# Calendar

Oct
6
Tue
Paul-Elliot Angles d’Auriac (University of Lyon)
Oct 6 @ 3:00 am – 4:00 am

Title: The computable strength of Milliken’s Tree Theorem and applications
by Paul-Elliot Angles d’Auriac (University of Lyon) as part of Computability theory and applications

Abstract
Devlin’s theorem and the Rado graph theorem are both variants of Ramsey’s theorem, where a structure is added but more colors are allowed: Devlin’s theorem (respectively the Rado graph theorem) states if S is ℚ (respectively G, the Rado graph), then for any size of tuple n, there exists a number of colors l such that for any coloring of [S]^n into finitely many colors, there exists a subcopy of S on which the coloring takes at most l colors. Moreover, given n, the optimal l is specified.

The key combinatorial theorem used in both the proof of Devlin’s theorem and the Rado graph theorem is Milliken’s tree theorem. Milliken’s tree theorem is also a variant of Ramsey’s theorem, but this time for trees and strong subtrees: it states that given a coloring of the strong subtrees of height n of a tree T, there exists a strong subtree of height ω of T on which the coloring is constant.

In this talk, we review the links between those theorems, and present the recent results on the computable strength of Milliken’s tree theorem and its applications Devlin and the Rado graph theorem, obtained with Cholak, Dzhafarov, Monin and Patey.

Nicolò Zava (University of Udine) @ Lecture held in Elysium
Oct 6 @ 6:00 am – 8:00 am

Title: Towards a unifying approach to algebraic and coarse entropy
by Nicolò Zava (University of Udine) as part of Topological Groups

Lecture held in Elysium.

Abstract
In each situation, entropy associates to a self-morphism a value that estimates the chaos created by the map application. In particular, the algebraic entropy $h_{alg}$ can be computed for (continuous) endomorphisms of (topological) groups, while the coarse entropy $h_c$ is associated to bornologous self-maps of locally finite coarse spaces. Those two entropy notions can be compared because of the following observation. If $f$ is a (continuous) homomorphism of a (topological) group $G$, then $f$ becomes automatically bornologous provided that $G$ is equipped with the compact-group coarse structure. For an endomorphism $f$ of a discrete group, $h_{alg}(f)=h_c(f)$ if $f$ is surjective, while, in general, $h_{alg}(f) eq h_c(f)$. That difference occurs because in many cases, if $f$ is not surjective, then $h_c(f)=0$.

In the first part of the talk, after briefly recalling the large-scale geometry of topological groups, we define the coarse entropy and discuss its relationship with the algebraic entropy. The second part is dedicated to the introduction of the algebraic entropy of endomorphisms of $G$-sets (i.e., sets endowed with group actions). We show that it extends the usual algebraic entropy of group endomorphisms and we provide evidence that it can represent a useful modification and generalisation of the coarse entropy that overcome the non-surjectivity issue.

Oct
13
Tue
Break (University of Hawaiʻi) @ Lecture held in Elysium
Oct 13 @ 6:00 am – 8:00 am

Title: Topological Groups Seminar Two-Week Hiatus
by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.
Abstract: TBA

Leszek Kołodziejczyk (University of Warsaw)
Oct 13 @ 10:00 am – 11:00 am

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.

Chris Conidis (CUNY-College of Staten Island)
Oct 13 @ 3:00 pm – 4:00 pm

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.

Oct
16
Fri
Data Science seminar
Oct 16 @ 2:00 pm – 3:00 pm

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.

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.

Oct
20
Tue
Alexandra Soskova (Sofia University)
Oct 20 @ 4:00 am – 5:00 am

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.

Break (University of Hawaiʻi) @ Lecture held in Elysium
Oct 20 @ 6:00 am – 8:00 am

Title: Topological Groups Seminar Two-Week Hiatus
by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.
Abstract: TBA