Adam Van Tuyl Department of Mathematics and Statistics McMaster University Hamilton, ON, Canada L8S 4L8 vantuyl@math.mcmaster.ca

Notes Supplementary Material People Home

My research is primarily in commutative algebra. My current research interests are: homological invariants, edge ideals, simplicial complexes, sets of points in (multi)-projective spaces, and toric ideals.

Lecture Notes (Classes and Workshops)

• In June-July 2017, I was one of the instructors for the PRAGMATIC summer school at the Universita di Catania, Italy. Here are a copy of my lecture notes for the workshop
  • PRAGMATIC Workshop Notes
  These lectures, along with my co-instructors, formed the basis of our monograph Ideal of Powers and Powers of Ideals
• During the Winter 2013 semester, J. Biermann and I ran a weekly seminar:
  • Applied Algebra and Discrete Math Seminar
  (You can only see the website via the Wayback Machine)
• In July 2011, I will be giving three one hour lectures on edge and cover ideals. These lectures are part of a workshop held at Castro Urdiales, Spain, as part of MONICA (Monomial Ideals, Compuations, and Applications). For more on this workshop, see
  • MONICA Workshop - July 2011
  These lecture notes will appear in Springer Lecture Notes in Mathematics sometime in Fall 2013.
• During the Fall 2007 semester, I co-taught a graduate course on Operations Research with Dr. W. Huang. As part of the course, the students and instructors typed up their lecture notes. You can see more about our seminar below:
  • Operations Research Seminar (2007)
• In July 2006, I will be giving one of the workshop lectures at the Fields Institute on Computational and Combinatorial Commutative Algbera. For more on this workshop, see
  • Fields Workshop -- July 2006
• During second sememster of the 2005-2006 school year, I ran a seminar on combinatorial commutative algebra. For more on this seminar, see
  • Combinatorial Commutative Algebra Seminar
  Included on this page are lecture notes typed up by my M.Sc. student Jing He.

Supplementary Research Material

• In two papers with Enrico Carlini and Elena Guardo, I studied problems related to star configurations. In both papers, we were required to compute a few special cases via computer. You can find the relevant code here:
  • Star configurations on generic hypersurfaces
  • Plane curves containing star configuration sets of points

• In the paper Revisiting the spreading and covering numbers, Ben Babock and I looked at the problem of computing some invariants of a graph that arises in a problem in algebraic geometry. As part of his summer NSERC USRA, Ben wrote a number of programs to compute and bound these invariants. You can find some this code here
  • Revisiting the spreading and covering numbers

• In my paper Zero-nonzero patterns for nilpotent matrices over finite fields with Kevin N. Vander Meulen, we looked at the problem of determining what zero-nonzero patterns allow nilpotence. As part of this project, we showed that one could use results from commutative algebra to elminate possible patterns as being potentially nilpotent. One can use computer programs, like CoCoA and Macaulay 2, to carry out the necessary computations. You can see worked out examples here:
  • Using computer algebra programs to find potentially nilpotent matrices

• With Chris Francisco and Andrew Hoefel, I wrote a Macaulay 2 package, entitled EdgeIdeals, that allows users to use experiment with hypergraphs and graphs via the edge ideal correspondence. See the papge
  • EdgeIdeals: a Macaulay 2 package for (hyper)graphs
  for more information.

• With Fabrizio Zanello, I introduced the notion of a levelable simplicial complex. We showed that a simplicial complex is levelable if and only if there existed a integral solution to a system of equations constructed from the facets of the complex. Please see
  • Programs for Levelable Simplicial Complexes
  for more on this, and a Mathematica program to determine if a complex is levelable.
• In my paper ACM sets of points in multiprojective space (joint with Elena Guardo) we looked for necessary and sufficient conditions for a set of points to be ACM. Our paper contains a number of examples to show that results in P1 x P1 cannot be easily extended to the general setting. To see the code CoCoA for our examples please see
  • Examples of ACM sets of points in multiprojective spaces.

• In a recent paper with Elena Guardo, we described an algorithm to compute the minimal resolution of double points in P¹ x P¹ with an ACM support. We have implemented this program into CoCoA and Macaulay 2. To see this code, please go to
  • Programs for Double Points in P¹ x P¹

People Here is a list of the many people I have worked with or supervised.

• Post-Docs
  • Jay Wang (jointly supervised with M. Harada and J. Rajchgot) 2021-2023
  • Sergio Da Silva (jointly supervised with M. Haradaand J. Rajchgot) 2020-2022
  • Jeremy Lane (jointly supervised with M. Harada) 2019-2022
  • Johannes Hofscheier (jointly supervised with M. Harada) 2017-2019
  • Federico Galetto (jointly supervised with M. Harada) 2015-2018
  • Jennifer Biermann 2011-2013

• PhD Students
  • Craig Kohne (co-supervised with A. Deza) Sept. 2017 - Aug. 2022 (expected)
    Thesis title: TBD
  • Graham Keiper, Sept. 2016 - December. 2021 (expected)
    Thesis title: Toric ideals of finite simple graphs
  • Eduardo Camps, Jan. 2020-June 2020
    (Visiting Ph.D. student from Escuela Superior de Fsica y Matemticas, IPN, Mexico)
  • Beatrice Picone (co-supervised with E. Guardo) Sept. 2015-December 2018 (Completed)
    Thesis title: Homological invariants of some special varieties.
    (Picone was a PhD student at the Universita di Catania, Italy.)
  • Miguel Eduardo Paczka, Jan. 2018-June 2018
    (Visiting Ph.D. student from Escuela Superior de Fsica y Matemticas, IPN, Mexico)

• MSc. Students
  • Maryam Nowroozi, MSc (completed May 2021) co-supervsied with Dr. M. Harada
    Thesis title: Virtual resolutions of sufficiently general points in P¹ x P¹.
  • Jarvis, Kennedy, MSc (completed August 2020) co-supervised with Dr. M. Harada
    Projecti title: An Algebraic Condition for a Complex to be Virtual
  • Michael Cox, MSc (completed December 2018) co-supervised with Dr. K.N. Vander Meulen
    Project Title: On condition numbers of companion matrices.
  • Jason Palombaro, MSc (completed December 2017)
    Project Title: On the Groebner bases of ideals of points in P¹ x P¹.
  • Michael Riddell, MSc (completed June 2017) co-supervised with Dr. K.N. Vander Meulen
    Project Title: Zero forcing number of circulant graphs.
  • Sam Budd, MSc (Completed June 2017)
    Project Title: A introduction to f-ideals and their complements.
  • Becky Hooper, MSc (Completed June 2017)
    Project Title: Shellability of the van der Waerden complex.
  • Ashwini Bhat, M.Sc. (Completed May 2012)
    Project: Associated Primes of Powers of the Alexander Dual of Path Ideals of Trees.
    (Update: A. Bhat, J. Biermann and myself turned this results into this journal article which appeared in the Journal of Pure and Applied Algebra.)
  • Kevin Jurcik, M.Sc. (Completed May 2009) co-supervised with W. Huang
    Project: Open Shop Scheduling to Minimize Makespan.
  • Jing (Jane) He, M.Sc. (Completed May 2007)
    Project: The path ideal of a tree and its properties.
    (Update: Jane and I turned her project into a this journal article which appeared in Comm. Alg.)

• NSERC USRA Students
  • Olivia Wdowiak, Summer 2003
    Project: Regularity of fat points.
  • John Kimball, (co-supervised with Greg Lee) Summer 2005
    Project: Graded Betti numbers of edge ideals.
  • Natalie Campbell, (co-supervised with Kevin N. Vander Meulen) Summer 2009
    Project: Nilpotent Patterns over Finite Fields.
  • Hannah Bergsma, (co-supervised with Kevin N. Vander Meulen) Summer 2010
    Project: Construction of Nilpotent Patterns.
  • Ben Babcock, Summer 2010
    Project: Revisiting the spreading and covering numbers.
  • Hannah Bergsma, (co-supervised with Kevin N. Vander Meulen) Summer 2011
    Project: Inertially artbitary patterns.
  • Ben Babcock, Summer 2011
    Project: Combinatorial Commutative Algebra.
  • Catriona Watt, (co-supervised with Kevin N. Vander Meulen) Summer 2012
    Project: Graded Betti numbers of some edge ideals.
  • Jonathan Earl, (co-supervised with Kevin N. Vander Meulen) Summer 2014
    Project: Independence complexes of circulant graphs.
  • Jonathan Baker, (co-supervised with Kevin N. Vander Meulen) Summer 2015
    Project: TBD
  • Jonathan Earl, (co-supervised with Kevin N. Vander Meulen) Summer 2015
    Project: TBD
• Honours Projects
  • Jessica Reinikka, Sept. 2011-April 2012
    Project: Mutually orthogonal Latin Squares and their applications.
  • Jesse Krauel, (co-supervised with Dr. J. Biermann) Sept 2012-April 2013
    Project: Some methods of primality testing.
  • Mitchell Gallinger, (co-supervised with Dr. J. Biermann) Sept 2012-April 2013
    Project: Groebner Bases: ideal membership and graph colouring.
  • Lindsey Daniels, (co-supevised with Dr. G. Lee) Sept 2013-April 2014
    Project: Group Theory and the Rubik's Cube.