( c), $${c \neq 0}$$ . In this paper we prove that if $${R_{\xi}(\phi A - A\phi) + (\phi A - A\phi)R_{\xi} = 0}$$ holds on M, then M is a Hopf hypersurface, where $${R_{\xi}}$$ is the structure Jacobi operator, A is the shape operator of M in M ( c) and $${\phi}$$ is the tangential projection of the complex structure of M ( c). We characterize such Hopf hypersurfaces of M ( c). [ABSTRACT FROM AUTHOR]

We study the variance of sums of the indicator function of square-full polynomials in both arithmetic progressions and short intervals. Our work is in the context of the ring $$\mathbb {F}_q[T]$$ of polynomials over a finite field $$\mathbb {F}_q$$ of q elements, in the limit $$q\rightarrow \infty $$ . We use a recent equidistribution result due to N. Katz to express these variances in terms of triple matrix integrals over the unitary group, and evaluate them. [ABSTRACT FROM AUTHOR]

In this paper we introduce a set of sufficient criteria for the construction of relative hemisystems of the Hermitian space $${\mathrm {H}}(3,q^2)$$ , unifying all known infinite families. We use these conditions to provide new proofs of the existence of the known infinite families of relative hemisystems. Reproving these results has allowed us to find new relative hemisystems closely related to an infinite family of Cossidente's, and develop techniques that are likely to be useful in finding relative hemisystems in future. [ABSTRACT FROM AUTHOR]

We introduce paired $$\mathrm{CR}$$ structures on 4-dimensional manifolds and study their properties. Such structures are arising from two different complex operators which agree in a 2-dimensional subbundle of the tangent bundle; this subbundle thus forms a codimension 2 $$\mathrm{CR}$$ structure. A special case is that of a strictly paired $$\mathrm{CR}$$ structure: in this case, the two complex operators are also opposite in a 2-dimensional subbundle which is complementary to the $$\mathrm{CR}$$ structure. A non-trivial example of a manifold endowed with a (strictly) paired $$\mathrm{CR}$$ structure is Falbel's cross-ratio variety $${\mathfrak {X}}$$ ; this variety is isomorphic to the $$\mathrm{PU}(2,1)$$ configuration space of quadruples of pairwise distinct points in $$S^3$$ . We first prove that there are two complex structures that appear naturally in $${\mathfrak {X}}$$ ; these give $${\mathfrak {X}}$$ a paired $$\mathrm{CR}$$ structure which agrees with its well known $$\mathrm{CR}$$ structure. Using a non-trivial involution of $${\mathfrak {X}}$$ we then prove that $${\mathfrak {X}}$$ is a strictly paired $$\mathrm{CR}$$ manifold. The geometric meaning of this involution as well as its interconnections with the $$\mathrm {CR}$$ and complex structures of $${\mathfrak {X}}$$ are also studied here in detail. [ABSTRACT FROM AUTHOR]