About Courtney

I am a fifth year graduate student in Mathematics at the University of Nebraska-Lincoln. I study commutative and homological algebra under the supervision of Lucho Avramov and Roger Wiegand.

Contact Information:

Office:	Avery Hall 312
Office Hours (Spring 2012):	By Appointment
Phone:	(402) 472 - 8239
Email:
Please try to contact me by email first.

If you'd like to schedule an appointment, please take a look at my schedule.

Background

Education:

• M.S. Mathematics; May 8, 2009
  • University of Nebraska-Lincoln
• B.A. Mathematics with distinction, Summa Cum Laude; May 22, 2006
  • Colorado College

Awards and Honor Societies:

• Outstanding Graduate Teaching Assistant, College of Arts and Sciences, UNL, Spring 2011
• The UNL Outstanding Graduate Teaching Assistant Award is a UNL-wide award given to a graduate student that demonstrates superior teaching skills, a focused interest in student learning and passionate interest in the professional development as a teacher.
• Emeritus Faculty Fellowship, UNL Math Department, Spring 2010
• The Emeritus Fellowship is given each academic year to support graduate student research.
• Outstanding Qualifying Exams, UNL Math Department, Fall 2008
• This award has recognized truly outstanding performance on the Qualifying Exams since 1989.
• Mentoring through Critical Transition Points (MCTP) Traineeship, UNL Math Department, Fall 2007-Summer 2008
• Florian Cajori Prize in Mathematics, Colorado College Math Department, 2006
• The Colorado College Mathematics Department presents the Cajori Prize as its top award for mathematical prowess, awarded most years since 1986.
• Phi Beta Kappa
• Golden Key Honor Society

Membership in Professional Societies:

• American Mathematics Society
• Association for Women in Mathematics

Teaching

Past Courses Taught:

• Courses taught as principal lecturer:
  • Math 101 (College Algebra)
  • Real numbers, exponents, factoring, linear and quadratic equations, absolute value, inequalities, functions, graphing, polynominal and rational functions, exponential and logarithmic functions, systems of equations.
  • Math 203 (Contemporary Mathematics)
  • Applications of quantitative reasoning and methods to problems and decision making in the areas of management, statistics, and social choice. Includes networks, critical paths, linear programming, sampling, central tendency, inference, voting methods, power index, game theory, and fair division problem
  • Math 300 (Math Matters)
  • Develop a deeper understanding of "number and operations". The importance of careful reasoning, problem solving, and communicating mathematics, both orally and in writing. Connections with other areas of mathematics and the need for developing the "habits of mind of a mathematical thinker".
• Other teaching experiences:
  • Super TA for Math 106 (Calculus I) and Math 107 (Calculus II)
  • Throughout the fall of 2011, I observed all first-time calculus teaching assistants and provided constructive feedback to each as to his or her teaching methods.
  • Teaching assistant for Math 850 (Discrete Mathematics).
  • Resident scientist for Project Fulcrum
  • Project Fulcrum was an NSF-funded K-12 outreach program, placing UNL graduate students in classrooms through the Lincoln, NE area. I served as an in-class resource for Mickle Middle School science teacher Joe Rousseau. My duties included creating original classroom content (available here).
  • Teaching assistant and mentor for Intensive Mathematics: a Mentoring, Education, and Research Summer Experience (IMMERSE).
  • For the summer 2008 program, I served as a graduate teaching assistant and a mentor to the undergraduate participants.

Research

Papers and Preprints:

• The cone of Betti diagrams over a hypersurface ring of low embedding dimension (with Christine Berkesch, Jesse Burke, and Daniel Erman) - arXiv 1109.5198Preprint
• Abstract: We give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form k[x,y]/<q>, where q is a homogeneous quadric. We also provide a finite algorithm for decomposing Betti diagrams, including diagrams of infinite projective dimension, into pure diagrams. Boij--Soederberg theory completely describes the cone of Betti diagrams over a standard graded polynomial ring; our result provides the first example of another graded ring for which the cone of Betti diagrams is entirely understood.
• Fixing numbers of graphs and groups (with Joshua Laison); Electronic Journal of Combinatorics - To Journal; Preprint
• Abstract: The fixing number of a graph G is the smallest cardinality of a set of vertices S such that only the trivial automorphism of G fixes every vertex in S. The fixing set of a group Γ is the set of all fixing numbers of finite graphs with automorphism group Γ. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label G so that the automorphism group of the labeled graph is trivial. The fixing number can be thought of as a variation of the distinguishing number in which every label may be used only once, and not every vertex need be labeled. We characterize the fixing sets of finite abelian groups, and investigate the fixing sets of symmetric groups.
• Critical pebbling numbers of graphs (with Joshua Laison and Erick Paul) - Preprint
  • Abstract: If some pebbles are distributed on the vertices of a graph, a pebbling step takes two pebbles from one vertex and replaces one at an adjacent vertex. A distribution D of pebbles is solvable if, starting from D, a pebble can be moved to any specified vertex by a sequence of pebbling steps. The pebbling number p(G) of a connected graph is the smallest number of pebbles such that every distribution with p(G) pebbles is solvable. If we restrict our attention to distributions which are minimally solvable or maximally unsolvable, we obtain three additional pebbling parameters on G, the r-, g-, and u-critical pebbling numbers of a connected graph. We investigate properties of the r-critical pebbling number and its relationship to the pebbling number.

Talks:

• "Matrix Factorizations Revisited", Commutative Algebra Reading Seminar, UNL. January 24, 2012.
  • Abstract: Last week we saw that matrix factorizations are awesome. This week we'll expand on that theme by constructing minimal free resolutions of MCMs over hypersurfaces. We'll end with a peek at another free resolution for arbitrary modules over a hypersurface.
    Notes (pdf)
• "A Seance to Summon Hilbert", Graduate Student Seminar, UNL. October 31, 2011
• Abstract: In 1900, David Hilbert remarked, "Who among us would not be happy to lift the veil behind which is hidden the future; to gaze at the coming developments of our science and at the secrets of its development in the centuries to come? What will be the ends toward which the spirit of future generations of mathematicians will tend? What methods, what new facts will the new century reveal in the vast and rich field of mathematical thought?" This Halloween, we'll reflect on some of the major developments in mathematics that would interest Hilbert's ghost (just in case he decides to join us).
• "New Directions in Boij-Soederberg Theory," AMS Midwestern Sectional Meeting, UNL, Fall 2011 (slides)
• "More on the homology of three-generated ideals," Commutative Algebra Reading Seminar, UNL. September 13, 2011
• Abstract: Bruns's paper "'Jede' endliche freie Auflösung ist freie Auflösung eines von drei Elementen erzeugten Ideals" ('Every' finite free resolution is the free resolution of a three-generated ideal) generalizes some of the ideas that Brian Johnson discussed last month. (English translation available: arXiv:1012.5551v1, translated by Soumya Deepta Sanyal.)
• Research Summary, MSRI, Summer 2011
• "One-dimensional hypersurfaces and the cone of Betti diagrams of finite length modules," Compact Commutative Algebra Reading Seminar, UNL; April 7, 2011
• "Solving Sudoku with Groebner Bases...Theoretically," Graduate Student Seminar, UNL, Spring 2011
• "Hilbert Series, Polynomials, and Functions," Commutative Algebra Reading Seminar, UNL, Spring 2011
• "Boij-Soederberg Theory I: Betti diagrams over polynomial rings" & "Boij-Soederberg Theory II: Betti diagrams over k[x,y]/(x²)," Commutative Algebra Seminar, UNL. September 14 and 15, 2010
• Abstract I: The Boij-Soederberg conjectures (now theorems, proved by Eisenbud and Schreyer in 2008) state that, given a graded, finitely generated Cohen-Macaulay module over a polynomial ring, its Betti diagram can be written uniquely as a postive rational linear combination of "special" Betti diagrams. These results allow us to use convex geometry tools to identify possible (or impossible) Betti diagrams up to rational multiples. In this talk, I'll describe the "special" Betti diagrams and an outline of the proof of the conjectures in the two variable case.
  Abstract II: One might hope to have a Boij-Soederberg type of decomposition theory for Betti tables of modules over other graded rings. Hypersurface rings are a natural place to start this investigation. For the ring k[x,y]/(x²), we provide characterizations of the cone of Betti diagrams in terms of extremal rays generating the cone, functionals determining the halfspaces cutting out the cone, and a simplicial fan structure guaranteeing uniqueness of the decomposition. In this talk, I'll describe these characterizations and outline a proof of the theorem. This is joint work with Christine Berkesch, Jesse Burke, and Daniel Erman.
• Commutative Algebra Reading Seminar, UNL, Fall 2010
• "The Determinant Trick", Math Department Seminar, Colorado College. April 23, 2010
• Abstract: Somewhere, buried in your mathematical tool box, you have a mathematical Swiss army knife. Somewhere in that mathematical Swiss army knife, you have the determinant trick. Courtney will introduce you to this handy tool and even show you some cute commutative algebra applications. You don't need to have seen much algebra to follow this talk, but having played with modules a little bit would be ideal.
• "Derivators," Commutative Algebra Reading Seminar, UNL, Fall 2009
• "Squaring the Circle," Graduate Student Seminar, UNL, Spring 2008

Posters:

• Undergraduate Poster Session, Joint MAA-AMS Meetings, New Orleans, LA, January 2007

Past Conferences Attended:

• AMS Sectional Meeting, Salt Lake City, October 2011
• AMS Sectional Meeting, UNL, October 2011
• KUMUNU (Conference in Algebra), University of Nebraska-Lincoln, April 2011, September 2009, September 2008
• Joint MAA-AMS Meetings, New Orleans, January 2011, 2007
• Joint MAA-AMS Meetings, San Francisco, January 2010
• Homotopy Theory and its Applications, University of Nebraska-Lincoln, April 2009
• Joint MAA-AMS Meetings, San Diego, January 2008

Summer Programs and Workshops Attended:

• Summer Graduate School on Commutative Algebra, Mathematical Sciences Research Institute, June 2011
• Macaulay2 Workshop, Colorado College, August 2010
• Mathematics Research Community (Commutative Algebra), Snowbird, Utah, July 2010
• School on Commutative Algebra and Interactions with Algebraic Geometry and Combinatorics, The Abdus Salam International Center for Theoretic Physics, Trieste, Italy, June 2010

Service

Committee Membership:

• Graduate Student Advisory Board, UNL Math Department, Fall 2007 through Fall 2009

Panel Discussion Panelist:

• NCUWM panel, Spring 2012
• "Tips from the Trenches" teaching assistant panel, Graduate Teaching Assistant Workshop, UNL College of Arts and Sciences, Summer 2011
• Panel on teaching, UNL Teaching Seminar, 2010

Other:

• Seminar Organizer for Commutative Algebra Reading Seminar, Fall 2011-present
• Annual Math Day at University of Nebraska - Lincoln
  Math Bowl Timer, 2007-2010
• Saturday Science, UNL Physics Department
  Teaching Assistant, Spring 2010

Resources

• A sample first-day handout (two tex files).
  Preamble file (tex) | Body text file (tex)
• Notes from Sri's class on Cohen-Macaulay modules over hypersurfaces (MA918, Spring 2012). Updated: 1/25/2012.< br/> Directory of notes
• My hyperlinked course notes with selected proofs and homework results (MA902, Spring 2009). Updated: 5/7/2009. (Incomplete.)
  Course Notes (pdf)
• Notes from the Algebraic Geometry Seminar (Fridays, Spring 2009). Updated: 5/7/2009.
  Notes (tex) | Notes (pdf) | Notes (zip)

Links

Math:

• KUMUNUjr 2012: The first annual KUMUNUjr conference website!
• Commutative Algebra Reading Seminar: The seminar line up for CARS (and other information about CARS and Compact CARS).
• Commutative Algebra Seminar: The seminar line up for CAS.
• UNL Math Department: Find out more about my department.
• Colorado College Math and Comp Sci Department: Find out more about my undergraduate department.
• Commalg.org: Database of upcoming conferences and events in commutative algebra.
• MathOverflow: A great way to lose an afternoon.
• Math World: An encyclopedia of mathematics!
• Wolfram|Alpha: An online mathematics computational tool.

Fun:

• Phutball, a great board game.
• SET, a great card game.
• Abstruse Goose, a math webcomic.
• Brown Sharpie, a math webcomic I (used to) write in my (former) spare time.