About Courtney

I am a sixth year graduate student in Mathematics at the University of Nebraska-Lincoln. I study commutative and homological algebra, and my primary research interest is the study of infinite free resolutions. My dissertation advisors are Lucho Avramov and Roger Wiegand.

Contact Information:

Office:	Avery Hall 344
Office Hours (Fall 2012):	By Appointment
Email:

Upcoming Trips: Jan 8-13 @JMM San Diego.
If you'd like to schedule an appointment, please take a look at my schedule: I am grateful for the grants that have partially supported my graduate studies: MCTP, GAANN, NebraskaMATH, Project Fulcrum, and my advisors' NSF grants.

Background

Education:

• Ph.D. Mathematics; Expected May, 2013
• 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
• A university-wide award given to two graduate teaching assistants each year.
• Emeritus Faculty Fellowship, UNL Math Department, Spring 2010
• Given each academic year by the department to top doctoral students
• Graduate Fellowship, UNL Graduate Fellowship Committee, Spring 2009
• In recognition of outstanding academic performance and scholarly potential; approximately 25 fellowships awarded each year
• Outstanding Qualifying Exams, UNL Math Department, Fall 2008
• Given each academic year by the department to the student who produces the most outstanding qualifying exams the previous spring
• 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 Top award for mathematical prowess
• Phi Beta Kappa
• Golden Key Honor Society

Membership in Professional Societies:

• American Mathematics Society
• Association for Women in Mathematics

Teaching

Current Courses:

• Fall 2012: Math 811T (Functions for High School Teachers);
  Section 700: Students, please rely on Blackboard for all course content.

Teaching Evalution Data:

• Student Evaluation Data and Comments (pdf); updated September, 2012.

Past Courses Taught:

• Courses taught as principal lecturer:
  • Math 101 (College Algebra)
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  • 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)
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  • 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 problems.
  • Math 300 (Math Matters)
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  • For future elementary educators. Develop a deeper understanding of "number and operation." 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:
  • Teaching assistant for Math 806T (Number Theory and Cryptology)
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  • The number theory required to understand RSA encryption, with a survey of other cryptographic techniques.
  • Teaching assistant for Math 850T (Discrete Mathematics for Middle Level Teachers)
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  • Concepts of discrete mathematics, as opposed to continuous mathematics, which extend in directions beyond, but related to, topics covered in middle-level curricula. Problems which build upon middle-level mathematics experiences. Logic, mathematical reasoning, induction, recursion, combinatorics, matrices, and graph theory. MATH 805T is intended for mid-level mathematics teachers.
  • Super TA for Math 106 (Calculus I) and Math 107 (Calculus II)
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  • The "Super TA" acts as a resource for first time recitation instructors. Throughout the fall semester, I observed these instructors, provided constructive feedback to each, and made follow-up visits upon request.
  • Resident scientist for Project Fulcrum
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  • 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, leading laboratory experiments, and facilitating teamwork among the seventh graders. (classroom content available here)
  • Teaching assistant and mentor for Intensive Mathematics: a Mentoring, Education, and Research Summer Experience (IMMERSE).
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  • These traineeships fully fund the first year of graduate school. MCTP Fellows also serve as TAs for the Intensive Mathematics: a Mentoring, Education, and Research Summer Experience (IMMERSE) program for students transitioning from undergraduate to graduate school
  • NebraskaMATH research assistant
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  • As a research assistant for the NebraskaMATH Noyce program, I participated in research activity designed to help strengthen curriculum for teachers in traditionally under-served schools, including group peer-review of research papers. I also acted as a mathematical resource for Nebraska teachers preparing "Dinner and a Math Problem" events for their colleagues.

Research

Papers and Preprints:

• Modules over short Gorenstein rings, in preparation
• Non-simplicial decompositions of Betti diagrams of complete intersections (with Jack Jeffries, Sarah Mayes, Claudiu Raicu, Branden Stone, and Bryan White) - arXiv 1301.3441 Preprint; Submitted January 2013.
• Abstract: We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij--Soederberg theory. That is, given a Betti diagram, we decompose it into pure diagrams; however we relax the requirement that the corresponding degree sequences for these diagrams be totally ordered. In doing so, we are able to define a multiplication law for Betti diagrams that respects the decomposition. In the more traditional sense, the decomposition of complete intersections of codimension at most 3 are also computed as given by the totally ordered decomposition algorithm obtained from (Eisenbud-Schreyer, 2009). In higher codimensions, obstructions arise that inspire our work on an alternative algorithm.
• The cone of Betti diagrams over a hypersurface ring of low embedding dimension (with Christine Berkesch, Jesse Burke, and Daniel Erman) - Journal of Pure and Applied Algebra, Volume 216, Issue 10, October 2012, Pages 2256--2268; arXiv 1109.5198
• 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
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• 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
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  • 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:

Invited Talks

• "Modules over short Gorenstein rings," AMS Western Sectional Meeting, University of Arizona in Tucson, Fall 2012
• "New Directions in Boij-Soederberg Theory," AMS Midwestern Sectional Meeting, UNL, Fall 2011 (slides)

Seminar Talks

University of Michigan algebra seminar

• November 28, 2012

Bard College seminar

• "Fun and (Sudoku) games with polynomials," November 16, 2012

William Jewell College

• "The German Tank Problem: statistics and the war effort," November 12, 2012

University of Utah algebra seminar

• "Betti numbers over a small quadric hypersurface," March 23, 2012

University of Texas at Arlington commutative algebra seminar

• "Betti numbers over a small quadric hypersurface," March 9, 2012

Colorado College mathematics department seminar

• "The Determinant Trick," April 23, 2010
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• 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.

University of Nebraska-Lincoln commutative algebra seminar

• "Boij-Soederberg Theory I: Betti diagrams over polynomial rings," September 14, 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.
• "Boij-Soederberg Theory II: Betti diagrams over k[x,y]/(x²)," September 15, 2010
• 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.

Unversity of Nebraska-Lincoln commutative algebra reading seminar

• "Matrix Factorizations Revisited," January 24, 2012
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  • 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)
• "More on the homology of three-generated ideals," September 13, 2011
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• 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.)
• "One-dimensional hypersurfaces and the cone of Betti diagrams of finite length modules," April 7, 2011
• "Hilbert Series, Polynomials, and Functions," Spring 2011
• "Derivators," Fall 2009

University of Nebraska-Lincoln graduate student seminar (mathematics)

• "A Mathemagical Card Trick," February 27, 2012
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• Abstract: A magician never reveals her tricks, but a mathematician will tell anyone who will listen! I'll demonstrate a neat card trick and then show you how it works. For this GSS, it will be helpful to brush up on your binary representations of the numbers 0 through 8. Feel free to bring along a deck of cards--we might have some time to practice.
• "A Seance to Summon Hilbert," October 31, 2011
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• 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).
• "Solving Sudoku with Groebner Bases...Theoretically," September 3, 2010
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• Abstract: Sudoku puzzles! I know you love them, because I watch you do them every week in GSS.* Why should computers be left out of the fun? In this talk, I'll explain two ways to embed Sudoku puzzles into polynomial rings (as ideals). Then I'll talk about Groebner bases and how they can be used to find the solution(s) to a given puzzle.
  *Except for Nathan, who does Kenken.
• "The Determinant Trick," September 30, 2009
• (click to expand/hide abstract)
• 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 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.
• "The History of the Abacus," October 8, 2008
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• Abstract: "Clown: Let me see. Every 'leven wether tods; every tod yields pound and odd shilling; fifteen hundred shorn--what comes the wool to? I cannot do't without counters."
  --Shakespeare's The Winter's Tale, 1623
  What is a wether, and how does it tod? How does one compute simple interest on an abacus? How does one compute *anything* on an abacus? Come to GSS find out!
• "Squaring the Circle," April 1, 2008
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• Abstract: "Three and a bit, that's pi. Only he said it was untidy, so he designed a wheel where pi was exactly three."
  "But that's impossible!" he said. "You can't do that! Pi is like...built in. You can't change it. You'd have to change the universe!"
  "Yes, sir. They tell me that's what happened."
  --Terry Pratchett, Going Postal

Posters:

• AWM Poster Session, Joint MAA-AMS Meetings, San Diego, CA, January 2013
• Undergraduate Poster Session, Joint MAA-AMS Meetings, New Orleans, LA, January 2007

Conferences:

• Joint MAA-AMS Meetings, San Diego, January 2013
• AMS Sectional Meeting, Tuscon, AZ, October 2012
• KUMUNUjr, University of Nebraska-Lincoln, April 2012
• AMS Sectional Meeting, Salt Lake City, UT, October 2011
• AMS Sectional Meeting, Lincoln, NE, 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:

• Joint Introductory Workshop: Cluster Algebras and Commutative Algebra, Mathematical Sciences Research Institute, August 2012
• 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:

• Nebraska Conference for Undergraduate Women in Mathematics graduate student panel, Spring 2012
• "Tips from the Trenches" teaching assistant panel, Graduate Teaching Assistant Workshop, UNL College of Arts and Sciences, August 2011
• Panel on teaching, UNL Teaching Seminar, 2010

Other:

• Conference organizer for KUMUNUjr, 2012-2013
• Seminar (co)organizer for Commutative Algebra Reading Seminar, Fall 2011-Spring 2012
• Annual Math Day at University of Nebraska - Lincoln volunteer, 2007-2012
• The purpose of Math Day is to stimulate interest in mathematics among Nebraska high school students, to encourage them to pursue mathematics or mathematics based science as a career and to recognize mathematical ability.
• Saturday Science teaching assistant, UNL Physics Department, Spring 2010
• During Saturday Science, about a hundred fifth and sixth-grade children spend two hours in the UNL Department of Physics and Astronomy for four consecutive Saturdays during the month of February. One hour is spent listening to a presentation by a university faculty member on a specific topic (spinning tops, electricity and magnetism, color and light, to name a few) and then the students spend the second hour working in small groups where they conduct hands-on experiments related to the same concept(s). Each week the lab activities include at least one project that the students make and take home with them.
• 2012 Campuswide Workshops for Graduate TAs, Disciplinary Breakout Session leader
• This session will focus on applying the principles of learning and effective teaching within the context of math and statistics discipline. Sessions will be facilitated by experienced faculty and award-winning teaching assistants.
• Pfizer–Novartis–LPS Science Fair judge, 2010, 2012

Resources

• SET problems and solutions: Solutions follow the problems (pdf)
• SET presentation: SET prezi (pdf) | SET prezi ("Prezi" at prezi.com)
• Contemporary Mathematics (Math 203) activities, journals, etc: email me!
• 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: 3/12/2012.
  Directory of notes
• Notes from the Algebraic Geometry Seminar (Fridays, Spring 2009). Updated: 5/7/2009.
  Notes (tex) | Notes (pdf) | Notes (zip)

Links

Math:

Seminars/Conferences:
• 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.

Departments:
• UNL Math Department: Find out more about my department.
• Colorado College Math and Comp Sci Department: Find out more about my undergraduate department.

Productive links:
• 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.

Etc

This is where I will eventually write things that make me sound like a human being. No one wants a robot for a TA.