Homological conjectures, commutative algebra and their connections with geometry

• Grant number: 25/20830-5
• Support Opportunities: Research Grants - Visiting Researcher Grant - Brazil
• Start date: July 30, 2026
• End date: July 29, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Josnei Antonio Novacoski
• Grantee: Josnei Antonio Novacoski
• Visiting researcher: Thiago Henrique de Freitas
• Visiting researcher institution: Campus Guarapuava/Utfprgpv/Utfpr , Brazil
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• City of the host institution: São Carlos

Abstract

We present here a comprehensive proposal for the activities of the applicant as a visiting researcher within the Graduate Program in Mathematics at UFSCar - São Carlos. Our objective is to simultaneously develop six research projects during the period of the proposed visit. These projects are organized under two overarching "umbrella projects," whose central pillars are commutative algebra and algebraic geometry.The first umbrella project, of an algebraic nature, encompasses three subprojects aimed at investigating the structure and relationships between the theory of fiber product rings and the theory of valuations of rings-both areas in which the applicant and the host supervisor possess recognized expertise. Furthermore, we intend to explore some of the most prominent homological conjectures: the Auslander-Reiten Conjecture, the Huneke-Wiegand Conjecture, and the Total Rank Conjecture, focusing on fiber product rings, quasi-fiber product rings, and connected sums. These long-standing open problems have received considerable attention over the past three decades. With a geometric perspective, now directed toward Algebraic Geometry and Singularity Theory, we propose our second umbrella project, which also comprises three subprojects. The first seeks to investigate the behavior of (quasi-)gluings and connected sums of complex analytic spaces and their invariants. Although at first sight this line of research may appear distant from the previous one, from a geometric standpoint (quasi-)gluings and connected sums of analytic spaces are deeply connected to the algebraic notion of fiber product rings. Thus, mastery of the algebraic tools developed in the first umbrella project will be crucial for achieving significant progress here. In addition, we aim to study the celebrated Betti numbers of gluings, quasi-gluings, and connected sums of complex analytic spaces, exploring the insights these invariants may provide regarding the geometric structure of the corresponding gluing. Finally, we propose a project that bridges the algebraic and geometric approaches developed in the previous studies. This project will focus on the investigation of Kähler differentials and derivations, with the goal of addressing some of the central open problems in valuation theory, as well as major geometric conjectures related to these topics, thereby establishing meaningful connections with the themes addressed in the preceding projects. (AU)