Recent zbMATH articles in MSC 11E https://zbmath.org/atom/cc/11E 2023-12-07T16:00:11.105023Z Werkzeug Book review of: A. Hatcher, Topology of numbers https://zbmath.org/1522.00080 2023-12-07T16:00:11.105023Z "Long, D. D." https://zbmath.org/authors/?q=ai:long.darren-d|long.d-darren Review of [Zbl 1502.11001]. Convolution sums engaging 1st and 5th powers of divisor functions and derived new identities https://zbmath.org/1522.11005 2023-12-07T16:00:11.105023Z "Ntienjem, Ebénézer" https://zbmath.org/authors/?q=ai:ntienjem.ebenezer Summary: Suppose that $$\mathbb{N}_0$$ denotes the set of positive integers, $$r,s,n$$, $$\alpha$$ and $$\beta$$ are elements of $$\mathbb{N}_0$$ such that $$\alpha$$ and $$\beta$$ are relatively prime, $$r$$ and $$s$$ are odd. For all levels $$\alpha\beta$$ and $$r+s=4$$, the evaluation of the convolution sums $\sum_{\substack{(k,l)\in\mathbb{N}^2_0\\ \alpha k+\beta l-n}}\sigma_{2r-1}(k)\sigma_{2s-1}(l)$ are carried out using in particular modular forms. Convolution sums belonging to this class of levels are then applied to determine formulae for the number of representations of $$n$$ by the quadratic forms in sixteen squares $$\sum^{16}_{i=1}x^2_i$$ if the level $$\alpha\beta$$ is a multiple of 4, and in sixteen variables $$\sum^8_{i=1}(x^2_{2i-1}+x_{2i-1}x_{2i}+x^2_{2i})$$ if the level $$\alpha\beta$$ is a multiple of 3. By evaluating the convolution sums for $$\alpha\beta=3,5,6,8,9,10,12,16,18,20,25,27,32$$, this approach is illustrated and explicit formulae for the number of representations of $$n$$ by quadratic forms in sixteen squares and in sixteen variables are determined. The confluence of the determination of a formula for the number of representations of a positive integer is used to bring new identities into existence. Estimates for certain shifted convolution sums involving Hecke eigenvalues https://zbmath.org/1522.11034 2023-12-07T16:00:11.105023Z "Hua, Guodong" https://zbmath.org/authors/?q=ai:hua.guodong Summary: In this paper, we obtain certain estimates for averages of shifted convolution sums involving Hecke eigenvalues of classical holomorphic cusp forms. This generalizes some results of \textit{G. Lü} and \textit{D. Wang} [J. Number Theory 199, 342--351 (2019; Zbl 1460.11055)] in this direction. Finite subgroups of automorphisms of $$K3$$ surfaces https://zbmath.org/1522.14050 2023-12-07T16:00:11.105023Z "Brandhorst, Simon" https://zbmath.org/authors/?q=ai:brandhorst.simon "Hofmann, Tommy" https://zbmath.org/authors/?q=ai:hofmann.tommy Authors' abstract: We give a complete classification of finite subgroups of automorphisms of $$K3$$ surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the $$K3$$ lattice. The moduli theory of $$K3$$ surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves Hermitian lattices over number fields. Reviewer: Jin-Xing Cai (Beijing) Exponential convergence of sum-of-squares hierarchies for trigonometric polynomials https://zbmath.org/1522.14072 2023-12-07T16:00:11.105023Z "Bach, Francis" https://zbmath.org/authors/?q=ai:bach.francis-r "Rudi, Alessandro" https://zbmath.org/authors/?q=ai:rudi.alessandro Sum-of-Squares (SoS) hierarchies provide a way to compute arbitrary close approximations of a general, global optimization problem over an algebraic or semialgebraic set. In the present paper, the hierarchies of lower bounds for the minimum of trigonometric polynomials on $$[0,1]^n$$ and of standard multivariate polynomials on $$[-1,1]^n$$ are considered. First, the authors consider the case of trigonometric polynomials on $$[0,1]^n$$, without any further assumption. They show that the SoS hierarchy of lower bounds has an error of at most $$O(1/s^2)$$, where $$s \in \mathbb{N}$$ is the order of the hierarchy. This result translates directly to the SoS hierarchy for standard polynomials on $$[-1,1]^n$$, which is based on Schmüdgen's Positivstellensatz. In the latter setting, a similar result with error bound $$O(1/s^2)$$ was previously obtained by \textit{M. Laurent} and \textit{L. Slot} [Optim. Lett. 17, No. 3, 515--530 (2023; Zbl 1515.90093)]. Second, the paper focuses on trigonometric polynomials on $$[0,1]^n$$ satisfying generic local optimality conditions. With this extra assumption, a better error bound of at most of the order $$O(\exp (-a \, s)^b)$$ is proven. Here, $$a > 0$$ and $$b > 1$$ are constants depending on $$n$$, on the degree $$d$$ of the trigonometric polynomial which is minimized, and on local properties of the trigonometric polynomial around its global minimizer. The technique used is based on a quantitative version of truncated Fourier approximation. As in the previous case, this result translates to the SoS hierarchy based on Schmüdgen's Positivstellensatz for standard polynomials on $$[-1,1]^n$$, which satisfies a local optimality condition. Under the same local optimality conditions, it was previously shown in [\textit{J. Nie}, Math. Program. 146, No. 1--2 (A), 97--121 (2014; Zbl 1300.65041)] that the SoS hierarchy for standard polynomials has finite convergence. This means that the lower approximation provided by the SoS hierarchy coincides with the true minimum for sufficiently big $$s$$. However, since no effective bound is known on $$s$$ for this finite convergence to happen, the error bound of the order $$O(\exp (-a \, s)^b)$$ remains an important explanation of the practical fast converge rate of the SoS hierarchies. Reviewer: Lorenzo Baldi (Paris) Hermitian K-theory for stable $$\infty$$-categories. I: Foundations https://zbmath.org/1522.19001 2023-12-07T16:00:11.105023Z "Calmès, Baptiste" https://zbmath.org/authors/?q=ai:calmes.baptiste "Dotto, Emanuele" https://zbmath.org/authors/?q=ai:dotto.emanuele "Harpaz, Yonatan" https://zbmath.org/authors/?q=ai:harpaz.yonatan "Hebestreit, Fabian" https://zbmath.org/authors/?q=ai:hebestreit.fabian "Land, Markus" https://zbmath.org/authors/?q=ai:land.markus "Moi, Kristian" https://zbmath.org/authors/?q=ai:moi.kristian-jonsson "Nardin, Denis" https://zbmath.org/authors/?q=ai:nardin.denis "Nikolaus, Thomas" https://zbmath.org/authors/?q=ai:nikolaus.thomas "Steimle, Wolfgang" https://zbmath.org/authors/?q=ai:steimle.wolfgang This paper is the first in a series. It is followed by [Hermitian $$K$$-theory for stable $$\infty$$-categories. II: Cobordism categories and additivity'', Preprint, \url{arXiv:2009.07224}] and [Hermitian $$K$$-theory for stable $$\infty$$-categories. III: Grothendieck-Witt groups of rings'', Preprint, \url{arXiv:2009.07225}] by the same authors. In this influential series, the authors develop a new framework for hermitian $$K$$-theory via Poincaré $$\infty$$-categories. In the next articles in the series, the authors define $$L$$- and Grothendieck-Witt spectra of Poincaré $$\infty$$-categories and construct a fibre sequence relating $$K$$-, $$L$$- and Grothendieck-Witt spectra. They use this to solve varies classical problems involving Grothendieck-Witt groups. The identification with Karoubi's classical hermitian and quadratic $$K$$-groups is obtained in [\textit{F. Hebestreit} and \textit{W. Steimle}, Stable moduli spaces of hermitian forms'', Preprint, \url{arXiv:2103.13911}]. The main role of this article is to develop the main concepts of Poincaré $$\infty$$-categories and Poincaré objects. Besides setting up the general framework, the article contains the following. \begin{itemize} \item[1.] The authors prove a version of Ranicki's algebraic Thom construction in their setting. \item[2.] The $$L$$-groups and zeroth Grothendieck-Witt group of a Poincaré $$\infty$$-category are defined. Furthermore, an exact sequence relating the zeroth $$K$$-, $$L$$- and Grothendieck-Witt groups is contructed. This is upgraded to the above mentioned fibre sequence in the next article in this series. \item[3.] For derived $$\infty$$-categories of rings, all Poincaré structures are classified. \item[4.] The example of visible Poincaré structures on $$\infty$$-categories of parametrised spectra is developed. This recovers the visible signature of a Poincaré duality space. \item[5.] The global structural properties of Poincaré $$\infty$$-categories are studied. In particular, the authors show that Poincaré $$\infty$$-categories form a bicomplete, closed symmetric monoidal $$\infty$$-category. \item[6.] The authors study the process of tensoring and cotensoring a Poincaré $$\infty$$-category over a finite simplicial complex. The authors use this for explicit constructions of the $$L$$- and Grothendieck-Witt spectra. \end{itemize} Reviewer: Daniel Kasprowski (Southhampton) Hodge operators and exceptional isomorphisms between unitary groups https://zbmath.org/1522.20193 2023-12-07T16:00:11.105023Z "Kramer, Linus" https://zbmath.org/authors/?q=ai:kramer.linus "Stroppel, Markus J." https://zbmath.org/authors/?q=ai:stroppel.markus-j Authors' abstract: We give a generalization of the Hodge operator to spaces $$(V,h)$$ endowed with a hermitian or symmetric bilinear form $$h$$ over arbitrary fields, including the characteristic two case. Suitable exterior powers of $$V$$ become free modules over an algebra $$K$$ defined using such an operator. This leads to several exceptional homomorphisms from unitary groups (with respect to $$h$$) into groups of semi-similitudes with respect to a suitable form over some subfield of $$K$$. The algebra $$K$$ depends on $$h$$; it is a composition algebra unless $$h$$ is symmetric and the characteristic is two.'' A map $$\lambda:V\to W$$ between vector spaces over a field $$F$$ is called $$F$$-semilinear if $$\lambda$$ is additive and there exists an automorphism $$\phi$$ of $$F$$ (called the companion of $$\lambda$$) such that $$\lambda(sv)=\phi(s) \lambda(v)$$ holds for each $$s\in F$$ and each $$v\in V$$. The paper develops semilinear counterparts of several constructions. In particular, if $$h$$ is a suitable form on the $$n$$ dimensional space $$V$$, then the Hodge operator $$J:\bigwedge^\ell V\to \bigwedge^{n-\ell}V$$ is constructed in semilinear fashion using a Pfaffian and a hermitian form on $$\bigwedge V$$. Now assume $$n=2\ell$$. On $$\bigwedge^\ell V$$ there is a form $$g$$ and the focus is now on a map from unitary groups (with respect to $$h$$) into groups of semi-similitudes with respect to $$g$$. It explains many `unexpected' homomorphisms in classification theorems. Reviewer: Wilberd van der Kallen (Utrecht)