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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Date/Time : 14 Aug 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Koh Heer Tern, Nanyang Technological University

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.

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Date/Time : 07 Aug 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Zhang Jing, University of Toronto

We expose an organizing framework to study higher dimensional infinitary combinatorics based on \v{C}ech 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 “aleph_n is incompact for (n+1)-dimensional combinatorics” and “aleph_{omega+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. Joint work with Jeffrey Bergfalk and Chris Lambie-Hanson.

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Date/Time : 31 Jul 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : George Barmpalias, Chinese Academy of Sciences

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.

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Date/Time : 17 Jul 2024 17:00
Venue : LT33 (S17 Level 2)
Speaker : Wojciech Woloszyn, University of Oxford

Two features of the Ultimate L are that it is forcing invariant and compatible with large cardinals. The resurrection principle stabilizes forcing out albeit mildly. The latter can be refuted from the existence of a large cardinal way below I0. What is the bearing of this on the status of large cardinals and the axiom V=Ultimate L? An interesting picture unfolds.

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