Recent zbMATH articles in MSC 53Dhttps://www.zbmath.org/atom/cc/53D2021-04-16T16:22:00+00:00WerkzeugThe group of symplectic birational maps of the plane and the dynamics of a family of 4D maps.https://www.zbmath.org/1456.530662021-04-16T16:22:00+00:00"Cruz, Inês"https://www.zbmath.org/authors/?q=ai:cruz.ines"Mena-Matos, Helena"https://www.zbmath.org/authors/?q=ai:mena-matos.helena"Sousa-Dias, Esmeralda"https://www.zbmath.org/authors/?q=ai:sousa-dias.esmeraldaSummary: We consider a family of birational maps \(\varphi_k\) in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family \(\varphi_k\) using Poisson geometry tools, namely the properties of the restrictions of the maps \(\varphi_k\) and their fourth iterate \(\varphi^{(4)}_k\) to the symplectic leaves of an appropriate Poisson manifold \((\mathbb{R}^4_+, P)\). These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product \(SL(2, \mathbb{Z})\ltimes\mathbb{R}^2\). The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for \(\varphi_k\) characterized by the parameter values \(k = 1\), \(k = 2\) and \(k\geq 3\).The method of averaging for Poisson connections on foliations and its applications.https://www.zbmath.org/1456.530642021-04-16T16:22:00+00:00"Avendaño-Camacho, Misael"https://www.zbmath.org/authors/?q=ai:avendano-camacho.misael"Hasse-Armengol, Isaac"https://www.zbmath.org/authors/?q=ai:hasse-armengol.isaac"Velasco-Barreras, Eduardo"https://www.zbmath.org/authors/?q=ai:velasco-barreras.eduardo"Vorobiev, Yury"https://www.zbmath.org/authors/?q=ai:vorobiev.yuriiSummary: On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.Deformation classes in generalized Kähler geometry.https://www.zbmath.org/1456.530572021-04-16T16:22:00+00:00"Gibson, Matthew"https://www.zbmath.org/authors/?q=ai:gibson.matthew-r"Streets, Jeffrey"https://www.zbmath.org/authors/?q=ai:streets.jeffrey-dSummary: We describe natural deformation classes of generalized Kähler structures using the Courant symmetry group, which determine natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. We show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.Shifted derived Poisson manifolds associated with Lie pairs.https://www.zbmath.org/1456.530652021-04-16T16:22:00+00:00"Bandiera, Ruggero"https://www.zbmath.org/authors/?q=ai:bandiera.ruggero"Chen, Zhuo"https://www.zbmath.org/authors/?q=ai:chen.zhuo"Stiénon, Mathieu"https://www.zbmath.org/authors/?q=ai:stienon.mathieu"Xu, Ping"https://www.zbmath.org/authors/?q=ai:xu.pingSummary: We study the shifted analogue of the ``Lie-Poisson'' construction for \(L_\infty\) algebroids and we prove that any \(L_\infty\) algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures from a purely algebraic perspective and, in particular, we establish a homotopy transfer theorem for derived Poisson algebras. As an application, we prove that, given a Lie pair \((L, A)\), the space \(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet (L/A))\) admits a degree \((+1)\) derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley-Eilenberg differential \(d_A^{\mathrm{Bott}}:\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\rightarrow\Omega^{\bullet+1}_A(\Lambda^\bullet (L/A))\) as unary \(L_\infty\) bracket. This degree \((+1)\) derived Poisson algebra structure on \(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\) is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley-Eilenberg hypercohomology \(\mathbb{H}(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet (L/A)),d_A^{\mathrm{Bott}})\) admits a canonical Gerstenhaber algebra structure.Reduced invariants from cuspidal maps.https://www.zbmath.org/1456.140702021-04-16T16:22:00+00:00"Battistella, Luca"https://www.zbmath.org/authors/?q=ai:battistella.luca"Carocci, Francesca"https://www.zbmath.org/authors/?q=ai:carocci.francesca"Manolache, Cristina"https://www.zbmath.org/authors/?q=ai:manolache.cristinaSummary: We consider genus one enumerative invariants arising from the Smyth-Viscardi moduli space of stable maps from curves with nodes and cusps. We prove that these invariants are equal to the reduced genus one invariants of the quintic threefold, providing a modular interpretation of the latter.A uniqueness theorem of complete Lagrangian translator in \(\mathbb C^2\).https://www.zbmath.org/1456.530772021-04-16T16:22:00+00:00"Li, Xingxiao"https://www.zbmath.org/authors/?q=ai:li.xingxiao"Liu, Yangyang"https://www.zbmath.org/authors/?q=ai:liu.yangyang"Qiao, Ruina"https://www.zbmath.org/authors/?q=ai:qiao.ruinaSummary: In this paper we study the complete Lagrangian translators in the complex 2-plane \(\mathbb C^2\). As the result, we obtain a uniqueness theorem showing that the plane is the only complete Lagrangian translator in \(\mathbb C^2\) with constant square norm of the second fundamental form. On the basis of this, we can prove a more general classification theorem for Lagrangian \(\xi\)-translators in \(\mathbb C^2\). The same idea is also used to give a similar classification of Lagrangian \(\xi\)-surfaces in \(\mathbb C^2\).Concise notes on special holonomy with an emphasis on Calabi-Yau and \(G_2\)-manifolds.https://www.zbmath.org/1456.530062021-04-16T16:22:00+00:00"Oliveira, Gonçalo"https://www.zbmath.org/authors/?q=ai:oliveira.goncaloSummary: These are notes for a very short introduction to some selected topics on special Riemannian holonomy with a focus on Calabi-Yau and \(G_2\)-manifolds. No material in these notes is original and more on it can be found in the papers/books of Bryant, Hitchin, Joyce and Salamon referenced during the text.Open and closed mirror symmetry.https://www.zbmath.org/1456.530692021-04-16T16:22:00+00:00"Amorim, Lino"https://www.zbmath.org/authors/?q=ai:amorim.linoSummary: Mirror symmetry predicts a deep correspondence between the symplectic geometry of a space and the algebraic geometry of its ``mirror''. There are different versions of this correspondence, from the equality of some numerical invariants, first predicted by physicists, to categorical versions proposed by Kontsevich. This paper reviews some of these versions and illustrates them on a relatively simple example: a sphere with three orbifold points (on the symplectic side). We explain how to construct the ``mirror'' space, state the mirror predictions and describe an approach to prove them.Linear Batalin-Vilkovisky quantization as a functor of \(\infty \)-categories.https://www.zbmath.org/1456.180182021-04-16T16:22:00+00:00"Gwilliam, Owen"https://www.zbmath.org/authors/?q=ai:gwilliam.owen"Haugseng, Rune"https://www.zbmath.org/authors/?q=ai:haugseng.runeThe authors consider a categorical construction of linear Batalin-Vilkovisky quantization in a derived setting.
The basic example that is the starting point for this article is the Weyl quantization, sending a symplectic vector space \(\mathbb R^{2n}\) to the Weyl algebra on \(2n\) generators. One can factor this construction as taking a vector space with a skew-symmetric form first to its Heisenberg Lie algebra and then to its universal envelopping algebra. The specalization at \(\hbar = 0\) of this universal envelopping algebra is a Poisson algebra and the specializiation at \(\hbar = 1\) is its quantizaiton.
The authors consider a special case of the shifted derived versions of this problem: Their starting point are chain complexes equipped with a 1-shifted symmetric pairing. Following the article we will call them quadratic modules for short.
They then construct \(\infty\)-categorical versions of both the Heisenberg Lie algebra (which is actually a shifted \(L_\infty\)-algebra) of a quadratic module, and the universal enveloping \(BD\)-algebra of a shifted Lie algebra. Both of these appear to be of independent interest.
The universal enveloping \(BD\)-algebra is a so-called Beilinson-Drinfeld algebra, a \(k[\hbar]\)-algebra over a certain operad that specialises to a shifted Poisson algebra at \(\hbar = 0\) and to an \(E_0\)-algebra at \(\hbar = 1\). (An \(E_0\)-algebra is just a pointed chain complex, but this is the correct edge case of the notion of \(E_n\)-algebras. The classical, unshifted case involves an unshifted Poisson algebra and an \(E_1\)-algebra (i.e.\ an associative algebra) as specializiations.)
Thus the authors are able to construct linear BV quantization as a symmetric monoidal \(\infty\)-functor from quadratic algebras to \(BD\)-algebras.
The proofs involve a mixture of categorical techniques (model, simplicial and \(\infty\)).
One upside of the \(\infty\)-categorical approach is that by using Lurie's descent theorem the author can consider linear BV quantization for sheaves of quadratic modules on derived stacks. Thus they are able to show that the graded vector bundle \(V \oplus V^\vee[1]\) with its obvious quadratic form quantizes to a line bundle. This is an explicit example of the BV formalism ``behaving like a determinant'', an idea the authors credit to K. Costello. The paper also provides an example that the behaviour for more general 1-shifted symplectic modules is more complicated and the quantization need only be invertible in the formal neighbourhood of a point.
The paper under review contains some interesting discussions in the introduction: Section 1.3 considers higher BV quantizations (which should arise from more general \((1-n)\)-shifted skew-symmetric forms) and a possible application to quantization of AKSZ field theories. Section 1.4 discusses the physical perspective on linear BV quantizations, providing useful context and motivation.
Reviewer: Julian Holstein (Hamburg)Categorical mirror symmetry on cohomology for a complex genus 2 curve.https://www.zbmath.org/1456.530702021-04-16T16:22:00+00:00"Cannizzo, Catherine"https://www.zbmath.org/authors/?q=ai:cannizzo.catherineSummary: Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs \(X\) and \(Y\) such that the complex geometry on \(X\) mirrors the symplectic geometry on \(Y\). It allows one to deduce symplectic information about \(Y\) from known complex properties of \(X\). \textit{A. Strominger} et al. [Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. \textit{M. Kontsevich} [in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)] conjectured that a complex invariant on \(X\) (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of \(Y\) (the Fukaya category, see [\textit{D. Auroux}, Bolyai Soc. Math. Stud. 26, 85--136 (2014; Zbl 1325.53001); \textit{K. Fukaya} et al., Lagrangian intersection Floer theory. Anomaly and obstruction. I. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53002); \textit{D. McDuff} et al., Virtual fundamental cycles in symplectic topology. New York, NY: American Mathematical Society (2019)]). This is known as homological mirror symmetry. In this project, we first use the construction of ``generalized SYZ mirrors'' for hypersurfaces in toric varieties following \textit{M. Abouzaid} et al. [Publ. Math., Inst. Hautes Étud. Sci. 123, 199--282 (2016; Zbl 1368.14056)], in order to obtain \(X\) and \(Y\) as manifolds. The complex manifold is the genus 2 curve \(\Sigma_2\) (so of general type \(c_1 < 0\)) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model \((Y, v_0)\) equipped with a holomorphic function \(v_0 : Y \to \mathbb{C}\) which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of \(D^b Coh(\Sigma_2)\) into a cohomological Fukaya-Seidel category of \(Y\) as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations in [\textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001); \textit{M. Abouzaid} and \textit{P. Seidel}, ``Lefschetz fibration methods in wrapped Floer cohomology'', in preparation].Contact and Frobenius solvable Lie algebras with abelian nilradical.https://www.zbmath.org/1456.170092021-04-16T16:22:00+00:00"Alvarez, M. A."https://www.zbmath.org/authors/?q=ai:alvarez.maria-alejandra"Rodríguez-Vallarte, M. C."https://www.zbmath.org/authors/?q=ai:rodriguez-vallarte.maria-c"Salgado, G."https://www.zbmath.org/authors/?q=ai:salgado.gilThe authors obtain that complex Frobenius Lie algebras are decomposable, while in the real case, there are exactly two that are indecomposable. Families of both Frobenius and contact solvable Lie algebras are characterized under certain conditions. When these Lie algebras have abelian nilradical, conditions are determined for which they are double extensions of Lie algebras of codimension 2. It is shown that these algebras have a natural Z\(_2\) grading.
Reviewer: Ernest L. Stitzinger (Raleigh)Symmetries of the simply-laced quantum connections and quantisation of quiver varieties.https://www.zbmath.org/1456.812722021-04-16T16:22:00+00:00"Rembado, Gabriele"https://www.zbmath.org/authors/?q=ai:rembado.gabrieleSummary: We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.On LA-Courant algebroids and Poisson Lie 2-algebroids.https://www.zbmath.org/1456.580042021-04-16T16:22:00+00:00"Jotz Lean, M."https://www.zbmath.org/authors/?q=ai:jotz.madeleineT.J. Courant discovered a skew-symmetric Lie bracket on \(TM \oplus T^* M\). The more general structure of a Courant algebroid, links the study of constrained Hamiltonian systems with generalised complex geometry. They were studied extensively throughout the 1990s by Zhang-Ju Liu, Alan Weinstein and Ping Xu, as well as Severa and Roytenberg. The associated integrability problem is an open question to this day.
To this end, it is important to understand better these structures. Courant algebroids are ``higher'' geometric structures. This can be made precise in the following ways: Roytenberg and Severa (independently) understood them in a graded sense, namely they described them as symplectic Lie 2-algebroids. On the other hand, Courant's example fits into \textit{K. C. H. Mackenzie}'s study of multiple structures, in particular it is an example of a double Lie algebroid [J. Reine Angew. Math. 658, 193--245 (2011; Zbl 1246.53112)]. \textit{D. Li-Bland} in his PhD thesis [LA-Courant algebroids and their applications. \url{arXiv:1204.2796}] introduced a more general class of Courant algebroids (LA-Courant algebroids) which are Courant algebroid structures in the category of vector bundles. They too fit in the multiple structures studied by Mackenzie.
The paper under review studies the correspondence between LA-Courant algebroids and Poisson Lie 2-algebroids (the latter generalize symplectic Lie 2-algebroids), using the author's earlier results on split Lie 2-algebroids and self-dual 2-representations.
Reviewer: Iakovos Androulidakis (Athína)Nonlinear flag manifolds as coadjoint orbits.https://www.zbmath.org/1456.370602021-04-16T16:22:00+00:00"Haller, Stefan"https://www.zbmath.org/authors/?q=ai:haller.stefan"Vizman, Cornelia"https://www.zbmath.org/authors/?q=ai:vizman.corneliaIn [Math. Ann. 329, No. 4, 771--785 (2004; Zbl 1071.58005)], the present authors introduced the notion of a nonlinear Grassmannian and studied the Fréchet manifold \(\mathrm{Gras}_n(M)\) of all \(n\)-dimensional oriented compact submanifolds of a smooth closed connected \(m\)-dimensional manifold \(M\).
They showed that every closed \((n+2)\)-form \(\alpha\) on \(M\) defines a closed 2-form \(\widetilde{\alpha}\) on \(\mathrm{Gras}_n(M)\), and if \(\alpha\) is integrable, then \(\widetilde{\alpha}\) is the curvature form of a principal connection on a principal \(S^1\)-bundle over \(\mathrm{Gras}_n(M)\). In the case \(\alpha\) is a closed, integrable volume form, then every connected component \(\mathcal{M}\) of \(\mathrm{Gras}_{m-2}(M)\), equipped with the symplectic form \(\widetilde{\alpha}\), is a prequantizable coadjoint orbit of some central extension of the Hamiltonian group \(\text{Ham}(M,\alpha)\) by \(S^1\).
In this paper, the authors generalize the notion of a nonlinear Grassmannian to the notion of a nonlinear flag manifold.
If \(M\) is a smooth manifold, \(S_1,\dots,S_r\) are closed smooth manifolds, then a sequence of nested embedded submanifolds \(N_1\subseteq\dots\subseteq N_r\subseteq M\) such that \(N_i\) is diffeomorphic to \(S_i\) for all \(i=1,\dots,r\) is called a nonlinear flag of type \(\mathscr{S}=(S_1,\dots,S_r)\) in \(M\).
The space of all nonlinear flags of type \(\mathscr{S}\) in \(M\) can be equipped with the structure of a Fréchet manifold in a natural way and is denoted by \(\mathrm{Flag}_{\mathscr{S}}(M)\).
The main goal of this paper is to study the geometry of this space.
A nonlinear Grassmannian is a special case of a nonlinear flag and corresponds to the case \(r=1\).
The authors present some applications of nonlinear flag manifolds by using them to describe certain coadjoint orbits of the Hamiltonian group.
If \(M\) is a closed symplectic manifold, \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) is the open subset in \(\mathrm{Flag}_{\mathscr{S}}(M)\) consisting of all symplectic flags of type \(\mathscr{S}\), then the symplectic form on \(M\) induces by transgression a symplectic form on the manifold of symplectic nonlinear flags. The Hamiltonian group \(\mathrm{Ham}(M)\) acts on \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) in a Hamiltonian fashion with equivariant moment map \(J:\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\to\mathfrak{ham}(M)^*\).
This moment map is injective and identifies each connected component of \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) with a coadjoint orbit of \(\mathrm{Ham}(M)\).
The main result of the paper states that the restriction of the moment map \(J:\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\to\mathfrak{ham}(M)^*\) to any connected component is one-to-one onto a coadjoint orbit cf the Hamiltonian group \(\mathrm{Ham}(M)\). The Kostant-Kirillov-Souriau symplectic form \(\omega_{\mathrm{KKS}}\) on the coadjoint orbit satisfies \(J^*\omega_{\mathrm{KKS}}=\Omega\), where \(\Omega\) is a natural symplectic form.
Reviewer: Andrew Bucki (Edmond)Categorical localization for the coherent-constructible correspondence.https://www.zbmath.org/1456.140472021-04-16T16:22:00+00:00"Ike, Yuichi"https://www.zbmath.org/authors/?q=ai:ike.yuichi"Kuwagaki, Tatsuki"https://www.zbmath.org/authors/?q=ai:kuwagaki.tatsukiKontsevich's homological mirror symmetry(HMS) conjecture states that two categories associated to
a mirror pair are equivalent. For a Calabi-Yau(CY) variety, a mirror is also Calabi-Yau and the conjecture is a
quasi-equivalence between the dg category of coherent sheaves over one and the derived Fukaya category of
the other. For non-CY's, mirrors do not need to be varieties. For a Fano toric variety, its mirror is a
Landau-Ginzburg (LG) model, which is a holomorphic function on \((\mathbb C^\times)^n\) which can be read from
the defining fan of the toric variety which is in fact the specialization of Lagrangian potential
function of the toric \(A\)-model that is the generating function of open Gromov-Witten invariants of
a toric fiber [\textit{C.-H. Cho} and \textit{Y.-G. Oh}, Asian J. Math. 10, No. 4, 773--814 (2006; Zbl 1130.53055); \textit{K. Fukaya} et al., Duke Math. J. 151, No. 1, 23--175 (2010; Zbl 1190.53078)].
For a smooth Fano, it has been proven for many special cases that the dg category of coherent sheaves
\(\mathbf{coh}\, X_\Sigma\) over the toric variety \(X_\Sigma\) associated to the fan \(\Sigma\) is quasi-equivalent to
the Fukaya-Seidel category \(\mathfrak{Fuk}(W_\Sigma)\) of the associated Laurent polynomial \(W_\Sigma\).
When a variety is not complete, \(\mathbf{coh}\, X_\Sigma\) is of infinite dimensional nature and its Fukaya-type
category also should have an infinite-dimensional nature. Such a construction is known to be
(partially) wrapped Fukaya categories. In this regard, the main theme of the present paper in review is
to establish a quasi-isomorphism \(\mathbf{coh}(X\setminus D)\cong \mathbf{coh} X/\mathbf{coh}_D X\) in some special cases
in the microlocal world: Here
\(X\setminus D\) is the complement of a divisor \(D\) and \(\mathbf{coh} X/\mathbf{coh}_D X\) is the dg category of
sheaves supported in \(D\) by relating the isomorphism to a similar isomorphism
\[
W_{\mathbf{s}\setminus\mathbf{r}}(M) \cong W_{\mathbf{s}}(M)/\mathfrak B_{\mathbf{r}}
\]
of \textit{Z. Sylvan} [J. Topol. 12, No. 2, 372--441 (2019; Zbl 1430.53097)] in the Fukaya-Seidel side: Here \(\mathbf{s}\) is a collection
of symplectic stops and \(\mathbf{r} \subset\mathbf{s}\) is a sub-collection thereof, and
\(\mathfrak B_{\mathbf{r}}\) is the full subcategory spanned by Lagrangians near the sub-stops \(\mathbf{r}\).
The paper extends a version of coherent-constructible correspondence [\textit{B. Fang} et al., Invent. Math. 186, No. 1, 79--114 (2011; Zbl 1250.14011); \textit{K. Bongartz} et al., Adv. Math. 226, No. 2, 1875--1910 (2011; Zbl 1223.16004)] to the dg category of
\emph{quasi-coherent shaves} over \(X_\Sigma\) in dimension 2.
Reviewer: Yong-Geun Oh (Pohang)Lifting semi-invariant submanifolds to distribution of almost contact metric manifolds.https://www.zbmath.org/1456.530632021-04-16T16:22:00+00:00"Bukusheva, A. V."https://www.zbmath.org/authors/?q=ai:bukusheva.aliya-vSummary: Let \(M\) be an almost contact metric manifold of dimension \(n = 2m + 1\). The distribution \(D\) of the manifold \(M\) admits a natural structure of a smooth manifold of dimension \(n = 4m + 1\). On the manifold \(M\), is defined a linear connection \(\nabla^N\) that preserves the distribution \(D\); this connection is determined by the interior connection that allows parallel transport of admissible vectors along admissible curves.
The assigment of the linear connection \(\nabla^N\) is equivalent to the assignment of a Riemannian metric of the Sasaki type on the distribution \(D\).
Certain tensor field of type \((1,1)\) on \(D\) defines a so-called prolonged almost contact metric structure.
Each section \(U\in\Gamma(D)\) of the distribution \(D\) defines a morphism \(U:M\to D\) of smooth manifolds. It is proved that if \(\tilde{M}\subset M\) a semi-invariant submanifold of the manifold \(M\) and \(U\in\Gamma(D)\) is a covariantly constant vector field with respect to the \(N\)-connection \(\nabla^N\), then \(U(\tilde{M})\) is a semi-invariant submanifold of the manifold \(D\) with respect to the prolonged almost contact metric structure.The globalization problem of the Hamilton-DeDonder-Weyl equations on a local \(k\)-symplectic framework.https://www.zbmath.org/1456.530732021-04-16T16:22:00+00:00"Esen, Oğul"https://www.zbmath.org/authors/?q=ai:esen.ogul"de León, Manuel"https://www.zbmath.org/authors/?q=ai:de-leon.manuel"Sardón, Cristina"https://www.zbmath.org/authors/?q=ai:sardon.cristina"Zając, Marcin"https://www.zbmath.org/authors/?q=ai:zajac.marcinSummary: In this paper, we aim at addressing the globalization problem of Hamilton-DeDonder-Weyl equations on a local \(k\)-symplectic framework and we introduce the notion of \textit{locally conformal k-symplectic (l.c.k-s.) manifolds}. This formalism describes the dynamical properties of physical systems that locally behave like multi-Hamiltonian systems. Here, we describe the local Hamiltonian properties of such systems, but we also provide a global outlook by introducing the global Lee one-form approach. In particular, the dynamics will be depicted with the aid of the Hamilton-Jacobi equation, which is specifically proposed in a l.c.k-s manifold.On the geometry and entropy of non-Hamiltonian phase space.https://www.zbmath.org/1456.820232021-04-16T16:22:00+00:00"Sergi, Alessandro"https://www.zbmath.org/authors/?q=ai:sergi.alessandro"Giaquinta, Paolo V."https://www.zbmath.org/authors/?q=ai:giaquinta.paolo-vQuantitative Tamarkin theory.https://www.zbmath.org/1456.530082021-04-16T16:22:00+00:00"Zhang, Jun"https://www.zbmath.org/authors/?q=ai:zhang.jun.8In 1980's, Kashiwara and Schapira developed a powerful theory, called the microlocal sheaf theory,
connecting analysis, symplectic geometry, and partial differential equations.
In symplectic geometry, a central topic is the non-displaceability problems.
In his pioneering work [Invent. Math. 82, 307--347 (1985; Zbl 0592.53025)], \textit{M. Gromov} proved the non-squeezing theorem,
which can be thought of as a classical result concerning non-displaceability.
It was \textit{D. Tamarkin} who first illustrated how to use the microlocal sheaf theory to solve non-displaceability problems [Springer Proc. Math. Stat. 269, 99--223 (2018; Zbl 1416.35019)].
Since then, aiming at translating more objects in symplectic geometry into the language of sheaves, extensive works have been done.
The purpose of the book under review is to provide an exposition of
the fast development of this topic, which focuses on the relations
between symplectic geometry and Tamarkin category theory, especially the Guillermou-Kashiwara-Schapira sheaf quantization
based on microlocal sheaf theory.
The book is divided into four parts.
The first part introduces the basic objects in symplectic geometry and
the key concept of singular support in microlocal sheaf theory.
The second part centers on the concepts of derived
category, persistence \textbf{k}-module, and singular support which serve as preparations
for the topics in later chapters.
The third part deals with the Tamarkin category theory.
The fourth part discusses various applications of Tamarkin categories in
symplectic geometry.
A more detailed review of the contents is given below.
The book starts with an introductory Chapter 1, that provides a quite readable overview of the whole book.
It contains a brief review of symplectic geometry and a sheaf-theoretic topics related to symplectic geometry, such as the singular support of a sheaf, the Tamarkin category, and the Hofer norm.
Chapter 2 is about the derived categories and the derived functors.
In particular, it includes an important result called the microlocal Morse lemma, a generalization of the classical Morse lemma to a microlocal formulation.
Based on the microlocal Morse lemma, the Tamarkin category is constructed at the beginning of Chapter 3.
This chapter devotes to a detailed study of the Tamarkin category theory.
Many symplectic-related topics are presented, for instance, the sheaf convolution and composition, Lagrangian Tamarkin categories and so on.
The last chapter is about the applications of Tamarkin categories in
symplectic geometry.
Starting with a presentation of the Guillermou-Kashiwara-Schapira sheaf quantization,
the author introduces many sheaf theoretic objects related to the symplectic geometry, especially, the symplectic geometry of the cotangent bundle.
At last, using sheaf invariants developed in the book, the author presents a new proof of Gromov's non-squeezing theorem.
The book contains an appendix, which presents some details on the relation between persistence modules and constructible sheaves, the computation of the sheaf hom, and the dynamical interpretation of the Guillermou-Kashiwara-Schapira sheaf quantization from the perspective of semi-classical analysis.
Reviewer: Xiaojun Chen (Chengdu)Floer cohomology, multiplicity and the log canonical threshold.https://www.zbmath.org/1456.140422021-04-16T16:22:00+00:00"McLean, Mark"https://www.zbmath.org/authors/?q=ai:mclean.markThe notions of multiplicity and log canonical threshold are the fundamental notions of
complex hypersurface \(H = \{f = 0\}\) defined by polynomials \(f\) on \(\mathbb C^{n+1}\). The former one is rather classical
which is defined by
\[
\mu_P(f) = \dim_{\mathbb C} \mathcal O_{\mathbb C^n,P}/(\delta f/\delta x_1, \ldots, \delta f/\delta x_n)
\]
at \(P \in H\). The latter notion of log canonical threshold is relative new which is given by
\[
\text{lct}_P(f) = \min \{(E_j) + 1/ \text{ord}_f(E_j): j \in S\}
\]
at \(P \in H\), where \((E_j)_{j \in S}\) are the \emph{resolution divisors} of a log resolution at \(0 \in H\) of the pair
\((\mathbb C^{n+1},H)\), whose precise current definition is
given by \textit{V. V. Shokurov} in birational geometry [Russ. Acad. Sci., Izv., Math. 40, No. 1, 95--202 (1992; Zbl 0785.14023); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 1, 105--201, Appendix 201--203 (1992)]. Both are algebraic invariants of hypersurface singularities
in complex algebraic geometry. The main result of the paper in review proves that these are indeed symplectic invariants of
the hypersurface in that when \(f, \, g: \mathbb C^{n+1} \to \mathbb C\) are two polynomials with isolated singular points at \(0\) with
embedded contactomorphic links, the multiplicity and the log canonical threshold of \(f\) and \(g\) are equal.
The main technical ingredient used to prove this result is to find formulas for the multiplicity and log canonical threshold
in terms of a sequence of fixed-point Floer cohomology groups in symplectic topology. The author does this by constructing a spectral sequence converging to the fixed-point Floer cohomology of any iterate of the Milnor monodromy map whose
\(E^1\) page is explicitly described in terms of a log resolution of \(f\). This spectral sequence is a generalization
of a forumla by \textit{N. A'Campo} [Comment. Math. Helv. 50, 233--248 (1975; Zbl 0333.14008)]. The author first carries out a rather detailed technical
symplectic massaging, called \(\omega\)-regularization, of a germ of the neighborhood of intersections of
the symplectic crossing divisor \((V_i)_{i\in S}\) which are transversally intersecting codimension 2 symplectic submanifolds.
Then he applies the geometric notions of Liouville domains and open-books to construct a contact open book that is
well-behaved such that the mapping torus of the Milnor monodromy map is isotopic to the mapping torus of a
symplectomorphism arising from the open book. The paper provides much details of basic constructions in symplectic topology
that is expected to be useful for other similar future applications of symplectic machinery to complex algebraic geometry.
Reviewer: Yong-Geun Oh (Pohang)Affine structures on Lie groupoids.https://www.zbmath.org/1456.220012021-04-16T16:22:00+00:00"Lang, Honglei"https://www.zbmath.org/authors/?q=ai:lang.honglei"Liu, Zhangju"https://www.zbmath.org/authors/?q=ai:liu.zhangju"Sheng, Yunhe"https://www.zbmath.org/authors/?q=ai:sheng.yunheAuthors' abstract: We study affine structures on a Lie groupoid, including affine \(k\)-vector fields, \(k\)-forms and \((p, q)\)-tensors. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the space of affine multivector fields with the Schouten bracket and the space of affine vector-valued forms with the Frölicher-Nijenhuis bracket are graded strict Lie 2-algebras, and affine \((1, 1)\)-tensors constitute a strict monoidal category. Such higher structures can be seen as the categorification of multiplicative structures on a Lie groupoid.
Reviewer: Iakovos Androulidakis (Athína)Chiral algebra, localization, modularity, surface defects, and all that.https://www.zbmath.org/1456.813682021-04-16T16:22:00+00:00"Dedushenko, Mykola"https://www.zbmath.org/authors/?q=ai:dedushenko.mykola"Fluder, Martin"https://www.zbmath.org/authors/?q=ai:fluder.martinThe authors study Lagrangian \(\mathcal{N} = 2\) superconformal field theories in four dimensions.
By employing supersymmetric localization on a rigid background of the form \(S^3 \times S^1_y\) they explicitly localize a given Lagrangian superconformal field theory and obtain the corresponding two-dimensional vertex operator algebra VOA (chiral algebra) on the torus \(S^1\times S^1_y\subset S^3\times S^1_y\). To derive the VOA the authors define the appropriate rigid supersymmetric \(S^3 \times S^1_y\) background reproducing the superconformal index. They analyze the supersymmetry algebra and classify the possible fugacities and their preserved subalgebras. Although the minimal amount of supersymmetry needed to retain the VOA construction is \(\mathfrak{su}(1|1)_\ell\times \mathfrak{su}(1|1)_r\) it appears that it is possible to turn on fugacities preserving an \(\mathfrak{su}(1|1)_\ell\times \mathfrak{su}(2|1)_r\) subalgebra which can be further broken to the minimal one by defects. Specifically, discrete fugacities \(M,N \in \mathbb{Z}\) can be turned on. The authors argue that these deformations do not affect the VOA construction but change the complex structure of the
torus and affect the boundary conditions (spin structure) upon going around one of the cycles, \(S^1_y\)
The authors address the two-dimensional theory corresponding to the localization of the \(\mathcal{N} = 2\) vector multiplets and hypermultiplets. In the latter case they show that the remnant classical piece in the localization precisely reduces to the two-dimensional symplectic boson theory on the boundary torus \(S^1\times S^1_y\). The authors show that in the presence of flavor holonomies, which appear as mass-like central charges in the supersymmetry algebra, vertex operators charged under the flavor symmetries fail to remain holomorphic while the sector that remains holomorphic is formed by flavor-neutral operators.
The authors study the modular properties of the four-dimensional Schur index. They introduce formal partition functions \(Z^{(\nu_1,\nu_2)}_{(m,n)}\), which are defined as the partition function in the given spin structure \((\nu_1,\nu_2)\), but with the modified contour of the holonomy integral in the localization formula, labeled by two integers \(m\) and \(n\). The authors suggest that the objects \(Z^{(\nu_1,\nu_2)}_{(m,n)}\) furnish an infinite-dimensional projective representation of \(\mathrm{SL}(2,\mathbb{Z})\).
Finally the authors comment on the flat \(\Omega\)-background underlying the chiral algebra.
Reviewer: Farhang Loran (Isfahan)Reduction of Nambu-Poisson manifolds by regular distributions.https://www.zbmath.org/1456.530672021-04-16T16:22:00+00:00"Das, Apurba"https://www.zbmath.org/authors/?q=ai:das.apurbaSummary: The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by \textit{R. Ibáñez} et al. [Rep. Math. Phys. 42, No. 1-2, 71--90 (1998; Zbl 0931.37024)]. In this paper we show that the reduction is always ensured unless the distribution is zero. Next we extend the more general Falceto-Zambon Poisson reduction theorem for Nambu-Poisson manifolds. Finally, we define gauge transformations of Nambu-Poisson structures and show that these transformations commute with the reduction procedure.Morse spectra, homology measures, spaces of cycles and parametric packing problems.https://www.zbmath.org/1456.530352021-04-16T16:22:00+00:00"Gromov, Misha"https://www.zbmath.org/authors/?q=ai:gromov.mikhaelA motivating question for this long and densely-written paper is the following: For an ensemble of moving particles in a space, what happens if the effectively observable number of states is replaced by the number of effective/persistent degrees of freedom? ``We suggest in this paper several mathematical counterparts to the idea of persistent degrees of freedom and formulate specific questions, many of which are inspired by Larry Guth's results and ideas on the Hermann Weyl kind of asymptotics of the Morse (co)homology spectra of the volume energy function on the spaces of cycles in a ball. And often we present variable aspects of the same ideas in different sections of the paper.''
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)Positive loops of loose Legendrian embeddings and applications.https://www.zbmath.org/1456.530622021-04-16T16:22:00+00:00"Liu, Guogang"https://www.zbmath.org/authors/?q=ai:liu.guogangThe article under review is concerned with positive loops of loose Legendrian embeddings. Let \((M, \xi = \ker \alpha)\) be a contact manifold, and \(L \subset M\) be a Legendrian submanifold. A Legendrian isotopy \(\phi: L \times [0,1] \to M\) is called positive if \(\alpha(\partial_t \phi_t)>0\). Meanwhile, the concept of loose Legendrian submanifolds in higher-dimensions was introduced by \textit{E. Murphy} in [``Loose Legendrian embeddings in high dimensional contact manifolds'', Preprint, \url{arXiv:1201.2245}]. The main result of this article is that for a contact manifold \((M, \xi)\) of dimension greater or equal to 5, any loose Legendrian submanifold \(L\) admits a contractible positive loop of Legendrian embeddings based at \(L\). Note that without the looseness assumption, \textit{F. Laudenbach} [in: New perspectives and challenges in symplectic field theory. Dedicated to Yakov Eliashberg on the occasion of his 60th birthday. Providence, RI: American Mathematical Society (AMS). 299--305 (2009; Zbl 1187.53080)] proves that \(L\) always admits positive loops of Legendrian immersions.
This main result can be regarded as an extension of certain flexibility, since the concept of higher dimensional loose submanifolds is a generalization of the stabilization of Legendrian submanifolds in dimension three. In particular, loose Legendrian submanifolds are flexible in the sense that they satisfy h-principle. As a comparison, a result from \textit{V. Colin} et al. [Int. Math. Res. Not. 2017, No. 20, 6231--6254 (2017; Zbl 1405.53108)] shows that the stabilization of the zero-section \(L\) of \(T^*S^1 \times \mathbb R\), denoted by \(S(L)\), admits a positive loop of Legendrian embeddings based at \(S(L)\).
Next, an application of this main result is a holomorphic curve free proof of the existence of tight contact structures in every dimension. Recall that the concept of an overtwisted or tight contact structure in higher dimension was studied in [\textit{M. S. Borman} et al., Acta Math. 215, No. 2, 281--361 (2015; Zbl 1344.53060)]. Explicitly, this article shows that for any \(n \geq 1\), the contact manifold \((S^{n-1} \times \mathbb R^n, \xi_{\mathrm{std}})\) is tight. Its proof is deeply rooted in the generating function theory.
Last but not least, this main result is clearly related to the orderability concept introduced by \textit{Y. Eliashberg} and \textit{L. Polterovich} [Geom. Funct. Anal. 10, No. 6, 1448--1476 (2000; Zbl 0986.53036)] on the universal cover \(\widetilde{\mathrm{Cont}}_0(M, \xi)\). In this article, a (possibly) different notation is introduced also on \(\widetilde{\mathrm{Cont}}_0(M, \xi)\), called strong orderability. It is defined via a canonical lift from a contact isotopy in \((M, \xi)\) to a Legendrian isotopy in the contact product \((M \times M \times \mathbb R, \pi_1^*\alpha - e^s \pi_2^*\alpha)\), together with a partial order on the level of Legendrian submanifolds [\textit{V. Chernov} and \textit{S. Nemirovski}, Geom. Topol. 14, No. 1, 611--626 (2010; Zbl 1194.53066)]. Moreover, this article shows that overtwisted contact manifolds are not strongly orderable. This result makes an effort towards the interesting study of whether all overtwisted contact manifolds are non-orderable (in Eliashberg-Polterovich's sense).
Reviewer: Jun Zhang (Montréal)Real hypersurfaces with isometric Reeb flow in Kähler manifolds.https://www.zbmath.org/1456.530422021-04-16T16:22:00+00:00"Berndt, Jürgen"https://www.zbmath.org/authors/?q=ai:berndt.jurgen"Suh, Young Jin"https://www.zbmath.org/authors/?q=ai:suh.young-jinThe paper under review consists of two main parts. In the first part of the article, the authors develop a general structure theory for real hypersurfaces in Kähler manifolds for which the Reeb flow preserves the induced metric. In the second part of the article, the authors apply this theory to classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type, obtain the following interesting classification result:
\textbf{Theorem.} Let \(M\) be a connected orientable real hypersurface in an irreducible Hermitian symmetric space \(\bar{M}\) of compact type. If the Reeb flow on \(M\) is an isometric flow, then \(M\) is congruent to an open part of a tube of radius \(0 < t < \pi/\sqrt{2}\) around the totally geodesic submanifold \(\Sigma\) in \(\bar{M}\), where
(i) \(\bar{M} = \mathbb{C} P^r = \mathrm{SU}_{r+1}/\mathrm{S}(\mathrm{U}_1\mathrm{U}_r)\) and \(\Sigma = \mathbb{C} P^k\), \(0 \leq k \leq r-1\);
(ii) \(\bar{M} = G_k(\mathbb{C}^{r+1}) = \mathrm{SU}_{r+1}/\mathrm{S}(\mathrm{U}_k\mathrm{U}_{r+1-k})\) and \(\Sigma = G_k(\mathbb{C}^r)\), \(2 \leq k \leq \frac{r+1}{2}\);
(iii) \(\bar{M} = G_2^+(\mathbb{R}^{2r}) = \mathrm{SO}_{2r}/\mathrm{SO}_{2r-2}\mathrm{SO}_2\) and \(\Sigma = \mathbb{C} P^{r-1}\), \(3 \leq r\);
(iv) \(\bar{M} = \mathrm{SO}_{2r}/\mathrm{U}_r\) and \(\Sigma = \mathrm{SO}_{2r-2}/\mathrm{U}_{r-1}\), \(5 \leq r\).
Conversely, the Reeb flow on any of these hypersurfaces is an isometric flow.
Reviewer: Gabriel Eduard Vilcu (Ploieşti)Linear instability for periodic orbits of non-autonomous Lagrangian systems.https://www.zbmath.org/1456.580132021-04-16T16:22:00+00:00"Portaluri, Alessandro"https://www.zbmath.org/authors/?q=ai:portaluri.alessandro"Wu, Li"https://www.zbmath.org/authors/?q=ai:wu.li"Yang, Ran"https://www.zbmath.org/authors/?q=ai:yang.ranTopological structure of spaces of stability conditions and topological Fukaya type categories.https://www.zbmath.org/1456.530722021-04-16T16:22:00+00:00"Qiu, Yu"https://www.zbmath.org/authors/?q=ai:qiu.yu|qiu.yu.1|qiu.yu.2Summary: This is a survey on two closely related subjects. First, we review the study of topological structure of `finite type' components of spaces of Bridgeland's stability conditions on triangulated categories \textit{J. Woolf} [J. Lond. Math. Soc., II. Ser. 82, No. 3, 663--682 (2010; Zbl 1214.18010)], \textit{A. King} and \textit{Y. Qiu} [Adv. Math. 285, 1106--1154 (2015; Zbl 1405.16021)], \textit{Y. Qiu} [Adv. Math. 269, 220--264 (2015; Zbl 1319.18004)], \textit{N. Broomhead} et al. [J. Lond. Math. Soc., II. Ser. 93, No. 2, 273--300 (2016; Zbl 1376.16006)], \textit{Y. Qiu} and \textit{J. Woolf} [Geom. Topol. 22, No. 6, 3701--3760 (2018; Zbl 1423.18044)]. The key is to understand Happel-Reiten-Smalø tilting as tiling of cells. Second, we review topological realizations of various Fukaya type categories \textit{Y. Qiu} [Adv. Math. 269, 220--264 (2015; Zbl 1319.18004)], \textit{Y. Qiu} and \textit{Y. Zhou} [Compos. Math. 153, No. 9, 1779--1819 (2017; Zbl 1405.16024)], \textit{Y. Qiu} [Math. Ann. 365, No. 1--2, 595--633 (2016; Zbl 1378.16027), Math. Z. 288, No. 1--2, 39--53 (2018; Zbl 1442.16017)], \textit{Y. Qiu} and \textit{Y. Zhou} [Trans. Am. Math. Soc. 372, No. 1, 635--660 (2019; Zbl 1444.16013)], \textit{F. Haiden} et al. [Publ. Math., Inst. Hautes Étud. Sci. 126, 247--318 (2017; Zbl 1390.32010)], namely cluster/Calabi-Yau and derived categories from surfaces. The corresponding spaces of stability conditions are of `tame' nature and can be realized as moduli spaces of quadratic differentials due to Bridgeland-Smith and Haiden-Katzarkov-Kontsevich [\textit{T. Bridgeland} and \textit{I. Smith}, ``Quadratic differentials as stability conditions'', Publ. Math., Inst. Hautes Étud. Sci. 121, 155--278 (2015; Zbl 1328.14025)], Haiden et al. [loc. cit.]; \textit{A. Ikeda} [Math. Ann. 367, No. 1--2, 1--49 (2017; Zbl 1361.14015)], \textit{A. King} and \textit{Y. Qiu} [Invent. Math. 220, No. 2, 479--523 (2020; Zbl 1457.13045)].
For the entire collection see [Zbl 1454.00056].Fukaya categories of two-tori revisited.https://www.zbmath.org/1456.530712021-04-16T16:22:00+00:00"Kajiura, Hiroshige"https://www.zbmath.org/authors/?q=ai:kajiura.hiroshigeSummary: We construct an \(A_\infty\)-structure of the Fukaya category explicitly for any flat symplectic two-torus. The structure constants of the non-transversal \(A_\infty\)-products are obtained as derivatives of those of transversal \(A_\infty\)-products.The \(\mathcal{N}_3=3\to\mathcal{N}_3=4\) enhancement of super Chern-Simons theories in \(D=3\), Calabi HyperKähler metrics and M2-branes on the \(\mathcal{C}(\mathrm{N}^{0,1,0})\) conifold.https://www.zbmath.org/1456.530742021-04-16T16:22:00+00:00"Fré, P."https://www.zbmath.org/authors/?q=ai:fre.pietro-giuseppe"Giambrone, A."https://www.zbmath.org/authors/?q=ai:giambrone.adam"Grassi, P. A."https://www.zbmath.org/authors/?q=ai:grassi.pietro-antonio"Vasko, P."https://www.zbmath.org/authors/?q=ai:vasko.petrSummary: Considering matter coupled supersymmetric Chern-Simons theories in three dimensions we extend the Gaiotto-Witten mechanism of supersymmetry enhancement \(\mathcal{N}_3=3\to\mathcal{N}_3=4\) from the case where the hypermultiplets span a flat HyperKähler manifold to that where they live on a curved one. We derive the precise conditions of this enhancement in terms of generalized Gaiotto-Witten identities to be satisfied by the tri-holomorphic moment maps. An infinite class of HyperKähler metrics compatible with the enhancement condition is provided by the Calabi metrics on \(T^\star\mathbb{P}^n\). In this list we find, for \(n=2\) the resolution of the metric cone on \(\mathrm{N}^{0,1,0}\) which is the unique homogeneous Sasaki-Einstein 7-manifold leading to an \(\mathcal{N}_4=3\) compactification of M-theory. This leads to challenging perspectives for the discovery of new relations between the enhancement mechanism in \(D=3\), the geometry of M2-brane solutions and also for the dual description of super Chern-Simons theories on curved HyperKähler manifolds in terms of gauged fixed supergroup Chern-Simons theories.Continuous singularities in Hamiltonian relative equilibria with abelian momentum isotropy.https://www.zbmath.org/1456.530682021-04-16T16:22:00+00:00"Rodríguez-Olmos, Miguel"https://www.zbmath.org/authors/?q=ai:rodriguez-olmos.miguelSummary: We survey several aspects of the qualitative dynamics around Hamiltonian relative equilibria. We pay special attention to the role of continuous singularities and its effect in their stability, persistence and bifurcations. Our approach is semi-global using extensively the Hamiltonian tube of Marle, Guillemin and Sternberg.A survey on \(g=1\) Gromov-Witten invariants via MSP.https://www.zbmath.org/1456.140712021-04-16T16:22:00+00:00"Chang, Huai-Liang"https://www.zbmath.org/authors/?q=ai:chang.huai-liang"Li, Wei-Ping"https://www.zbmath.org/authors/?q=ai:li.wei-pingFor the entire collection see [Zbl 1454.00056].