In this document we solve some local connectivity problems in matrix representations of the form C(T^N) -> M_n and C(T^N) -> M_n <- C([-1, 1]^N) using the so called toroidal matrix links, which can be interpreted as normal contractive matrix analogies of free homotopies in algebraic topology. In order to deal with the locality constraints, we have combined some techniques introduced in this document with several versions of the Basic Homotopy Lemma L.2.3.2, T.2.3.1 and C.2.3.1 obtained initially by Bratteli, Elliot, Evans and Kishimoto in  and generalized by Lin in  and . We have also implemented some techniques from matrix geometry, combinatorial optimization and noncommutative topology developed by Loring [24, 27], Shulman , Bhatia , Chu , Brockett , Choi [7, 6], Effros , Exel , Eilers , Elsner , Pryde [31, 30], McIntosh  and Ricker .
Level of Degree
Mathematics & Statistics
Loring, Terry A.
First Committee Member (Chair)
Second Committee Member
Third Committee Member
Matrix homotopy, relative lifting problems, matrix representation, noncommutative semialgebraic sets, K-theory, amenable C*-algebra, joint spectrum.
Vides Romero, Fredy Antonio. "Toroidal Matrix Links: Local Matrix Homotopies and Soft Tori." (2016). https://digitalrepository.unm.edu/math_etds/51