Blog Archives

Topic Archive: ultrafilters

Set theory seminarModels of PAThursday, June 18, 20155:00 pm4214-03Math Thesis room

Mauro Di Nasso

Ultrafilters and nonstandard methods in combinatorics of numbers

Universita di Pisa

In certain areas of Ramsey theory and combinatorics of numbers, diverse non-elementary methods are successfully applied, including ergodic theory, Fourier analysis, (discrete) topological dynamics, algebra in the space of ultrafilters. In this talk I will survey some recent results that have been obtained by using tools from mathematical logic, namely ultrafilters and nonstandard models of the integers.

On the side of Ramsey theory, I will show how the hypernatural numbers of nonstandard analysis can play the role of ultrafilters, and provide a convenient setting for the study of partition regularity problems of diophantine equations. About additive number theory, I will show how the methods of nonstandard analysis can be used to prove density-dependent properties of sets of integers. A recent example is the following theorem: If a set $A$ of natural numbers has positive upper asymptotic density then there exists infinite sets $B$, $C$ such that their sumset $C+B$ is contained in the union of $A$ and a shift of $A$. (This gives a partial answer to an old question by Erdős.)

The slides are here.

Set theory seminarFriday, October 18, 201310:45 amGC 6417Two talks for set theory seminar on this day

Natasha Dobrinen

Survey on the structure of the Tukey theory of ultrafilters

University of Denver

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.

Natasha Dobrinen
University of Denver
Professor Dobrinen earned her Ph.D. at the University of Minnesota under Karel Prikry in 1996, afterwards holding post-doctoral positions at Penn State and the University of Vienna before moving to the University of Denver. Her research interests mainly fall under the broad category of logic and foundations of Mathematics, and includes research in set theory, Ramsey theory, Boolean algebras, and measure theory. She has investigated relationships between random reals, eventually dominating functions, measure, generalized weak distributive laws, infinitary two-player games, and complete embeddings of the Cohen algebra into complete Boolean algebras. Currently, she is working on problems in Ramsey theory, problems regarding the structure of the Tukey types of ultrafilters, and problems involving both.
CUNY Logic WorkshopFriday, October 18, 20132:00 pmGC 6417

Paul B. Larson

Generic choice functions and ultrafilters on the integers

Miami University of Ohio

We will discuss a question asked by Stefan Geschke, whether the existence of a selector for the equivalence relation $E_0$ 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 $mathbb{P}_{mathrm{max}}$. This proof shows that, under suitable hypotheses, if $E$ is a universally Baire equivalence relation on the reals, with countable classes, then forcing over $L(E,mathbb{R})$ to add a selector for $E$ does not add a nonprincipal ultrafilter on the integers.