Location: MALL
Title: Hilbert polynomials for finitary matroids
Eventual polynomial growth is a common theme in combinatorics and commutative algebra. The quintessential example of this phenomenon is the Hilbert polynomial, which eventually coincides with the linear dimension of the graded pieces of a finitely generated module over a polynomial ring. A later result of Kolchin shows that the transcendence degree of certain field extensions of a differential field is eventually polynomial. More recently, Khovanskii showed that for finite subsets $A$ and $B$ of a commutative semigroup, the size of the sumset $A+tB$ is eventually polynomial in $t$. I will present a common generalization of these three results in terms of finitary matroids (also called pregeometries). I’ll discuss other instances of eventual polynomial growth (like the Betti numbers of a simplicial complex) as well as some applications to bounding model-theoretic ranks. This is joint work with Antongiulio Fornasiero.
Location: Social Sciences SR (14.33)
Title: Generalised measurability and bilinear forms
NOTE location change: Social Sciences SR (14.33)
In this talk I will go over measurable and generalised measurable structures, giving examples and non-examples. I will then go on to consider the two sorted structure $(V,F,β)$ where $V$ is an infinite dimensional vector space over $F$ an infinite field, and $β$ a bilinear form on this vector space. In particular I will consider the interaction of different notions of independence when this structure is pseudo finite. I will finish with some questions around generalised measurable structures.
Location: MALL
Title: Ordinal arithmetic and subgroups of Thompson's group
The class of finitely generated groups embeddable into Richard Thompson's group $F$ is both restrictive and rich at the same time. We show that there is a family of groups within this class which is pre-well-ordered in type $\epsilon_0$ by the embeddability relation. Moreover, the operations of addition and multiplication on the ordinals translate into natural group-theoretic operations—direct sum and a type of permutational wreath product. This talk will give a description of this correspondence. This is joint work with Collin Bleak and Matt Brin.
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.
