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The mathematics department will be hosting an internal research experience for undergraduates and early graduate students in Summer 2026. 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 2026ÌýProjects

Optimizing Vascularization in Bioprinted Skin Grafts: Computational Modeling, Calibration, and Optimal Design

Bioprinting of skin grafts is an emerging technology with strong potential for applications in wound healing and reconstructive medicine. One of the main challenges limiting its clinical use is achieving reliable vascularization, i.e., formation of blood vessel networks that supply nutrients and oxygen to the engineered tissue. Recent research by Dr.ÌýBukshtynov focuses on developing computational models that describe these vascularization processes and using optimization techniques to improve their performance. A full description of the underlying research can be found here: .

This REU(G) summer research project will focus on computational optimization and model calibration for vascularized skin grafts. Students will begin by estimating unknown model parameters using data and optimization-based techniques. If time permits, the project will extend to optimizing the overall vascularization process to identify conditions that lead to improved tissue perfusion and uniform vessel growth. Students will work with numerical simulations and implement algorithms in MATLAB and C/C++, gaining experience in scientific computing, optimization, and interdisciplinary research at the interface of mathematics, engineering, and biomedical science.
Prerequisites: Background in linear algebra, multivariable calculus, and differential equations. Experience with MATLAB or C/C++ is preferred. Prior exposure to optimization, numerical methods, or biomedical applications is helpful but not required.
Mentor: Vladislav Bukshtynov
Dates:Ìý May 11th – July 3rd.

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Constructing and visualizing moduli spaces of cubic surfaces

A cubic surface is a surface in three-dimensional space defined by a cubic polynomial. In the mid-19th century, Cayley and Salmon famously showed that every smooth cubic surface contains exactly 27 lines (over the complex numbers). In this project, we will study the parameter space for cubic surfaces, a four dimensional geometric object that keeps track of four parameters indexing every cubic surface, up to change of coordinates in three space. ÌýThe parameter space actually depends on a choice of 27 other numbers called weights, and we will study how the parameter space changes as we change the weights. ÌýThis builds off of a number of recent results in the literature due to Gallardo—Kerr—Schaffler, Schock, and Kim.

The project has two main goals (emphasis on the different goals to be determined by the participants):
(1) Investigate changes in the parameter space corresponding to new choices of weights not already explored in literature.
(2) Write computer code to create a visualization tool keeping track of how the parameter space changes with changes in the weights.
Prerequisites: (MATH 2001) and (MATH 2130 or 2135) or APPM equivalents. The project will be geared towards the backgrounds of each of the participants, and participants with more mathematical background than the above prerequisites (such as MATH 3001 and MATH 3140) will be given material to work on that corresponds to their background. Also, please let us know if you have coding experience in Python, Sage, or html (coding experience is not required, but we are interested to know who in the group will have coding experience).
Mentors: Sebastian Casalaina-Martin and Jon Kim
Dates: June 15 -- July 24

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Extreme Covering Systems

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Critical Points of Random Polynomials

Polynomials are fundamental mathematical objects that appear in all areas of math and science. ÌýA random polynomial is one whose coefficients are random variables. ÌýThese are interesting in their own right and, in some ways, express the behavior of a "typical" polynomial. ÌýIn this project, we will explore the relationship between the roots of a random polynomial and its critical points for various models of random polynomials. ÌýWe will draw on tools from analysis, combinatorics, geometry, and probability. ÌýPart of the project will involve numerical simulations, but no previous programming experience is required. Ìý
Prerequisites:ÌýSome experience with probability and/or analysis
Mentors: Nhan Nguyen and Sean O’Rourke
Dates:ÌýMay 11 -- June 26

Computational Methods in Linear, Commutative and Homologial Algebra

Whether in quantum chemical computations of molecular properties, in google's page rank algorithm to rank the importance of web pages, in topological data analysis or in the numerical treatment of PDE's the computation of eigenvalues of largeÌý matrices plays a crucial role. In this REU, the underlying mathematical methodsÌý in numerical linear, commutative and homological algebra will be studied. The acquired knowledge will then be applied to create software tools to implement efficient algorithms forÌý the computation of the eigenvalues of a Hermitian matrix, the singular value decomposition of a complex matrix or the Smith normal form of an integral matrix. One particular goal in creating these software tools is to write GPU accelerated code via the ArrayFire package. Moreover, the resulting software shall then be used to improve existing software packages for computations in algebraic topology or quantum chemistry.

As programming languages the student may use python, C/C++, Haskell, or julia.
Prerequisites: fundamentals of linear algebra
Mentor: Markus Pflaum
Dates: May 4 — June 26
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The representation theory of unipotent polynomial groups

The representation theory of loop groups appears in many of the current compelling problems in mathematics: for example, p-adic representation theory of groups of Lie type, and Macdonald polynomials in algebraic combinatorics. These infinite groups may be viewed as infinitely periodic matrices or as matrices with Laurent polynomials entries.ÌýÌý

Last summer’s REU/G investigated how the representation theory of a family of finite unipotent groups lift to this infinite setting.Ìý Over the course of the project it became clear that we had selected the wrong family of finite groups to generalize.Ìý The correct family of finite groups is a family that has seen far less study, and many of the underlying representation theory remains mysterious.ÌýÌý The goal of this project is to tease out more of the underlying combinatorics of the algebraic structure of these finite groups.Ìý
Prerequisites: Math 3140. Some representation theory is desirable, but optional.
Mentor: Nat Thiem
Dates: May 4 — June 19

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Celestial holography and its classical limits

In recent years, physicists have discovered deep connections between symmetry structures in quantum field theory and conformal field theory. In particular, certain limits of scattering processes, known as collinear limits, turn out to be closely related to infinite-dimensional symmetry algebras. These ideas have led to the development of an active research program called celestial holography, which uses conformal field theory techniques to study scattering amplitudes in quantum field theory.

From the mathematical perspective, collinear limits naturally give rise to interesting Lie algebras. We will explore how collinear limits (and logarithmic collinear limits) lead to the appearance of infinite dimensional algebras. The main objective of this REU is to understand the classical limitsÌýof some of these algebras from a mathematical viewpoint.
Prerequisites:ÌýMath 2130,Ìý Math 3140 or 4140 are considered a plus.
Mentor:ÌýJuan Villarreal
Dates:ÌýMay 4 — June 19

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Quantitative S-unit Theorem

This REU project concerns the quantitative S-unit theorem, which gives explicit bounds for the number of solutions to S-unit equations and is a key input in effective results for Diophantine problems such as Thue–Mahler equations. The project will analyze the main ingredients of the proof, including height functions on multiplicative groups, the logarithmic embedding of S-units into Euclidean space, and auxiliary constructions via Padé approximation that produce linear relations with controlled error terms. A further component is the use of height lower bounds on the algebraic torus, including results in the direction of Zhang’s theorem, which provide uniform control on points of small height outside proper algebraic subgroups.

A central goal of the project is to examine a standard proof of the quantitative S-unit theorem that contains a subtle gap in one of the key quantitative estimates. The work will focus on isolating this point and refining the Padé approximation or the subsequent geometric or height-theoretic estimates so that the argument becomes fully rigorous.
Prerequisites: Willingness to learn some elementary number theory. Some basic knowledge of groups . Some exposure toÌýalgebraic geometry and algebraic number theory is helpful but not required.
Mentor: Zheng Xiao
Dates: May 4 -- June 5

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How do I apply?

Applications for the above Summer 2026ÌýREU are now open, and due Monday, March 2, 2026.Ìý 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. ÌýThis 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 (MUST be signed in to Google with your CU identikey). ÌýThis application includes

• A .pdf copy of your (un)official transcript (note that you can't have graduated before Summer 2026)
• 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. ÌýYou 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?

The ³Ô¹ÏÍø 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.

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