# Seminars & Colloquia Calendar

## The tree property

#### Dima Sinapova (Univ. of Illinois at Chicago)

Location: ** Hill 705**

Date & time: Friday, 05 April 2019 at 4:00PM - 5:00PM

Abstract: Two central themes in logic are how much the universe of sets resembles Gödel's constructible universe \(L\) versus what is possible from forcing and large cardinals. Both are addressed by using infinite combinatorics to investigate how much compactness can be obtained in the universe. Compactness is the phenomenon when a given property holding for every smaller substructure of some object implies that property holds for the object itself. This is usually a consequence of large cardinals, and tends to fail in \(L\).

A key instance of compactness is the tree property, which states that every tree of height \(\kappa\) and levels of size less than \(\kappa\) has a cofinal branch. Informally, this principle is a generalization of König's infinity lemma to uncountable cardinals. It turns out that the tree property and certain strengthenings capture the combinatorial essence of large cardinals. An old project in set theory is to force the tree property (and some strengthenings) at every regular cardinal greater than \(\aleph_1\). I will go over the background and then discuss some recent results giving the state of the art of this project.

R. Shapiro Organizer's Page

Chiara Damiolini, Ian Coley and Franco Rota -Charles Weibel Organizer's Page

Brooke Logan

Wujun Zhang Organizer's webpage

Ziming Shi, Sagun Chanillo, Xiaojun Huang, Chi Li, Jian Song Seminar website Old seminar website

Swastik Kopparty, Sepehr Assadi Seminar webpage

Jeffry Kahn, Bhargav Narayanan, Jinyoung Park Organizer's webpage

Brooke Ogrodnik, Website

Robert Dougherty-Bliss and Doron Zeilberger --> homepage

Paul Feehan, Daniel Ketover, Natasa Sesum Organizer's webpage

Lev Borisov, Emanuel Diaconescu, Angela Gibney, Nicolas Tarasca, and Chris Woodward Organizer's webpage

Jason Saied Seminar webpage

Brian Pinsky, Rashmika Goswami website

Quentin Dubroff Organizer's webpage

James Holland; Organizer website

Edna Jones Organizer's webpage

Brooke Ogrodnik website

Yanyan Li, Zheng-Chao Han, Jian Song, Natasa Sesum Organizer's Webpage

Organizer: Luochen Zhao

Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song Organizer's Page

Lisa Carbone, Yi-Zhi Huang, James Lepowsky, Siddhartha Sahi Organizer's webpage

Simon Thomas website

Kasper Larsen, Daniel Ocone and Kim Weston Organizer's page

Joel Lebowitz, Michael Kiessling

Yanyan Li, Haim Brezis Organizer's Webpage

Stephen D. Miller, John C. Miller, Alex V. Kontorovich, Alex Walker seminar website

Stephen D. Miller

Brooke Ogrodnik, Website

Organizers: Yanyan Li, Z.C. Han, Jian Song, Natasa Sesum

Yael Davidov Seminar webpage

Kristen Hendricks, Xiaochun Rong, Hongbin Sun, Chenxi Wu Organizer's page

Fioralba Cakoni Seminar webpage

Ebru Toprak, Organizer

Organizer's webpage: Organizer's webpage

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