Trace Diagrams

My primary research interest is trace diagrams and their applications to other areas of mathematics. What is a trace diagram? Fundamentally, it's a combinatorial construction that can be identified in a precise way with multilinear functions. If you're a physicist, it's much like a spin network or birdtrack. If you're a knot theorist, it's much like skein algebra, except without the quantization. It's also sometimes called a tensor diagram. The big difference between a trace diagram and these classical constructions is that trace diagrams have edges labeled by matrices, and so the objects of study are almost always functions on matrices. Applications Diagrammatic notation is extremely powerful and has the potential to simplify many classical proofs. Some examples of this capability (for linear algbera) are here. Some other applications include: Invariant Theory: Diagrams can be used to describe the functions on matrices which are invariant under simultaneous conjugation (think traces of products of matrices). Trace diagrams are very powerful for demonstrating relations among these functions. Geometric Structures: Because invariant theory is vital for understanding character varieties, trace diagrams can be used to describe the moduli space of geometric structures on a surface. (This was the primary direction of my thesis.) History Diagrammatic techniques are notoriously difficult to research from a historical standpoint. The costs of typesetting were extremely high until recently, so many of the early references are buried in PhD theses and hard-to-find references. Roger Penrose invented the term “spin network” in the early 1970s for graphs labeled by representations of SU(2) in his work on combinatorial space-time. Today, “spin network” is a general term for any graph whose edges represent representations of a particular group and whose nodes are interwiners, or functions, between tensor powers of these representations. References Here are the best books on the topic: Birdtracks, a webbook by the physicist Predrag Cvitanovic available at http://birdtracks.eu/. This is by far the most thorough reference, but unfortunately is not well-known, especially in the mathematics community. It has some fascinating results for anyone who is interested in the classification of Lie groups. Section 4.9 contains a history of diagrammatic notation which is definitely worth reading. Group Theory, by G.E. Stedman (also a physicist). This book is very hard to find and extremely expensive to purchase. Most of the material is also included in Cvitanovic's book.

Papers and Preprints "Unshackling Linear Algebra from Linear Notation" , submitted. arXiv:0910.1362 "Trace Diagrams, Signed Graph Colorings, and Matrix Minors" (with Steve Morse), accepted pending revisions. arXiv:0903.1373 "On a Diagrammatic Proof of the Cayley-Hamilton Theorem" , submitted. arXiv:0907.2364 "Computing SL(2,C) Central Functions with Spin Networks" (with Sean Lawton), submitted. arXiv:0903.2372 (see also this mathematica notebook with combinatorial recurrence implementation) "Spin Networks and SL(2,C) Character Varieties" (with Sean Lawton), Handbook of Teichmuller Theory Volume II (Chapter 16), 2009, Athanase Papadopoulos, ed., EMS Publishing House. arXiv: math.QA/0511271 "Trace Diagrams, Representations, and Low-Dimensional Topology" , PhD Thesis at the University of Maryland (advised by Bill Goldman), 2006. PDF Talks "Trace Diagrams: Unshackling Linear Algebra from Linear Notation" , Bard College Mathematics Seminar, Nov 12, 2009. PDF Slides need to update "Unshackling Linear Algebra from Linear Notation" , USMA Math Department Research Seminar, Sep 9, 2009. PDF Slides "Trace Diagram Recurrences and Central Functions of SL(2,C)-Character Varieties" , AMS Special Session on Geometry, Algebra, and Topology of Character Varieties, JMM, Jan 8, 2009. PDF Slides "Signed Graph Coloring, the Art of Linear Algebra, and a Theorem of Jacobi" (with Steve Morse), MathFest, July 31, 2008. PDF slides "The Character Variety's New Clothes" , AMS Session on Algebra and Number Theory, JMM 2008, Jan 7, 2008. PDF slides "The Art of Linear Algebra" , USMA Department of Mathematical Sciences Research Seminar, Dec 5, 2007. PDF slides and PDF handout "Trace Diagrams, Spin Networks, and Spaces of Graphs" , first of two invited lectures at the 7th KAIST Geometric Topology Fair in Gyeongju, Korea, July 10, 2007. PDF slides "Diagrammatic Central Functions" , second of two invited lectures at the 7th KAIST Geometric Topology Fair in Gyeongju, Korea, July 10, 2007. PDF slides "Trace Diagrams, Surfaces, and Character Varieties" , Kansas State University Mathematics Colloquium, April 15, 2007. PDF slides "Trace Diagrams, Surfaces, and Low-Dimensional Topology" , PhD Defense at the University of Maryland, Apr 25, 2006. PDF slides I use LaTeX for typesetting, including the MikTeX package, TeXnicCenter for editing/compiling, Beamer for slides, and PGF/TiKZ for figures.