## SPUR PROGRAM - Summer Program for Undergraduate Research
REU PROGRAM - Research Experience for Undergraduates
These
summer programs provide the opportunity for undergraduate students of
mathematics to participate in leading-edge research. This year, one
project is designated "REU" and the other project is designated "SPUR."
The difference between REU and SPUR projects is the funding support
that is available.
We
welcome all students to apply including U.S., Permanent Resident, and
International students whether your home institution is within the U.S.
or outside the U.S. Please read carefully the information below under
details and support/costs. Certain criteria apply for international
students with their home institution outside the U.S. Please refer to the programs https://math.cornell.edu/undergraduate-research for more information on housing, costs, and other important details.
**REU and SPUR PROGRAM PROJECTS**
**REU Project 1 Title: Periodic non-intersecting random walks and random matrices**
Directed by: Andrew Ahn
Project Details: The project is focused on a non-interesting random walk model on a cylinder with connections to a variety of statistical mechanics models. The marginals of this random walk can be interpreted as a random matrix model with a bias on the eigenvectors, known as the Moshe-Neuberger-Shapiro (MNS) model. It can also be realized as a diffuse limit of lozenge tiling models of the cylinder. The goal of the project is to find limit shapes for his random walk and describe the fluctuations from the limit shape. The fluctuations are believed to relate to th Gaussian free field, a 2 dimensional random field which appears in a variety of statistical mechanics model, with some additional features inherited from the geometry of the model. The MNS model possesses a rich algebraic structure due to its connection with symmetric functions, which will serve as an entry point for analysis. **REU Project 2 Title: Nonintegrable Constraints in Mechanics**
Directed by: William Clark Project details: Newton's laws state that a body subjected to zero force will experience zero acceleration. A geometric interpretation of zero acceleration is straight line motion. This observation leads to Lagrangian mechanics where physical motion satisfies the principle of least action - straight lines minimize the distance between two points and physical motion, analogously, minimizes the action. Complexities can be introduced by imposing constraints: the bob on a pendulum must remain a fixed distance from the pivot, a bicycle cannot move sideways, and a billiard ball must remain within the confines of the tabletop. The first constraint only depends on the position of the pendulum (holonomic), the second depends on the velocity of the bike (nonholonomic), and the third is an inequality constraint on the position of the ball (unilateral). One particularly strange facet of nonholonomic systems is that they fail to obey the principle of least action. Students participating in this project will investigate properties of nonholonomic and unilaterally constrained systems and how they inter-relate. Students for this program should have a strong understanding in linear algebra, exposure to qualitative reasoning of ordinary differential equations, and good computer programming skills. Background in the following areas will be helpful but not required: Newtonian/Lagrangian/Hamiltonian mechanics and exposure to manifolds. **REU Project 3 Title: Optimality & Uncertainty**
Directed by: Alex Vladimirsky Project details: Equations describing optimal behavior often present serious computational challenges. (The quickest driving directions? The most energy-efficient trajectory for a Mars-rover? The risk-of-detection-minimizing flight-plan for a spy plane? The best schedule for a drug therapy for cancer patients? The best strategy for a predator to hunt its prey?) The need for efficient algorithms becomes particularly obvious once you add to the mix the uncertainty about our environment, conflicting goals, and multiple (competing or cooperating) participants. Students participating in this project will investigate the theoretical properties and build fast algorithms for optimal control and differential games. We will also explore the usefulness of such algorithms for problems arising in robotic navigation, traffic engineering, environmental crime modeling, ecological management, and design of adaptive drug therapies. Successful candidates will need good programming skills, previous exposure to ordinary differential equations and numerical computing. Some background in the following areas will be also helpful, but is not expected or required: partial differential equations, game theory, probability theory. More on this project can be found at https://pi.math.cornell.edu/~vlad/reu. **SPUR Project 4 Title: Symplectic embeddings of 4D toric domains**
Directed by: Morgan Weiler Project details: Our project will focus on symplectic embeddings of toric domains, which are regions in four-dimensional space equipped with a "symplectic form", a gadget which generalizes the concept of conservation energy. Symplectic embeddings of 4D domains encode coordinate transformations of the phase spaces of 2D physical systems. These problems are very hands-on, and students will be able to use cutting-edge geometric tools via combinatorial and number-theoretical methods. We are committed to making this an inclusive experience, and students from all backgrounds are encouraged to apply. Prerequisites: At least one proof-based mathematics course. Applications from students who have taken (or will take in Spring 2022) at least one proof-based geometry or topology course will be given preference. If students have prior experience with mathematical software (e.g. Mathematica), that will be useful, but is not necessary. The most important thing is an interest in solving geometrically motivated problems via hands-on methods. If you have questions regarding the project, please feel free to contact Morgan Weiler.
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