Wednesday, October 16, 2013 6:30 pm GC 4214.03
Speaker: Whanki Lee Queensborough Community College, CUNY
Title: Cofinal extensions of recursively saturated ordered structures
Friday, October 18, 2013 9:30 am GC 6417 Two talks for set theory seminar on this day
Speaker: Marcin Sabok Instytut Matematyczny Uniwersytetu Wrocławskiego, Instytut Matematyczny Polskiej Akademii Nauk
Title: Canonical Ramsey theory on Polish spaces
I would like to give an overview of recent results in canonical Ramsey theory in the context of descriptive set theory. This is the subject of a recent monograph joint with with Vladimir Kanovei and Jindra Zapletal. The main question we address is the following. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? Canonical Ramsey theory stems from finite combinatorics and is concerned with finding canonical forms of equivalence relations on finite (or countable) sets. We obtain canonization results for analytic and Borel equivalence relations and in cases when canonization is impossible, we prove ergodicity theorems. For a publisher’s book description see:
Friday, October 18, 2013 10:15 am GC 5382
Speaker: Tom Scanlon University of California – Berkeley
Title: D-Fields as a Common Formalism for Difference and Differential Algebra
In a series of papers with Rahim Moosa, I have developed a theory of D-rings unifying and generalizing difference and differential algebra. Here we are given a ring functor D whose underlying additive group scheme is isomorphic to some power of the additive group. A D-ring is a ring R given together with a homomorphism f : R → D(R). A first motivating example is when D(R) = R[ε]/(ε2), so that the data of D-ring is that of an endomorphism σ:R → R and a σ-derivation ∂:R → R (that is, ∂(rs) = ∂(r)σ(s)+σ(r)∂(s)). Another example is when D(R) = R, where a D-ring structure is given by an endomorphism of R.
We develop a theory of prolongation spaces, jet spaces, and of D-algebraic geometry. With our most recent paper, we draw out the model theoretic consequences of this work showing that in characteristic zero, the theory of D-fields has a model companion, which we call the theory of D-closed fields, and that many of the refined model theoretic theorems (eg the Zilber trichotomy) hold at this level of generality. As a complement, we show that no such model companion exists in characteristic p under a mild hypothesis on D.
Friday, October 18, 2013 10:45 am GC 6417
Speaker: Natasha Dobrinen University of Denver
Title: Survey on the structure of the Tukey theory of ultrafilters
The Tukey order on ultrafilters is a weakening of the well-studied Rudin-Keisler order, and the exact relationship between them is a question of interest. In second vein, Isbell showed that there is a maximum Tukey type among ultrafilters and asked whether there are others. These two questions are the main guiding forces of the current research. In this talk, we present highlights of recent work of Blass, Dobrinen, Mijares, Milovich, Raghavan, Todorcevic, and Trujillo (in various combinations for various papers). Further information about results mentioned in this talk can be found in a recent survey article by the speaker.
There will be two talks for the set theory seminar on this day.
Friday, October 18, 2013 12:30 pm GC6417
Speaker: Lynn Scow Vassar College
Title: Ramsey Transfer Theorems
We survey some of the known approaches to transfer a Ramsey theorem for one class of finite structures to another. We will isolate some easy consequences and point to further directions.
Friday, October 18, 2013 2:00 pm GC 6417
Speaker: Paul B. Larson Miami University of Ohio
Title: Generic choice functions and ultrafilters on the integers
We will discuss a question asked by Stefan Geschke, whether the existence of a selector for the equivalence relation E0 implies the existence of a nonprincipal ultrafilter on the integers. We will present a negative solution which is undoubtedly more complicated than necessary, using a variation of Woodin’s mathbbPmathrmmax. This proof shows that, under suitable hypotheses, if E is a universally Baire equivalence relation on the reals, with countable classes, then forcing over L(E,mathbbR) to add a selector for E does not add a nonprincipal ultrafilter on the integers.
Friday, October 18, 2013 2:00 pm GC 4419
Speakers: Haim Gaifman (Columbia) and Rohit Parikh (CUNY)
Title: The Columbia-CUNY Workshop in Logic, Probability, and Games
There will be a meeting of this seminar on October 18 from 2 to 4 PM in room 4419. Haim Gaifman (Columbia) and Rohit Parikh (CUNY) will speak. Details will be announced next week.
This is a meeting of a joint CUNY-Columbia research group on Logic, Probability and Games.
Description: This workshop is concerned with applying formal methods to fundamental issues, with an emphasis on probabilistic reasoning decision theory and games. In this context “logic” is broadly interpreted as covering applications that involve formal representations. The topics of interest have been researched within a very broad spectrum of different disciplines, including philosophy (logic and epistemology), statistics, economics, and computer science. The workshop is intended to bring together scholars from different fields of research so as to illuminate problems of common interest from different perspectives. Throughout each academic year, meetings are regularly presented by the members of the workshop and distinguished guest speakers and are held alternatively at Columbia University and CUNY Graduate Center.