Exploring Advanced Topics in Applied Mathematics: Linear Algebra and Matrix Computations

As a Bachelor's student in Mathematics, you may be looking for a thesis topic that delves into the fascinating world of linear algebra, particularly focusing on advanced applications. This article explores various topics that can serve as the basis for your thesis, including the use of Goldie rings, the application of noncommutative computer algebra in linear algebra, and generalized theories of homology. Additionally, we will discuss the importance of computational methods in matrix computations and their relevance to big data and machine learning.

Goldie Rings in Linear Algebra

Goldie rings are a significant area of study in ring theory, which has profound implications for linear algebra. A Goldie ring is a right Goldie ring, which is a type of ring in which every right ideal is finitely generated. While not as extensively applied in the standard curriculum, understanding Goldie rings can provide a deeper insight into the structure of matrices and their applications.

Thesis Idea: Investigate the role of Goldie rings in the context of matrix theory and linear algebra. Specifically, explore how concepts such as finitely generated ideals in Goldie rings can be translated into the language of linear algebra problems, such as solving systems of linear equations or analyzing the properties of matrices.

Noncommutative Computer Algebra in Linear Algebra

Noncommutative computer algebra is a growing field that merges the power of algebraic methods with modern computational techniques. In the context of linear algebra, this means harnessing noncommutative algebra to solve problems in matrices that go beyond the realm of commutative algebra.

Thesis Idea: Examine the applications of noncommutative computer algebra in linear algebra. Focus on specific algorithms and their implementations, such as the computation of noncommutative determinants or the analysis of noncommutative polynomial ideals in matrix theory. Compare and contrast these methods with traditional linear algebra techniques and explore their advantages in terms of computational efficiency and accuracy.

Computational Methods in Matrix Computations

The computational methods for big data and machine learning rely heavily on matrix computations. Efficient and accurate algorithms are crucial for handling large datasets and performing complex operations such as PCA (Principal Component Analysis) and sparsity-based clustering.

Thesis Idea: Devote your research to the development and application of computational methods in matrix computations for big data and machine learning. Specifically, focus on stochastic methods inspired by PageRank and Spark sparse clustering. Analyze the performance of these methods and compare them with classical linear algebra techniques.

Generalized Theories of Homology

Homology is a fundamental concept in algebraic topology that has found applications in various fields, including linear algebra. Generalized theories of homology extend this concept to more complex algebraic structures, providing a richer framework for understanding the underlying geometry of matrices and linear transformations.

Thesis Idea: Explore the generalized theories of homology in the context of linear algebra. Investigate how homology can be used to analyze the structure of matrices and their transformations, and apply this theory to solve practical problems in big data and machine learning.

Conclusion

By selecting one or more of these thesis topics, you can contribute valuable insights to the field of applied mathematics and leverage your skills in linear algebra. Whether you focus on the intricacies of Goldie rings, the cutting-edge methods of noncommutative computer algebra, the computational challenges of big data, or the theoretical advancements in homology, your research can have a significant impact on the future of these fields.