Title: A conceptual overview of forcing

Abstract: Paul Cohen—who visited UH Mānoa in the 1990s—introduced the method of forcing to prove that the failure of the continuum hypothesis is consistent with ZFC, the standard base axioms for set theory. Since then it has become a cardinal tool within set theory, being the main method for proving independence results and even enjoys use in proving ZFC results. In this talk I will give an introduction to forcing, focusing on the big picture ideas.

This talk is a sequel to my previous talk and a prequel to my next talk.

Title: Forcing as a computational process

Abstract: In this talk we will consider computable structure theoretical aspects of forcing. Given an oracle for a countable model of set theory $M$, to what extent can we compute information about forcing extensions $M[G]$? The main theorem I will present gives a robustly affirmative answer in several senses.

* Given an oracle for the atomic diagram of a countable model of set theory $M$, then for any forcing notion $\mathbb P \in M$ we can compute an $M$-generic filter $G \subseteq \mathbb P$.

* From the $\Delta_0$ diagram for $M$ we can moreover compute the atomic diagram of the forcing extension $M[G]$, and indeed its $\Delta_0$ diagram.

* From the elementary for $M$ we can compute the elementary diagram of the forcing extension $M[G]$, and this goes level by level for the $\Sigma_n$ diagrams.

On the other hand, there is no functorial process for computing forcing extensions.

* If ZFC is consistent then there is no computable procedure (nor even a Borel procedure) which takes as input the elementary diagram for a countable model $M$ of ZFC and a partial order $\mathbb P \in M$ and returns a generic $G$ so that isomorphic copies of the same input model always result in the same corresponding isomorphic copy of $G$.

This talk is a sequel to my previous talk. The work in this talk is joint with Joel David Hamkins and Russell Miller.

Speaker: Yuriy Mileyko (UHM)

Title: Another look at recovering local homology from samples of stratified sets.

Abstract

Recovering homological features of spaces from samples has become one of

the central themes in topological data analysis, which has led to many

successful applications. Most of the results in this area focus on

global homological features, whose recovery predicates on imposing well

understood geometric conditions on the underlying space. Existing work

on recovering local homological information of a space from samples has

been much less abundant, since the required local geometric conditions on

the underlying space may vary from point to point and are not easily

integrated into a global condition, unless the space is a smooth

manifold. In this talk, we show that such global conditions for

recovering local homological information can be obtained for a fairly

large class of stratified sets.

Prof. Ross will speak about Nathanson’s “Generalized Egyptian Fractions.” Usual time and place (2:30, K314).

Abstract:

In a paper earlier this year Mel Nathanson generalized the notion of

“Egyptian Fraction” and extended some results of Sierpinski to sets of

these generalized Egyptian fractions. I’ll give short nonstandard proofs of

further generalizations of these results.

Speaker: Ricardo Teixeira (U. Houston-Victoria)

Title: Teaching Precalculus in Hawaii

Abstract: Most students enjoy learning about applications of math concepts. In

this talk, we will show how Precalculus concepts can motivate students

in the University of Hawai`i at Mānoa. Some interesting and recreational

use of certain concepts will be explored. At the end, we will cover

other ideas such as how to develop and maintain an effective culture of

assessment that may be used for future improvement and more.

Speaker: B. Kjos-Hanssen (joint work with Lei Liu)

Abstract:

Campeanu and Ho (2004) stated that it is “very difficult” to compute the number $m_n$ of maximally complex languages (in a finite automata sense) consisting of binary words of length $n$. We show that $m_n=O_{i,n}$, the number of functions from $[2^i]$ to $[2^{2^{n-i}}]$ whose range contains $[2^{2^{n-i}}-1]$, for the least $i$ for which $O_{i,n}>0$. Here, $[a]=${1,…,a}.