### Abstracts

**Joerg Brendle** (KOBE University)

*Methods in iterated forcing *

We present some techniques for iterating forcing constructions.For
example, we discuss Shelah's method of iterating by repeatedly
taking ultrapowers of a forcing notion. We will also give a brief
outline of Shelah's technique of iterating along templates. While
we shall mention some applications, the focus will be on illustrating
the basic ideas underlying these techniques.

**Moti Gitik** (Tel-Aviv University)

*A weak generalization of SPFA to higher cardinals. *
We apply a form of the Neeman iteration to finite structures
with pistes. This allows to formulate a certain weak analog of
SPFA for higher cardinals.

**Martin Goldstern** (Technische Universität
Wien)

*Cichon's diagram and large continuum *
I will sketch a forcing construction of a model in which several
well-known cardinal characteristics if the continuum (in particular:
continuum itself, cofinality of null, uniformity of null, uniformity
of meager, covering of meager) all have different values.

Joint work with Arthur Fischer, Kellner, Shelah. (Work in progress.)

**John Krueger**
*Forcing with Models as Side Conditions*
I describe a comparison of elementary substructures which allows
for a uniform method of forcing with models as side conditions
on $\omega_2$.

**Heike Mildenberger** (Albert-Ludwigs-Universität
Freiburg)

*Forcings with block sequences*
I will discuss some new preservation theorems for forcings with
block sequences.

**Tadatoshi Miyamoto** (Nanzan University)

*A study of iterating semiproper forcing *

I would like to introduce a way to iterate semiproper forcing.
Suppose we have an initial segment, of limit length, of an iterated
forcing. We consider the set of conditions that have sort of traceable
countable stages. It turns out that this set of conditions forms
a limit which sits between the direct and full limits. If we keep
iterating semiproper p.o. sets under this limit, then every tail
of the iteration is semiproper in the intermediate stage. In particular,
the iteration itself is semiproper. This is a generalization of
an iteration lemma on proper forcing under countable support.

**Itay Neeman** (University of California,
Los Angeles)

*Higher analogs of the proper forcing axiom*
I will present a higher analogue of the proper forcing axiom, and
discuss some of its applications. The higher analogue is an axiom
that allows meeting collections of $\aleph_2$ maximal antichains,
in specific classes of posets that preserve both $\aleph_1$ and
$\aleph_2$.

This talk will include more details and proofs than my talk in
the workshop on Forcing Axioms and their Applications. I will quickly
survey the previous talk for audience members who were not present
in the

previous workshop.

**Ralf Schindler **(WWU Münster)

*An axiom.*
We propose and discuss a new strong axiom for set theory.

**Xianghui Shi** (Beijing Normal University)

*Some consequences of I0 in Higher Degree Theory*
We present some consequences of Axiom I0 in higher degree theory.
These results indicate some connection between large cardinals
and general degree structures. We shall also discuss more evidences
along this direction, raise some open questions. This is a joint
work with W. Hugh Woodin.

**Matteo Viale** (University of Torino)

*Absoluteness of theory of $MM^{++}$*
Assume $\delta$ is a limit ordinal.

The category forcing $\mathbb{U}^\mathsf{SSP}_\delta$ has as objects
the stationary set preserving partial orders in $V_\delta$ and
as arrows the complete embeddings of its elements with a stationary
set preserving quotient.

We show that if $\delta$ is a super compact limit of super compact
cardinals and $\mathsf{MM}^{++}$ holds, then

$\mathbb{U}^\mathsf{SSP}_\delta$ completely embeds into a pre
saturated tower of height $\delta$.

We use this result to conclude that the theory of $\mathsf{MM}^{++}$
is invariant with respect to stationary set preserving posets
that preserve this axiom.

For additional information contact thematic@fields.utoronto.ca

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