Kelli-Jean Chun, REU 2011

# Kelli-Jean Chun

## Math REU 2011

General InformationProject DescriptionProject LogResources

General Information

 Email: kchun@berkeley.edu Office: 628 Hill, Rutgers Busch Campus School: University of California, Berkeley Project: Computing the Equivariant K-Theory Class of the Subvariety of Triality-Symmetric Maps

Project Description

My project is in the field of combinatorial/enumerative algebraic geometry, specifically K-Theory. The goal of the project is to compute the K-Theory Class of a subvariety given a condition known as the triality-symmetric condition. I am expanding on work done by Dave Anderson in his paper Degeneracy of Triality-Symmetric Morphisms (2009) . The main problem is to compute the free resolution of a subvariety that characterizes the space of all triality-symmetric 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 K-theory 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 K-theory classes for Torus representations. In addition, I proved that if V is a ℂ\{0}-representation, V* = Hom(V,ℂ), and V = ℂa1 ⊕ ... ⊕ ℂan, then V* = ℂ-a1 ⊕ ... ⊕ ℂ-an Furthermore, I proved that ℂa1 ⊗ ℂa2 = ℂa1+a2
• Week 5: This week, I continued my work from Week 4, but represented the K-Theory classes in a more general way. In addition, I worked more with triality-symmetric maps. I found a basis for H = Hom(V,End(V) ⊕ V*) when dim(V) = 2. In addition I found a basis for Ωr = {φ ∈ H | φ is triality-symmetric, rank(φ) ≤ r}
• Week 6: July 4th was Monday! I found the equations I need to compute the set Ωr of triality-symmetric 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 K-Theory Class for Ω1. 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 K-Theory Class in terms of new variables, and read Dave Anderson's paper Degeneracy of Triality-Symmetric Morphisms. Program ends July 22, 2011.

Resources

Text
• Adkins, W., Weintraub, S., Algebra: An Approach via Module Theory , Springer-Verlag, 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.

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