Blog Archives

Topic Archive: descriptive set theory

Set theory seminarFriday, October 23, 201510:00 amGC 3212

Scott Cramer

Tree representations from very large cardinals

Rutgers University

We will discuss the propagation of certain tree representations in the presence of very large cardinals. These tree representations give generic absoluteness results and have structural consequences in the area of generalized descriptive set theory. In fact these representations give us a method for producing models of strong determinacy axioms.

CUNY Logic WorkshopFriday, February 20, 20152:00 pmGC 6417

Linda Brown Westrick

Effective dimension in subshifts

University of Connecticut

A subshift is a subset of Cantor space that is both topologically closed and closed under the shift operation. Any element of a subshift can be viewed as a trajectory in a discrete dynamical system. One way a trajectory can be complicated is to have a large effective dimension. We consider the effective dimension spectrum of a subshift X, defined as { dim x : x ∈ X }, where dim is the effective dimension. Most commonly, the dimension spectrum of X is [ 0, h(X) ], where h(X) is the entropy of X. We give examples where the dimension spectrum does not follow this pattern, and discuss partial results and open questions related to the problem of characterizing the dimension spectra of subshifts. No prior knowledge about subshifts will be assumed.

Linda Brown Westrick
University of Connecticut
Linda Brown Westrick received her doctorate in 2014 from the University of California at Berkeley, under the supervision of Ted Slaman. Currently she holds a postdoctoral position at the University of Connecticut. She works in computability theory and effective descriptive set theory, applying techniques from these areas to questions in analysis, symbolic dynamics, and chaos.
Set theory seminarFriday, December 12, 201412:00 pm

Sandra Uhlenbrock

Producing $M_n^{\#}$ from Boldface Level-wise Projective Determinacy

University of Münster

Projective determinacy is the statement that for certain infinite games, where the winning condition is projective, there is always a winning strategy for one of the two players. It has many nice consequences which are not decided by ZFC alone, e.g. that every projective set of reals is Lebesgue measurable. An old so far unpublished result by W. Hugh Woodin is that one can derive specific countable iterable models with Woodin cardinals, $M_n^{\#}$, from this assumption. Work by Itay Neeman shows the converse direction, i.e. projective determinacy is in fact equivalent to the existence of such models. These results connect the areas of inner model theory and descriptive set theory. We will give an overview of the relevant topics in both fields and, if time allows, sketch a proof of the result that for the odd levels of the projective hierarchy boldface $\Pi^1_{2n+1}$-determinacy implies the existence of $M_{2n}^{\#}(x)$ for all reals $x$.

Model theory seminarFriday, November 21, 201410:45 am

Countable model theory and the complexity of isomorphism

University of Maryland

We discuss the Borel complexity of the isomorphism relation (for countable models of a first order theory) as the “right” generalization of the model counting problem. In this light we present recent results of Dave Sahota and the speaker which completely characterize the complexity of isomorphism for o-minimal theories, as well as recent work of Laskowski and Shelah which give a partial answer for omega-stable theories. Along the way, we introduce a few open problems and barriers to generalizing the existing results.

Tuesday, May 13, 20143:50 pm750 Schapiro CEPSR, Columbia University Morningside Campus

Descriptive Graph Combinatorics and Computability

Cal Tech
CUNY Logic WorkshopFriday, March 14, 20142:00 pmGC6417

Robin Tucker-Drob

Structurability and countable Borel equivalence relations

Rutgers University

The dynamical and descriptive set theoretic complexity of a countable Borel equivalence relation E can often be understood in terms of the kinds of countable first order structures which are compatible with E in a suitable sense. In this talk I will make this suitable sense precise by discussing the notion of Borel structurability. I will also discuss some recent joint work with Brandon Seward in which we show that the equivalence relation generated by the free part of the translation action of a countable group G on its powerset is structurably-universal among equivalence relations generated by free Borel actions of G.

CUNY Logic WorkshopFriday, March 7, 20142:00 pmGC 6417

Scott Cramer

Generalized descriptive set theory with very large cardinals

Rutgers University

We will discuss the structure L(Vλ+1) and attempts to generalize facts of descriptive set theory to this structure in the presence of very large cardinals. In particular we will introduce an axiom called Inverse Limit Reflection which we will argue is analogous to the Axiom of Determinacy in this context. The slides for this talk are available here.

Grigor Sargsyan
Rutgers University
Grigor Sargsyan is a professor of mathematics at Rutgers University. He received his Ph.D. at UC Berkeley, 2009. His research interests are in logic, set theory, and foundations: descriptive set theory, inner model theory, large cardinals, and forcing.
Set theory seminarFriday, October 18, 20139:30 amGC 6417Two talks for set theory seminar on this day

Marcin Sabok

Canonical Ramsey theory on Polish spaces

Instytut Matematyczny Uniwersytetu Wrocławskiego, Instytut Matematyczny Polskiej Akademii Nauk

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:

Marcin Sabok
Instytut Matematyczny Uniwersytetu Wrocławskiego, Instytut Matematyczny Polskiej Akademii Nauk
Dr. Sabok (Ph.D. 2009 University of Wrocław) undertakes research in set theory, especially descriptive set theory, forcing and its applications.