We will finish explaining the minimal model program for surfaces and introduce minimal models of higher dimensional varieties. In particular, we will discuss how singularities can arise when contracting external rays in higher dimensions and how we should expand our category of minimal models to account for these singularities.
Introduction to the Minimal Model Program IV (Final part)
We ascertain properties of the algebraic structures in towers of codes, lattices, and vertex operator algebras (VOAs) by studying the associated subobjects fixed by lifts of code automorphisms. In the case of sublattices fixed by subgroups of code automorphisms, we identify replicable functions that occur as quotients of the associated theta functions by suitable eta products. We show that these lattice theta quotients can produce replicable functions not associated to any individual automorphisms. Moreover, we show that the structure of the fixed subcode can induce certain replicable lattice theta quotients and we provide a general code theoretic characterization of order doubling for lifts of code automorphisms to the lattice-VOA. Finally, we prove results on the decompositions of characters of fixed subVOAs. This talk is based on joint work with Jennifer Berg, Eva Goedhart, Hussain M. Kadhem, Allechar Serrano López, and Stephanie Treneer.
Starting from the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and the Dirac operator on the total space and on the base space of the bundle. An example of the noncommutative three-torus viewed as a U(1) bundle over the noncommutative two-torus gives all connections compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori is found, which arise from the base-space Dirac operator and a suitable connection. Based on a joint work with Ludwik Dabrowski.
Noncommutative circle bundles and new Dirac operators
Introduction to the Minimal Model Program IV (Final part)