me
Briana Foster-Greenwood
Teaching Fellow
Department of Mathematics
University of North Texas
Denton, Texas
GAB 407
BrianaFoster-Greenwood@my.unt.edu
   
Research
I'm a graduate student working with advisor Anne Shepler.  Here's a bit about what I study...

Level curves of D_4 invariant polynomialComplex reflection groups and invariant theory
Invariant theory simultaneously studies functions on a space and the geometry of a space via group actions.  At left see the symmetry in the level curves of a polynomial invariant under the order eight dihedral group.  Reflection groups (generated by reflections each fixing a hyperplane) are characterized by their extremely nice invariant theory.  Among these are Coxeter groups, acting on real space, which have a beautiful theory in terms of geometry and combinatorics.  Unfortunately(?), components of Coxeter theory do not always have clear analogues for general complex reflection groups—e.g. reflecting hyperplanes partition real space into chambers, but removing a hyperplane from complex space leaves another connected space!  I am currently interested in various partial orderings of reflection groups which have applications to the deformation theory of algebras arising from group actions.

Noncommutative algebras and deformation theory
We're used to yx=xy in a polynomial ring, but what if instead yx=xy+1?  This is the relation of the Weyl algebra, a noncommutative deformation of a commutative polynomial ring.  In deformation theory, we start with an associative algebra and then try to construct families of algebras with the same vector space structure but different rules for multiplication.  Hochschild cohomology is a measuring tool for detecting potential associative deformations.  I am interested in computing Hochschild cohomology of skew group algebras, which turns out to be a computation in invariant theory.
  
Preprint Comparing codimension and absolute length in complex reflection groups. Submitted. arXiv:1202.0266v1
 
Talks
Graded Hecke algebras and reflection length versus codimension, presented at AMS Fall Western Section Meeting, Salt Lake City, Utah, October 2011

Invariant theory and Hochschild cohomology of S(V)#G, presented at UNT and TAMU Workshop on Algebras, UNT, April 2011

Deformations of skew group algebras, presented at UNT and TAMU Workshop on Deformations, TAMU, October 2010
 
Teaching Spring 2012: Math 1710.007, Calculus I
Fall 2011: Math 1650.002, Precalculus
Spring 2011: Math 1710.007, Calculus I
Fall 2010: Math 1650.001, Precalculus
Spring 2010: Math 1100.009, College Algebra
Fall 2009: Math 1100.026, College Algebra
 
Fun Stuff
Polynomial Lemniscates (interactive Mathematica notebook)