Date/Time : 06 Nov 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Michael Takaaki Leong,

A weakening of a Suslin tree, known as a Suslin lattice, was introduced by Dilworth, Odell, and Sari in 2007, and subsequently investigated by Raghavan and Yorioka in 2012. In this talk, we will show that the compatibility of a Suslin lattice with Martin’s Axiom and its variants mirrors that of a Suslin tree by showing that a fragment of Martin’s Axiom suffices to imply the non-existence of a Suslin lattice. We will also discuss the possible consistency of a Suslin lattice with the P-ideal Dichotomy.

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Date/Time : 30 Oct 2024 16:45
Venue : S17 #04-06 (SR1)
Speaker : Desmond Lau,

We develop a forcing framework based on the idea of amalgamating language fragments into a theory with a canonical term model. We then demonstrate the usefulness of this framework by applying it to variants of the extended Namba problem, as well as to the analysis of models of certain theories with constraints in interpretation (TCIs). Separately, we look at small extensions of V as generalised degrees of computability over V. Using TCIs, we formalise and investigate the complexity of certain methods one can use to define, in V, subclasses of degrees over V. Finally, we give a characterisation of the complexity of forcing.

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Date/Time : 23 Oct 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Ellen Hammatt, Victoria University of Wellington

In this talk we investigate what happens when we take concepts from computable structure theory and forbid the use of unbounded search. In other words, we discuss the primitive recursive content of structure theory. The central definition is that of punctual structures, introduced by Kalimullin, Melnikov and Ng in 2017. We investigate various concepts from computable structure theory in the primitive recursive case. A common theme is that new techniques are required in the primitive recursive case. In particular we will focus on topics such as finite punctual dimension, punctual 1-decidability and the punctual degrees. Where the punctual degrees is a degree structure within punctual presentations of a fixed structure which is induced by primitive recursive isomorphisms. I will present various results from my PhD thesis as well as pose some open questions in the area.

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Date/Time : 09 Oct 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Athipat Thamrongthanyalak, Chulalongkorn University

In this talk, we study a converse of the Banach fixed point theorem and its connection to tameness in expansions of a real closed field. Let R be a definably complete expansion of a real closed field.

We say that R has the BFPP (short for, Banach fixed point property) when, for every locally closed definable set E, if every contraction on E has a fixed point, then E is closed. In this talk, we prove that if R has an o-minimal open core, then R has the BFPP; and if R has the BFPP, then R has a locally o-minimal open core.

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Date/Time : 18 Sep 2024 16:45
Venue : S17 #04-04 (SR3)
Speaker : Le Quy Thuong,

Since 1995, motivic integration has been a powerful tool in algebraic geometry and other branches of mathematics. In particular, it has many important applications to singularity theory. For instance, Denef-Loeser around 2000 gave a breakthrough point of view in the study of singularities, by introducing the so-called motivic Milnor fiber, with the philosophy that this is a motivic incarnation of the classical Milnor fiber. One shows that many singularity invariants can be easily recovered from motivic zeta function and motivic Milnor fiber employing an appropriate Hodge realization. Furthermore, there are important problems concerning singularity theory such as monodromy conjecture, the integral identity conjecture, and the Thom-Sebastiani theorem that are waiting for new methods in motivic integration to have a solution.

In this talk, we will describe some surprising interactions between motivic integration, model theory and singularity theory that lead to our proofs for the integral identity conjecture, and the motivic Thom-Sebastiani theorem, as well as other applications to singularities. The talk will avoid technical aspects and emphasize key ideas in motivic integration and singularity theory, which may be friendly to a general audience.

Note that this week the seminar starts 15 minutes earlier than usual.

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Date/Time : 11 Sep 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Kihara Takayuki, Nagoya University

The theories of degrees of unsolvability and realizability interpretation both have long histories, having both been born in the 1940s. S. C. Kleene was a key figure who led the development of both theories. Despite having been developed by the same person, there seems to have been little deep mixing of these theories until recently. In this talk, we will reconstruct the theory of degrees of unsolvability from the perspective of realizability theory.

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Date/Time : 04 Sep 2024 16:45
Venue : S17 #04-04 (SR3)
Speaker : Chong Chi Tat, National University of Singapore

Let $\alpha$ be an admissible ordinal. A set $A\subset \alpha$ is of minimal $\alpha$-degree if it is non -recursive and every set of lower $\alpha$-degree is recursive. In 1956 Spector introduced the technique of forcing with perfect sets to prove the existence of a minimal $\omega$-degree. Sacks posed the problem of the existence of a minimal $\alpha$-degree for all admissible $\alpha$. This problem, poised in the early 1970’s, remains unsolved today. The best result to-date is that of Shore (1972) in which he showed that there is a positive solution if $\alpha$ is $\Sigma_2$-admissible. There has been no progress made on this problem since. In this talk we present a recent result on the minimal $\alpha$-degree problem for singular cardinals of uncountable cofinality which are not $\Sigma_2$-admissible.

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Date/Time : 28 Aug 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Linus Richter, National University of Singapore

I will outline a few connections between various notions of definability (which vary in degree of logical formality), give examples, and describe some open questions at the intersection of logic and classical mathematics.

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Date/Time : 21 Aug 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Vo Ngoc Thieu, National University of Singapore

The main aim of this talk is to introduce our recent results on computational problems related to DAEs, including the effective differential Nullstellensatz, effective differential elimination, and finding general solutions of low-order algebraic ODEs. The effective differential Nullstellensatz involves finding a positive integer N for a given DAE system, such that one can check the consistency of the system by performing N differentiations and polynomial eliminations. Differential elimination involves removing independent variables from a DAE system. Differential Nullstellensatz and elimination are two fundamental problems in differential algebra and differential algebraic geometry. Since the number N represents the computational complexity of the effective differential Nullstellensatz and elimination, finding an upper bound for N is crucial. We will present our recent investigations into the problem of determining an upper bound for N. In addition, our results on the problem of determining algebraic/rational general solutions of first-order algebraic ODEs, as well as their connection with the Poincaré problem, will also be presented.

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