Titles and Short Summaries of the Talks

8 Foundation of Mathematics and History of Mathematics
9:30–11:40
March 21st (Thu) Conference Room VII
Final: 2013/2/7
13 Kohtaro Tadaki (Chuo Univ.) ♯ The generic group model and algorithmic randomness · · · · · · · · · · · · · · 20
Summary: In modern cryptography, the generic group model is used as an imaginary framework
in which the security of a cryptographic scheme is discussed. In particular, the generic group
model is often used to discuss the computational hardness of problems, such as the discrete
logarithm problem, which is used as a computational hardness assumption to prove the security of a
cryptographic scheme. In this talk, we apply the concepts and methods of algorithmic randomness
to the generic group model, and consider the secure instantiation of the generic group, i.e., a
random encoding of the group elements. We show that the generic group can be instantiated by
a specific computable function while keeping the computational hardness originally proved in the
generic group model.
14 Kenshi Miyabe (Kyoto Univ.) ♯ Van Lambalgen’s Theorem for uniform Kurtz randomness · · · · · · · · · · · 15
Summary: The theory of algorithmic randomness defines some randomness notions such as
Martin–Löf randomness, Schnorr randomness, Kurtz randomness, Demuth randomness and so
forth. Van Lambalgen’s theorem is one of the most important theorems in the theory of algorithmic
randomness and talks about relative Martin–Löf randomness. The usual relativization works well
for Martin–Löf randomness (thus, van Lambalgen’s theorem holds), but it does not work well for
other randomness notions. In my previous work, I proposed uniform relativization and proved
van Lambalgen’s theorem for uniform Schnorr randomness. In this talk, I will talk about van
Lambalgen’s theorem for uniform Kurtz randomness.
15 Akitoshi Kawamura (Univ. of Tokyo) ♯ On representations of analytic functions and polynomial-time com-
Norbert Müller (Univ. Trier) putability of operators · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 15
Carsten Rösnick (TU Darmstadt)
Martin Ziegler (TU Darmstadt)
Summary: We study the computational complexity of operators on analytic functions. We work
in the framework of Recursive Analysis, the theory of real computation by approximation up to
guaranteed error bounds. We introduce and compare several representations of analytic functions.
We then study the polynomial-time computability of basic operators on analytic functions, and
discuss extensions to larger classes of smooth functions.
16 Takayuki Kihara (JAIST) ♯ An application of Kumabe–Slaman forcing to the ω-decomposability
problem on Borel functions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 20
Summary: We show some variants of the Jayne–Rogers Theorem at finite and transfinite levels of
the hierarchy of Borel functions, by using the Shore–Slaman Join Theorem on the Turing degrees.
This can be viewed as a partial solution to Andretta’s conjecture and Motto Ros’ conjecture on the
strong generalizations of the Jayne–Rogers Theorem.
17 Tatsuya Miyazaki (Nagoya Univ.) ♯ On rigid Souslin trees and their preservation · · · · · · · · · · · · · · · · · · · · · · 15
Summary: The author investigated the rigid structure of Souslin trees. A rigidity notion “free”,
for Souslin trees, is known to be preserved under forcings with the property (K). This can be used to
obtain the consistency result that there is a rigid Souslin tree and simultaneously the CH fails. The
author introduces a new rigidity notion for ω1-trees and proves an analogous preservation result.
The investigation includes a few structural analysis for various rigid Souslin trees.