Date/Time : 26 Aug 2026 17:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Rod Downey, Victoria University of Wellington

A computable Banach space X = (X, || · ||, +, ·) is a Banach space with a computable dense sequence of points which makes X a computable metric space using the norm, for which the operations are computable functions.  Most, but not all, familiar spaces of analysis and physics commonly met are such spaces. In the same way as classical vector space theory revolves around (Hamel) bases, Banach spaces are best understood using what is called a Schauder basis. There are many, many questions around this concept, both classically and in the effective content.

In this talk I will overview many of the questions and results we have so far, and particularly report on some recent work seeking to understand the effective content of Kadets’ Theorem, which states (more or less) that all infinite dimensional Banach spaces look the same if we forget the norm: any two infinite-dimensional, separable Banach spaces are homeomorphic as topological spaces. Surprisingly, this result can be proven for computable Banach spaces using only the halting problem, which seems rare in effective topology.

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Date/Time : 19 Aug 2026 17:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Clement Yung, University of Toronto

In the past century, the insight behind the original Ramsey’s theorem proved to be applicable to a wide range of mathematics, such as number theory, functional analysis and topology. This spurred two particular directions of Ramsey theory: The first one is known as topological Ramsey theory, a general procedure developed by Carlson and Simpson to prove many seemingly unrelated Ramsey’s theorem-like results. The second one is the Ramsey theory of Banach spaces, kickstarted by Gowers’ unexpected application of Ramsey theory to resolve a long-standing open problem in Banach space theory. In this talk, I introduce the theory of weak A2 spaces, which serves as a possible intersection between these two Ramsey theories and discuss how several infinite games that appeared in these Ramsey theories (the Kastanas game, the Gowers game and the asymptotic game) are closely related.

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Date/Time : 12 Aug 2026 17:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Wang Wei, Sun Yat-Sen University

A (binary) tree is positive iff its infinite paths form a set with positive Lebesgue measure. It is a well-known phenomenon in algorithmic randomness that positive trees contain infinite paths with weak computational power. By Cantor, positive trees always have perfect subtrees and even positive perfect subtrees. Chong et al. [1] studied computational power of perfect subtrees of positive trees, and to some extend showed that positive trees also contain perfect subtrees of weak computational power. This work was followed up by Barmpalias and Wang [2], by Barmpalias and Zhang [3], and also by Greenberg, J. Miller and Nies [4]. This talk will survey the main results in the above works, and also sketch a recent result that every ∆n positive tree contains a lown perfect subtree.

[1] Chong, Chitat and Li, Wei and Wang, Wei and Yang, Yue, On the com-putability of perfect subsets of sets with positive measure, Proceedings of the Amer-ican Mathematical Society, vol. 147 (2019), no. 9, pp. 4021–4028.
[2] Barmpalias, George and Wang, Wei, Pathwise-randomness and models of second-order arithmetic, Information and Computation, vol. 299 (2024), 105181.
[3] Barmpalias, George and Zhang, Xiaoyan, Growth and irreducibility in path-incompressible trees, Information and Computation, vol. 297 (2024), 105136.
[4] Greenberg, Noam and Miller, Joseph S. and Nies, André, Highness prop-erties close to PA completeness, Israel Journal of Mathematics, vol. 244 (2021), pp. 419–465.

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Date/Time : 08 Apr 2026 17:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Frank Stephan, National University of Singapore

Many-one degrees and one-one degrees are basic degree structures in recursion theory and since Young’s result 1966, that every nonrecursive many-one degree either coincides with a one-one degree or embeds every possible countable linear order inside the structure of its one-one degrees, the question whether this is also true for partial orders received attention. Dëgtev showed in 1976 that this is true for all nonrecursive and recursively enumerable many-one degrees. Odifreddi in 1981 made the special case whether every nonrecursive many-one degree either contains an antichain of one-one degrees or coincides with a single one-one degree formally an open question. Since then for long time no progress was made until Batyrshin confirmed this for all limit-recursive many-one degrees. The present work shows this then also for all many-one degrees except for the greatest recursive many-one degree which is a special case by having a two-element antichain but not a longer one in its one-one degrees. Furthermore, the talk will also look at degree structures between one-one and many-one, namely the finite-one and bounded finite-one degrees. Finite-one degrees were introduced by Maslova 1979, but up to now did not find the attention they deserved.

This is joint work of Linus Richter, Frank Stephan and Xiaoyan Zhang.

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Date/Time : 25 Mar 2026 17:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Josiah Jacobsen-Grocott, Nanyang Technological University

Hyperenumeration reducibility was first introduced by Sanchis. A set $A$ is hyperenumeration reducible to $B$, $A\le_{he} B$, if there is a c.e.\ set $W$ such that
$x\in A \iff \forall f\in \omega^{\omega}\exists v,n \in \omega [\langle f{\restriction} n,x,v\rangle\in W \land D_v\subseteq B]$. The relationship between hyperenumeration and hyperarithmetic reducibility shares many parallels with the relationship between enumeration and Turing reducibility, like the fact that the hyperarithmetic degrees properly embed into the hyperenumeration degrees. However, not all properties transfer from the enumeration degrees; for instance, Selman’s Theorem is known to be false in the hyperenumeration degrees.

In this talk we look the downwards density property of the enumeration degrees, and show that it does transfer to the hyperenumeration degrees. This involves transferring injury arguments from a finite context into constructions in $L_{\omega_1^{CK}}$, and finding tools to overcome some of the limitations of hyperenumeration operators.

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Date/Time : 18 Mar 2026 17:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Stefan Ludwig, University of Freiburg

What is the common theory of all finite fields equipped with an additive and/or multiplicative character? Hrushovski answered this question in the additive case working in (a mild version of) continuous logic. Motivated by natural number-theoretic examples we generalise his results to the case allowing for both (non-trivial) additive character and (sufficiently generic) multiplicative character. Apart from answering the above question we obtain a quantifier elimination result and a generalisation of the definability of the Chatzidakis-van den
Dries-Macintyre counting measure to this context. The proof relies on classical results on bounds of character sums following from the work of Weil.

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Date/Time : 11 Mar 2026 17:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Zhang Xiaoyan, National University of Singapore

Chaitin’s Omega is defined to be the halting probability of a randomly constructed program. Equivalently it can be defined by the sum of Kolmogorov weights of all binary strings. In this talk we investigate the Omega function f that maps a real x to the sum of Kolmogorov weights of all strings on the left of x. We establish properties of f and compare it with previously defined Omega-related functions. We present that (i) f is strictly increasing, continuous and differentiable precisely at density random reals; (ii) The range of f is a null, nowhere dense, perfect Pi01(0′) class with Hausdorff dimension 1; (iii) there are continuum many x such that f(x) is not random.

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Date/Time : 28 Jan 2026 17:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Chong Chi Tat, National University of Singapore

This talk is a continuation and concluding part of the talk given in 2024/25. It concerns the problem of definable solutions of combinatorial principles such as the Cohesive Set Theorem, Ramsey’s Theorem and the Halpern-Lauchli Theorem over models of subsystems of second-order arithmetic. In a joint work with Wong Tin Lok reported here, we give a complete solution of this problem, as well as problem of the existence of a definable minimal degree over such models.

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Date/Time : 21 Jan 2026 17:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Andre Nies, University of Auckland

Let R be the structure of r.e. many-one degrees. Lachlan showed that each finite distributive lattice is embeddable in R as an initial segment, thereby showing its theory is hereditarily undecidable. In this retrospective of some of my results from the 1990s, I will discuss how this proof in fact leads to undecidability of the Sigma-3 theory. I showed that the whole theory is Turing equivalent to true arithmetic. The technique used there paved the way for the similar result for the theory of the lattice of r.e. sets (with Harrington).

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