Algebra Group
Lenny Chastkofsky
Faculty: Brian Boe and Dan Nakano
Graduate students: Bobbe Cooper, Kenyon Platt, Wenjing Li, Carrie Wright, Irfan Bagci, Ben Connell, Jae-Ho Shin
Post-Doc, Ben Jones
We will be studying Cohomology of Lie Algebras in characteristic p. Kostant's Theorem gives a description of the cohomology in characteristic 0. Previously, the group had come up with a new proof that the cohomology is the same as in characteristic 0 for large enough primes, and that there is always extra cohomology for small enough primes. By using Magma to compute many examples, the group is close to coming up with a conjecture as to precisely when these exceptions occur. We will work on refining this conjecture, and think of how we might prove it.
Algebraic Geometry Group
Elham Izadi Graduate Students: Maxim Arap, Tyler Kelly, Al Lapointe, Jeremy Praissman.
Undergraduates: Kyle Istvan and Jasmine Mathis
This group will work on desingularizing elementary examples of algebraic varieties. The examples considered will include plane curves, subvarieties of Grassmannians such as those obtained as Fano varieties of hypersurfaces in projective space. If time permits, we will also work on some rationality problems.
Circle Packing Group
Faculty: Will Kazez and Malcolm Adams
Graduate Students: Jennifer Belton, Xiaoyan Hu, Yang Liu, Matt Mastin, Whitney Montgomery
Undergraduate students: Nona Dowling,Casey Murphy, Meredith Perrie, Josh Wood.
Circle Packing VIGRE group will be first studying the Koebe-Andreev-Thurston Theorem. We will start with some basic facts about Circle Packing and constant curvature geometries in dimension two. We will then learn some foundational tools in the field, basic examples and explore applications and further research directions.
Combinatorics Group
Dino Lorenzini
Faculty: Sybilla Beckmann, Lenny Chastkofsky
Graduate Students: Michael Berglund, Brian Cook, Jennifer Muskovin, Brandon Samples
Undergraduates: Grant Fiddyment,George Vulov
Algebraic graph theory
One associates to a graph G on n vertices two (n x n)-matrices, the adjacency matrix A and the Laplacian matrix L. Both A and L have a set of eigenvalues, and a Smith normal
form over the integers. Much has been written on the relationships between the eigenvalues and the combinatorics/topology of the graph. In this seminar, we will start by investigating
the information encoded in the Smith normal form of the Laplacian of a graph.Equivalent to the Smith normal form of the Laplacian of a graph is a finite abelian groupthat has
'appeared' independently in several different fields, and is known under several names, such as the component group, the critical group, the Picard group, or the sandpile group.
This interesting group is the main motivation for studying the Smith normal form of the laplacian. Its order is the number of spanning trees of the graph.
Computations on CM elliptic curves Pete Clark and Patrick Corn Graduate students: Steve Winburn, Ben Wyser, Nathan Walters, Steve Lane, Jim Stankewicz Undergraduate: Alex Rice Computations on elliptic curves with complex multiplicationDescription: It is known (due to work of L. Merel) that the torsion subgroup of an elliptic curve over a number field K is finite and has order bounded by a function depending only on the degree d of K over Q. However, this function grows exponentially with d whereas the known lower bounds are (very nearly) linear in d. For d at least 4 there is little in the way of explicit examples. The goal our group is to devise and implement an algorithm to enumerate the possible torsion subgroups over number fields of degree d -- not for all elliptic curves (no one has any idea how to do that) but for a restricted class of elliptic curves with complex multiplcation (CM). What is attractive about the class of CM curves is that on the one hand they are easier to compute with but on the other hand there is reason to believe that for sufficiently large d the largest torsion subgroups will come from CM elliptic curves. Later we may attack some other interesting algorithmic problems on CM curves as well
Tropical Geometry Group
Faculty: Dino Lorenzini, Gordana Matic, Robert Varley
Graduate students: Lev Konstantinovskiy, Maxim Arap, Kevin Kennedy, Kate Thompson, Maury LeBlanc
Undergraduate: Eric Cho(not for credit)
The aim of the group is to study this new and rapidly developing and its applications in the fields of interest to the participants: algebraic geometry, arithmetic geometry, topology, mathematical physics, quantum theory. We also aim to work on a circle of problems related to curves and to moduli spaces, motivated by several open problems in classical algebraic geometry and number theory.