Unless otherwise noted, all seminar meetings are on Thursdays at 4:00PM in Room 901 Van Vleck. Cookies and juice precede, starting at about 3:45PM.
- Thursday, Sept. 13: Organizational meeting
- Thursday, Sept. 20: David Milovich, Diamond and ultrafilters
- Thursday, Sept. 27: Dan Turetsky, Eta-representations
- Thursday, Oct. 4: David Milovich, Two proofs of the Delta-system Lemma
- Thursday, Oct. 11: Dilip Raghavan, A basic Introduction to Small Cardinals
- Thursday, Oct. 18: Konstantinos Beros, A topological characterization of the Cantor set
- Thursday, Oct. 25, 4:30PM: Andrea Medini, Algebraically closed vs. real closed
- Thursday, Nov. 1: Rob Owen, Class forcing
- Thursday, Nov. 8: Dan McGinn, Truth is a Lattice
- Thursday, Nov. 15: James Hunter, The Church-Rosser Theorem for the untyped lambda-calculus
- Thursday, Nov. 22: No talk (Thanksgiving)
- Thursday, Nov. 29: Ben Ellison, Adjoining a Boolean algebra to a Model
- Thursday, Dec. 6: (Talk cancelled)
- Thursday, Dec. 13: Esra Yeniaras, S-closed spaces
David Milovich, Diamond and ultrafilters
Jensen's diamond principle implies that there is a ultrafilter on omega that is not a P-point, yet every uncountable subset of the ultrafilter has an infinite subset with pseudointersection in the ultrafilter. Diamond also implies there is a nonprincipal ultrafilter on omega such that every uncountable subset of the ultrafilter has an infinite subset with (actual) intersection in the ultrafilter.
David Milovich, Two proofs of the Delta-system lemma
I shall prove the Delta-system Lemma in two different ways. Time permitting, I may also prove the Pressing Down Lemma in two different ways. In each case, one of the proofs uses elementary substructures.
Dilip Raghavan, A basic Introduction to Small Cardinals
This talk will be aimed at "beginners" in set theory. I will
give an accessible introduction to the small cardinals, also known as
Cardinal Characteristics of the Continuum. I plan to discuss several of
the combinatorial small cardinals, as well as some topological ones. I
will introduce them, prove some ZFC inequalities between them and
sketch the proof that certian inequalites are not provable. So I will
assume some familiarity with forcing, up to some iterated forcing.
Dan McGinn, Truth is a Lattice
I'll present a proof system that emphasizes the lattice structure of provability. I suspect most people have figured out that vaguely the meet corresponds to and, join to or, etc. but this system makes this quite connection quite clear. Should be very accessible (I'll do a brief intro to basic proof theory, and I don't know any lattice theory), and likely not surprising to anyone who has thought about these issues. I thought I'd present it because it's the kind of big picture idea that is sometimes hard to find in books (and I thought it was cool when I saw it).
James Hunter, The Church-Rosser Theorem for the untyped lambda-calculus
Computer scientists use the untyped lambda calculus as a theoretical model for computability--numbers and functions are coded as lambda expressions; to apply a function one applies string-replacement rules to perform a reduction. The key is that every lambda expression is reducible to exactly one normal-form expression; uniqueness was proved (in 1936) by Alonzo Church (Joel Robbin's advisor) and former UW Madison Computer Science Prof. J. Barkley Rosser. The uniqueness proof uses induction on terms, and although it is messy, the underlying idea will look familiar to a logic student.
I will give some background on the untyped lambda calculus and then sketch a proof of the Church-Rosser theorem. I think it is interesting to see how computer science has diverged from mathematical logic and to see how ideas from logic can be applied to what is now considered a "computer science" theorem.
Ben Ellison, Adjoining a Boolean algebra to a Model
Wheeler's conjecture states that if a universal theory has a model companion, than its universal Horn subtheory also has a model companion. This was proven false by Glass and Pierce. My research is centered around finding a characterization which will show when a generalization of Wheeler's conjecture holds. In my talk, I will present the current direction of my research. The talk should be accessible to most students, as I will go over most of the basic definitions and concepts I will be dealing with. I hope my talk will provide an example of how one conducts research.