Date/Time : 03 Mar 2021 17:00
Venue : via Zoom
Speaker : Desmond Lau,

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract: Martin’s Maximum^{++} and Woodin’s axiom (*) are two statements independent of, but consistent with, ZFC. I will present the common reasons they are appealing as set-theoretic axioms, before comparing the sense in which they are “maximal”. I will also run through an exposition of the recent work by Aspero and Schindler, which shows Martin’s Maximum^{++} implies (*), effectively “unifying” the statements.

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Date/Time : 17 Feb 2021 17:00
Venue : via Zoom
Speaker : Xiao Ming, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract: Order dimension is a classical combinatorial object and has been widely studied by set theorists, combinatorists and computer scientists since its introduction by Dushnik and Miller in 1941. We focus on the partial orderings that are definable as a Borel subsets in a Polish space and analyze the order dimension that can be realized by Borel definable orders and show that there are some interesting behaviors that can be quite different from the classical order dimension using arbitrary realization. This is a joint work with Dilip Raghavan.

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Date/Time : 03 Feb 2021 17:00
Venue : via Zoom
Speaker : Wong Tin Lok, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract: Arithmetic generally does not admit any non-trivial quantifier elimination. I will talk about one exception, where the negation of an induction axiom is included in the theory. Here the Weak Koenig’s Lemma from reverse mathematics arises as a model completion. This work is joint with Marta Fiori-Carones, Leszek Aleksander Kolodziejczyk and Keita Yokoyama.

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Date/Time : 11 Nov 2020 17:00
Venue : via Zoom
Speaker : Philipp Schlicht, University of Vienna

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract:  Similarities between c.e., Pi^1_1 and Sigma^1_2 sets can be explained by representing Pi^1_1 and Sigma^1_2 sets as c.e. with respect to a transfinite enumeration procedure. This representation is important for classical results about Pi^1_1 and Sigma^1_2 sets, but also for recent results, for instance about randomness at the level of Pi^1_1 by Hjorth and Nies and structural results about Sigma^1_2 sets by Chong, Wu and Yu. In the main result, we calculate the lengths of enumerations. More precisely, we isolate a certain countable ordinal tau and show that the length of an enumeration of a Pi^1_1 or Sigma^1_2 set either equals omega_1 or is below tau, and tau is optimal. This ordinal has many other interesting characterisations. For example, we show that tau is the optimal bound for the countable ranks of wellfounded Sigma^1_2 relations, in analogy with the Kunen-Martin theorem. We will further touch on related structural problems such as calculating Borel ranks of Pi^1_1 and Sigma^1_2 Borel sets. This is joint work with Philip Welch and Merlin Carl.

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Date/Time : 04 Nov 2020 17:00
Venue : via Zoom
Speaker : Andre Nies, University of Auckland

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract: The K-trivial sets are antirandom in the sense that the initial segment complexity in terms of prefix-free Kolmogorov complexity K grows as slowly as possible. Chaitin introduced this notion in about 1975, and showed that each K-trivial is Turing below the halting set. Shortly after, Solovay proved   that a K-trivial set can be noncomputable. In the past two decades, many alternative characterisations of this class have been found: properties such as  being low for K, low for Martin-Löf (ML) randomness, and a basis for  ML randomness, which state in one way or the other that the set is close to computable. Initially, the class of noncomputable K-trivials appeared to be  amorphous. More recently, evidence of an internal structure has been found. Most of these results can be phrased in the language of a mysterious  reducibility on the K-trivials which is weaker than Turing’s: A is ML-below B if each ML-random computing B also computes A. Bienvenu, Greenberg, Kucera, Nies and Turetsky (JEMS 2016) showed that there an ML-complete K-trivial set. Greenberg, Miller and Nies  (JML, 2019) established   a dense hierarchy of subclasses of the K-trivials based on fragments of Omega computing the set, and each such subclass is an initial segment for ML.  More recent results  generalise these approaches using so-called cost functions.  They also show that each K-trivial set  is ML-equivalent to a c.e. K-trivial. Alternative reducibilities on the K-trivials will be considered near the end of the talk. One of them is related to the extreme lowness notion of strong jump traceability, and appears to be orthogonal to ML-reducibility. Very recent work with Greenberg and Turetsky provides manyfold characterisations in terms of cost functions, and random oracles.

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Date/Time : 28 Oct 2020 17:00
Venue : via Zoom
Speaker : Liao Yuke, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract:  TTE-computable is one of the major definitions of computable real functions. We focus on a generalisation of TTE  and check the basic cases of computability of Julia sets. We prove quadratic polynomials with Siegel disc incomputable in TTE is still incomputable in generalisation.

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Date/Time : 21 Oct 2020 17:00
Venue : via Zoom
Speaker : Rupert Hoelzl, Universitaet der Bundeswehr Muenchen

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract:  We introduce Monte Carlo computability as a probabilistic concept of computability on infinite objects and prove that Monte Carlo computable functions are closed under composition. We then mutually separate the following classes of functions from each other: the class of multi-valued functions that are non-deterministically computable, that of Las Vegas computable functions, and that of Monte Carlo computable functions. We give natural examples of computational problems witnessing these separations. As a specific problem which is Monte Carlo computable but neither Las Vegas computable nor non-deterministically computable, we study the problem of sorting infinite sequences that was recently introduced by Neumann and Pauly. Their results allow us to draw conclusions about the relation between algebraic models and Monte Carlo computability.

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Date/Time : 14 Oct 2020 17:00
Venue : via Zoom
Speaker : Tran Chieu-Minh,

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract:  Zarankiewicz’s problem in graph theory asks to determine the largest possible number of edges |E| in a bipartite graph G = (V_1, V_2; E) with the parts V_1 and V_2 containing n_1 and n_2 vertices, respectively, and such that G contains no complete bipartite subgraphs on k vertices. Graphs definable in o-minimal (or more generally distal structures) enjoy stronger bounds than general graphs, providing an abstract setting for the Szemeredi-Trotter theorem and related incidence bounds. We obtain almost optimal upper and lower bounds for hypergraphs definable in locally modular o-minimal structures, along with some applications to incidence counting (e.g. the number of incidences between points and boxes with axis parallel sides on the plane whose incidence graph is K_{k,k}-free is almost linear). We explain how the exponent appearing in these bounds is tightly connected to the trichotomy principle in o-minimal structures. Joint work with Abdul Basit, Artem Chernikov, Sergei Starchenko and Terence Tao.

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Date/Time : 30 Sep 2020 17:00
Venue : via Zoom
Speaker : Vasco Brattka, Universtiaet der Bundeswehr, Muenchen

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract:   We discuss the question whether there is a weakest unsolvable mathematical problem. In recent years the Weihrauch lattice has been established as a suitable computability theoretic framework to analyze the uniform computational content of problems from many different fields of mathematics. Here we answer a question by Schroeder, whether there is a weakest discontinuous problem with respect to the continuous version of Weihrauch reducibility. We introduce the discontinuity problem, and we show that it is reducible exactly to the effectively  discontinuous problems, defined in a suitable way. However, in which sense this answers Schroeder’s question sensitively depends on the axiomatic framework that is chosen and it is a positive answer if we work in Zermelo- Fraenkel set theory with dependent choice and the axiom of determinacy. On the other hand, using the axiom of choice, one can construct problems which are discontinuous, but not effectively so.   Hence, the exact structure of the “bottom” of the Weihrauch lattice sensitively depends on the axiomatic setting that we choose.   We prove our result using Wadge games for mathematical problems and while the existence of a winning strategy for player II characterizes continuity of the problem (as already shown by Nobrega and Pauly), the existence of a winning strategy for player I characterizes effective discontinuity of the problem. We also provide further insights into the algebraic nature of the discontinuity problem. For one we show that the parallelization of the discontinuity problem is exactly the non-computability problem that was studied before. One the other hand, we introduce a new algebraic operation in the Weihrauch lattice that we call summation and which is the dual operation to parallelization. While parallelization can be seen as an analogue of the bang operator in linear logic, summation can be seen as an analogue of the question mark operator. It turns out that the discontinuity problem can be obtained as summation of a number of well-known problems in the Weihrauch lattice, such as the (lesser) limited problem of omniscience and variants thereof. More generally, we study the action of the monoid formed by parallelization and summation on the Weihrauch lattice, and we prove that this action can lead to at most five different Weihrauch degrees, which (in the maximal case) are always organized in a pentagon. We show that the discontinuity problem appears as the bottom of several natural such pentagons. This leads to further interesting characterizations of the discontinuity problem.

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