ALBERT

Description: Hjelmslev–Moufang (HM) planes are point-line geometries related to the exceptional algebraic groups of type $mathsf{E}_6$. More generally, point-line geometries related to spherical Tits buildings—Lie incidence geometries—are the prominent examples of parapolar spaces: axiomatically defined geometries consisting of points, lines and symplecta (structures isomorphic to polar spaces). In this paper we classify the parapolar spaces with a similar behaviour as the HM planes, in the sense that their symplecta never have a non-empty intersection. Under standard assumptions, we obtain that the only such parapolar spaces are exactly given by the HM planes and their close relatives (arising from taking certain restrictions). On the one hand, this work complements the algebraic approach to HM planes using Jordan algebras and due to Faulkner in his book ‘The Role of Nonassociative Algebra in Projective Geometry’, published by the American Mathematical Society in 2014; on the other hand, it provides a new tool for classification and characterization problems in the general theory of parapolar spaces.

Description: In this paper, we determine the order of magnitude of the 2 q-th pseudomoment of powers of the Riemann zeta function $zeta(s)^{alpha}$ for $0 lt qle 1/2$ and $0 lt alpha lt 1$, completing the results of Bondarenko, Heap and Seip, and Gerspach. Our results also apply to more general Euler products satisfying certain conditions.

Description: In the present paper we study $mathbf{S}^2!imes!mathbf{R}$ and $mathbf{H}^2!imes!mathbf{R}$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the ‘surface of a geodesic triangle’. Using the above Apollonius surfaces we develop a procedure to determine the centre and the radius of the circumscribed geodesic sphere of an arbitrary $mathbf{S}^2!imes!mathbf{R}$ and $mathbf{H}^2!imes!mathbf{R}$ tetrahedron. Moreover, we generalize the famous Menelaus’s and Ceva’s theorems for geodesic triangles in both spaces. In our work we will use the projective model of $mathbf{S}^2!imes!mathbf{R}$ and $mathbf{H}^2!imes!mathbf{R}$ geometries described by E. Molnár in [6].

Description: We explicitly construct the Dirichlet series $$ egin{equation*}L_{mathrm{Tam}}(s):=sum_{m=1}^{infty}frac{P_{mathrm{Tam}}(m)}{m^s},end{equation*}$$ where $P_{mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/mathbb{Q}$ in short Weierstrass form with Tamagawa product m. Although there are no $E/mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{mathrm{Tam}}(1)={0.5053dots}$. As a corollary, we find that $L_{mathrm{Tam}}(-1)={1.8193dots}$ is the average Tamagawa product for elliptic curves over $mathbb{Q}$. We give an application of these results to canonical and Weil heights.

Description: Inspired by the idea of blurring the exponential function, we define blurred variants of the j-function and its derivatives, where blurring is given by the action of a subgroup of $mathrm{GL}_2({mathbb{C}})$. For a dense subgroup (in the complex topology) we prove an Existential Closedness theorem which states that all systems of equations in terms of the corresponding blurred j with derivatives have complex solutions, except where there is a functional transcendence reason why they should not. For the j-function without derivatives we prove a stronger theorem, namely, Existential Closedness for j blurred by the action of a subgroup which is dense in $mathrm{GL}_2^+(mathbb{R})$, but not necessarily in $mathrm{GL}_2({mathbb{C}})$. We also show that for a suitably chosen countable algebraically closed subfield $C subseteq {mathbb{C}}$, the complex field augmented with a predicate for the blurring of the j-function by $mathrm{GL}_2(C)$ is model theoretically tame, in particular, ω-stable and quasiminimal.

Description: We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Nash-Moser implicit function theorem.

Description: We study regularity in the context of connective ring spectra and spectral stacks. Parallel to that, we construct a weight structure on the category of compact quasi-coherent sheaves on spectral quotient stacks of the form $X=[operatorname{Spec} R/G]$ defined over a field, where R is a connective ${{mathcal{E}}_infty}$-k-algebra and G is a linearly reductive group acting on R. Under reasonable assumptions, we show that regularity of X is equivalent to regularity of R. We also show that if R is bounded, such a stack is discrete. This result can be interpreted in terms of weight structures and suggests a general phenomenon: for a symmetric monoidal stable $infty$-category with a compatible bounded weight structure, the existence of an adjacent t-structure satisfying a strong boundedness condition should imply discreteness of the weight-heart. We also prove a gluing result for weight structures and adjacent t-structures, in the setting of a semi-orthogonal decomposition of stable $infty$-categories.

Description: We conjecture that the class of frame matroids can be characterized by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterization for the class of bicircular matroids. The proof does not depend on an excluded-minor characterization.

Description: We show three basic properties of the image Milnor number µI(f) of a germ $fcolon(mathbb{C}^{n},S) ightarrow(mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond’s conjecture, which states that µI(f) = 0 if and only if f is stable. Finally, we show a conjecture by Houston that any family $f_tcolon(mathbb{C}^{n},S) ightarrow(mathbb{C}^{n+1},0)$ with $mu_I(,f_t)$ constant is excellent in Gaffney’s sense. For technical reasons, in the last two properties, we consider only the corank 1 case.

Description: For a germ of a variety $mathcal{V}, 0 subset mathbb C^N, 0$, a singularity $mathcal{V}_0$ of ‘type $mathcal{V}$’ is given by a germ $f_0 : mathbb C^n, 0 o mathbb C^N, 0$ which is transverse to $mathcal{V}$ in an appropriate sense so that $mathcal{V}_0 = f_0^{,-1}(mathcal{V})$. If $mathcal{V}$ is a hypersurface germ, then so is $mathcal{V}_0 $, and by transversality ${operatorname{codim}}_{mathbb C} {operatorname{sing}}(mathcal{V}_0) = {operatorname{codim}}_{mathbb C} {operatorname{sing}}(mathcal{V})$ provided $n gt {operatorname{codim}}_{mathbb C} {operatorname{sing}}(mathcal{V})$. So $mathcal{V}_0, 0$ will exhibit singularities of $mathcal{V}$ up to codimension n. For singularities $mathcal{V}_0, 0$ of type $mathcal{V}$, we introduce a method to capture the contribution of the topology of $mathcal{V}$ to that of $mathcal{V}_0$. It is via the ‘characteristic cohomology’ of the Milnor fiber (for $mathcal{V}, 0$ a hypersurface), and complement and link of $mathcal{V}_0$ (in the general case). The characteristic cohomology of the Milnor fiber $mathcal{A}_{mathcal{V}}(,f_0; R)$, and respectively of the complement $mathcal{C}_{mathcal{V}}(,f_0; R)$, are subalgebras of the cohomology of the Milnor fibers, respectively the complement, with coefficients R in the corresponding cohomology. For a fixed $mathcal{V}$, they are functorial over the category of singularities of type $mathcal{V}$. In addition, for the link of $mathcal{V}_0$ there is a characteristic cohomology subgroup $mathcal{B}_{mathcal{V}}(,f_0, mathbf{k})$ of the cohomology of the link over a field $mathbf{k}$ of characteristic 0. The cohomologies $mathcal{C}_{mathcal{V}}(,f_0; R)$ and $mathcal{B}_{mathcal{V}}(,f_0, mathbf{k})$ are shown to be invariant under the $mathcal{K}_{mathcal{V}}$-equivalence of defining germs f0, and likewise $mathcal{A}_{mathcal{V}}(,f_0; R)$ is shown to be invariant under the $mathcal{K}_{H}$-equivalence of f0 for H the defining equation of $mathcal{V}, 0$. We give a geometric criterion involving ‘vanishing compact models’ for both the Milnor fibers and complements which detect non-vanishing subalgebras of the characteristic cohomologies, and subgroups of the characteristic cohomology of the link. Also, we consider how in the hypersurface case the cohomology of the Milnor fiber is a module over the characteristic cohomology $mathcal{A}_{mathcal{V}}(,f_0; R)$. We briefly consider the application of these results to a number of cases of singularities of a given type. In part II, we specialize to the case of matrix singularities and using results on the topology of the Milnor fibers, complements and links of the varieties of singular matrices obtained in another paper allow us to give precise results for the characteristic cohomology of all three types.