اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA Hofer-like metric on the group of symplectic diffeomorphisms
122
0
0.0
(
0
)
تأليف
Augustin Banyaga
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Using a Hodge decomposition of symplectic isotopies on a compact symplectic manifold $(M,omega)$, we construct a norm on the identity component in the group of all symplectic diffeomorphisms of $(M,omega)$ whose restriction to the group $Ham(M,omega)$ of hamiltonian diffeomorphisms is bounded from above by the Hofer norm. Moreover, $Ham(M,omega)$ is closed in $Symp(M,omega)$ equipped with the topology induced by the extended norm. We give an application to the $C^0$ symplectic topology. We also discuss extensions of Ohs spectral distance.
قيم البحث
اقرأ أيضاً
We generalize the hamiltonian topology on hamiltonian isotopies to an intrinsic symplectic topology on the space of symplectic isotopies. We use it to define the group $SSympeo(M,omega)$ of strong symplectic homeomorphisms, which generalizes the grou
p $Hameo(M,omega)$ of hamiltonian homeomorphisms introduced by Oh and Muller. The group $SSympeo(M,omega)$ is arcwise connected, is contained in the identity component of $Sympeo(M,omega)$; it contains $Hameo(M,omega)$ as a normal subgroup and coincides with it when $M$ is simply connected. Finally its commutator subgroup $[SSympeo(M,omega),SSympeo(M,omega)]$ is contained in $Hameo(M,omega)$.
In this work, we prove that any symplectic matrix can be factored into no more than 9 unit triangular symplectic matrices. This structure-preserving factorization of the symplectic matrices immediately reveals two well-known features that, (i) the de
terminant of any symplectic matrix is one, (ii) the matrix symplectic group is path connected, as well as a new feature that (iii) all the unit triangular symplectic matrices form a set of generators of the matrix symplectic group. Furthermore, this factorization yields effective methods for the unconstrained parametrization of the matrix symplectic group as well as its structured subsets. The unconstrained parametrization enables us to apply faster and more efficient unconstrained optimization algorithms to the problems with symplectic constraints under certain circumstances.
We prove that the Gromov width of coadjoint orbits of the symplectic group is at least equal to the upper bound known from the works of Zoghi and Caviedes. This establishes the actual Gromov width. Our work relies on a toric degeneration of a coadjoi
nt orbit to a toric variety. The polytope associated to this toric variety is a string polytope arising from a string parametrization of elements of a crystal basis for a certain representation of the symplectic group.
We study the existence of symplectic resolutions of quotient singularities V/G where V is a symplectic vector space and G acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the form $K r
times S_2$ where $K < SL_2(C)$, for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for $dim V eq 4$, we classify all quotient singularities $V/G$ admitting a projective symplectic resolution which do not decompose as a product of smaller-dimensional quotient singularities, except for at most four explicit singularities, that occur in dimensions at most 10, for whom the question of existence remains open.
For $(mathbb{C} P^2 # 5{overline {mathbb{C} P^2}},omega)$, let $N_{omega}$ be the number of $(-2)$-symplectic spherical homology classes.We completely determine the Torelli symplectic mapping class group (Torelli SMCG): the Torelli SMCG is trivial if
$N_{omega}>8$; it is $pi_0(Diff^+(S^2,5))$ if $N_{omega}=0$ (by Paul Seidel and Jonathan Evans); it is $pi_0(Diff^+(S^2,4))$ in the remaining case. Further, we completely determine the rank of $pi_1(Symp(mathbb{C} P^2 # 5{overline {mathbb{C} P^2}}, omega)$ for any given symplectic form. Our results can be uniformly presented regarding Dynkin diagrams of type $mathbb{A}$ and type $mathbb{D}$ Lie algebras. We also provide a solution to the smooth isotopy problem of rational $4$-manifolds.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
