For more information about this meeting, contact Becky Halpenny.

Title: | "Rigidity of periodic cyclic homology under certain smooth deformations" |

Seminar: | Ph.D. Thesis Defense |

Speaker: | Allan Yashinski, Adviser: Nigel Higson, Penn State |

Abstract: |

Given a formal deformation of an algebra, Getzler defined a connection on the periodic cyclic homology of the deformation, which he called the Gauss-Manin connection. We define and study this connection for smooth one-parameter deformations. Our main example is the smooth noncommutative n-torus viewed as a deformation of the algebra of smooth functions on the n-torus. In this case, we use the Gauss-Manin connection to give a parallel translation argument that shows that the periodic cyclic homology groups of noncommutative tori are the same as in the commutative case. As a consequence, we obtain differentiation formulas relating various cyclic cocycles on noncommutative tori.
By considering the properties leveraged in the case of noncommutative tori, we generalize to a larger class of deformations, including nontrivial crossed product algebras by the group of real numbers. The algebras of such a deformation extend naturally to differential graded algebras, and we show that they are fiberwise isomorphic as A_infinity-algebras. As a corollary, periodic cyclic homology is preserved under this type of deformation. In particular, this gives yet another calculation of the periodic cyclic homology of noncommutative tori and a proof of the Thom isomorphism in cyclic homology. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 05 / 20 / 2013 |

Time: | 10:30am - 12:30pm |