Title: Effective embeddings and interpretations

by Alexandra Soskova (Sofia University) as part of Computability theory and applications

Abstract

Friedman and Stanley introduced Borel embeddings as a way of comparing classification problems for different classes of structures. Many Borel embeddings are actually Turing computable. The effective decoding is given by a uniform effective interpretation. Part of the effective interpretation is actually Medvedev reduction.

The class of undirected graphs and the class of linear orderings both lie on top under Turing computable embeddings. We give examples of graphs that are not Medvedev reducible to any linear ordering, or to the jump of any linear ordering. For any graph, there is a linear ordering, that the graph is Medvedev reducible to the second jump of the linear ordering. Friedman and Stanley gave a Turing computable embedding $L$ of directed graphs in linear orderings. We show that there do not exist $L_{omega_1omega}$-formulas that uniformly interpret the input graph $G$ in the output linear ordering $L(G)$. This is joint work with Knight, and Vatev.

We have also one positive result — we prove that the class of fields is uniformly effectively interpreted without parameters in the class of Heisenberg groups.

The second part is joint work with Alvir, Calvert, Goodman, Harizanov, Knight, Miller, Morozov, and Weisshaar.

Title: Topological Groups Seminar Two-Week Hiatus

by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract: TBA

Title: Abelian Varieties as Algebraic Protori?

by Wayne Lewis (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract

An outcome of the structure theory of protori (compact connected abelian groups) is their representability as quotients of $mathbb{A}^n$ for the ring of adeles $mathbb{A}$. $mathbb{A}$ does not contain zeros of rational polynomials, but rather representations of zeros. Investigating the relations between algebraicity of complex tori and algebraicity of protori leads one to the problem of computing the Pontryagin dual of $mathbb{A}/mathbb{Z}$. Applying an approach by Lenstra in the setting of profinite integers to the more general $mathbb{A}$ leads to a definition of the closed maximal $Lenstra$ $ideal$ $E$ of $mathbb{A}$, whence the locally compact field of $adelic$ $numbers$ $mathbb{F}=mathbb{A}/E$, providing a long-sought connection to $mathbb{C}$ enabling one to define a functor from the category of complex tori to the category of protori – is it possible to do so in a way that preserves algebraicity? While $mathbb{F}$ marks tentative progress, much work remains…

Title: Fickleness and bounding lattices in the recursively enumerable Turing degrees

by Li Ling Ko (University of Notre Dame) as part of Computability theory and applications

Abstract

The ability for a recursively enumerable Turing degree $d$ to bound certain

important lattices depends on the degree’s fickleness. For instance, $d$

bounds $L_7$ (1-3-1) if and only if $d$’s fickleness is $>omega$

($geqomega^omega$). We work towards finding a lattice that characterizes

the $>omega^2$ levels of fickleness and seek to understand the challenges

faced in finding such a lattice. The candidate lattices considered include

those that are generated from three independent points, and upper

semilattices that are obtained by removing the meets from important

lattices.

Title: On the descriptive complexity of Fourier dimension and Salem sets

by Manlio Valenti (Università di Udine) as part of Computability theory and applications

Abstract

It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets.

In this talk, we explore the descriptive complexity of the family of closed Salem subsets of the Euclidean space. We also show how these results yield a characterization of the Weihrauch degree of the maps computing the Hausdorff or the Fourier dimensions.

Title: Accounting with $mathbb{QP}^infty$

by Wayne Lewis (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract

Rational projective space provides a useful accounting tool in engineering decompositions of $mathbb{Q}[x]$ for desired effect. The device is useful for defining a correspondence between summands of such a decomposition and elements of a partition of $mathbb{A}$. This mechanism is applied to a decomposition of $mathbb{Q}[x]$ relative to which the correspondence gives the $Lenstra$ $ideal$ $E$, a closed maximal ideal yielding the $adelic$ $numbers$ $mathbb{F}=frac{mathbb{A}}{E}$.

Title: The interplay between randomness and genericity

by Laurent Bienvenu (Université de Bordeaux) as part of Computability theory and applications

Abstract

In computability theory, one often think of (Cohen)-genericity and algorithmic randomness as orthogonal notions: a truly random real will look very non-generic, and a truly generic real will look very non-random. This orthogonality is best incarnated by the result of Nies, Stephan and Terwijn that any 2-random real and 2-generic real form a minimal pair for Turing reducibility. On the other hand, we know from the Kucera-Gacs theorem that for any n there is a 1-random real which computes an n-generic one, but also (and more surprisingly), by a result of Kautz that every 2-random real computes a 1-generic real. These last two results tell us that the interplay between randomness and genericity is rather complex when “randomness” is between 1-random and 2-random or “genericity” between 1-generic and 2-generic. It is this gray area that we will discuss in this talk (based on the paper of the same title, joint work with Chris Porter).