Abstract: Milnor-Witt K-theory arises in the Morel-Voevodsky homotopy theory over a field and plays a role in the classification of vector bundles over smooth schemes. Morel in collaboration with Hopkins discovered a nice presentation of these groups, which has been recently generalized by Changlong Zhong, Stephen Scully and myself to semilocal rings which contain an infinite field. In my talk I will discuss this result and also present some applications of these groups.

Abstract: "Down, down, down. Would the fall never come to an end! I wonder how many miles I have fallen by this time? she said aloud. I must be getting somewhere near the center of the earth", wrote Charles Dodgson, the English writer and mathematician, of Alice's fantastic adventure into a bizarre world of underground creatures. I shall retell the story of our centennial adventure into the relativistic wonderland, a tall tale of non-sensical creatures that includes blunders, black holes, Schroedinger's cats, and a very vibrant vacuum!