The cotangent complex and derived deRham complex in the h-topology
Dienstag, 2.8.16, 15:00-16:00, Raum 404, Eckerstr. 1
I will give a generalization of the notion of topological spaces to the categorical setting. In particular we will see the h-topology, which has the property that everything is locally smooth. It is well known that the Kähler differentials are not well-behaved for singular varieties and there are several competing generalizations. We will compare two of them, namely the h-differentials and the cotangent complex, and see that they are basically the same.
Local Morse homology with finite-cyclic symmetry
Montag, 22.8.16, 16:15-17:15, Raum 404, Eckerstr. 1
Morse theory is concerned with the relationships between the\nstructure of the critical set of a function and the topology of the\nambient space where the function is defined. The applications of Morse\ntheory are ubiquitous in mathematics, since objects of interest\n(geodesics, minimal surfaces etc) are often critical points of a\nfunctional (length, area etc). In this talk I will review basic\nconcepts in Morse theory, and will focus on Hamiltonian dynamics where\nthe applications emerge from the fact that periodic solutions of\nHamilton's equations are critical points of the action functional. I\nwill explain how to define a local Morse homology of the action\nfunctional at an isolated periodic orbit which takes into account the\nsymmetries associated to time-reparametrization, and serves a\nwell-defined alternative to local contact homology. Then I will\nexplain dynamical applications. This is joint work with Doris Hein\n(Freiburg) and Leonardo Macarini (Rio de Janeiro).
Classification of 4-qubit entanglement, based on the singularities of the GIT-quotient map
Mittwoch, 24.8.16, 10:15-11:15, Raum 404, Eckerstr. 1
For \(L\)-qubits a state can be represented as\n\(\bsbr{\bpsi} \bin \bmathbb{P} (V)\) with \(V = \bbigotimes_{i=1}^L V_i\) and\n\(V_i \bcong \bmathbb{C}^2\). On this projective space4 there is a linear\ngroup action by \(G = (SL_2 (\bmathbb{C}))^{\btimes L}\) which is a complex\nreductive group whose maximal compact subgroup \(K=(SU_2)^{\btimes L}\).\nFor such a situation there is a unique moment \(\bmu : \bmathbb{P} (V) \bto\n\bmathfrak{k}^{*}\), whose zero set \(\bmu^{-1}(0)\) is of the particular\ninterest due to the isomorphism \(\bmathbb{P} (V)_{ss}//G \bcong \bmu^{-1}\n(0)/K\) where \(\bmathbb{P} (V)_{ss}//G\) is the GIT quotient of the set of\nsemistable points. It turns out, that these quotients maps need to be\nsingular in some points, which are exactly the points of interest, i.e.\nentangled states. These singular points would thus have some non-trivial\nisotropy \(H\). We provide full classification of the families with\nnon-trivial isotropy for the case of \(4\)-quibits.