Date/Time : 15 Mar 2023 17:00
Venue : S17 #05-11 (SR5)
Speaker : Liu Shixiao ,

In the first half of the talk I shall be proving the properness of Mathias forcing with lower density requirement. In the second half of the talk I shall demonstrate the difficulty in applying this method (and some other commonly used methods) to Silver forcing with lower density requirement.

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Date/Time : 08 Mar 2023 17:00
Venue : S17 #05-11 (SR5)
Speaker : Chong Chitat, National University of Singapore

Let T be the full infinite binary tree (i.e. the Cantor space). An infinite subtree S of T is a strong subtree if (i) S is isomorphic to T, (ii) if sigma is a node in S, and sigma_0, sigma_1 are its immediate successors in S, then sigma_0 and sigma_1 have the same length in T and furthermore (iii) the intersection of sigma_0 and sigma_1 is sigma.

The Halpern-Lauchli Theorem is a theorem in combinatorial mathematics which states that for any finite number of infinite full binary trees T_1,T_2,…, T_d, and any finite coloring of the d-row vectors in T_1 x
T_2 x … x T_d, there exist strong subtrees S_1 subseteq T_1, S_2 subseteq T_2, …, S_d subseteq T_d for which all d-row vectors in S_1 x S_2 x … x S_d have the same color.

This Ramsey type theorem is yet another striking demonstration of the existence of order within chaotic disorder. There is no known simple proof of this theorem in the published literature.

In this talk we discuss the proof-theoretic strength of the Halpern-Lauchli theorem from the reverse mathematics point of view. In particular, we give a characterisation of the inductive strength of this theorem as well as its conservation strength over a base theory weaker than Sigma_2-induction.

This is joint work with Li Wei and Yang Yue.

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Date/Time : 01 Mar 2023 17:00
Venue : via Zoom
Speaker : Linus Richter, Victoria University of Wellington

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Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

A recent “point-to-set principle” of Jack Lutz and Neil Lutz characterises the Hausdorff dimension of any subset of Euclidean space in terms of the Kolmogorov complexity of its individual points.
Sets with particular fractal properties can now be constructed point-by-point, by coding “enough” information into each point, bit-by-bit.

After introducing the point-to-set principle, I will present a new result in fractal geometry: under V=L (the axiom of constructiblity), I will outline the construction of co-analytic sets of Euclidean space which fail Marstrand’s Projection Theorem, a classical result in fractal geometry concerning the dimension of orthogonal projections of analytic plane sets onto lines.

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Date/Time : 15 Feb 2023 17:00
Venue : S17 #05-11 (SR5)
Speaker : David Belanger, National University of Singapore

We present a system of functionals that can serve as Skolem functions, so that any arithmetical formula can be rewritten in terms of them, in a model of BSigma_n + not ISigma_n. A distinguishing feature of our construction is that each functional is coded as a natural number within the model, and there is an upper bound on the codes. The method has a number of interesting applications.

This is joint work with Chong, Li, Wong and Yang.

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Date/Time : 08 Feb 2023 17:00
Venue : via Zoom
Speaker : Will Johnson, Fudan University

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Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

Is every NIP integral domain a henselian local ring? In this talk, I will give evidence for this conjecture, sketching why it holds in the positive characteristic case and the finite dp-rank case. I will also discuss how this “generalized henselianity conjecture” is related to the standard conjectures on NIP fields and valued fields. For example, the henselianity conjecture for NIP valued fields implies the generalized henselianity conjecture for Noetherian NIP integral domains. Lastly I will discuss how this conjecture fits into the general problem of classifying NIP fields and NIP commutative rings. For example, it implies that NIP fields with definable field topologies must be ample/large in the sense of Pop, and NIP commutative rings must be finite products of henselian local rings.

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Date/Time : 01 Feb 2023 17:00
Venue : S17 #05-11 (SR5)
Speaker : Yu Liang, Nanjing University

I shall present some recent results in set theory which are based on proof methods and results from recursion theory.

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Date/Time : 25 Jan 2023 17:00
Venue : via Zoom
Speaker : Leonardo Mainardi,

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Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

When the Canonical Ramsey’s Theorem by Erdös and Rado is applied to regressive functions, one obtains the Regressive Ramsey’s Theorem by Kanamori and McAloon. Taylor proved a “canonical” version of Hindman’s Theorem, analogous to the Canonical Ramsey’s Theorem. We introduce the restriction of Taylor’s Canonical Hindman’s Theorem to a subclass of the regressive functions, the λ-regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman’s Theorem and of natural restrictions of it. In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that this same principle reduces the well-ordering-preservation principle for base-ω exponentiation by a uniform reduction.

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Date/Time : 18 Jan 2023 17:00
Venue : S17 #04-05 (SR2) via Zoom
Speaker : Bakhadyr Khoussainov, University of Auckland

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Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

We aim to describe the isomorphism types of algorithmically presented structures in the language of the first order logic.

This is one of the key research topics related to the expressive power of the first order logic. The focus is on two classes of structures. The first class is the class of structures for which the positive atomic diagrams are computably enumerable.

We call these structures positive structures. The second class is the class of structures for which the negative atomic diagrams are computably enumerable. We call these structures negative structures. We investigate definability of positive and negative structures by sets of sentences quantified with Existential, Universal, Existential-Universal and Universal-Existential quantifiers using expansions of languages.

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Date/Time : 16 Nov 2022 16:00
Venue : S15 #03-15 (Seminar Room L3)
Speaker : Cristian Calude, University of Auckland

Abstract: We construct a class of 3-dimensional photonic quantum random number generators and prove that each generates maximally unpredictable digits via measurements that are robust to errors. In particular, every sequence generated is strongly incomputable; hence its quality is provable better than that of every pseudo-random sequence. We also briefly contrast 2-dimensional and 3-dimensional quantum random number generators, discuss photonic implementations and show the superiority of the latter ones. These results suggest that incomputability in physics is real and practically useful.

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