General Information •
Project Description •
Project Log •
Resources
General Information
Project Description
My project is in the field of combinatorial/enumerative algebraic geometry, specifically KTheory.
The goal of the project is to compute the KTheory Class of a subvariety given a condition known as the trialitysymmetric condition.
I am expanding on work done by Dave Anderson in his paper Degeneracy of TrialitySymmetric Morphisms (2009) .
The main problem is to compute the free resolution of a subvariety that characterizes the space of all trialitysymmetric maps in the vector space L(V,End(V) ⊕ V*) where L(V,W) denotes the set of all linear maps from V into W and dim(V) = 2.
Presentation 1 (pdf)
Presentation 2 (pdf)
Project Log
 Week 1: The first week of the REU, I met my advisor and was introduced to my topic.
In order to understand what I would be researching, I read about tensors and dual spaces.
I also learned about the second exterior product, which I discussed in presentation1.
Furthermore, I computed the conditions necessary for φ to be triality symmetric. (φ is an element of L(V,End(V) ⊕ V*),
V a vector space of dimension 2.)
 Week 2: This week, I am reading up on modules. I haven't worked with them before, so I am familarizing myself with their properties.
In addition, I started reading about graded modules/rings and Grothendieck rings.
 Week 3: I learned about free resolutions and read more about graded modules. I calculated some Ktheory classes for the origin of vector spaces that
are direct sums of ℂ.
I also proved that an mxn matrix, A, is of rank ≤ r iff all (r+1)x(r+1) minors = 0, and rank(A) = r iff there is a rxr minor ≠ 0.
 Week 4: I read about Koszul complexes, and through this newfound knowledge, I was able to compute some Ktheory classes for Torus representations.
In addition, I proved that if V is a ℂ\{0}representation, V* = Hom_{ℂ}(V,ℂ), and V = ℂ_{a1} ⊕ ...
⊕ ℂ_{an}, then V* = ℂ_{a1} ⊕ ... ⊕ ℂ_{an}
Furthermore, I proved that ℂ_{a1} ⊗ ℂ_{a2} =
ℂ_{a1+a2}
 Week 5: This week, I continued my work from Week 4, but represented the KTheory classes in a more general way. In addition, I worked more with
trialitysymmetric maps.
I found a basis for H = Hom(V,End(V) ⊕ V*) when dim(V) = 2. In addition I found a basis for Ω_{r} =
{φ ∈ H  φ is trialitysymmetric, rank(φ) ≤ r}
 Week 6: July 4th was Monday! I found the equations I need to compute the set Ω_{r} of trialitysymmetric maps of rank r.
In addition, I found the free resolution of k(Ω) = k[V]/I(Ω). I also finished the slides for my powerpoint.
 Week 7: I started practicing for my final presentation on Tuesday. Also, I found the KTheory Class for Ω_{1}. Finally, I
gave my final presentations around 11:20 am on Thursday.
Here are the slides: Presentation2.
 Week 8: This week I wrote up my final report. In addition, I rewrote the equation I found for the
KTheory Class in terms of new variables, and read Dave Anderson's paper Degeneracy of TrialitySymmetric Morphisms.
Program ends July 22, 2011.
Resources
Text
 Adkins, W., Weintraub, S., Algebra: An Approach via Module Theory , SpringerVerlag, New York, 1992.
 Cox, D., Little, J., O'Shea, D., Ideals, Varieties, and Algorithms , 3rd ed., Springer, New York, 2007.
 Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry , Springer, New York, 1995.
 Roman, S., Advanced Linear Algebra , 2nd ed., Springer, New York, 2005.
 Rotman, J., Advanced Modern Algebra , 2nd ed., American Mathematical Society, 2010.
 Stanley, R., Combinatorics and Commutative Algebra , 2nd ed., Birkhauser, Boston, 1996.
Web
 Anderson, D., Degeneracy of TrialitySymmetric Morphisms, http://arxiv.org/pdf/0901.1347.pdf , 2009.
 Driessel, K., Appendix: Tensor Products, http://homepage.mac.com/driessel/IAState/Groups&Physics/tensors.pdf, 2009.
 Evens, L., Chapter V: Modules, http://www.math.northwestern.edu/~len/d70/chap5.pdf, 1999.
 Evens, L., Chapter VI: Hom and Tensor, http://www.math.northwestern.edu/~len/d70/chap6.pdf, 1999.
 Kamnitzer, J., Representation theory of compact groups and complex reductive groups, http://www.math.toronto.edu/jkamnitz/courses/reptheory/reptheory.pdf, 2011.
 Lent, C., Representation Theory, http://math.berkeley.edu/~teleman/math/RepThry.pdf, 2005.
 Murayama, H., Notes on Tensor Product, http://hitoshi.berkeley.edu/221a/tensorproduct.pdf, 2006.
 Murfet, D., Graded Rings and Modules, http://therisingsea.org/notes/GradedModules.pdf, 2006.
