Summer Research Undergraduate

The mathematics department will be hosting an internal research experience for undergraduates and early graduate students in Summer 2025. 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 $25/hr for 20 hours a week (that is, $500/week).

Offers to undergraduate students will be made by late March

Summer 2025 Projects

Special solutions to the Euler & the Navier-Stokes equations

The Euler and the Navier-Stokes equations are the fundamental equations of fluid mechanics.  Each equation is a system for the vector field, the velocity of the fluid, and for the scalar, which is the pressure.  Both systems admit special solutions.  In this project, we investigate if the special solutions can still be obtained when the domain is changed.  In both cases, we can derive a system of ODEs and investigate the solutions using analytical and/or numerical methods.

Prerequisites: All students interested in Analysis and PDE are encouraged to apply: Math 3430 and 3001 are required prerequisites, and Math 4470 is highly recommended.  Bonus points for Math 4001.

Mentors: Siddhant Agrawal and Magdalena Czubak

Dates: May 12 – June 20 

Neural nets and polymorphic learning

This project is a continuation of the discrete neural nets projects I've run at REUs at the University of Rochester in 2021 and 2023. The initial idea was introduced in this video, where I propose studying neural nets whose activation functions can only take on a finite set of values. This prevents us from using established techniques like gradient descent but opens up new avenues not available for continuous functions. Unlike in discretization methods (which start with a continuous model and truncate to make it discrete), we start with discrete activation functions, which are chosen to be polymorphisms of a relational structure encoding some information about our learning task. This guarantees that we will not overfit our training data, but makes training more difficult.

Initial coding for the project was done in part by Rachel Dennis in her 2023 senior thesis, and the current state of the code can be found here. I wrote a paper describing the basic setup, and you can find a talk I gave about a category-theoretic treatment of the same ideas here. A paper on the categorical formulation is forthcoming.

In this year's project, we will explore how effectively this technique can be applied to performing classification and generation of images via diffusion. Whether you're more interested in the relevant algebra, the machine learning theory, or building on the existing code, there's something for everyone in this polymorphic learning project.

Prerequisites: Ideally 3140 and some facility with Python. Math 2001 is a minimum and 2135 is probably sufficient.

Mentor: Charlotte Aten

Dates: June 16 — July 25

Cancer Detection via Computational Optimization Applied to Electrical Impedance Tomography (EIT)

EIT is a rapidly developing non-invasive imaging technique gaining popularity in just a few decades in various medical applications involving screening for cancer detection. Different parts of the human body react differently to the electrical current that appears due to the voltages applied by placing electrodes. A very well-known fact is that the electrical properties, e.g., electrical conductivity or permittivity, of different tissues change if tissue status changes from healthy to cancer-affected. It allows the development of simple techniques, aiming to produce images of the biological tissues by interpreting their response to applied voltages, or injected currents, of single or multiple frequencies. This imaging approach has become very popular due to almost no harm to the tissues – no electrolytic effects and nerve stimulation from contrast agents or ionizing radiation applied. It is also easier and cheaper to implement in comparison with conventional computed tomography and magnetic resonance imaging technologies. However, EIT has still considered a pilot technique as not having enough trust in cancer screening procedures. A new level of reliability for EIT may be achieved by advancing computational techniques to bring the proven facts into practice(http://www.bukshtynov.xyz/research_EIT.html).

Dr. Bukshtynov and his former students at Florida Tech developed sophisticated computational algorithms for solving the EIT inverse problem by dealing with multiple complications caused by the mathematical and physical complexities to describe the EIT phenomena. Increased computational performance and reliability are achieved by adding novel numerical techniques for proper regularization, optimal space reduction, and multiscale parameter estimation(http://www.bukshtynov.xyz/research_EIT_MS.html). They also developed the in-house computational software EIT-OPT, an all-purpose open-structure optimization framework for solving numerically 2D/3D inverse EIT problems. This software has been used actively as an open platform in (under)graduate research and for preparing multiple research papers. Several REU(G) students will be assigned to work on this topic, focusing on various computational aspects of early cancer detection through EIT-based techniques using EIT-OPT as a robust research simulator: multi-sample PCA, multiscale and sample-based parameterization, dynamical mesh refinement, and use of various approaches for performing numerical computations for EIT-based optimization, including gradient-based/derivative-free/stochastic/machine learning techniques.

Prerequisites: Proficiency in linear algebra, multivariable calculus, differential equations, and some familiarity with partial differential equations. Previous experience in coding in either MATLAB or C/C++ is preferred. Experience with numerical methods in optimization will be helpful but not required.

Mentor: Vladislav Bukshtynov

Dates:  May 12th – June 27th

Enumeration and Properties of Latin Tableaux

Consider a diagram consisting of left-aligned rows of boxes that weakly decrease in length from top to bottom. We associate this shape with a partition λ=(λ₁,λ₂,…,λ_k) where λ_i is the length of row i. To create a Latin tableau on such a diagram, we fill the boxes in row i with the numbers 1 through λ_i for each row so that no column contains the same number twice. Latin tableaux are generalizations of Latin squares, which represent the case where the base diagram is a square. Latin squares famously appear in Sudoku puzzles, although there they have additional restrictions.

Interestingly, it is not possible to make a Latin tableau on every shape. A longstanding conjecture suggests that Latin tableau can only be created on a special class of shapes called wide partitions. This conjecture has close ties to Rota's basis conjecture, integer multiflow problems, and graph coloring. 

Our primary focus will be the more general question of enumerating Latin tableau of shape $\lambda$. We may also study analogues of properties of Latin squares such as orthogonality, equivalence classes, and the existence of transversals. There may be interesting generalizations of the algebraic relationship between Latin squares and quasigroups.

Prerequisites: Discrete math strongly suggested, experience with python beneficial

Mentor: Spencer Daugherty

Dates:  May 19th  - June 27th

Dynamical systems on flat manifolds

Informally, a dynamic system is any physical system that evolves with time (e.g., a pendulum, a planet orbiting the sun, the weather, etc). From a more mathematically precise perspective, one can consider a function mapping a space to itself. For example, f(x)=x^2 defined on the set of real numbers. Using this formulation, time is represented by iterating the function. In the example f(x)=x^2, if the initial value is 2, then after one unit of time, the value is f(2)=4, after two units of time, the value is f(f(2))=f(4)=16 and so on.

We will study two classes of dynamical systems on flat manifolds. The first are minimal dynamical systems, meaning every orbit is dense in the space. The prototypical example is irrational rotation on the circle. The second class of dynamical systems are chaotic. Roughly speaking chaos is characterized by the property that "the present determines the future, but the approximate present does not approximately determine the future." Our investigation will be example based. Our main objective will be to understand when these systems exist. In addition, we will aim to explicitly compute invariants associated with these dynamical systems. These invariants will allow us to distinguish different dynamical systems. A prototypical example of an invariant is the length of time it takes a planet to orbit the sun, which can be used to distinguish the planets in the solar system. 

Prerequisites: Math 2135 (or another linear algebra class), Math 3001, and Math 3140 (in particular, no experience with flat manifolds or dynamical systems is required). 

Mentor: Robin Deeley

Dates: May 12-June 20

Spatial partial isometries on finite-dimensional spaces. 

The Banach-Lamperti theorem is an important result that characterizes the partial isometries acting on L^p-spaces when p is not equal to 2 (i.e, not a Hilbert space). This was originally proven by Banach for L^p([0,1], Lebesgue measure) and later generalized by Lamperti for sigma-finite measures. A way more general version for localizable measures is now known and used widely in the current research. Roughly, this result states that any such isometry actually comes from a Boolean algebra homomorphism of the underlying sigma-algebras. 

The first goal of this project is to make sense of the Banach-Lamperti result in the finite dimensional case (i.e. such isometries can now be represented by matrices), so that we can have a statement of the Banach-Lamperti theorem that does not need any measure theory background. Once we have this, we will write a computer program that generates all the partial isometries between any two given finite dimensional L^p-spaces, for p not 2. Finally, we will try to decide whether these partial isometries can be used to compute certain operator norms, which might require us to develop some extra computer programs in addition to the aforementioned one. 

This is a project that will be well suited for graduate students with interests in measure theory and functional analysis

Prerequisites: Linear algebra (2130 or 2135) is required. Familiarity with proofs will be preferred (2001, 2135, or 3001 will satisfy this requirement). At least one course in either Analysis (3001-4001) or Abstract Algebra (3140) will be a great bonus. Previous experience coding in either MATLAB, Mathematica, or Python will be helpful but not required.

Mentor: Alonso Delfin

Dates: May 19-July 11. 

Real roots of random polynomials

Random polynomials arise when the coefficients of a polynomial are treated as random variables. This seemingly simple change transforms a fundamental mathematical object into a powerful tool connecting mathematics, physics, and other fields. Since the coefficients are random, the number of real roots of these random polynomials itself becomes a random variable.  A central challenge in this area concerns the behavior of this random variable: How many real roots are there typically? How are these roots distributed along the real number line? And how do their positions relate to each other? This project investigates these questions for a specific type of random polynomial: generalized Kac polynomials. We will explore the correlations between the real roots of these polynomials by combining theoretical insights from analysis and probability with hands-on computer simulations. The work is designed to be accessible, even for participants with no prior programming experience.

Prerequisites: Some experience with probability and/or analysis.

Mentor: Nhan Nguyen

Dates: May 12 – June 27, 2025.

Steenrod Algebras and Graph Theory

The Steenrod algebra is a foundational algebraic input for studying topological spaces. One way to study the Steenrod algebra (or its dual) is through graph theory. Recently, the trees and Hamilton cycles appearing in graphs which arise from quotients of the dual Steenrod algebra have been characterized and further algebraic structures, such as the coproduct and antipode on certain monomials, have been described in terms of graphs.      

Motivated by the Steenrod algebra's applications to the study of spaces, we can also choose to record more information. Specifically, we can study a space along with its symmetric reflection across an axis. The action of flipping a space along an axis of symmetry is an example of an action by the cyclic group of order two, C2. In this setting, there is a more complicated version of the Steenrod algebra called the C2-equivariant Steenrod algebra. In this project, we will extend Wood's graph interpretation to quotients of the C2-equivariant dual Steenrod algebra, which is a Hopf algebroid rather than a Hopf algebra. We will then study algebraic structures such as the left and right unit of the Hopf algebroid in this graphical setting. If time allows, we will extend our graphical representations to certain topologically significant comodules.

This project would be a good fit for students interested in algebra, graph theory, or topology. We will start by introducing foundational definitions from algebra and quickly get into studying graph theoretic interpretations. For more advanced students, there is also the possibility to learn about the topological context for the Steenrod algebra. 

Prerequisites: MATH 2001 or equivalent

Mentor: Sarah Petersen

Dates: May 19 - July 11

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 physics and chemistry.   In particular it is planned to study energy landscapes of molecules 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. As for data we will either use those we can obtain from  some experimental physics or chemistry groups or generated them via some computational  software for quantum mechanics or quantum chemistry (usually open source).  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.

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

Mentor: Markus Pflaum

Dates: May 12 — June 30

The supercharacter theory of loop groups

The finite symmetric groups (permutations of finite sets) are an essential tool in algebra and combinatorics.  There are numerous ways to study infinite analogues of these groups (permutations of infinite sets, for example).  This project takes the following approach which allows us to create an infinite version of permutations of an n-set: consider the set of permutations w of all integers that have the periodicity property w(i+n)=w(i) for all integers i.  The resulting group is known as the affine symmetric group, and a similar technique can be used to create infinite analogues of other finite groups.   

This summer’s favorite group will be the group of uppertriangular n by n matrices with 1’s on the diagonal (this is also known as the p-group analogue of the symmetric group).  These groups have a rich combinatorial representation theory built on set partitions (think Bell numbers, Catalan numbers, etc).  However, as far as I know, no one has investigated how this structure lifts to its affine (or loop group) analogue.

Prerequisites: Knowledge of linear algebra is necessary and Math 3140 is highly recommended. Bonus points for Math 3170 and/or Math 4140.

Mentor: Nat Thiem

Dates: May 12 — June 27

Some recent applications of Lie algebras in physics

Prerequisites: Linear algebra. Math 3140 is highly recommended

Mentor: Juan Villareal

Dates: May 12 — June 27. 

Compact moduli of genus 2 curves

A Riemann surface is a compact complex manifold of dimension 1.  The genus is the number of "handles" when viewed as a topological space.  For each genus, there is a space $M_g$, called the moduli space of curves, whose points are in one-to-one correspondence with Riemann surfaces, but this space is not "compact" : there are sequences of Riemann surfaces of the same genus that do not converge to a Riemann surface.  Deligne and Mumford proved in 1969 that there is a compact enlargement $\bar M_g$ of the moduli space of curves where the points correspond to generalized Riemann surfaces that are permitted to have nodal singularities.  However, Deligne and Mumford's solution is not the only one : other compactifications exist, allowing different classes of singularities.  Five years ago, an REU characterized *all* ways of compactifying $M_1$.  In this REU, we will try to construct new compactifications in genus 2.

It is possible to describe a compactification of $M_2$ by giving a set of graph theoretic rules, subject to certain conditions.  The main task will be to identify examples of, and eventually classify, all legal such sets of rules.  Students will need to be comfortable with graphs and maps between them.  Depending on the participants' interests and backgrounds, we may also try to learn about how these graphs are connected to Riemann surfaces.  That will require understanding of topological spaces and commutative rings.

Prerequisites: Graduate student only

Mentor: Jonathan Wise

Dates:May 19 - June 13

How do I apply?

Applications for the above Summer 2025 REU are now open; the deadline to apply is March 3, 2025.  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 here (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 2025)
• A ranking of the projects you'd be interested in participating in
• A statement describing your experiences in ways that are not captured by your transcript.  You might want to include details on the following (as applicable):
• • What makes you especially interested in your top choice(s)?
  • How do your experiences and/or background make you a strong candidate with potential for significant contributions in the program?
  • If you do not meet the prerequisites for one of your top projects, what are the mitigating circumstances that make you feel nevertheless prepared for that project?
  • What else would you like us to know about you as a candidate?

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