# Undergraduate Research Opportunities

Current undergraduate research opportunities in the Mathematics Department are described below. For further information, contact the listed faculty member.

## Mathematical Biology

### Project Description

Undergraduate projects could involve studying various aspects of evolution, ecology, or disease transmission in populations where interactions are restricted by spatial location or some form of contact network (perhaps governed by behavior, social status, etc.).

### Project Prerequisites

Some familiarity with rates of change (calculus or differential equations) and experience with a coding language would be helpful. Some knowledge of biology is also useful. Interest and the willingness to work hard are the most important prerequisites.

### Faculty Contact

Steve Krone, krone@uidaho.edu, 208-885-6317.

## Discrete Mathematics

### Project Description

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 2*n*/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.

### Project Prerequisites

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.

### Faculty Contact

Hong Wang, hwang@uidaho.edu, 208-885-6550.

## 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.

### Project Ideas

- 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.

### Faculty Contact

Somantika Datta, sdatta@uidaho.edu, 208-885-6692.

## Deep Learning

### Project Description

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.

### Project Prerequisites

Multivariable Calculus (275) and Python programming skills.

### Faculty Contact

Frank Gao, fuchang@uidaho.edu, 208-885-5274.## Mathematics of Biomedical Imaging

Topics include photo-acoustic tomography, SAR (Synthetic Aperture Radar Imaging), SONAR (Sound Navigation and Ranging), etc.

### Project Prerequisites

Ordinary Differential Equations, Partial Differential Equations, Numerical Methods/Numerical Linear Algebra, and Programming Skills.

*Please note: Interested students should contact Dr. Nguyen, even if they do not have the listed prerequisites or if they are interested in other applied math topics and would like to have his guidance.

### Faculty Contact

Linh Nguyen, lnguyen@uidaho.edu, 208-885-6629.

## Data Sciences

Topics include machine learning (especially neural network), data science for economics and technology.

### Project Prerequisites

Probability, Statistics, Numerical Linear Algebra, and Programming Skills.

*Please note: Interested students should contact Dr. Nguyen, even if they do not have the listed prerequisites or if they are interested in other applied math topics and would like to have his guidance.