Graduate Participation Seminar in Logic (GPSLogic)

Spring 2005 Schedule (All talks at 4:00PM in VV901 unless otherwise noted; Cookies at 3:45PM)
Slides, if available, can be accessed by clicking on the talk title.

27 January 2005: Philosophy Department Seminar - Will not meet.
3 February 2005: Chris Alfeld on "Non-branching degrees in the Medvedev Lattice of Pi01 classes"
10 February 2005: Workshop on Quantum Computation - Will not meet.
17 February 2005: Ramiro de la Vega on "Countable tightness, Elementary submodels and Homogeneity"
24 February 2005: Alex Raichev on "Computability and Measure"
3 March 2005: Ben Ellison on "Nonstandard Analysis"
10 March 2005: Discussion (Starring Tom Kent): "The Good, the Bad, and the Ugly: A (Un)Successful Graduate Career
17 March 2005: Asher Kach on "Tournaments and March Madness"
24 March 2005: Spring Break - Will not meet.
31 March 2005: Derek Moffitt on "Bohr Topologies and Hypergraphs"
7 April 2005: Erik Andrejko on "Basic S and L"
14 April 2005: James Hunter on "Baire Category Theorem in Reverse Mathematics"
21 April 2005: Eugene Tsai on "An Introduction to (omega_1,omega_1) Gaps"
28 April 2005: Jaime Posada on "Productive $[\kappa,\lambda]$-Compactness and Regular Ultrafilters"
5 May 2005: Dilip Raghavan on "Generalized Iteration of Forcing"

Abstracts (In presentation order)
Slides, if available, can be accessed by clicking on the talk title.

"Non-branching degrees in the Medvedev Lattice of Pi01 classes" (Chris Alfeld): A Pi01 class is the set of infinite paths through a computable tree. A Medvedev reduction is a computable function from one into another. The Pi01 classes under this reduction form a lattice. I will present several (never before seen) results about non-branching (meet-irreducible) degrees.
Countable tightness, Elementary submodels and Homogeneity (Ramiro de la Vega): We will show (in ZFC) that the cardinality of a compact homogeneous space of countable tightness is no more than the size of the continuum.
Computability and Measure (Alex Raichev): I'll present a few cool arguments in computability theory involving measure. For example, we'll do Sacks's Theorem, which says if the measure of {Y: A <=T Y} is > 0, then A is computable.
Nonstandard Analysis (Ben Ellison): I will present the model theoretic construction of the hyperreals and compare the construction, field properties, and other properties of the hyperreals and the real numbers.
Tournaments and March Madness (Asher Kach): Although often overlooked, the structure of a tournament plays a large role in determining who the winner is. After introducing some terminology, basic results about single-elimination and round-robin tournaments will be given. Infinite tournaments will then be considered, and a characterization of the countable, homogeneous ones will be described.
Basic S and L (Erik Andrejko): I will introduce the basic definitions of S and L spaces. Kunen in 1977 proved that strong S and L spaces do not exist under MA and $\neg$ CH. Szentmiklossy also showed in the same year that compact S spaces don't exist under MA and $\neg$ CH. I will present the proof of Szentmiklossy's 1978 result of the consistency of "MA + \neg CH + S spaces exist".
Baire Category Theorem in Reverse Mathematics (James Hunter): One standard formulation of the Baire Category Theorem--"The intersection of a countable number of dense open sets is dense"--is provable in RCA_0; the proof follows directly from the definition of a real number in RCA_0.
However, a perhaps more useful formulation of the Baire Category Theorem based on "separably closed" (vs. "closed") sets is not provable in RCA_0 nor in the stronger system WKL_0. These two definitions for closed sets are not equivalent under the more restrictive subsystems of second-order arithmetic; Prof. Lempp has presented in Math 975 a proof that "separably closed" and "closed" sets being equivalent requires the even stronger system ACA_0.

Douglas Brown, from whose thesis much of this talk comes, and his advisor Stephen Simpson introduced additional axioms, construction a system they called RCA_0^+. Within RCA_0^+ they were able to prove this other formulation of the Baire Category Theorem (which they called B.C.T. II). Michael Mytilinaios and Ted Slaman subsequently showed that there was a model in which B.C.T. II held but RCA_0^+ failed; so RCA_0^+ is not equivalent to B.C.T. II; the rest of this talk comes from their proof.
An Introduction to (omega_1,omega_1) Gaps (Eugene Tsai): I will introduce the idea of gaps. Then I will present Hausdorff's proof of the existence of (omega_1,omega_1) gaps followed by an axiom of Abraham and Todorcevic which preserves (omega_1, omega_1) gaps in any omega_1-preserving forcing extension.
Productive $[\kappa,\lambda]$-Compactness and Regular Ultrafilters (Jaime Posada): Weak compactness properties as the Lindelof property are not preserved by products. A characterization due to X. Caicedo of $[\kappa,\lambda]$-compactness will be provided using the notion of $[\kappa,\lambda]$-regular ultrafilter.
Generalized Iteration of Forcing (Dilip Raghavan): Iterating forcings along well-ordered sets has proved to be a very versatile method for proving consistency results. However it has several limitations. Iterated forcings along non-well-ordered sets has been recently studied in an attempt to overcome these limitations. I will discuss these generalized methods of iteration.

Links