Evaluations of Polynomials in Algebras: Images, Identities, Gradings and Connections with Geometry

• Grant number: 25/02457-5
• Support Opportunities: Scholarships abroad - Research
• Start date: August 11, 2025
• End date: August 10, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Thiago Castilho de Mello
• Grantee: Thiago Castilho de Mello
• Host Investigator: Lucio Centrone
• Host Institution: Instituto de Ciência e Tecnologia (ICT). Universidade Federal de São Paulo (UNIFESP). Campus São José dos Campos. São José dos Campos , SP, Brazil
• Institution abroad: Università degli Studi di Bari - Aldo Moro, Italy

Abstract

The project will be dedicated to various aspects related to evaluation of polynomials in algebras. The goal is to continue the research already conducted by the proponent while also expanding their field of work by incorporating new aspects into the study of polynomial identities. Below, we highlight two main approaches in this project.One approach will focus on polynomial images in algebras with different signatures, a topic in which the proponent already has considerable experience and aims to obtain new results. This will involve studying graded algebras and algebras with involution, searching for results related to the well-known Lvov-Kaplansky conjecture. We will also consider generalizations of the theorem of Shoda, Albert, and Muckenhoupt for broader classes of associative algebras.Another aspect to be addressed is the problem of embeddings and the relationship between algebras with polynomial identities and geometry. This is a topic that has recently begun to be developed, where the host researcher already has interesting results, and the proponent intends to specialize further.