Talks during the summer of 2021

April 16th 10:00am  Liang Yu (Nanjing University). A basis theorem for \(\Pi^1_1\)sets.
Abstract: It was claimed by Harrington, but never published, that every nonthin \(\Pi^1_1\)set ranges over an upper cone of hyperarithmetic degrees. We shall give a proof via a full approximation argument.

April 21st  Ralf Schindler. Generation of Grounds.
Abstract: A ground is an inner model of which \(V\) is a generic extension. We will present a general method for producing grounds of a given model of set theory via the HOD direct limit system. This may be used to analyze the \(<\kappa\) mantle of \(L[x]\), \(x\) any real above \(M_1^\sharp\) and \(\kappa\) any cardinal. This is joint work with G. Sargsyan and F. Schlutzenberg. 
April 29th  Farmer Schlutzenberg. Local mantles of \(L[x]\).
UPDATE: This Talk has been moved to Thursday, April 29th at 4:15pm.
Abstract: Recall that for a cardinal \(\kappa\), a \(<\kappa\)ground is an inner model \(W\) of ZFC such that \(V\) is a setgeneric extension of \(W\), as witnessed by a forcing of size \(<\kappa\), and the \(\kappa\)mantle is the intersection of all \(<\kappa\)grounds. We will start with a brief overview
of some known facts on the \(\kappa\)mantle. Following this, assuming sufficient large cardinals, we will analyze the \(\kappa\)mantle \(M\) of \(L[x]\), where \(x\) is a real of sufficiently high complexity, and \(\kappa\) is a limit cardinal of uncountable cofinality in \(L[x]\). We will show in particular that \(M\) models ZFC + GCH + "There is a Woodin cardinal". We will also discuss a variant, joint with John Steel, for the \(\kappa\)cc mantle, where \(\kappa\) is regular uncountable in \(L[x]\) and \(\kappa\leq\) the least Mahlo of \(L[x]\). The proof relies on Woodin's analysis of \(\mathrm{HOD}^L[x,G]\) and Schindler's generation of grounds, and is motivated by work of Fuchs, Sargsyan, Schindler and the author on Varsovian models and the mantle.
References: arXiv:2006.01119, arXiv:2103.12925

May 5th  Jindrich Zapletal (University of Florida). Chromatic numbers of distance graphs on Euclidean spaces.
Abstract: Let \(G_n\) be the graph on \(n\)dimensional Euclidean space connecting points of rational distance. I will show that it is consistent relative to an inaccessible cardinal that ZF+DC holds, chromatic number of \(G_3\) is countable, yet the chromatic number of \(G_4\) is uncountable. I will use the opportunity to explain the basic concepts,methods, and results of geometric set theory, as contained in a recent book with Paul Larson.
In the first lecture, I will provide a broad outline of geometric set theory. I will define balanced forcing, a class of partial orders which can be used to prove numerous independence results in ZF+DC, and prove its central theorems. In its usage and flexibility, balanced forcing is a parallel to proper forcing in the context of choiceless set theory. In the second lecture, I will discuss chromatic numbers of algebraic hypergraphs in general and the rational distance graphs in particular. Finally, I will construct a coloring poset which yields the consistency result mentioned in the first paragraph.

May 12th  Jindrich Zapletal (University of Florida). Chromatic numbers of distance graphs on Euclidean spaces.
Continuation 
June 2nd  Gunter Fuchs (CUNY). TBA

June 30th  Diana Montoya (University of Vienna). TBA

July 7th  Omer BenNeria (Hebrew University, Jerusalem). TBA