2020年度のセミナー

Venue: Complex A 802.

Date/Time: March 18, 2021 (Thursday) / 13:00 - 15:00.

Abstract: One of main difference between ℚ and ℝ is the completeness. However, there are a lot of representations for the completeness. In this talk, we will compare some of these from the viewpoint of reverse mathematics.

Date/Time: February 19, 2021 (Friday) / 15:00 - 16:00.

Venue: Zoom.

Abstract: In this presentation I sketch a few results about Lipschitz Determinacy in reverse mathematics. I also comment on possible ways to extend these results.

Date/Time: November 27, 2020 (Friday) / 15:00 - 16:00.

Venue: Room 802, Science Complex A, Tohoku University.

Abstract: Blackwellゲームを二階算術で考えるにあたって最も簡単な場合から考える。それは利得集合が有限個の開基の和になっている場合だ。行列ゲームの構成的な証明を用いることによって証明することを試みる。

Date/Time: November 20, 2020 (Friday) / 15:00 - 16:00.

Venue: Room 802, Science Complex A, Tohoku University.

Abstract: Lipschitz and Wadge games were introduced by William W. Wadge as a tool for studying the complexity of subsets of real numbers. A. Louveau and J. Saint Raymond proved Second order arithmetic 𝗭_２ can prove that all Borel Wadge and Lipschitz games are determined. In this talk, we will introduce definition of Lipschitz and Wadge game and evaluate strength of subsystem of second order arithmetic which suffice to prove the determinacy of Lipschitz and Wadge games for first levels of Borel hierarchy. And we will see some reversal and open problems.

Date/Time: November 20, 2020 (Friday) / 16:00 - 17:00.

Venue: Room 802, Science Complex A, Tohoku University.

Prof. Tanaka proved that decent & eigen-distribution implies PID of the tree. However, the concept of decent is somewhat awkward and complicated in the field of game tree therefore difficult to determine as we need derivatives to compute. In this talk, we will explain about how we could consider the weaker condition of decent, or some conditions we could think of. Also, we would mention the generalized cost, introduced by Peng and see how much work has done, and some works to be done.

Date/Time: November 6, 2020 (Friday) / 15:00 - 17:00.

Venue: Room 802, Science Complex A, Tohoku University.

Abstract: It is a well-known theorem that Weak König Lemma and Brouwer's Fixed Point Theorem are mutually equivalent over the system RCA₀ of second order arithmetic, which was proved by Shioji and Tanaka in 1990. In the proof, they constructed a retraction map from the square 𝕀² to its boundary under the assumption that Weak König Lemma fails. At this moment, the theorem Invariance of Domain also fails. In 2020, under the same assumption, Kihara proved that the invariance theorem does not hold not only squares but also higher dimension. In particular, he constructed a topological embedding of ℝ⁴ into ℝ³ . However, he left the following open question: Does RCA₀ prove that there exists no topological embedding of ℝ³ into ℝ² .
In this talk, we will present some researches about this question.

Date/Time: November 6, 2020 (Friday) / 15:00 - 17:00.

Venue: Room 802, Science Complex A, Tohoku University.

Abstract: We already know that Heine Borel compactness for [0,1] is equivalent to WKL₀ over RCA₀. But this "compactness" has a restricted meaning, i.e., coutable compactness. We talk about compactness for uncoutable covers and the relation between compactness and the Lindelöf property, based on the result of Dag Normann and Sam Sanders.

Date/Time: October 23, 2020 (Friday) / 15:00 - 16:00.

Venue: Room 802, Science Complex A, Tohoku University.

Abstract: Higher order reverse math has a strong connection with higher order computability, as the ordinary reverse math. In the former part of this talk, we'll study one of reasonable higher order computability notions, which was introduced by S. C. Kleene. That notion allows us to use some theorems similar with ones in 1st order computability theory.
In the later part, we'll see some typical theories and models observed in higher order reverse math.

Date/Time: October 16, 2020 (Friday) / 15:00 - 16:00.

Venue: Room 802, Science Complex A, Tohoku University.

Abstract: Hindman's theorem is a Ramsey-type theorem which was proved by N. Hindman. Galvin-Glazer gave a proof of it which uses an ultrafilter on ℕ with specific properties. In the context of reverse mathematics, Galvin-Glazer's proof was analysed by H. Towsner, and it was shown that the proof can be formalized in Π¹₁-TR₀ .
Gowers' theorem is a generalization of Hindman's theorem which is provable in a similar way of Galvin-Glazer's proof of Hindman's theorem. In this talk, we will compare those proofs and then consider how the proof of Gowers' theorem be formalized in second order arithmetic.

Date/Time: October 9, 2020 (Friday) / 15:00 - 16:00.

Venue: Rm 802, Science Complex A, Tohoku Univ.

Abstract: The Wadge degrees were defined by William Wadge in his PhD thesis. Wadge also gave two classifications of the Wadge degrees below 𝚺⁰₃ . The first of these was extended up to the Borel sets by Louveau, and the second was also extended up to the Borel sets by Duparc. Later, Fournier extended both of the classifications up to the collection of increasing differences of 𝝥¹₁ sets.
In this presentation we sketch these two classifications and the extentions given by Fournier.

Date/Time: August 30, 2019 (Friday) / 13:00 - 14:00.

Venue: Rm 205, Science Complex A, Tohoku Univ.

Abstract: The set-theoretic geology, which was initiated by Fuchs-Hamkins-Reitz, is a study of the global structure of all ground models of the universe. It attempts a study of the nature of the forcing method. In this talk, we present a framework of set-theoretic geology and basic results. Meanwhile, the standard geology is a research on ZFC, so the axiom of choice is assumed. However in the current set theory, the forcing method over choiceless model become a common tool. We also present the set-theoretic geology without the axiom of choice.