# Category Archives: Seminars

## Ashutosh Kumar: On possible restrictions of null and meager ideal

BIU seminar in Set Theory

On 04/05/2017, 10-12, Building 604, Room 103

Speaker:  Ashutosh Kumar

Title: On possible restrictions of null and meager ideal

Abstract. Fremlin asked if the null ideal restricted to a non null set of reals could be isomorphic to the non stationary ideal on omega_1. Eskew asked if the null and the meager ideal could both be somewhere countably saturated. We’ll show that the answer to both questions is yes. Joint work with Shelah.

## Alessandro Vignati: Set theoretical dichotomies in the theory of continuous quotients

Place: Fields Institute (Room 210)

Date: April 28, 2017 (13:30-15:00)

Speaker: Alessandro Vignati, York University

Title: Set theoretical dichotomies in the theory of continuous quotients

Abstract: We state and (depending on time) prove some dichotomies of set theoretical nature arising in the theory of continuous quotients. In particular we show that the assumption of CH on one side, and of Forcing Axioms on the other, affects the nature of possible embeddings of certain corona algebras, as well as the behavior of their automorphisms group. This is partly joint work with P. McKenney.

## Alberto Marcone: Some results about the higher levels of the Weihrauch lattice

KGRC Research Seminar – 2017‑04‑27 at 4pm

Speaker: Alberto Marcone (Università di Udine, Italy)

Abstract: In the last few years Weihrauch reducibility and the ensuing Weihrauch lattice have emerged as a useful tool for studying the complexity of mathematical statements viewed as “problems” or multi-valued functions. This approach complements nicely the reverse mathematics approach, and has been very successful for statements which are provable in ${\mathsf{ACA}_0}$. The study the Weihrauch lattice for functions arising from statements laying at higher levels, such as ${\mathsf{ATR}_0}$, of the reverse mathematics spectrum is instead in its infancy. We will present some results (work in
progress with my graduate student Andrea Cettolo).

In some cases we obtain the expected finer classification, but in other we observe a collapse of statements that are not equivalent with respect to provability in subsystems of second order arithmetic. This is in part due to the increased syntactic complexity of the statements. Our preliminary results deal with comparability of well-orderings, $\Sigma^1_1$-separation, and
$\Delta^1_1$-comprehension.

## Matteo Viale: Useful axioms

KGRC Research Seminar – 2017‑04‑26 at 4pm

Speaker: Matteo Viale (Università di Torino, Italy)

Abstract: ￼I overview several aspects of forcing axioms which (in my eyes) give solid mathematical arguments explaining why these axioms are so useful in establishing new (consistency) results and/or theorems.

• The first aspect outlines that forcing axioms are natural strengthenings not only of Baire’s category theorem, but also of the axiom of choice (these are two of the most useful non-constructive principles in mathematics), and also strengthenings of most large cardinal axioms (at least for cofinally many of them).
• The second aspect outlines that Shoenfield’s absoluteness, Cohen’s forcing theorem, and Los theorem for standard ultrapowers of a first order structure by a non principal ultrafilter are all specific instances of a more general form of Los theorem which can be declined for what I call boolean ultrapowers.
• The third aspect outlines how strong forcing axioms and Woodin’s generic absoluteness results are two sides of the same coin and will try to explain how stronger and stronger forms of generic absoluteness can be obtained by asserting stronger and stronger forcing axioms. In this context category theoretic ideas start to play a role and we are led to analyze forcings whose conditions are (certain classes of) forcing notions and whose order relation is given by (certain classes of) complete embeddings.

There is a surprising analogy between the theory of these class forcings, the theory of towers of normal ideals, and many of the classical arguments yielding generic absoluteness results. For what concerns the first two aspects of my talk, I do not claim authorship of essentially none of the result I will be talking about, nonetheless it is hard to attribute correctly the relevant results.

## Nicholas Ramsey: NSOP_1 Theories

BGU seminar in Logic, Set Theory and Topology.

Time: Tuesday, April 25th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Nicholas Ramsey (UC Berkeley)

Title: NSOP_1 Theories

Abstract:
The class of NSOP_1 theories was isolated by Džamonja and Shelah in the mid-90s and later investigated by Shelah and Usvyatsov, but the theorems about this class were mainly restricted to its syntactic properties and the model-theoretic general consensus was that the property SOP_1 was more of an unimportant curiosity than a meaningful dividing line. I’ll describe recent work with Itay Kaplan which upends this view, characterizing NSOP_1 theories in terms of an independence relation called Kim-independence, which generalizes non-forking independence in simple theories.  I’ll describe the basic theory and describe several examples of non-simple NSOP_1 theories, such as Frobenius fields and vector spaces with a generic bilinear form.

## Marcin Michalski: Luzin’s theorem

Tuesday, April 25, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Marcin Michalski (Wroclaw University of Science and Technology)

Title: Luzin’s theorem

Abstract:

In 1934 Nicolai Luzin proved that each subset of the real line can be decomposed into two full subsets with respect to ideal of measure or category. We shall present the proof of this result partially decoding his work and we will also briefly discuss possible generalizations.

## Ari Brodsky: Distributive Aronszajn trees

Place: Fields Institute (Room 210)

Date: April 21, 2017 (13:30-15:00)

Speaker: Ari Brodsky, Bar-Ilan University

Title: Distributive Aronszajn trees

Abstract: We address a conjecture asserting that, assuming GCH, for every singular cardinal $\lambda$, if there exists a $\lambda^+$-Aronszajn tree, then there exists one which is moreover $\lambda$-distributive.A major component of this work is the study of postprocessing functions and their effect on square sequences. This is joint work with Assaf Rinot

## Andrés Caicedo: Real-valued measurability and Lebesgue measurable sets

University of Notre Dame, Logic Seminar • 125 Hayes-Healy Hall
Tue May 2, 2017 2:00PM – 3:00PM

Speaker: Andres Caicedo – Mathematical Reviews

Title: Real-valued measurability and Lebesgue measurable sets

Abstract: I will show that the existence of atomlessly measurable cardinals does not settle the range of Lebesgue measure on the projective sets.

## James Cummings: Definable subsets of singular cardinals

Mathematical logic seminar – Apr 11 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     Definable subsets of singular cardinals

Abstract:

Shelah proved the surprising result that if μ is a singular strong limit cardinal of uncountable cofinality, then there is a subset X of μ such that all subsets of μ are ordinal-definable from X. We will give a proof and discuss some complementary consistency results.

## David J. Fernández Bretón: mathfrak p=mathfrak t, III

Tuesday, April 18, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: David J. Fernández Bretón (University of Michigan)

Title: mathfrak p=mathfrak t, III

Abstract:

This is the third and last talk in the series (reasonably self-contained for those who missed any number of previous parts). I will continue to present the proof, due to Maryanthe Malliaris and Saharon Shelah in 2012, that the cardinal invariants p and t are equal, which constitutes an extremely important result in the theory of Cardinal Characteristics of the Continuum.