The mathematics department will be hosting an internal research experience for undergraduates and early graduate students in Summer 2022. If your idea of fun is thinking about mathematics, then this may be your chance for getting paid for having fun.

What's the job?

- You will be part of a group of researchers working on math problems, guided by the official mentor.
- You will be expected to be around most of the time, attend official functions, and be expected to work about 20 hours per week.
- Pay is $15/hr for 20 hours a week (that is, $300/week).

## Current Projects

*Offers to undergraduate students will be made by late March*

### (1) Topological Data Analysis

Mentor: Markus Pflaum

Project Description:

Two of the main challenges of modern data analysis are how to separate noise and outliers from significant parts of data samples and the high dimensionality of data. One of the most recent and powerful methods to attack these problems comes from topology. It is the goal of this REU to study methods from topological data analysis (TDA) and apply them to examine data sets coming from the sciences. In particular it is planned to consider energy landscape as they appear in chemistry and see in how far TDA can help to better understand them.

During the REU an review of simplicial complexes and their homology will be given, afterwards we will study the main tool of topological data analysis, persistent homology. We will also do some hands-on experimental mathematics using software tools for TDA and computational homology. Part of the REU will also be to learn the software/computational tools we need (from MATLAB, python, topology toolkit etc...). In the end we hopefully can apply all this to some more realistic data scenarios coming from chemistry.

Proposed Dates: May 9 – July 1 (8 weeks)

Prerequisites: knowledge of linear algebra, fundamentals of simplicial complexes and willingness to learn some abstract math and do computer assisted computations.

### (2) Topology and lattice models

Mentors: Agnès Beaudry and Juan Moreno

Project Description:

In this project, we will explore the literature for examples of lattice models and create a nice repository of examples explained for the more topology inclined mathematician. We may also try to look at common features the models have or look at simplified situation to see if we can learn something about topological classifications.

Proposed Dates: May 16 – June 24 (6 weeks)

Prerequisites: 2135 (Linear algebra for math majors)

### (3) Roots and critical points of random polynomials

Mentor: Sean O'Rourke

Project Description:

A random polynomial is a polynomial whose coefficients are random variables. One goal of studying random polynomials is to understand the "typical" (or "average") behavior of a polynomial. For example, the roots of a random polynomial describe the typical behavior of roots of all polynomials.

In this project, we will study the roots and critical points of random polynomials. We will consider questions such as: How do the roots of a polynomial affect the behavior of its critical points? What does a typical critical point look like? We will use tools from analysis and probability theory. Part of the project will involve numerical simulations using software such as Mathematica, but no prior programming knowledge is required.

Proposed Dates: May 9 -- June 17 (6 weeks)

Prerequisites: Some experience with probability and/or analysis.

### (4) Numerical analysis for a PDE model on Chemotaxis

Mentor: Padi Fuster

Project Description:

From the Greek, taxis meaning movement, chemotaxis is the movement of an organism in response to a chemical stimulus. The movement of sperm towards an egg during fertilization or of bacteria attracted towards a source of food are examples of chemotaxis. This phenomenon is usually modeled by a type of PDE systems called reaction-diffusion equations, which also have applications in epidemiology, ecology, and the social sciences.

In this REU, we will study a particular PDE model for chemotaxis with logarithmic sensitivity and logistic growth and understand the qualitative behavior of its solution. We will then perform numerical simulations with MatLab to learn more about how its non-dimensional parameters affect the behavior of solutions. By the end of this project, students should have a good grasp on both the basics on qualitative analysis of PDEs and how to perform numerical simulations to obtain information about the PDE.

Proposed Dates: Probably on Zoom: May 9 - June 17 (6 weeks)

Prerequisites: ODEs, Linear Algebra.

Preferred: PDE, Real Analysis, some knowledge of MatLab.

### (5) Perturbation theory in differential equations

Mentor: Divya Vernerey

Project Description:

Besides using computers to numerically solve differential equations that are nonlinear, inhomogeneous, or multidimensional, one can also use asymptotic and perturbation methods which are analytical in nature. By doing this, one can derive an understanding of the physics of the problem. In particular, we can use matched asymptotic expansion to analyze problems with layers.

For example, for supersonic air flow over a wedge, the high-speed flow results in a shock layer in front of the wedge across which the pressure undergoes a rapid transition. Because of its position in the flow, the shock is an example of an interior layer. There are also boundary layers present. These can be seen near the surface of the wedge, and they are thin regions where the flow drops rapidly to zero (which is the speed of the wedge). We will study these phenomena.

Proposed Dates: May 9 – June 24 (7 weeks)

Prerequisites: ODEs

## How do I apply?

Applications for the above Summer 2022 REU projects will open in February 2022. If you have any questions, please contact Divya E. Vernerey.

A completed application includes:

- This completed Google form (MUST be signed in to Google with your CU identikey)
- A .pdf copy of your (un)official transcript (note that you can't have graduated before Summer 2022)
- A ranking of the projects you'd be interested in participating in
- A statement describing why you would like to do mathematics research, and if applicable why your favorite project is especially compelling to you. Other things you might add to your statement include:
- If your experience does not meet the prerequisites of a project, but you nevertheless feel prepared, then please explain why
- If you unsuccessfully applied to this REU in previous years, please let us know (in general this will positively affect your application)

**Applications are due Monday, March 7th**

The University of Colorado Boulder is committed to building a culturally diverse community of faculty, staff, and students dedicated to contributing to an inclusive campus environment. We are an Equal Opportunity employer. Human diversity includes, but is not limited to ethnicity, race, gender, age, socio-economic status, sexual orientation, religion, disability, political viewpoints, veteran status, gender identity or expression, and health status.