The mathematics department will be hosting an internal research experience for undergraduates and early graduate students in Summer 2024. 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 $20/hr for 20 hours a week (that is, $400/week).

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

## Summer 2024 Projects

### Chains of Invariant Subspaces

The Google Code Jam problem ASeDatAb has an elegant solution if you can construct a chain of nested subspaces of the eight-dimensional vector space over the field of order two, with one subspace in each dimension, and all of them invariant under cyclic permutation. (Don’t worry if those words aren’t familiar to you!) Here’s what the chain of subspaces looks like if we make the problem easier by only using four bits instead of eight:

Dimension 0: {0000}

Dimension 1: {0000, 1111}

Dimension 2: {0000, 1111, 0101, 1010}

Dimension 3: {0000, 1111, 0101, 1010, 0011, 0110, 1100, 1001} (Can you see what these eight strings all have in common?)

Dimension 4: {all four-digit strings of 0s and 1s}

There’s at least one such chain of subspaces for the eight-dimensional space, but are there more? If so, how many? Are there any chains of invariant subspaces for the 16-dimensional space over the field of order two? The 32-dimensional space? 64? What about dimensions that aren’t powers of 2? What if we allow not just 0s and 1s but also 2s? (This would be the field of order three.) There are many interesting questions that can be investigated with minimal background knowledge required.

**Dates:** May 13—June 28

**Prerequisites:** Linear algebra would be helpful but is not necessary; anyone who is interested should apply.

**Mentor:** Justin Barhite

### Numerical computations of p-operator norms

The operator norm on complex valued m x n matrices comes by regarding them as linear transformations from C^n to C^m. In particular, if n=m and C^n is equipped with the usual p-norm (that is ||z||_p = (sum_j |z_j|^p)^(1/p) for p in [1,infinity) and ||z||_infinity = max_j{|z_j|}), then the p-operator norm of an n x n matrix A is defined as the the radius of the smallest ||-||_p-circle that contains the image of the unit ball of C^n under the transformation A : C^n \to C^n. If p is either 1 or infinity, this is easily found using the entries of A. For p = 2 this norm is equal to the largest singular value of A. However, for all the other values of p, the p-operator norm of a matrix is NP-hard to compute (there is no general formula for it in terms of the entries of the matrix A). Fortunately there are particular cases for which this norm is easy to compute and there are also computational algorithms that give good approximations.

The goal of this project is to test two conjectures that need computations of the p-operator norm for matrices described above. It is known that both conjectures are true when p = 2. Part of the project will consist in writing an elementary proof for the case p = 2 (i.e. a proof that doesn’t need C*-algebraic techniques). We will then focus on investigating what happens when p is not 2 by first conducting numerical experiments to find out whether a counterexample exists for 2x2 or 3x3 matrices. In particular, the simplex method might be useful for the case p = 1. For interested graduate students: There is a bigger picture behind proving/disproving these conjectures that will shed some light on the the C*-likeness for analogues of Hilbert modules on Lp spaces.

**Dates:** May 13th - July 6th.

**Prerequisites:** Linear Algebra (MATH 2130 or MATH 2135), Analysis I (MATH 3001), and familiarity with complex numbers. Previous experience coding in either MATLAB or Mathematica will be helpful but not required.

**Mentor:** Alonso Delfin

### Derivatives of Random Polynomials

Polynomials are fundamental mathematical objects that appear in all areas of math and science. A random polynomial is one whose coefficients are random variables. These are interesting in their own right and, in some ways, express the behavior of a "typical" polynomial. In this project, we will explore the relationship between the roots of a random polynomial and its derivative for various models of random polynomials. We will draw on tools from analysis, combinatorics, geometry and probability. Part of the project will involve numerical simulations, but no previous programming experience is required.

**Dates:** May 13 — June 28

**Prerequisites:** Some experience with probability and/or analysis

**Mentors:** Kyle Luh and Sean O’Rourke

### Graded Pseudo-traces for the universal Virasoro algebra

The pseudo-trace of a matrix is computed as the trace of an off-diagonal submatrix. Graded pseudo-traces are built by taking the generating function of a collection of pseudo-traces together with a modification arising from the nilpotent part of a conformal vector action. The graded pseudo-trace can be thought of as a complex valued function and this yields interesting number theoretical properties as well as connections with algebraic structures. The goal of this REU is to compute concrete examples of graded pseudo-traces for the universal Virasoro vertex algebra and study their number theoretical properties.

**Dates:** May 13 — June 28.

**Prerequisites:** Proficiency in linear algebra as well as enthusiasm to learn and work with others.

**Mentor**: Flor Orosz Hunziker

### A geometric equivalence of $\sigma$-delooping machines

One way to gain knowledge about a pointed topological space $X$ is to consider the space of all maps from a simpler space into that space. The loop space $\Omega X,$ the set of continuous pointed maps from the pointed circle $S^1$ to $X$ equipped with the compact-open topology, is an example with many applications. A natural question to ask is, given a space $X$, is there a space $Y$ such that $\Omega Y \simeq X$?

In this project, we will study two delooping constructions that are known to produce equivalent spaces $Y$. Specifically, we will work towards producing an explicit a geometric proof of the equivalence of the monoidal twisted bar construction and the two sided monadic bar construction. Given its geometric motivation, much of this project can be readily visualized. For instance, points in loop spaces can be concretely represented as paths in $X$ beginning and ending at the same point. Additionally, both the monoidal and monadic bar constructions can be modeled using rooted trees so finding a proof of equivalence will amount to writing a homotopy between the two tree models.

This project would be a good fit for students looking for a concrete entry point to working with a number of foundational algebraic topology concepts: loop spaces, point set level models, and homotopies. For more advanced students, there is also the possibility to learn and work with operads, classifying spaces, and a bit of category theory.

**Dates:** May 13 - June 21

**Prerequisites:** MATH 2001 or equivalent

**Mentor:** Sarah Petersen

### Topological Data Analysis, Computational Algebraic Geometry and Applications in the Sciences

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 corresponding methods from topological data analysis (TDA) and computational algebraic geometry. We will then apply these methods to examine data sets coming from the sciences. In particular it is planned to study energy landscapes of molecules as they appear in physics and chemistry and see in how far TDA and computational algebraic geometry can help to better understand them. During the REU a review of simplicial complexes and their homology will be given, afterwards we will study methods from computational algebraic geometry. 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, the topology toolkit etc...). In the end we hopefully can apply all this to some more realistic data scenarios coming from physics and chemistry.

**Dates:** May 9 — June 27

**Prerequisites:** knowledge of linear algebra and willingness to learn some abstract math and do computer assisted computations.

**Mentor:** Markus Pflaum

### Hopf structures on matchings and their generalizations

A matching between two sets A and B is a set of pairings between elements of A and elements of B such that every element is in at most one pair. These gadgets appear in all kinds of contexts: non-attacking rooks on chessboards, permutations, a family of integral polytopes, etc.

We can construct an infinite dimensional vector space with basis given by the set of all matchings on A={1,2,..., n} and B={n+1,..., 2n} for all n. We obtain a Hopf structure by giving rules for combining two matchings and breaking a matching apart in a compatible way. In general, there are many ways of accomplishing this, but our rules will be inspired by the representation theory of a family of finite groups. The goal is to understand these Hopf algebras combinatorially and algebraically, and how they relate to their surrounding structures.

**Dates:** May 13 — June 28

**Prerequisites:** Minimally, a strong comfort with linear algebra. Preference will be given to students who have taken Math 3140 with bonus points for Math 4140.

**Mentor:** Nat Thiem

**How do I apply?**

Applications for the above Summer 2024 REU are live and will be due March 4, 2024. Offers to students will be made by late March. If you have any questions, please contact Nat Thiem.

Women and other underrepresented groups are encouraged to apply. This program is not funded with federal money, so all students are welcome to apply regardless of eligibility for federal support or immigration status.

To apply, please fill out the form https://forms.gle/BkrFAKS9ocn1dvPN7 (MUST be signed in to Google with your CU identikey). This application includes

- A .pdf copy of your (un)official transcript (note that you can't have graduated before Summer 2024)
- 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)

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.