Geometry - Topology Working Day IV
23 Eylül 2016
Speaker: Yıldıray Ozan, METU
Geometri, Topoloji Etkinliğin Web Sayfası
Title: An Introduction to Symplectic Geometry and Topology
Abstract: First we will give a quick review of symplectic linear algebra. Then we will define symplectic manifolds, symplectic and Lagrangian submanifolds and give basic examples. After defining symplectomorphisms we will talk about Darboux and Weinstein Theorems and the Moser trick, the basic tool used in proving these theorems. Next we will talk about compatible almost complex structures on symplectic manifolds and the relations between symplectic, complex and Kahler geometries. Finally, we will introduce symplectic and Hamiltonian fows, symplectic reduction and talk about Arnold Conjecture if time permits.
Speaker: Sinem Onaran, Hacettepe University
Title: Techniques to built 4-manifolds, Symplectic 4-manifolds
Abstract: In this talk we will define and study some techniques to built 4-manifolds such as Generalized Log transforms, Normal sums, a special type of normal sum called Knot Surgery and Branch Covers.
Speaker: Mehmet Akif Erdal, Bilkent University
Title: Burnside rings of monoids
Abstract: Let M be a monoid. By an M-set we understand a set with a two sided action of M on it, and by an M-equivariant function we understand a function that is equivariant simultaneously on both sides. An M-set is called reversible if M acts by bijections from both sides. There is a subclass of the class of M-sets that forms a category together with M-equivariant functions, which we call category of M actions. This category carries a non-trivial homotopical structure, determined by the maximal reversible parts of the M-sets. We prove that this homotopical structure, restricted to the full-subcategory of left M-actions on finite sets, is 'saturated'; that is, weak equivalences in the category of finite M-sets are completely determined by isomorphisms in its homotopy category. By using this property we introduce the Burnside ring of a monoid and construct the Burnside mark homomorphism. By this way, we get the theory of Burnside rings for the monoids. We show that for a group this construction of Burnside ring coincides with the usual one, and for a commutative monoid it is equal to Burnside ring of its Gröthendieck group. This is a joint work with Özgün Ünlü.