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講演者：Slawomir Solecki (University of Illinois, USA)

題目：An abstract approach to finite Ramsey theory

時間：14:00-15:00

アブストラクト：

I will present an abstract approach to fnite Ramsey theory.

In the abstract context, I will isolate a pigeon-hole principle and prove that it implies

a general Ramsey-type result. It turns out that the classical Ramsey theorem,

the Hales-Jewett lemma (with Shelah's bounds), the Graham-Rothschild theorem, the

versions of these results for partial functions are all iterative consequences of this general result.

Moreover, using this approach, I will prove a new, self-dual Ramsey theorem,

which naturally generalizes both the classical Ramsey theorem and the

dual Ramsey theorem of Graham and Rothschild.

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講演者：S. M. Srivastava (Indian Statistical Institute, India)

題目：A result on Extensions of Translation Probabilities using Borel Selections

時間：15:30-16:30

アブストラクト：

For a measurable space (Y, B), let P(Y, B) denote the set of all proba-

bility measures on (Y, B). We equip P(Y, B) with the weak σ-algebra,

i.e., the smallest σ-algebra on P(Y, B) making each µ → µ(B), B ∈ B,

measurable. A transition probability is a measurable function

P : (X, A) → P(Y, B), (X, A) a measurable space.

In this talk, we shall present a proof of the following surprising the-

orem.

Theorem. Let Y be a Polish space and B a countably generated sub

σ-algebra of the Borel σ-algebra BY of Y . The following conditions are

equivalent:

(a) For every measurable space (X, A), every transition probability

P : (X, A) → P(Y, B) can be extended to a transition probability

Q : (X, A) → P(Y, BY ).

(b) The statement (a) only for X Polish and A = BX , the Borel

σ-algebra of X.

This is one of the series of results proved on transition probabilities

needed to develop a stochastic analogue of modal logic jointly with E.

E. Doberkat, Dortmund, Germany.

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講演者：Tamas Matrai

題目：On typical properties of Hilbert space operators

時間：17:00-18:00

アブストラクト：

What are the typical properties of Hilbert space operators in the sense of Baire category? Or,

at least, is Baire category a useful tool for studying operators?

The answer to these questions, obviously, may depend on the underlying topology. So in

collaboration with Tanja Eisner, we investigated the typical behavior of Hilbert space operators in

the norm topology and in four important separable topologies (a property Φ of operators is typical

if the operators satisfying Φ form a co-meager set).

We obtained that in the separable topologies, from the point of view of Baire category, the

theory of Hilbert space operators reduces to the theories of very particular classes of operators,

e.g. unitary operators, positive self-adjoint operators or even one single operator. In the norm

topology, the theory of Hilbert space operators does not trivialize; however, this topology is so

ﬁne that every property we studied, e.g. various mapping and spectral properties, holds for a non-

meager set of operators. In a sense, our results outline some limitations of Baire category methods

in operator theory.

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交通：阪急六甲駅またはJR六甲道駅から神戸市バス36系統「鶴甲団地」
行きに乗車，「神大本部工学部前」停留所下車，徒歩すぐ．
http://www.kobe-u.ac.jp/info/access/rokko/rokkodai-dai2.htm

連絡先：ブレンドレ　ヨーグ　 brendle@kurt.scitec.kobe-u.ac.jp