[FOM] 3 Logic Tutorials in Helsinki January/February

[jkennedy at mappi.helsinki.fi]

THREE WINTER TUTORIALS IN MATHEMATICAL LOGIC

in the Department of Mathematics and Statistics at the University of Helsinki.

Lecturers: Andres Villaveces (Bogota), John Baldwin (Chicago) and  
Daisuke Ikegami (Berkeley).

Sponsored by: Finnish Graduate School in Mathematics and its Applications

TUTORIAL 1: Model Theory and Geometry: old and new directions. Given  
by Andres Villaveces, January 28-30.

SCHEDULE: January 28th, 16-18. January 29th 14-16, Room TBA.

Lecture 1: Historical remarks on the interaction between Model Theory  
and Geometry. Model theory on sheaves (from Macintyre to recent  
developments).

Lecture 2: Modular invariants for elliptic curves. Nonstandard  
methods. From elliptic curves to quantum tori. Sheaves for the modular  
invariant. Transfer principles.

TUTORIAL 2:  Stationary Tower Forcing, January 29th-30th, given by  
Daisuke Ikegami.

SCHEDULE:  January 30th, 12-14, C124, and  January 31st 10-12. Room TBA.

ABSTRACT: Stationary Tower Forcing was invented by Hugh Woodin to  
produce various kinds of generic elementary embeddings (i.e.,  
elementary embeddings from a ground model into a transitive model of  
set theory defined in a generic extension). It has been extensively  
used in modern set theory to obtain generic absoluteness, theorems of  
ZFC + large cardinals (e.g., regularity properties in L(R)), and to  
analyze certain models of set theory (e.g., derived models). In this  
tutorial, we introduce some basics of Stationary Tower Forcing and its  
applications.

TUTORIAL 3: Constructing Atomic Models in the Continuum, given by John Baldwin

SCHEDULE:  February 6-8th, 12-14. Room TBA.

Lecture 1. Forcing to get model theoretic results in ZFC. The  
absoluteness of basic properties of first order logic was a cornerstone  
of late 20th century model theory. Recent analysis of similar problems  
for infinitary
logic places the focus on the issue of whether aleph_1-categoricity  
forces amalgamation
and ω-stability in aleph_0. We will consider several uses of set  
theoretic forcing to
establish results in model theory that are provable in ZFC. This  
includes work with
Larson extending Keisler’s proofs that for certain infinitary logics  
the existence of
uncountably many types over the empty set implies the existence of the maximal
number of models in aleph_1 and work with Shelah showing that a strong  
failure of
exchange for a natural notion of algebraicity implies the existence of  
the maximal
number of models in aleph_1.

Lecture 2. Few models in aleph_1 implies a form of exchange. We will  
use the methodology of the first talk to expound the following theorem  
of Shelah: If a sentence of L_{omega_1, omega} has fewer than  
2^{aleph_1} models in aleph_1, then pseudo-closure satisfies exchange  
(locally). (This requires understanding the notion of pseudo-closure  
and then a forcing argument.)

This exposition is joint with Laskowski and depends heavily on  
discussions with Koerwien and Larson as well as Shelah.

Lecture 3. Pseudominimal Atomic Classes have models in the continuum.  
I will discuss several applications of a method of Shelah to build a  
model in the continuum from a countable model satisfying certain  
geometric conditions. In particular, this provides a streamlined  
argument for the first part of a Ackerman, Freer, and Patel paper.

This exposition is joint with Laskowski and depends heavily on  
discussions with Shelah.

WWW-page of the tutorials:  
http://www.math.helsinki.fi/logic/opetus/TUTORIALS2013.htm

Contact Person. Juliette Kennedy (Juliette.kennedy at helsinki.fi)

-- 
Department of Mathematics and Statistics
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FI-00014 University of Helsinki, Finland
tel. (+358-9)-191-51446, fax (+358-9)-191-51400
http://www.helsinki.fi/science/logic