Application deadline: February 15, 2016.
Eligibility: Any Columbia / Barnard student (independent of U.S. residency).
Students graduating before summer 2016 will only be accepted in exceptional circumstances.
Application process: e-mail completed application form to
ccliu@math.columbia.edu, with "Application for 2016 REU" in the subject line.
Program dates: May 31 - August 5, 2016.
Stipend: $3500. Summer session housing is also paid for by the program.
Projects:
Constructions and Obstructions of Symplectic Embeddings
Jo Nelson
A symplectic manifold is an even dimensional manifold which admits a nondegenerate closed 2-form; this allows one to measures 2-dimensional space within the manifold, akin to dxdy or rdrd\theta from multivariable calculus. A major question in the field is when one symplectic manifold can be symplectically embedded into another. Gromov's nonsqueezing theorem tells us that one cannot symplectically squeeze a large 2n-dimensional ball into a thin 2n-dimensional cylinder. Thus while symplectic diffeomorphisms are volume-preserving, it is much more restrictive for a diffeomorphism to be symplectic than volume preserving. This research project will involve exploring constructions and obstructions to 4 dimensional symplectic embeddings by exploiting a relationship between symplectic embedding problems and lattice point counts in polygonal regions.
Interacting particle systems and representations of quantized Lie Algebras
Jeffrey Kuan
The asymmetric simple exclusion process (ASEP) is a central
object of study in interacting particle systems and non-equilibrium statistical
mechanics. Despite its analytical and physical description, it can also be studied
algebraically: namely, it satisfies a symmetry with the fundamental representa-
tion of the quantized Lie algebra U_q(sl_2), which can be used to prove stochastic
duality.
In this project, we investigate a natural extension to different classes of
simple Lie algebras of higher rank.
What does a random variety look like?
Daniel Halpern-Leistner and
Daniel Litt
Much recent work in mathematics (by Poonen, Vakil, Matchett-Wood, Ellenberg, and others)
has focused on understanding the properties of "random" systems of polynomial equations over a finite
field. The most basic question one can ask is: how many solutions are there to a random system of
equations? What is the distribution of the number of solutions? More geometrically, what is the
probability that the set of solutions to such a system (also known as a variety) is smooth?
The answers to these sorts of questions often have deep arithmetic or geometric significance.
This project will study some variant questions--for example, what is the size of the automorphism group of a "random" hypersurface? How many hypersurfaces have a given automorphism group? Given time, we may also study more arithmetic invariants--for example, the average rank or torsion subgroup of an elliptic fibration, or the average size of the Brauer group of a hypersurface.
Some background in programming, number theory, or algebraic geometry will be useful; for example, students should know what a finite field is.