Date | Time | Speaker | Title | Cookies, dinner, etc. |

Tuesday, October 13 | 4:00 p.m. | Joe Miller, UW | Subclasses of the K-trivial degrees (see abstract) | cookies/juice at 3:30 |

Tuesday, October 20 | 4:00 p.m. | Julia Knight, University of Notre Dame, Indiana | Lengths of roots of polynomials in a Hahn field (see abstract) | cookies/juice at 3:30/ dinner at 6 at Crandall's Peruvian & American Cuisine, (334 State St.) |

Tuesday, October 27 | 4:00 p.m. | Reese Johnston, UW | TBA | cookies/juice at 3:30 |

**Prerequisites:** Math 770, and Math 773 or concurrent registration
in Math 773

**Time and Place:** MWF 13:20-14:10

**Textbook:** none

**Course Description:**
The Medvedev and Muchnik degrees are extensions of the Turing degrees studied
in computability theory. Both of these structures also have nice substructures,
namely, the degrees of the so-called effectively closed sets, which form an
analogue to the computably enumerable degrees studied in the Turing degrees.

We will discuss various aspects of these structures. In particular, we will cover the following topics, including their necessary prerequisites:

- Definability of substructures
- Automorphisms
- Embeddability
- Filters and ideals
- Density
- Connections to intuitionistic logic
- Complexity

I will finish by describing a further generalization of k/n-bases. In general, arbitrary families of projections (along the coordinate axes) do not give us new subideals of the K-trivial sets.

An **integer part** for a real closed field R is a discrete
ordered subring appropriate for the range of a floor function. Mourgues and
Ressayre [MR] showed that every real closed field R has an integer part.
To do this, they showed that there is a "truncation closed'' embedding d into
the Hahn field k((G)), where G is the value group of R and k is the residue
field. Given a residue field section k and a well ordering < of R,
the M-R procedure becomes canonical. Knight and Lange wanted to measure the
complexity of the Mourgues and Ressayre procedure. To do this they needed to
bound the lengths of the power series in the range of the embedding.
In [KL], they conjectured that if < has order type ω, then the
elements of d(R) have length less than
ω^{ωω}.
The conjecture follows from the technical theorem.

Bibliography:

[KL] J. F. Knight and K. Lange, " Complexity of structures associated with real closed fields", Proc. London Math. Soc., vol. 107 (2013), pp. 177-197.

[M] S. Maclane, "The universality of formal power series fields", Bull. Amer. Math. Soc., vol. 45 (1939), pp. 888-890.

[MR] M. H. Mourgues and J.-P. Ressayre, "Every real closed field has an integer part", J. Symb. Logic, vol. 58 (1993), pp. 641-647.

Prepared by Steffen Lempp (@math.wisc.edu">lemppmath.wisc.edu)