In the Spring of 2017, the seminar will be run as a UCLA math course 290 with the students of the course speaking for credit. We will meet wednesday 4-6 in MS 6221. The recent/current talks are listed below in reverse chronological order.
This page contains only the somewhat recent history of the seminar. Information about the seminar from before the creation of this page can be found here.
Thomas Gilton. A generalization of the Semi-Open Coloring Axiom to aleph_2. June 7.
In this talk, we'll define and prove the consistency of a generalization of SOCA
to \aleph_2. In order to do so, we review some of the machinery
that Neeman has developed for his generalization of Baumgartner's theorem on
\aleph_1-dense sets of reals, and then we show how the original argument for
SOCA at \aleph_1, found in Abraham-Rubin-Shelah, can be generalized using
Neeman's machinery. If time permits, we'll give a few applications to the
bounding number, chains and antichains in P(\omega), and monotonic subfunctions
of injective real-valued functions.
Tyler Arant. Effectively decomposing Borel functions. May 31.
Abstract: When can a Borel function be decomposed into "simple" pieces? One
answer to this (admittedly vague) question is given by the celebrated
Jayne-Rogers Theorem; roughly, this theorem states that a Borel function can be
decomposed into countably many continuous pieces each with closed domain if and
only if the preimage under f of any \Sigma^0_2 set is again \Sigma^0_2. This
theorem has inspired a lot of research, with the goal of proving similar
decomposability results for other subclasses of Borel functions. In this talk
we will explore recent work of Kihara which establishes a partial solution to an
effective analogue of the so-called decomposability conjecture. Interestingly,
at the heart of the proof is an application of the Shore-Slaman Join Theorem
from degree theory. Our discussion will be limited to functions on the Baire
Space, so as to avoid the full abstract machinery of Effective Descriptive Set
Theory.
Zach Norwood. Madness, Ramsey stuff, and ideals. May 24.
We'll begin by exploring a connection between mad families and the Ramsey
property. Using this connection, we will be able to sketch a few arguments for
nonexistence of definable mad families. The middle third of the talk is devoted
to proving a neat recent theorem of Hrusak et al. about the Ramsey theory of
ideals on omega. We will conclude, depending on time and interest, with a
selection of the following activities: proving my characterization of filters
with the Baire property, proving the Matthias property for the forcing used in
Itay & my study of the Solovay model, and chocolate-eating.
I intend this to be pretty accessible. Light on the serious set theory.
Alex Mennen. Zariski geometries. May 17.
I will describe Zariski geometries and their relationship to algebraic
geometry (including intersection theory) and stability.
Madeline Barnicle. Ideals of Polynomial Rings and Power Series. May 10,
2017.
We examine the definability and special classes of ideals (such as
primes) in polynomial rings and power series, such as those given by p-adics,
from Herman to the present day. Along the way, we mention several results of van
den Dries.
Riley Thornton. Effective dimension and fractal geometry. May 3, 2017.
I will give a proof of Lutz's "point-to-set principle," relating effective
dimension and Hausdorff dimension, and recover some classical results with very
short proofs.
Assaf Shani. The effect of a Sacks real on forcing extensions. April 26,
2017.
I will present the following theorem of Carlson: Assuming PFA, after adding a
single Sacks real, the resulting model satisfies MA. Time permitting, I will
sketch Todorcevic's proof that PFA fails after adding a single Sacks real. To
put things in prerspective: Roitman proved that MA fails in the extension by a
single Cohen real. Shelah then strengthened the result, showing that there is
in fact a Souslin tree in a the Cohen real extension. Kunen noted that
Roitman's construction also works to show that MA fails in a single Random real
extension. On the other hand, Laver proved that, assuming MA, Souslin's
hypothesis remains true after adding a Random real.
Wednesday April 19th. Zach Norwood. Coding and trees.
We'll outline a proof of Kunen's theorem that the absoluteness of L(R)
by ccc forcing is equiconsistent with a weakly compact cardinal. After a
discussion of trees and weak compactness, we'll sketch the routine parts of the
proof and then turn our focus to the cleverest part of the lower-bound argument.
Wednesday April 12th. Thomas Gilton. Open coloring axioms and the continuum.
Monday March 6th. Nadja Hempel. Division rings with ranks, continued.
Monday February 27th. Nadja Hempel. Division rings with ranks.
In this talk I will first give a brief introduction on fields and division rings
which fall into different well studied model theoretic classes (stable, simple,
NIP). The goal is to analyze division rings which admit a well-behaved ordinal
valued rank function on definable sets that behaves like a rudimentary notion of
dimension. These are called superrosy division rings. Examples are the
quaternions, any superstable division ring (which are known to be algebraically
closed fields by a theorem of Macintyre/Cherlin-Shelah) and more generally
supersimple division rings (which are commutative by a result of Pillay, Scanlon
and Wagner). We show that any superrosy division ring has finite dimension over
its center, generalizing the aforementioned results.
Monday February 6th. Spencer Unger. Borel circle squaring part 3.
Monday January 30th. Spencer Unger. Borel circle squaring part 2.
Monday January 23rd. Spencer Unger. Borel circle squaring.
Monday November 28th. Omer Ben-Neria. Weak prediction principles.
I plan to talk about Weak Prediction Principles and present some results from my
joint paper with Shimon Garti and Yair Hayut. I will review the history of the
weak diamond principle introduced by Shelah and Devlin in the 70's and discuss
its relations with cardinal arithmetic.
Monday November 21th. Tyler Arant. Continuation of the previous week.
Monday November 14th. Tyler Arant. Barwise Compactness and Effective
Descriptive Set Theory.
The compactness theorem fails for infinitary logic; however, in his 1967 thesis,
Jon Barwise proved a compactness result for so called admissible fragments of
infinitary logic. In addition to being a fundamental result in admissible set
theory, the so called Barwise compactness theorem has had many applications,
from providing a simple proof of Sacks' theorem to more recent applications in
the effective theory of equivalence relations on Polish spaces. In this talk,
we will briefly discuss the rudiments of the theory of admissible sets, we will
prove the Barwise compactness theorem using game theoretic techniques (avoiding
the need for the proof theory of infinitary logic), and (time permitting) we
will explore some applications of the theorem to the study of effectively Borel
equivalence relations.
Monday November 7. John Susice. Suslin trees part 2.
Monday October 24. John Susice. Suslin trees.
Monday October 17. Thomas Gilton. A Generalization of MA.
In this talk we will present a generalization of MA due to Aspero and Mota in a
recent paper of theirs. This forcing axiom, for so-called "finitely proper"
forcings, is strictly stronger than MA; we will discuss an example witnessing
this, the failure of "Weak Club Guessing" on \omega_1. However, unlike PFA, a
model of this forcing axiom can be obtained without large cardinals. We will
introduce the technology necessary to prove the iteration theorem, that of
symmetric systems of models as side conditions, and discuss why they are useful.
Additionally, if time permits, we will gesture towards a proof of the iteration
theorem itself.
Monday October 10. Spencer Unger. Successive failures of weak square continued.
Monday October 3. Spencer Unger. Successive failures of weak square.
Abstract: We will work towards the construction of a model where weak square
fails up to aleph_{omega^2+2} and aleph_{omega^2} is strong limit. The proof
involves generalizations of old work of Mitchell and techniques from an
unpublished proof of Woodin that GCH can fail first at aleph_omega.
Monday May 16. Zach Norwood. Are the Cohen & random posets special?
Abstract: Prikry asked whether it's consistent that every ccc forcing adds a
Cohen or a random real. Even modest subquestions have proved to be quite
difficult. I'll survey some known results and mention some related open
questions. Should be a pretty light talk.
Monday May 9. Zach Norwood. TBA.
Monday May 2. Zach Norwood. How difficult is it to change the theory of L(R)? Abstract: We'll prove the theorem of Neeman-Zapletal that, under large cardinals, the theory of L(R) with real and ordinal parameters cannot be changed by proper forcing. Along the way, we'll see why the theory of L(R) with only real parameters cannot be changed by any small forcing.
Monday April 25. John Susice. Diamond.
Monday April 18th. Spencer Unger. Gitik's gap 2 short extender forcing. Third and final talk.
Monday April 11th. Spencer Unger. Gitik's gap 2 short extender forcing. Second talk.
Monday April 4th. Spencer Unger. Gitik's gap 2 short extender forcing. First talk.
Wednesday March 9th. Andre Nies. See the Cabal seminar page
Wednesday March 2nd. Bill Chen. Shelah's revised GCH theorem.
Abstract: In this talk, I will state and prove Shelah's revised GCH theorem,
which he declared in humor to be the positive solution to Hilbert's first
problem. This is a ZFC theorem which says that a certain cardinal arithmetic
function assumes the least possible value above uncountable strong limit
cardinals, and has been used to replace GCH in some combinatorial applications.
The talk will not require previous knowledge of pcf theory, but the proofs will
use the "possible cofinality" idea and some nice elementary submodel arguments.
Wednesday February 24th. Thomas Gilton, The failure of GCH at a measurable from the optimal hypothesis.
Wednesday February 17th. Noah Schweber, The reverse mathematics of determinacy.