Logic Seminar in AY 2024/2025 at NUS

• Wednesday 31.07.2024, 17:00 hrs, Department of Mathematics, Room S17#04-05.
  George Barmpalias. Questions and progress in Algorithmic Randomness.
  I will discuss current challenges and progress in algorithmic randomness, focusing on Chaitin's halting probability, almost everywhere domination and measures of relative randomness. I will offer conjectures, partial results and benchmark problems toward solving the main questions.
  
  
• Wednesday 07/08/2024, 17:00 hrs, Week 1, Department of Mathematics, Room S17#04-04.
  Zhang Jing. Higher dimensional combinatorics.
  We expose an organizing framework to study higher dimensional infinitary combinatorics based on Čech cohomology, originating from works by Barry Mitchell, Barbara Osofsky and others. Key combinatorial notions include n-coherence and n-triviality for sequences of functions. We will use some recent vanishing and non-vanishing results to demonstrate "ℵ_n is incompact for (n+1)-dimensional combinatorics" and "ℵ_ω+1 can be compact for n-dimensional combinatorics for all n". Time permitting, we will also discuss the possibility of generalizing classical 2-dimensional properties like being special or being Suslin to higher dimensions. The talk will be purely combinatorial.
  This is joint work with Jeffrey Bergfalk and Chris Lambie-Hanson.
  
  
• Wednesday 14/08/2024, 17:00 hrs, Week 1, Department of Mathematics, Room S17#04-04.
  Koh Heer Tern. Effective Polish spaces.
  We explore a recent program in comparing effective topological spaces up to (not necessarily computable) homeomorphism. We provide a survey of some earlier results separating some of the common notions of presentations for Polish spaces. By pushing existing definability techniques and effective versions of Stone duality, we also provide some new counterexamples. Most interestingly, that there exists a Polish space both left-c.e. and right-c.e. presentable but not computably presentable.
  
  
• Wednesday 21/08/2024, 17:00 hrs, Week 2, Department of Mathematics, Room S17#04-04.
  Vo Ngoc Thieu. Some Computational Aspects of Differential-Algebraic Equations (DAEs).
  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.
  
  
• Wednesday 28/08/2024, 17:00 hrs, Week 3, Department of Mathematics, Room S17#04-04.
  Linus Richter. Definable (Classical) Mathematics.
  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.
  
  
• Wednesday 04/09/2024, 16:45 hrs, Week 4, Department of Mathematics, Room S17#04-04.
  Chong Chitat The minimal α-degree revisited.
  Let α be an admissible ordinal. A set A ⊂ α is of minimal α-degree if it is non-recursive and every set of lower α-degree is recursive. In 1956 Spector introduced the technique of forcing with perfect sets to prove the existence of a minimal ω-degree. Sacks posed the problem of the existence of a minimal α-degree for all admissible α. This problem, posed 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 α is Σ₂-admissible. There has been no progress made on this problem since. In this talk we present a recent result on the minimal α-degree problem for singular cardinals of uncountable cofinality which are not Σ₂-admissible.
  Note that this week the seminar starts 15 minutes earlier than usual.
  
  
• Wednesday 11/09/2024, 17:00 hrs, Week 5, Department of Mathematics, Room S17#04-04.
  Kihara Takayuki. Degrees of unsolvability of natural problems: A realizability-theoretic approach.
  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.
  
  
• Wednesday 18/09/2024, 16:45 hrs, Week 6, Department of Mathematics, Room S17#04-04.
  Le Quy Thuong. Motivic integration in valued fields and applications to singularity theory.
  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.
  
  
• Wednesday 02/10/2024, 17:00 hrs, Week 7, Department of Mathematics, Room S17#04-04.
  No Talk.
  
  
• Wednesday 09/10/2024, 17:00 hrs, Week 8, Department of Mathematics, Room S17#04-04.
  Athipat Thamrongthanyalak. Tame expansions of real closed fields and Banach fixed point property.
  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.
  
  
• Wednesday 16/10/2024, 17:00 hrs, Week 9, Department of Mathematics, Room S17#04-04.
  No talk.
  
  
• Wednesday 23/10/2024, 17:00 hrs, Week 10, Department of Mathematics, Room S17#04-04.
  Ellen Hammatt. Arriving on time: punctuality in structures, isomorphisms and 1-decidability.
  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.
  
  
• Wednesday 30/10/2024, 16:45 hrs, Week 11, Department of Mathematics, Room S17#04-04.
  Desmond Lau. Forcing with language fragments ... and without.
  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.
  Note the early start of 16:45 hrs for the logic seminar talk.
  
  
• Wednesday 06/11/2024, 17:00 hrs, Week 12, Department of Mathematics, Room S17#04-04.
  Leong Michael Takaaki. A weakening of a Suslin tree with variants of Martin's Axiom.
  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.
  
  
• Wednesday 13/11/2024, 17:00 hrs, Week 13, Department of Mathematics, Room S17#04-04.
  Dilip Raghavan. Nowhere dense ultrafilters and weak forms of selectivity.
  I will present some recent consistency results on nowhere dense ultrafilters and a weakening of selectivity.
  
  
• Tuesday 26/11/2024, 14:00 hrs. Faculty of Science, Room S16#05-18.
  Patrick Lutz. Complexity of oracles for packing dimension. 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 Π⁰₁ sets the 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 Π⁰₁ 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 Π⁰₁ 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.
  
  
• Wednesday 18/12/2024, 17:00 hrs. Department of Mathematics, Room S17#05-11.
  Manlio Valenti. On the density of the Weihrauch degrees.
  Recall that, in a partial order P, b is a minimal cover of a 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 ≤ 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.