Research

Higher Order Finite Elements and Optimal Assembly Procedures

Description: Research Project

Advisor: Mark Ainsworth, PhD

Overview: This project focuses on implementing higher-order Raviart–Thomas (RT) finite elements using the Bernstein–Bézier basis to achieve optimal computational complexity. The Bernstein–Bézier basis exhibits several properties, making it well-suited for high-order FEM applications. The approach ensures optimal assembly and solution complexity while maintaining accuracy in divergence-conforming problems. Additionally, post-processing techniques are incorporated.

Keywords: Finite Element Analysis, Raviart-Thomas (RT), Bernstein-Bezier Techniques 

Computable Finite Factorization Domains

Description: M.Sc. Thesis

Advisor: Victor Ocasio González, Ph.D

Overview: This work classifies Strongly Computable Strong Finite Factorization Domains by showing the existence of a computable norm that possess certain properties as being able to solve norm-form equations computably and allow us to extend the notion of strongly computability to Computable Finite Factorization Domains in general. We use the  technique of constructions by priority requirements and examples using quadratic extensions are provided using the algorithmic properties of the Pell Equation.

Keywords: Computable Structures, Quadratic Extensions, Computable Norms, Finite Factorization, Norm-Form Equations.

f-Factorizations

Description: Research Supported by (PR-LSAMP)

Fall 2018-Spring 2020

Advisor: Reyes Ortiz Albino, Ph.D

Overview: This work study factorizations when a new relation is imposed to the multiplication. This theory was formalized by Dan Anderson and widely studied when the relation was symmetric. We, on the other hand, replace the relation by a function over the integral domain and provided the foundations to study this kind of factorizations. Furthermore, when the function is a monomial, we classified irreducible and primes elements in this context.

Keywords: Factorizations, Integral Domains, UFD, irreducible, primes

Classification of Bi-Inverstible Connections Comming from Subfactor Classifications

Description: Research Experience for Undergraduates

Summer 2019

Advisor: David Penneys, Ph.D.

Overview: The small index subfactor classification led to interesting examples of ‘exotic’ quantum symmetries. However, many results remain mysterious in the language of tensor categories. We used computational and combinatorial methods to construct and classify bimodule categories over Temperley-Lieb-Jones categories using a non-unitary version of Ocneanu's theory of connections on square-partite graphs in order to help bridge this gap.

Keywords: Bimodule categories, Temperley-Lieb-Jones, Square-Partite Graphs

Peer Reviewed Articles

• Geraldo Soto-Rosa, Victor Ocasio-González. A Characterization of Strongly Computable Finite Factorization Domains. Arch. Math. Logic (2024). https://doi.org/10.1007/s00153-024-00941-6

Poster Sessions and Talks

[Feb 25 2023] Inter-Institutional Mathematics Research Seminars (SIDIM)

• Soto-Rosa,G and Ocasio-Gonzalez, V. Computable Strong Finite Factorization Domains. Concurrent Talk, Mayagüez, PR. 2022  [Presentation (PDF) ]

[2019 - 2020] Inter-Institutional Mathematics Research Seminars (SIDIM)

• Soto-Rosa, G. and Ortíz-Albino, R. On the Study of f-factorizations over Unique Factorization Domains with f(x)=x^r. Poster session presented, Cayey, PR. 2020.

• Soto-Rosa, G. and Ortíz-Albino, R. Introduction to f-factorizations. Poster session presented, Humacao, PR. 2019.

[Aug 2019] Young Mathematician Conference (YMC): Ohio, United States.

• Soto-Rosa, G. Badillo-Rosario, M. Ferrer-Suarez,G. On the Classification of Bi-Invertible Connections Arising From Small Index Subfactor Classification. Poster session presented, Columbus, OH. 2019.

[May 2019] Puerto Rico Interdisciplinary Scientific Meeting and JTM

• Soto-Rosa, G. and Ortíz-Albino, R. Introduction to f-factorizations. Concurrent Talk, Mayagüez, PR. 2019.