### Research Projects

#### Introduction

Invasive species are alien species capable of propagating themselves in non-native environments. Once they are established in such non-native environments, they become very difficult to eradicate and costly to control. Examples of invasive species include the Eurasian Watermilfoil, Brook Trout fish, Cane toad, zebra mussel, Lanternfly etc. Research done by Seebens et al. [1] in 2017 revealed that, the rate of introduction of alien species continues to rise unchecked. Often, control methods are dependent on chemical treatments which are detrimental to the ecosystem. There are also biocontrol methods which involves the introduction of natural enemies to predate and compete with the invasive species. Other biocontrol strategies involve disrupting the mating system of these invasive species. An example of such strategy is the Trojan Y Chromosome (TYC) eradication strategy currently being implemented, in Idaho State, skews the sex ratio of these species towards males. Therefore, invasive species control is of essence in ecology and viable strategies to curb their spread will have economic and societal value.

#### The Project

This project will begin with identifying and understanding how some invasive species spread uncontrollably and what control methods are currently being implemented to curb their invasion. Upon gaining understanding, we will study some already existing models for invasive species control. Later, we will focus on developing mathematical models with optimal bio-control strategies to achieve faster eradication of the invasive species.

The project also involves learning how to carry out a dynamical system analysis of our developed model and numerical simulations to validate each theoretical result obtained. In all, students will be introduced to the fundamentals of population modeling, analysis and writing of simple algorithms for the dynamics of a single specie population and later, an extension to multi-level food chains. We will also extend the model to include spatial effects where we can study the possible occurrence of Turing patterns. These patterns provide useful information to stakeholders in adopting efficient management strategies for invasive species control. Above all, the direction of this project is flexible and can take a detour when interesting and great ideas pop up.

#### Background

Interested students will be introduced to the relevant concepts in Differential Equations and Linear Algebra. Student should at least have taken Calculus I. Programming skills in Matlab/Python/Mathematica/R is desirable but not required. Students will learn on the job.

#### References

[1] Seebens, Hanno, et al. “No saturation in the accumulation of alien species worldwide.” *Nature communications* 8.1 (2017): 1-9.

#### How to Apply

#### Introduction

Wind farms produce renewable energy, but there is still a cost to the environment. Birds and bats are known to collide with operating wind turbines, resulting in their injury and death. However, given the finite nature of many other sources of energy, and the associated negative impacts on the environment, there is a need to balance the development of renewable energy sources against the potential cost to wildlife. Of particular concern are eagle collisions with wind turbines, as these birds are protected by US law and have proven to be particularly vulnerable to collisions. In response to this concern, the US Fish and Wildlife Service has created the Eagle Permit Rule, which is intended to help balance eagle conservation and wind energy development. We will use statistical models to aid the US Fish and Wildlife Service in the further development and implementation of the Eagle Permit Rule.

#### The Project

The project will mainly focus on developing statistical models to understand the distribution of golden and bald eagles across the continental United States. Not only do we want to know where the birds can be found, but also how much time they spend there. Knowledge of how much time the eagles spend in a particular part of the landscape can help us understand the birds’ exposure to any wind farms built in that location. We will investigate whether data sets reporting the relative abundance of eagles can be combined with data sets on eagle use of an area to create maps of eagle exposure that can inform the Eagle Permit Rule. Relative abundance is a measure of how common (or rare) a species is in relation to other species in a defined area. Relative abundance can be used to give a broad picture of where eagles can be found on the landscape, but doesn’t provide enough detail to inform the Eagle Permit Rule directly. The project will begin by introducing students to the basics of spatial modeling, and the application of these methods to the two data sets individually. We will then build up to the use of more complex methods, such as species distribution modelling. The ultimate aim is to combine the two data sets in one analysis to improve our ability to map eagle exposure to wind farms on the landscape. We will work directly with individuals in the US Fish and Wildlife Service to build the model and implement it within the Eagle Permit Rule.

#### Background

Interested students will be introduced to the relevant concepts of spatial modelling. Students should have at least two semesters of statistics. Programming in R is desired but not required, as students will be provided with training as part of the project.

#### How to Apply

#### Introduction

The theory of integer partitions is an important area of number theory that has overlap with many other areas of mathematics and physics, in particular with representation theory and conformal field theory. Briefly, given a positive integer n, a *partition* of n is a way of writing n as a sum of smaller positive integers (called *parts*) where order doesn’t matter. For example, a partition of 8 is 4+2+1+1. Many interesting identities for integer partitions exist where certain restrictions are placed on the parts. One of the earliest such results, proven by Euler using generating functions (i.e. power series), is that the number of partitions of n into odd parts is equal to the number of partitions of n into distinct parts. Since Euler’s time, the use of generating functions has become fundamental in the study of many integer partition identities, including identities such as the Rogers-Ramanujan identities, Göllnitz-Gordon identities, and many others. Interestingly, the generating functions used in the study of integer partition identities also arise in representation theory as the graded-dimensions of modules for various algebraic objects. In particular, proofs of these identities often suggest ideas about the underlying structure of these modules.

#### The Project

The aim of this project is to extend the ideas in Andrews and Baxter’s “motivated proof” of the Rogers-Ramanujan identities. The Rogers-Ramanujan identities state the following:

- The number of partitions of n into parts with difference of 2 or more = the number of partitions of n into parts = 1,4 mod 5
- The number of partitions of n into parts > 2 with difference of 2 or more = the number of partitions of n into parts = 2, 3 mod 5

The Rogers-Ramanujan identities and their extensions and generalizations are often written in the form of “infinite sum = infinite product,” where, in this case, the sum represents the difference 2 conditions and the product represents the mod 5 conditions. It is apparent from the left-hand side of each identity (the “sum-sides”) that there are more partitions in 1. than in 2. Leon Ehrenpreis posed the following question: Can one deduce, from the right-hand sides of each identity and without knowledge of the identities, that there are more partitions in 1. than in 2.? The answer is not at all obvious. Andrews and Baxter answered this question in the affirmative and in doing so discovered a new proof of the Rogers-Ramanujan identities.

In this project, we find aim to find a new proof, in the sense of Andrews and Baxter, of a set of partition identities related to the Göllnitz-Gordon-Andrews identities. In particular, starting with only the product side of the identities we will consider, we extend the ideas developed by Andrews, Baxter, Lepowsky, Zhu, and many others, to the set of partition identities which we will consider. Interestingly, this project will likely involve all the machinery developed in earlier works on motivated proofs of partition identities.

#### Background

Interested students should have a background manipulating power series (for example, as is typically covered in Calculus 2) and in the basics of proof-writing, in particular working with modular arithmetic. Knowledge of basic combinatorics will be helpful but is not strictly necessary.

#### How to Apply

#### Introduction

Integer sequences often come up in number theory. A natural question and an active research area is to study the divisibility properties of such sequences. We can do so through the framework of the p-adic valuations. For a given prime p, the p-adic valuation of a non-zero integer n is the highest power of p that divides n. For example, for p=2 the 2-adic valuation of n=40 would be 3 because 40=2^3*5. Now consider the sequences of valuations generated by polynomials with integer coefficients by successively computing the p-adic valuation of f(n).

We can visualize the sequence above by using p-nary tree diagrams. In order to construct the tree, begin with the top node. Let n be an arbitrary natural number. If the valuation is constant for every natural n in that node, then we stop. The result is the simplest of tree diagrams: a *dot tree*. If the valuation depends on n, then the node splits into p branches. Now look at each node on the next level and try to determine its valuation. Is the p-adic valuation constant for each n in that node (i.e. the node is terminating), or does it vary with n (i.e. the node splits)? Proceed… The constructed trees could be finite or infinite.

For example, for p=2 the polynomial f(n)=15n^2+1142n+25559 has a 2-adic tree with an infinite branch, while the tree of g(n)=n^2+27 is a finite two-level tree. More than that, it is a short proof that if f(n)=an^2+bn+c where both a and b are congruent to 1 mod 3 and c is congruent to 2 mod 3, then its 3-adic valuation tree is a dot tree with one node.

#### The Project

The goal of the project is to investigate the properties of the p-adic valuation trees

depending on the coefficients of the polynomial. The problem has deep connections to the roots of the polynomial: it is well-known that the tree has an infinite branch if and only if the polynomial has a root in the field of p-adic integers. Hence, first we will begin by acquiring background on the p-adic numbers. We will study the conditions for the coefficients of a quadratic polynomial that allow to predict whether the tree is finite or infinite. Then we will dive into investigating the properties of the trees. Can we predict the valuations at the terminating nodes, especially for infinite trees? Are there any patterns or a symmetry in the tree? How can we define the condition of symmetry in a meaningful way? For an infinite tree, can we predict whether the tree will turn left or right? And finally, what if the tree is generated by other type of function, not necessarily a polynomial?This project is a continuation of the work done at REU at Ursinus during Summer 2021. It originated from the research completed in https://cs.uwaterloo.ca/journals/JIS/VOL24/Trulen/trulen5.pdf

#### Background

Interested students should have a background in proof-writing and Calculus 2 (we will encounter series). To determine the valuations, we will make frequent use of properties of primes and polynomials. Hence, knowledge of basic number theory will be helpful. But students who have never taken number theory are most welcome to apply since they can learn the necessary concepts on the job.

#### How to Apply

Please see the “How to apply” page

*This REU program is supported by the National Science Foundation (NSF), grant number 1851948.*