Date on Master's Thesis/Doctoral Dissertation

8-2019

Document Type

Doctoral Dissertation

Degree Name

Ph. D.

Department

Mathematics

Degree Program

Applied and Industrial Mathematics, PhD

Committee Chair

Kulosman, Hamid

Committee Co-Chair (if applicable)

Biro, Csaba

Committee Member

Biro, Csaba

Committee Member

Li, Jinjia

Committee Member

Riedel, Thomas

Committee Member

Hindy, Nicholas

Author's Keywords

commutative algebra; neural code; partial code; monomial morphisms

Abstract

The neural rings and ideals as algebraic tools for analyzing the intrinsic structure of neural codes were introduced by C. Curto, V. Itskov, A. Veliz-Cuba, and N. Youngs in 2013. Since then they have been investigated in several papers, including the 2017 paper by S. G\"unt\"urk\"un, J. Jeffries, and J. Sun, in which the notion of polarization of neural ideals was introduced. We extend their ideas by introducing the polarization of motifs and neural codes, and show that these notions have very nice properties which allow the studying of the intrinsic structure of neural codes of length $n$ via the square-free monomial ideals in $2n$ variables. As a result, we can obtain minimal prime ideals in $2n$ variables which do not come from the polarization of any motifs of length $n$. For this reason, we introduce the notions for a partial code, including partial motifs and inactive neurons. With these notions, we are able to relate those non-polar primes back to the original neural code. Additionally, we reformulate an existing theorem and provide a shorter, simpler proof. We also give intrinsic characterizations of neural rings and the homomorphisms between them. We characterize monomial code maps as the composition of basic monomial code maps. This work is based on two theorems, introduced by C. Curto and N. Youngs in 2015, and the notions of a trunk and a monomial map between two neural codes, introduced by R. A. Jeffs in 2018.

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