Colloqiuim-The dimension of a partial order

Title: The dimension of a partial order
Speaker: Steffen Lempp, University of Wisconsin,
Date and time: Friday, November 15, 2:00pm-3pm
Place: Rome Hall (801 22nd Street), Room 206

Abstract: A “linearization” of a partial order is any linear extension of a given partial order. The “dimension” of a partial order is the smallest number of linearization of the partial order such that their intersection is the original partial order. (So, e.g., the dimension of a linear order is clearly 1, and the dimension of a nontrivial antichain is 2.) We call a partial order “locally finite”/”locally countable” if any element in the partial order bounds at most finitely/countably many other elements.

I will first present an old result of Kierstead and Milner, characterizing the precise upper bound on the dimension of an infinite locally finite partial order. We will then present a partial result (joint with Higuchi, Raghavan and Stephan) bounding the dimension of a locally countable partial order and apply it to the partial order of the Turing degrees: Combining this result with a recent result of Kumar and Raghavan, we can show that ZFC (the usual axioms of set theory) does not determine the dimension of the Turing degrees.

My talk will not assume any knowledge of logic, and only easy fact about linear and partial orders.

Short Bio: Steffen Lempp received his Ph.D. in 1986 from the University of Chicago under the direction of Bob Soare and has been on the faculty of the University of Wisconsin-Madison since 1988. He is the author of almost 100 research publications, has been an invited speaker at over 100 conferences, workshops and seminars, and has advised 18 doctoral students. For many years, he served first as the editor of the Journal of Symbolic Logic and then the Transactions of the AMS, and he has co-organized four Oberwolfach workshops.