2010-01-07Jakob Kellner (KGRC)Precipitous ideals
2010-01-14Asger Törnquist (KGRC)Introduction to operators algebras
I will give an overview of the basic concepts of operator algebras. Specifically, I will cover topics such as: What are C* algebras and von Neumann algebras? And what is a state or a representation of such an algebra? The aim is to give enough background to discuss recent applications of set theory to problems in operator algebras in a future talk (next week, say.)

2010-01-21Asger Törnquist (KGRC)Intro to operator algebras, II
Today I will outline a proof of a result of Akemann and Weaver from 2004 that Diamond implies that there is a unital C* algebra generated by aleph_1 elements, which has the property that all irreducible representations (irreps) are equivalent (i.e. unitarily conjugate.) This shows the consistency of a negative solution to a problem posed by Naimark in 1948, asking if K(H) is the only C* algebra with the property that all irreps are equivalent. (In that 1948 paper Naimark showed that all irreps of K(H) are equivalent, hence the question.)

2010-01-28Luca Motto Ros (KGRC)On the complexity of the relations of isomorphism and bi-embeddability
Given a pair of analytic equivalence relations E and F, we determine under which conditions it is possible to find an Lω1-elementary class C such that E and F are Borel equivalent to, respectively, isomorphism and bi- embeddability on C: quite surprisingly, it turns out that, apart from the obvious limitations, such an elementary class always exists. This result gives an (almost complete) answer to previous questions of Louveau-Rosendal (2005) and Friedman-Motto Ros (2009).

2010-03-04Marcin Sabok (KGRC)Iteration of idealized forcing and infinite-dimensional perfect set theorems
I will show some recent results about the countable-support iteration of idealized forcing. As application, I will prove infinite-dimensional versions of some "perfect-set" theorems.

2010-03-11Giovanni Panti (Udine)Kakutani-von Neumann maps on simplexes
A Kakutani-von Neumann map is the push-forward of the group rotation (Z2,+1) to a unit simplex via an appropriate topological quotient. The usual quotient towards the unit interval is given by the base 2 expansion of real numbers, which in turn is induced by the doubling map.

We replace the doubling map with an n-dimensional generalization of the tent map; this allows us to define Kakutani-von Neumann transformations in simplexes of arbitrary dimensions. The resulting maps are piecewise-linear bijections (not just mod 0 bijections), whose orbits are all uniformly distributed; in particular, they are uniquely ergodic w.r.t. the Lebesgue measure.

The forward orbit of a certain vertex provides an enumeration of all points in the simplex having dyadic coordinates, and this enumeration can be translated via the n-dimensional Minkowski function to an enumeration of all rational points. In the course of establishing the above results, we introduce a family of {+1,-1}-valued functions, constituting an n-dimensional analogue of the classical Walsh functions.

Paper reference: []

2010-03-18Tetsuo Ida (Tsukuba)Symbolic and Algebraic Methods in Computational Origami
The art of paper folding, known as ''origami'', provides the methodology of constructing a geometrical object out of a sheet of paper solely by means of folding by hands. Computational origami studies the mathematical and computational aspects of origami, including geometrical theorem proving and visualization. By the assistance of software tools for modeling, reasoning and verifying properties of origami, we expect to be able to formalize origami with rigor and capability beyond the methods performed by hands.

In this talk I will show the importance of symbolic and algebraic meth- ods in computational origami, that are employed by our computational origami system called ''Eos'' (E-Origami System). I discuss (1) Huzita's axiomatization of origami,(2) application of Gobner bases method and the cylindrical and algebraic decomposition, and (3) the algebraic graph rewriting of abstract origami. Issue (1) is discussed with relation to the algorithmic treatment of origami foldability, issue (2) for origami geomet- rical theorem proving, and issue (3) for modeling origami fold. On the whole, I would like to emphasize the importance of symbolic and algebraic computations on discrete geometrical objects, and of the separation of the domains of concern between symbolic and numeric computations. It leads to clearer and more abstract formulation of origami theories.
2010-03-18Masahiko Sato (Kyoto)A Constructive Theory of Objects
A Constructive Theory of Objects
Masahiko Sato
Graduate School of Informatics, Kyoto University

We present a constructive theory of objects in which a Turing complete functional programming language, named Z, can be formally specified. Our theory is developed in two stages. In the first stage, we introduce a set of symbolic expressions generated from two initial objects by two binary operators. This set is used as the meta-level universe in which the object language defined in the second stage is interpreted. In the second stage, we define the syntax of Z, by inductively defining a set of symbolic objects. Z is Lisp-like in the sense that both data and programs of Z are symbolic objects. We give both operational and denotational semantics to Z by interpreting Z data and programs as symbolic expressions.

We propose a new principle which we call the fundamental principle of object creation. Both symbolic expressions and symbolic objects are created inductively by following the fundamental principle. In this way, we see that the set of symbolic objects provides a natural framework for developing a constructive theory of objects, which contain both data objects and programs.

2010-03-25Vincenzo Dimonte (KGRC)Non-proper Elementary Embeddings Beyond L(Vλ+1)
In a recent work Woodin has defined new axioms stronger than I0 (the existence of an elementary embedding j from L(Vλ+1) to itself), that involve elementary embeddings between slighter large models. While the correspondence between I0 and Determinacy carries on without further hypotheses, for these new axioms we need the embeddings to be proper. This initially seemed a common property, but during the seminar there will be presented two essentially different cases of non-proper elementary embeddings. These results fill a gap in a Theorem by Woodin and legitimate the definition of properness.
2010-04-15Miguel Angel Mota (KGRC)Divorces in the PFA family
Using some variants of weak club guessing we separate some fragments of the proper forcing axiom: we show that for every two indecomposable ordinals α < β, the forcing axiom for the class of all the β-proper posets does not imply the bounded forcing axiom for the class of all the α-proper posets. This is joint work with S. Friedman and M. Sabok.

2010-04-22Benjamin Miller (KGRC)A generalization of Silver's theorem
We give a classical proof of the natural generalization of Silver's dichotomy theorem to omega-universally Baire, co-kappa-Souslin equivalence relations on Hausdorff spaces.

2010-04-29David Aspero (KGRC)On Π2 maximality and CH
I will show the existence of two Π2 sentences about H(ω2), both of which can separately be forced to hold together with CH (assuming, for one of them, the existence of an inaccessible limit of measurable cardinals), but whose conjunction implies the failure of CH. This solves a well-known problem of Woodin. This is joint work with P. Larson and J. Moore.

2010-05-06Ajdin Halilović (KGRC)The tree property at the double successor of a singular cardinal
I will present the proof that relative to the existence of something called a weakly compact hypermeasurable cardinal the tree property can hold at the double successor of a singular cardinal.

2010-05-20Andrew Brooke-Taylor (Bristol University and the Heilbronn Institute for Mathematical Research)Fraïssé limits for a kind of "locally finite" Fraïssé class
One of the nicest results about Fraïssé limits is the fact that, when the language is finite, one obtains a 0-1 law for members of the Fraïssé class. We introduce a notion of locally finite Fraïssé class, for which the language is infinite, but the Fraïssé class acts like one for a finite language. Examples include the Fraïssé classes of finite hypergraphs, finite simplicial complexes, and antichains on P(n) for finite n. Amongst other results, we derive a 0-1 law for such structures; notably, ours differs from the known 0-1 law for simplicial complexes of Blass & Harary, as we use a different by still very natural choice of measure guided by the Fraïssé limit considerations.

2010-05-27Sy-David Friedman (KGRC)What Andrew and I talked about last week
Actually I will only tell you about the mathematics we discussed, saving our discussion of kangaroos for a later time. We looked at the interplay between Square/Stationary Reflection on the one hand, and Large Cardinal Axioms on the other. Solovay showed ages ago that supercompactness kills Square, improved recently by Jensen, who showed that if κ is Subcompact then Squareκ fails. We extend Jensen's result to: If κ is α+-Subcompact (and κ is at most α) then Squareα (indeed Squareα,<κ fails. Moreover, forcing shows that this is best possible: One can preserve all instances of α-Subcompactness as well as very LARGE cardinals (like ω-Superstrongs) and force Squareα to hold everywhere not ruled out by the previous result. We found similar results with Squareα replaced by Stationary Reflection at α+ (on small cofinalities) and with α+ Subcompactness replaced by α++ Subcompactness. The proof of this latter result, unlike the proofs of the earlier ones, caused us some worries.

2010-06-10Lyubomyr Zdomskyy (KGRC)Projective wellorders and mad families with big continuum
We shall show that b=c=ω_3 is consistent with the existence of a Δ13 denable wellorder of the reals and a Π12 denable ω-mad subfamily of [ω]ω.

This is joint work with Vera Fischer and Sy-David Friedman.

2010-06-17Anand Pillay (University of Leeds)Measures in first order theories
In the first part of the talk I will discuss (for a general audience) issues around the classification of first order theories, and reasons that studying measures in place of types have become popular, especially for theories without the independence property.

In the second part which will be a bit more technical I will discuss how Shelah's notion of "strongly dependent" (or "strong NIP") can be characterized in terms of weight with respect to average measures.

2010-06-24Moti Gitik (Tel Aviv)On Precipitous Ideals
2010-06-29Denis Saveliev (Moscow)Groupoids of Ultrafilters
There exists a natural way to extend the operation of any groupoid (in fact, any universal algebra) to ultrafilters; the extended operation is right topological in the standard compact Hausdorff topology on the set of ultrafilters; the extensions of semigroups are semigroups. Semigroups of ultrafilters are used to obtain various deep results of number theory, algebra, dynamics, etc. The main tool is idempotent ultrafilters. They exist by a general theorem establishing the existence of idempotents in compact Hausdorff right topological semigroups.

Expanding this technique to non-associative groupoids, we isolate a class of formulas such that any satisfying them compact Hausdorff right topological groupoid has an idempotent, and a class of formulas that are stable under passing from a given groupoid to the groupoid of ultrafilters. If a formula belongs to both classes (like associativity), any satisfying it groupoid carries an idempotent ultrafilter. Results on semigroups following from the existence of idempotent ultrafilters (like Hindman's Finite Sums Theorem) remain true for such groupoids.

Another generalization concerns infinitary analogs of these results. The main obstacle here is that non-principal idempotent ultrafilters cannot be σ-additive. We define ultrafilters with two weaker properties (ultrafilters near κ-additive subgroupoids and κ-additive ultrafilters near subgroupoids) and show that their existence suffices to obtain desired infinitary theorems.

2010-10-14Luca Motto Ros (Albert-Ludwigs-Universität Freiburg)Invariantly universal analytic quasi-orders
We introduce a strengthening of the notion of completeness for analytic quasi-orders called "invariant universality" (roughly speaking, an analytic quasi-order is invariantly universal if it contains in a "natural" way a copy of any other analytic quasi-order), and we then show that, quite surprisingly, many natural examples of complete analytic quasi-orders arising in various areas of mathematics are indeed invariantly universal.

This is joint work with Riccardo Camerlo and Alberto Marcone.

2010-10-21Iskander Kalimullin (Kazan State University)Algorithmic reducibilities of algebraic structures
We will see how the notion of Medvedev's mass problem allows to extend the classical algorithmic reducibilities to the algebraic structures. In particular, the studies of degree spectra of structures can be considered as a development of the structural theory of the non-uniform reducibility of mass problems of presentability.
2010-10-28Wu Liuzhen (KGRC)Set forcing for strong condensation on H(\omega_2)
2010-11-04Clinton Conley (KGRC)Measurable colorings of graphs, I
We discuss how the study of graph colorings changes when definability restrictions are placed upon the coloring functions. In this talk, we focus on "positive" results, i.e., reworking classical (nondescriptive) theorems to fit into this context. In particular, we analyze colorings of hyperfinite graphs and also study the Brooks bound: the chromatic number of a graph is at most the maximum degree of a vertex. This is joint work with Alekos Kechris.
2010-11-11Marcin Sabok (Instytut Matematyczny Uniwersytetu Wrocławskiego, Wrocław, Poland, Instytut Matematyczny Polskiej Akademii Nauk, Warszawa, Poland, and KGRC)A dichotomy for σ-ideals generated by closed sets
We say that a σ-ideal I on a Polish space X has the "1-1 or constant" property if every Borel function defined on a Borel I-positive subset of X can be restricted to a Borel I-positive set, on which it is 1-1 or constant. In other words, this means that the forcing PI adds a minimal real degree. The classical well-known examples of σ-ideals with the 1-1 or constant property are the σ-ideal of countable sets (Sacks forcing) and the σ-ideal of σ-compact subsets of the Baire space (Miller forcing). On the other hand, an example for which this property drastically does not hold is the σ-ideal of meager sets (or any I for which PI adds a Cohen real). During the talk I will prove the following dichotomy: if I is a σ-ideal generated by closed sets, then

(i) either I has the 1-1 or constant property,<br/> (ii) or else PI adds a Cohen real.

This is joint work with Jindra Zapletal.
2010-11-18Misha Gavrilovich (KGRC)A homotopy approach to some questions in set theory
of PCF theory.

Further we observe a similarity between homotopy theory ideology/yoga and "artificially/naturality thesis" of Shelah (Logical Dreams, S5) claiming that "the various cofinalities are better measures" of size.

We shall argue that the formalism is curious as it suggests to look at a homotopy-invariant variant of Generalised Continuum Hypothesis about which more can be proven within ZFC and first appeared in PCF theory independently but with a similar motivation.

This is joint work with Assaf Hasson.
2010-11-25Peter Holy (KGRC)Condensation, Large Cardinals and the consistency strength of PFA
Gödel's constructible universe L satisfies the strongest possible form of condensation: if M is an elementary submodel of any Lα, then M is isomorphic to Lβ for some β which is at most α. But L does not allow for very large cardinals, ω1-Erdös cardinals already cannot exist within L. We will generalize and then weaken Gödel's condensation principle to obtain new condensation principles (= fragments of condensation) and investigate whether those principles are consistent with the existence of (very) large cardinals. We will introduce (and deal with) the following principles:

Strong Condensation: strong, inconsistent with ω1-Erdös cardinals

Stationary Condensation: consistent with ω-superstrong cardinals but pretty weak

Local Club Condensation: pretty strong, consistent with ω-superstrong cardinals

Acceptability: incomparable to the above, weak, consistent with ω-superstrong cardinals

A very interesting open question is whether Local Club Condensation and Acceptability are (simultaneously) consistent with the existence of an ω-superstrong cardinal. If this question has a positive answer, we would probably be able to prove the following:

Conjecture: Let S(κ) denote any large cardinal property of κ consistency-wise weaker than supercompactness. It is then consistent that there exists κ such that S(κ) holds but no proper forcing extension satisfies PFA.

Since any reasonable way to obtain a model of PFA seems to be starting with a model with large cardinals and to then obtain PFA in a proper forcing extension, this would be a strong hint towards the consistency strength of PFA actually being that of a supercompact cardinal.

This is joint work with Sy Friedman.
2010-12-02Miguel Angel Mota (KGRC)Small proper forcing and the size of the continuum
In a recent work with David Asperó I proved that the forcing axiom for the class of all the proper posets of small cardinality does not impose any bound on the size of he continuum. The corresponding proof is quite technical and uses some new ideas regarding forcing iteration. During this talk, however, I will prove something more modest: the consistency of the forcing axiom for the class of all the finitely proper posets together with a large continuum. On the one hand, this will give me the opportunity to explain the main ingredients of our main result. On the other, this is near to be optimal since it is consistent to assume that all the small proper posets are finitely proper.
2010-12-09Jindrich Zapletal (University of Florida, Gainesville and Czech Academy of Sciences, Prague)Solution to the pinned conjecture
The notion of a pinned equivalence relation is useful in proofs of irreducibility between Borel equivalence relations. Kechris conjectured that the smallest unpinned equivalence relation is equality of countable sets of reals. I will show that the answer is negative using old ideas of Shelah regarding Borel sets in the plane containing large squares. The talk will include the necessary background.
2010-12-16Stefan Geschke (Hausdorff Center for Mathematics, Universität Bonn, Germany)Basis theorems for continuous colorings
The collection of ''n''-element subsets of a Hausdorff space carries a natural topology. A continuous ''n''-coloring on a Polish space ''X'' is a continuous map that assigns to each ''n''-element subset of ''X'' one of two colors.

An ''n''-coloring is uncountably homogeneous if the underlying space ''X'' is not the union of countably many sets on which the coloring is constant.

Generalizing a previous result about ''2''-colorings (i.e., graphs) and answering a question of Ben Miller, it is shown that the class of uncountably homogeneous, continuous ''n''-colorings on Polish spaces has a finite basis.

I.e., there is a finite collection of uncountably homogeneous, continuous ''n''-colorings on the Cantor space such that every uncountably homogeneous, continuous ''n''-coloring on any Polish space contains a copy of one of the finitely many colorings.

This complements some recent results of Lecomte and Miller on the nonexistence of small bases for uncountably chromatic analytic graphs.