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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Date/Time : 09 Nov 2022 17:15
Venue : S17 #04-05 (SR2)
Speaker : Benjamin T Castle, Ben Gurion University of the Negev

Abstract: Motivated by recent progress toward Zilber’s Restricted Trichotomy Conjecture, we study reducts of the complex field up to interdefinability over parameters. Precisely, we will consider structures of the form (C, P_1,…,P_n), where C is set of complex numbers and the P_i are polynomial maps of potentially different arities.   Somewhat surprisingly, our main result states that almost all such structures (in a precise sense) are interdefinable. The proof uses a mix of tools from geometric stability theory, combinatorics, and algebraic geometry.

This is joint work with Chieu-Minh Tran.

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Date/Time : 02 Nov 2022 17:00
Venue : S17 #04-05 (SR2)
Speaker : Wu Guohua, Nanyang Technological University

Abstract: Many constructions in rings involve applications of various axioms, which guarantee the existence of wanted objects. In this talk, I will present examples of such constructions, and show how these axioms are applied and really needed.

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Date/Time : 26 Oct 2022 17:00
Venue : S17 #04-05 (SR2)
Speaker : Sun Mengzhou, National University of Singapore

Abstract: Paris and Kirby showed that a countable model satisfies B Sigma_{n+2} if and only if it has an (n+2)-elementary proper end extension. Later Kaufman asked whether we can always end extend countable models of B Sigma_{n+2} to some model of B Sigma_{n+1}. We briefly discuss what we have now related to this question and what is the difficulty here.

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Date/Time : 19 Oct 2022 17:00
Venue : S17 #04-05 (SR2)
Speaker : Abdul Basit,

Abstract: The shatter function, a combinatorial function associated to a family of sets, is an important measure of its complexity. For example, it is related to the popular notions of VC dimension and VC density. We show that the shatter function of a semilinear family (i.e., a family definable in (R, + , <)) is asymptotic to a polynomial. This implies, in particular, that any semilinear family has integer VC density, which confirms a conjecture by Artem Chernikov.

This is joint work with Tran Chieu Minh

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