My research

$Description: Description: Description: C:\Tai's Documents\public_html\images\write.gif$ Research

Last updated: November 2021

Publications and Preprints

The reviews of my published papers can be found on MathSciNet. Copies of manuscripts are on the arXiv.

Papers submitted and accepted
1. Newton-Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals (with Thai Thanh Nguyen)
  Submitted.
2. Depth functions and symbolic depth functions of homogeneous ideals (with Ngo Viet Trung)
  Submitted.
3. Saturation bounds for smooth varieties (with Lawrence Ein and Robert Lazarsfeld)
  To appear in Algebra and Number Theory.
4. The Alexander-Hirschowitz theorem and related problems (with Paolo Mantero)
  Submitted.
5. Density of f-ideals and f-ideals of mixed small degrees (with Graham Keiper, Hasan Mahmood and Jonathan O'Rourke)
  To appear in Mathematica Scandinavica.
6. Chudnovsky's Conjecture and the stable Harbourne-Huneke containment (with Sankhaneel Bisui, Eloísa Grifo and Thai Thanh Nguyen)
  To appear in Transactions of American Mathematical Societ.
Published in 2021-present
1. Demailly's Conjecture and the containment problem (with Sankhaneel Bisui, Eloísa Grifo and Thai Thanh Nguyen)
  Journal of Pure and Applied Algebra 226 (2022), no. 4, Paper No. 106863, 21 pp.
2. Resurgence numbers of fiber products of projective schemes (with Sankhaneel Bisui, A.V. Jayanthan and Abu C. Thomas)
  Collectanea Mathematica 72 (2021), no. 3, 605-614.
3. Symbolic powers of cover ideals of graphs and Koszul property (with Yan Gu and Joseph Skelton)
  International Journal of Algebra and Computation 31 (2021), no. 5, 865-881.
4. Max Min vertex cover and the size of Betti tables (with Takayuki Hibi)
  Annals of Combinatorics 25 (2021), no. 1, 115-132.
5. Depth functions of powers of homogeneous ideals (with Hop D. Nguyen, Ngo Viet Trung and Tran Nam Trung)
  Proceedings of American Mathematical Society 149 (2021), no. 5, 1837-1844.
Published in 2016-2020
1. Ideals of powers and powers of ideals: intersecting algebra, geometry, and combinatorics (with Enrico Carlini, Brian Harbourne and Adam Van Tuyl)
  Lecture Notes of the Unione Matematica Italiana, 27. Springer, Cham, 2020. xix+159 pp. ISBN: 978-3-030-45246-9; 978-3-030-45247-6.
2. Regularity of powers of edge ideals: from local properties to global bounds (with Arindam Banerjee and Selvi Beyarslan)
  Algebraic Combinatorics 3 (2020), no. 4, 839-854.
3. Depth of powers of squarefree monomial ideals (research) (with Louiza Fouli and Susan Morey)
  Advances in Mathematical Sciences, 161-171.
  Association for Women in Mathematics Series, 21, Springer, Cham, 2020.
4. Symbolic powers of edge ideals of graphs (with Yan Gu, Jonathan L. O'Rourke and Joseph W. Skelton)
  Communications in Algebra 48 (2020), no. 9, 3743-3760.
5. Initially regular sequences and depths of ideals (with Louiza Fouli and Susan Morey)
  Journal of Algebra 559 (2020), 33-57.
6. Fiber invariants of projective morphisms and regularity of powers of ideals (with Sankhaneel Bisui and Abu C. Thomas)
  Acta Mathematica Vietnamica 45 (2020), no. 1, 183-198..
7. Symbolic powers of sums of ideals (with Dang Hop Nguyen, Ngo Viet Trung and Tran Nam Trung)
  Mathematische Zeitschrift.
8. Edge ideals of oriented graphs (with Kuei-Nuan Lin, Susan Morey, Enrique Reyes and Rafael H. Villarreal)
  International Journal of Algebra and Computation 29 (2019), no. 3, 535-559.
9. Membership criteria and containments of powers of monomial ideals (with Ngo Viet Trung)
  Acta Mathematica Vietnamica 44 (2019), no. 1, 117-139.
10. Algebraic properties of toric rings of graphs (with Selvi Beyarslan and Augustine O'Keefe)
  Communications in Algebra 47 (2019), no. 1, 1-16.
11. Regularity of edge ideals and their powers (with Arindam Banerjee and Selvi Beyarslan)
  Advances in Algebra, 17-52.
  Springer Proceedings in Mathematics and Statistics 277, Springer, Cham, 2019.
12. Depth and regularity modulo a principal ideal (with Giulio Caviglia, Jürgen Herzog, Manoj Kummini, Naoki Terai and Ngo Viet Trung)
  Journal of Algebraic Combinatorics 49 (2019), no. 1, 1-20.
13. Symbolic powers of monomial ideals (with Susan M. Cooper, Robert J.D. Embree and Andrew H. Hoefel)
  Proceedings of Edinburg Mathematical Society (2) 60 (2017), no. 1, 39-55.
14. Depth and regularity of powers of sums of ideals (with Ngo Viet Trung and Tran Nam Trung)
  Mathematische Zeitschrift 282 (2016), no. 3-4, 819-838.
15. Growth of graded families of ideals (with Pham An Vinh)
  Journal of Algebra, 452 (2016), 311-323.
Published in 2011-2015
1. Regularity of powers of forests and cycles (with Selvi Beyarslan and Tran Nam Trung)
  Journal of Algebraic Combinatorics 42 (2015) no. 4, 1077-1095.
2. Betti numbers of subgraphs (with Duc Ho)
  Australasian Journal of Combinatorics 63 (2015), 182-195.
3. Normal 0-1 polytopes (with Kuei-Nuan Lin)
  SIAM Journal of Discrete Mathematics 29 (2015), no. 1, 210-223.
4. Partial colorings, vertex decomposability, and sequentially Cohen-Macaulay simplicial complexes (with Jennifer Biermann, Chris Francisco and Adam Van Tuyl)
  Journal of Commutative Algebra 7 (2015) no. 3, 337-352.
5. Squarefree monomial ideals that fail the persistence property and non-increasing depth (with Mengyao Sun)
  Acta Mathematica Vietnamica, 40 (2015), no. 1, 125-137.
6. Results on the regularity of squarefree monomial ideals (with Russ Woodroofe)
  Advances in Applied Mathematics, 58 (2014), 21-36.
7. Regularity of squarefree monomial ideals
  In S. Cooper and S. Sather-Wagstaff (Ed.) Connections between Algebra, Geometry and Combinatorics, 251-276. Springer Proceedings in Mathematics and Statistics, Vol. 56, 2014.
8. Eventual shape of Betti tables of powers of ideals (with Amir Bagheri and Marc Chardin)
  Mathematical Research Letters 20 (2013), no. 6, 1033-1046.
9. Pure O-sequences and matroid h-vectors (with Fabrizio Zanello and Erik Stokes)
  Annals of Combinatorics 17 (2013), no. 3, 495-508.
10. Powers of squarefree monomial ideals and combinatorics (with Chris A. Francisco and Jeff Mermin)
  In I. Peeva (Ed.) Commutative Algebra, 373-392, Springer, 2013.
11. Hilbert functions of double point schemes in P^2 (with A. V. Geramita)
  Vietnam Journal of Mathematics 39 (2011), 327-342.
12. Path ideals of rooted trees and their graded Betti numbers (with Rachelle R. Bouchat and Augustine O'Keefe)
  Journal of Combinatorial Theory, Series A 118 (2011), 2411-2425.
13. Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals (with Francisco and Van Tuyl)
  Journal of Algebra 331 (2011), 224-242.
14. Asymptotic linearity of regularity and a*-invariant of powers of ideals
  Mathematical Research Letters 18 (2011), no. 1, 1-9.
Published in 2006-2010
1. A conjecture on critical graphs and connections to the persistence of associated primes (with C. A. Francisco and A. Van Tuyl)
  Discrete Math. 310 (2010), 2176-2182.
2. Associated primes of monomial ideals and odd holes in graphs (with Chris A. Francisco and Adam Van Tuyl)
  Journal of Algebraic Combinatorics 32 (2010), no. 2, 287-301.
3. Embedded associated primes of powers of square-free monomial ideals (with Susan Morey)
  Journal of Pure Applied Algebra 214 (2010), no. 4, 301-308.
4. Splittings of monomial ideals (with Chris A. Francisco and Adam Van Tuyl)
  Proc. Amer. Math. Soc. 137 (2009), 3271-3282.
5. Cohen-Macaulay admissible clutters (with Susan Morey and Rafael H. Villarreal)
  Journal of Commutative Algebra 1 (2009), 463-480.
6. Cohen-Macaulay multigraded modules (with C-Y. Jean Chan and Christine Cumming)
  Illinois Journal of Mathematics 52 (2008), no. 4, 1147-1163.
7. Monomial ideals, edge ideals of hypergraphs, and their minimal graded Betti numbers (with Adam Van Tuyl)
  Journal of Algebraic Combinatorics 27 (2008), no. 2, 215-245.
8. Whiskers and sequentially Cohen-Macaulay graphs (with Chris A. Francisco)
  Journal of Combinatorial Theory, Series A 115 (2008), no. 2, 304-316.
9. Resolution of square-free monomial ideals via facet ideals: a survey (with Adam Van Tuyl)
  Contemporary Mathematics 448 (2007), 91-117.
10. Adjoint line bundles and syzygies of projective varieties
  Vietnam Journal of Mathematics 35 (2007), no. 2, 135-151.
11. Minimal free resolutions and asymptotic behaviour of multigraded regularity (with Brent Strunk)
  Journal of Algebra 311 (2007), no. 2, 492-510.
12. Multigraded regularity, a*-invariants and the minimal free resolution
  Journal of Algebra 310 (2007), no. 1, 156-179.
13. Splittable ideals and the resolutions of monomial ideals (with Adam Van Tuyl)
  Journal of Algebra 309 (2007), no. 1, 405-425.
14. Homology multipliers and the relation type of parameter ideals (with Ian M. Aberbach and Laura Ghezzi)
  Pacific Journal of Mathematics 226 (2006), no. 1, 1-40.
15. Toroidalization of generating sequences in dimension two function fields (with Olga Kashcheyeva and Laura Ghezzi)
  Journal of Algebra 301 (2006), no. 2, 838-866.
Published in 2001-2005
1. Asymptotic behaviour of the length of local cohomology (with S.Dale Cutkosky, Hema Srinivasan and Emanoil Theodorescu)
  Canadian Journal of Mathematics 57 (2005), no. 6, 1178-1192.
2. Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups (with N.V. Trung)
  Transactions of American Mathematical Society 357 (2005), no. 9, 3655-3672.
3. Arithmetic Macaulayfication of projective schemes (with S.Dale Cutkosky)
  Journal of Pure and Applied Algebra. 201 (2005), no. 1-3, 49-61. (in the special volume for Vasconcelos).
4. The depth of the associated graded ring of ideals with any reduction number (with Ian M. Aberbach and Laura Ghezzi)
  Journal of Algebra 276 (2004), 168-179.
5. Projective embeddings of projective schemes blown up at subschemes
  Mathematische Zeitschrift 246 (2004), no 1-2, 111-124.
6. The regularity of points in multiprojective spaces (with A. VanTuyl)
  Journal of Pure and Applied Algebra 187 (2004), no 1-3, 153-167.
7. On the Rees algebra of certain codimension two perfect ideals
  Manuscripta Mathematica 107 (2002), 479-501.
8. Box-shaped matrices and the defining ideal of certain blown up surfaces
  Journal of Pure and Applied Algebra 167 (2002), no 2-3, 203-224.
9. Computing the Spreading and Covering numbers (with E. Carlini and A. Van Tuyl)
  Communications in Algebra 29 (2001), no 12, 5687-5699.

Publications in non-refereed journals

The graph and the image of a rational map from Pⁿ to P^m. The Curves Seminar at Queen's. Vol. XII. 141-162, Queen's Papers in Pure and Appl. Math. 114, Queen's Univ., Kingston, ON, 1998. (with A. Van Tuyl).

Tutorial 2: A Chess Puzzle. Proceedings of COCOA VI International School, Queen's Papers in Pure and Appl. Math. 120 (2001), 215-221. (with E. Carlini and A. Van Tuyl).

Tutorial 3: Hilbert Function of Points. Proceedings of COCOA VI International School, Queen's Papers in Pure and Appl. Math. 120 (2001), 227-237. (with E. Carlini and A. Van Tuyl).

Tutorial 5: The Ideal Generation Conjecture. Proceedings of COCOA VI International School, Queen's Papers in Pure and Appl Math. 120 (2001), 245-262. (with E. Carlini and A. Van Tuyl).

Tutorial 6: The Minimal Resolution Conjecture. Proceedings of COCOA VI International School, Queen's Papers in Pure and Appl. Math. 120 (2001), 263-273. (with E. Carlini and A. Van Tuyl).

Thesis and Dissertation

PhD Thesis. Rational surfaces from an algebraic perspective. (PDF, PS).
Queen's University, Kingston, Canada.
Honours Dissertation. Cycles in graphs.
Curtin University of Technology, Perth, Australia.

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