The studies of Weihrauch degrees and reverse mathematics share many ideas, and many similar results and close relations are known. The studies of Weihrauch degrees of the arithmetical transfinite recursion (ATR) and their relation to reverse mathematics are developed, e.g., in [1,2,3]. Typically, principles which are provable from $ATR_0$ (in the setting of reverse mathematics) by way of the pseudo-hierarchy method have various strengths. In this talk, we overview these situations and study the structure between ATR and the choice principle on the Baire space. This is joint work with Yudai Suzuki.
T. Kihara, A. Marcone and A. Pauly. Searching for an analogue of $ATR_0$ in the Weihrauch lattice. J. Symb. Log., 85(3):1006–1043, 2020.
Jun Le Goh. Some computability-theoretic reductions between principles around ATR0. arXiv preprint arXiv:1905.06868, 2019.
Paul-Elliot Anglès d'Auriac. Infinite Computations in Algorithmic Randomness and Reverse Mathematics. PhD thesis, Université Paris-Est, 2019.
Y. Suzuki and K. Yokoyama. Searching problems above arithmetical transfinite recursion. In preparation.
Location: MALL
Title: Formalising Erdős and Larson: Ordinal Partition Theory
NOTE date and time change: Tue 11am
A number of results in infinitary combinatorics have been formalised in the proof assistant Isabelle/HOL. These include results on ordinal partition relations by Erdős–Milner, Specker, Larson and Nash-Williams, leading to Larson’s 1973 proof that for all $m$ in $ℕ$, $ω^ω → (ω^ω,m)$. This material is available online; here we discuss some of the most challenging aspects of the formalisation process, and wider issues in the formalisation of research-grade mathematics. See also the paper in Experimental Mathematics 31:2, 2022.
Location: MALL
Title: Definable refinements of classical algebraic invariants
In this talk I will explain how methods from logic allow one to construct refinements of classical algebraic invariants that are endowed with additional topological and descriptive set-theoretic information. This approach brings to fruition initial insights due to Eilenberg, Mac Lane, and Moore (among others) with the additional ingredient of recent advanced tools from logic. I will then present applications of this viewpoint to invariants from a number of areas in mathematics, including operator algebras, group theory, algebraic topology, and homological algebra.
Location: MALL
Title: Forcing over choiceless models
Forcing over models of ZF set theory without the axiom of choice has been studied in particular for L(ℝ) in work of Steel, Van Wesep, Woodin and more recently Larson and Zapletal. However, the axiom of choice can fail in much stronger ways than in L(ℝ). For example, in Gitik’s celebrated model all uncountable cardinals are singular. Since virtually all known forcing techniques fail in this situation, it is interesting to understand what forcing does to such models. We develop a toolbox for forcing over arbitrary choiceless models. We further introduce very strong absoluteness principles and show their relation with Gitik’s model. This is joint work with Daisuke Ikegami and in part with W. Hugh Woodin.
Fixpoint operators are tools to reason on recursive programs and infinite data types obtained by induction (e.g. lists, trees) or coinduction (e.g. streams). They were given a categorical treatment with the notion of categories with fixpoints. An important result by Plotkin and Simpson in this area states that provided some conditions on bifree algebras are satisfied, we obtain the existence of a unique uniform fixpoint operator. This theorem allows to recover the well-known examples of the category Cppo (complete pointed partial orders and continuous functions) in domain theory and the relational model in linear logic. In this talk, I will present a categorification of this result where the 2-dimensional framework allows to study the coherences associated to the reductions of λ-calculi with fixpoints i.e. the equations satisfied by the program computations steps.
Location: MALL
Title: Big Ramsey degrees for internal colorings
In this talk, I will define what it means for a coloring of substructures of an ultraproduct structure to be "internal", and a notion of finite big Ramsey degree for internal colorings. I will also present a certain Ramsey degree transfer theorem from countable sequences of finite structures to their ultraproducts, assuming AC and some additional mild assumptions. The big Ramsey degree of a finite structure in an ultraproduct can differ markedly from its internal big Ramsey degree, as demonstrated by the example of the class of all finite linear orders, which I will explain.
This is joint work with Dana Bartošová, Mirna Džamonja and Rehana Patel.
Location: MALL
Title: Jordan permutation groups and limits of treelike structures
A transitive permutation group $G$ on a set $X$ is a Jordan group if there is a subset $A$ of $X$ (a 'Jordan set') with $|A|>1$ such that the subgroup of $G$ fixing the complement of $A$ is transitive on $A$ (+ a non-degeneracy condition that if $G$ is $k$-transitive on $X$ then $|X \setminus A|\geq k$.) So if $X$ carries some first-order structure, this is a bit like saying all elements of $A$ realise the same type over $X \setminus A$. Work of Adeleke, myself and Neumann in the 1990s gave a kind of classification of Jordan groups which are 'primitive', i.e. preserve no proper non-trivial equivalence relation on $X$. Many key examples can be seen as Fraïssé limits.
I will discuss examples, and also sketch recent work in Asma Almazaydeh's thesis (and subsequent work with her) on certain mysterious $\omega$-categorical structures which are limits of treelike structures. This relates to earlier work with Meenaxi Bhattacharjee, and to a recent preprint of David Bradley-Williams and John Truss.
Location: MALL
Title: Adelic Geometry via Geometric Logic
NOTE time change: 3pm
On a syntactic level, the peculiarity of geometric logic can be seen from the choice of logical connectives used (in particular, we allow for infinitary disjunctions but do not allow negation), but there are deep ramifications of this seemingly innocuous move. One, geometric logic is incomplete if we restrict ourselves to set-based models, but is complete if we also consider models in all toposes (i.e. not just $\mathrm{Set}$) — as such, geometric logic can be viewed as an attempt to pull our mathematics away from a fixed set theory. Two, there is an intrinsic continuity to geometric logic, which is furnished by the definition of the classifying topos. Indeed, since every Grothendieck topos is a classifying topos of some geometric theory, this provides yet another way of viewing Grothendieck toposes as generalised spaces.
Both insights will inform the content of this talk. We shall start by giving a leisurely introduction to the theory of geometric logic and classifying toposes, before introducing a new research programme (joint with Steven Vickers) of developing a version of adelic geometry via topos theory.
The first step of this programme is to define the geometric theory of absolute values of $\mathbb{Q}$ and provide a point-free account of exponentiation. The next step is to construct the classifying topos of places of $\mathbb{Q}$, which incidentally provides a topos-theoretic analogue of the Arakelov compactification of $\operatorname{Spec}(\mathbb{Z})$. Interestingly, whereas the classical picture views the Archimedean place as a single point "at infinity", our picture reveals that the Archimedean place resembles a blurred interval living below $\operatorname{Spec}(\mathbb{Z})$. This raises challenging questions to our current understanding of the number theory, particularly in regards to reconciling the Archimedean vs. the non-Archimedean aspects.
