The Higson corona and ultrafilters

• Grant number: 17/04355-9
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: May 08, 2017
• End date: August 07, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Artur Hideyuki Tomita
• Grantee: Yasser Fermán Ortiz Castillo
• Supervisor: Takamitsu Yamauchi
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Institution abroad: Ehime University, Japan
• Associated to the scholarship: 14/16955-2 - Outer measures by weak selections and hyperspaces., BP.PD

Abstract

This is a joint work proposed by Professor Yamauchi as a continuation of a collaboration what begin in 2012 with the paper "Insertion theorems for maps to linearly ordered topological spaces"' (Topology App. 188, (2015), pag. 74-81). The Higson corona and its compactification was introduced by Higson to analyzing the index theorem for non compact complete Riemannian manifolds. It is fundamental concept in coarse geometry in the sense that if two metric spaces are coarsely equivalent, then their Higson coronas are homeomorphic. The topic that we are interested in is the covering dimension of Higson coronas, especially in Dranishnikov's problem, the Higson corona of the real line and the problem of whether every Higson compactification is a Wallman type. Recently Professor Yamauchi found that the coarse disjoint union of a sequence of graphs with large girth may be a candidate of a counterexample of Dranishnikov's problem. So, the next step will be calculating the covering dimension of the Higson corona of the coarse disjoint union. There are several approaches to study Higson coronas but we think that by using the properties of ultrafilters it is possible to find some partial answers to this problem. I have experience in the study of ultrafilters and compactifications so our conjecture is that using the language and properties of ultrafilters we will clarify topological properties of Higson coronas described above. Also I will use the visit to work with Professor Professor D. Shakhmatov and Dr. A. Dorantes Aldama about selective pseudo compactness which is related with one point of my research project: Generalized pseudo compact properties by filters (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

ORTIZ-CASTILLO, YASSER F.; TOMITA, ARTUR HIDEYUKI; YAMAUCHI, TAKAMITSU. HIGSON COMPACTIFICATIONS OF WALLMAN TYPE. TSUKUBA JOURNAL OF MATHEMATICS, v. 42, n. 2, p. 233-250, DEC 2018. (14/16955-2, 16/23134-0, 17/04355-9, 16/26216-8)