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

Tuesday, April 29 (Midwest Computability Seminar, University of Chicago)
| 1:00 p.m. | Rod Downey, Victoria University of Wellington, New Zealand | Integer valued randomness (see abstract) | depart at 8:30 a.m. from Van Vleck loading dock/ lunch at noon in Ryerson/ dinner at 6 TBA |

2:00 p.m. | Greg Igusa, University of Notre Dame | TBA | ||

3:10 p.m. | Noam Greenberg, Victoria University of Wellington, New Zealand | Ordinals in computability (see abstract) | ||

4:10 p.m. | Kyle Riggs, Indiana University, Bloomington | The decomposability problem for torsion-free Abelian groups is analytic-complete (see abstract) | ||

4:50 p.m. | Sasha Mel'nikov, Victoria University of Wellington, New Zealand | TBA | ||

Wednesday, April 30
| 3:30 p.m./room B131
| Paul Tveite, UW | TBA (specialty exam) | |

Thursday, May 1
| 4:00 p.m./room B239
| Ashutosh Kumar, UW | Avoiding rational distances (thesis defense, see abstract) | |

Tuesday, May 6 | 4:00 p.m. | Rutger Kuyper, Radboud University of Nijmegen, Netherlands | TBA | cookies/ beverages at 3:30/ dinner at 6 TBA |

Wednesday, May 7
| 3:30 p.m./room B131
| Brian Rice, UW | The Thin Set Theorem for pairs and substructures of the Muchnik lattice (thesis defense) | |

Thursday, May 8
| TBA strength | Mushfeq Khan, UW | Some results on algorithmic randomness and computability-theoretic strength (thesis defense, see abstract) | |

Tuesday, May 13 | 4:00 p.m. | Anush Tserunyan, University of Illinois at Urbana-Champaign | TBA | cookies/ beverages at 3:30/ dinner at 6 TBA |

**Course Description:**
I will be talking about the length of Borel hierarchies.

My interest in this topic started with my thesis in 1979:

On the length of
Borel hierarchies

Annals of Math Logic, 16(1979), 233-267.

and continues to this day:

Universal
Functions

(with P.Larson, J.Steprans, W.Weiss)

eprint Apr 2012 last revised Feb 2013.

There is no textbook, however see:

Descriptive
Set Theory and Forcing:
How to prove theorems about Borel sets the hard way

Lecture Notes in Logic 4(1995), Springer-Verlag.

This will lead us into the second topic, diagonally noncomputable (or DNC) functions. With Joe Miller, I showed that there are arbitrarily slow-growing DNC functions that do not compute any Kurtz random real. I also extended Kumabe's result that there is a DNC function of minimal degree by showing that given any oracle X, there is a function that is DNC relative to X and of minimal degree. I will briefly sketch the ideas involved, focusing on how we add to Kumabe's proof for the second theorem.

The last topic is (Lebesgue) density and how it interacts with randomness and computational strength. Bienvenu, Hölzl, Miller, and Nies have shown that the ML-random positive density points are exactly the ones that do not compute the halting problem. Using this theorem as a starting point, I will discuss my own results on the interactions between density, domination properties, minimality, completeness, 1-genericity, and randomness.

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