Date/Time : 05 Feb 2025 17:00
Venue : S17 #04-04 (SR3)
Speaker : Frank Stephan, National University of Singapore

The talk presents joint work with David Belanger (Singapore), C.T.~Chong (Singapore), Rupert Hoelzl (Munich), Gordon Hoi (Singapore), Sanjay Jain (Singapore), Dilip Raghavan (Singapore), Frank Stephan (Singapore), Daniel Turetsky (Wellington) and Jing Zhang (Toronto) published in four papers. The first one dealt with comparing the growth behaviour of functions or families of functions partial-recursive relative to an oracle and classified oracles with respect to this measure. The second, third and fourth papers are about reverse mathematics and study weakly represented families, that are families of functions computed relative to some oracle in the second order part of the model (all functions in the family use the same oracle). These families are used as a tool to define notions like dominated (every weakly represented family is dominated by some function in the model), hyperimmune, biimmune and so on. Furthermore, one can such families also use as a hypothesis space for classes to be learnt – this extends the study to inductive inference. This is a survey talk with sample proofs.

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Date/Time : 18 Dec 2024 17:00
Venue : S17 #05-11 (SR5)
Speaker : Manlio Valenti, Swansea University

Recall that, in a partial order P, b is a minimal cover of b if the interval (a,b) is empty. A strengthening of this notion is the one of strong minimal cover, namely b is a strong minimal cover of a if c<b implies c\le a. In this talk, we provide a complete characterization of how minimal covers and strong minimal covers in the Weihrauch degrees look like. As a consequence, we obtain that the Weihrauch degree of the identity problem id can be defined as the highest top of a strong minimal cover, or the least degree whose upper cone is dense. In other words, “being (uniformly) computable” is a first-order definable property in (W,<). This implies that the first-order theory of the Weihrauch degrees (below id) is recursively isomorphic to the third-order theory of true arithmetic.

This is joint work with Steffen Lempp, Joe Miller, Arno Pauly, and Mariya Soskova.

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Date/Time : 26 Nov 2024 14:00
Venue : S16 #05-18
Speaker : Patrick Lutz, University of California, Berkeley

Recently, there has been a spate of work applying tools from computability theory to prove theorems about Hausdorff dimension and packing dimension. Central to these applications are computability-theoretic analogues of Hausdorff dimension and packing dimension, known respectively as effective Hausdorff dimension and effective packing dimension. A fact which is useful for many applications is that for sufficiently simple sets—in particular, for Pi-0-1 sets—Hausdorff dimension and effective Hausdorff dimension agree. Unfortunately, the corresponding statement for packing dimension is known to fail. In particular, an example due to Conidis shows that there is a Pi-0-1 set of packing dimension 0 and effective packing dimension 1. In this talk, I will consider the question of exactly how bad this failure is. In particular, given a Pi-0-1 set E, what is the minimum complexity of an oracle A for which the packing dimension of E is equal to the effective packing dimension of E relative to A? Surprisingly, it turns out that there is not always even a hyperarithmetic oracle. I will also discuss to what extent this affects potential applications of effective packing dimension.

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Date/Time : 13 Nov 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Dilip Raghavan, National University of Singapore

I will present some recent consistency results on nowhere dense ultrafilters and a weakening of selectivity.

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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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