Undergraduate Research Opportunities
Current undergraduate research opportunities in the Department of Mathematics and Statistical Science are described below. For further information, contact the listed faculty member.
Machine Learning in Engineering and Biology
Are you interested in the limitless possibilities of Artificial Intelligence and want to be at the forefront of cutting-edge technology? Are you eager to develop or apply mathematical knowledge to solve real-world problems? If so, we invite you to join our exciting machine learning projects such as regularized neural networks, modeling bacterial resistnace, or developing topological data algorithms. In addition to the research experience during this project, we offer economic support to develop research activities from our NSF project.
Strong scientific ambition, programming skills in Python or willingness to learn them, junior or senior year. Underrepresented minorities in STEM are very welcome to apply.
Interested students should contact Dr. Vargas, even if they do not have the listed prerequisites or if they are interested in other applied math topics and would like his guidance.
This project is in discrete mathematics, particularly in graph theory. The topic is about the existence of substructures of a graph. Given a graph G = (V, E) of order n and a subset W of V , let every vertex of W have degree at least 2n/3. Conditions on vertices of V - W are unknown. What will be the impact of W on the structure of G? For example, are there disjoint cycles of G passing through prescribed numbers of vertices of W? When W = V ,this topic has been investigated extensively. When W is a proper subset of V, i.e., W is a subset of (or is included in) V and W is not equal to V , this topic is introduced in the following publication:
H. Wang, Partial Degree Conditions and Cycle Coverings, Journal of Graph Theory, 78(2015), 295-304.
Substructures other than cycles can be considered as research projects, too. Some weak conditions can be imposed on the vertices which are not in W for the research purpose.
Students should have strong interests in discrete mathematics. It is desirable that students have some knowledge of discrete mathematics, for instance, having taken Math 176 (Discrete Mathematics) or other combinatorial courses. It is also desirable that students have taken Calculus I, Calculus II, and Linear Algebra.
Applied Linear Algebra: Frame Theory
In linear algebra, one of the basic ideas is to express a given object in some space in terms of elements in a representative set like a basis. When dealing with different kinds of data sets, the structure of this representative set becomes crucial for efficient storage and transmission of data. Frames are representative sets like bases but are redundant. The redundancy allows more flexibility and freedom of choice. Frames have now become standard tools in signal processing due to their resilience to noise and transmission losses.
- Study and compare the effect of different kinds of frames in signal (image) reconstruction. Certain frames turn out to be better than others. Determine properties or characteristics of frames that perform better.
- Frame design: In some situations, one might seek a “sparse” representation of a signal. In other situations, one might have to use a subset of a frame to approximate a signal or to deal with loss. Given the goal, properties like “equiangularity”, “equal-norm”, or “tightness” might be desirable. Consequently, one wishes to construct frames having some “desirable” properties.
- Frame transformations: Starting with a frame, investigate the action of certain “transformations” on the given frame. Which properties of the starting frame are preserved? Determine transformations that can add some property missing in the original frame.
Deep learning is a class of machine learning algorithms that use convolutions on a cascade of many layers of nonlinear processing units to extract features from data. It is the current state-of-the-art approach to achieve artificial intelligence, and has been successfully applied to a vast variety of difficult tasks. This project aims to get a better understanding of the role of convolutions in the deep learning architecture by testing various mathematically-guided designs.
Multivariable Calculus (275) and Python programming skills.