Title: Topological Groups Seminar Two-Week Hiatus

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

Lecture held in Elysium.

Abstract: TBA

Title: Reverse mathematics of combinatorial principles over a weak base theory

by Leszek Kołodziejczyk (University of Warsaw) as part of Computability theory and applications

Abstract

Reverse mathematics studies the strength of axioms needed to prove various

mathematical theorems. Often, the theorems have the form $forall X exists

Y psi(X,Y)$ with $X, Y$ denoting subsets of $mathbb{N}$ and $psi$

arithmetical, and the logical strength required to prove them is closely

related to the difficulty of computing $Y$ given $X$. In the early decades

of reverse mathematics, most of the theorems studied turned out to be

equivalent, over a relatively weak base theory, to one of just a few typical

axioms, which are themselves linearly ordered in terms of strength. More

recently, however, many statements from combinatorics, especially Ramsey

theory, have been shown to be pairwise inequivalent or even logically

incomparable.

The usual base theory used in reverse mathematics is $mathrm{RCA}_0$, which

is intended to correspond roughly to the idea of “computable mathematics”.

The main two axioms of $mathrm{RCA}_0$ are: comprehension for computable

properties of natural numbers and mathematical induction for c.e.

properties. A weaker theory in which induction for c.e. properties is

replaced by induction for computable properties has also been introduced,

but it has received much less attention. In the reverse mathematics

literature, this weaker theory is known as $mathrm{RCA}^*_0$.

In this talk, I will discuss some results concerning the reverse mathematics

of combinatorial principles over $mathrm{RCA}^*_0$. We will focus mostly on

Ramsey’s theorem and some of its well-known special cases: the

chain-antichain principle CAC, the ascending-descending chain principle ADS,

and the cohesiveness principle COH.

The results I will talk about are part of a larger project joint with Marta

Fiori Carones, Katarzyna Kowalik, Tin Lok Wong, and Keita Yokoyama.

Title: Non-arithmetic algebraic constructions

by Chris Conidis (CUNY-College of Staten Island) as part of Computability theory and applications

Abstract

We examine two radical constructions, one from ring theory and another from module theory, and produce a computable ring for each construction where the corresponding radical is $Pi^1_1$-complete.

Join the Hawai‘i Data Science Institute for another Data Science Friday seminar titled “Bayesian Topological Learning for Complex Data Analysis” presented by Assistant Professor of Mathematics Dr. Farzana Nasir on October 16, 2020 at 2 pm on Zoom.

Please find more information below and on the attached flyer.

**Zoom registration:** http://go.hawaii.edu/39f

**Abstract:** Persistent homology is a tool in topological data analysis for learning about the geometrical/topological structures in data by detecting different dimensional holes and summarizing their appearance disappearance scales in persistence diagrams. However, quantifying the uncertainty present in these summaries is challenging. In this talk, I will present a Bayesian framework for persistent homology by relying on the theory of point

processes. This Bayesian model provides an effective, flexible, and noise-resilient scheme to analyze and classify complex datasets. A closed form of the posterior distribution of persistence diagrams based on a family of conjugate priors will be provided. The goal is to introduce a

supervised machine learning algorithm using Bayes factors on the space of persistence diagrams. This framework is applicable to a wide variety of datasets. I will present an application to filament networks data classification of plant cells.**Bio:** Farzana Nasrin graduated from Texas Tech University with a Ph.D. in Applied Mathematics in August 2018. Her research interests span algebraic topology, differential geometry, statistics, and machine learning. Currently, she is holding an assistant professor position at UH Manoa in the Department of Mathematics. Before coming to UHM, she was working as a postdoctoral research associate funded by the ARO in mathematical data science at UTK. She has been working on building novel learning tools that rely on the shape peculiarities of data with application to biology, materials science, neuroscience, and ophthalmology. Her dissertation involves the development of analytical tools for smooth shape reconstruction from noisy data and visualization tools for utilizing information from advanced imaging devices.

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.