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Research

msri   lie   oberwolfach   banff

RESEARCH INTERESTS: 

Commutative algebra, computational algebra, algebraic geometry, coding theory.

My main area of research is commutative algebra. I am especially interested in graded free resolutions and homological questions that relate geometry, topology and combinatorics. 

PAPERS

  1. "G-quadratic, LG-quadratic, and Koszul quotients of exterior algebras." submitted. (with Zach Mere) [arXiv]
  2. "Subadditivity of syzygies of ideals and related problems." submitted. [pdf]
  3. "Canonical modules and class groups of Rees-like algebras." to appear in Michigan Journal of Math.  (with Paolo Mantero and Lance Miller) [pdf]
  4. "Singularities of Rees-like algebras." Mathematische Zeitschrift, Volume 297 (2021), no. 1-2, 535–555. (with Paolo Mantero and Lance Miller) [arXiv] [journal link]
  5. "On the maximal graded shifts of ideals and modules." Journal of Algebra (Special volume in honor of Craig Huneke), Volume 571, Number 1 (2021), Pages 121-133. [arXiv] [journal link]
  6. "The regularity conjecture for prime ideals in polynomial rings." EMS Surveys in Mathematical Sciences, Volume 7 (2020), no. 1, 173-206. (with Irena Peeva) [pdf]
  7. "Green-Lazarsfeld condition for toric edge ideals of bipartite graphs." Journal of Algebra, Volume 562 2020), 1-27.  (with Zach Greif) [arXiv] [journal link]
  8. "Quadratic Gorenstein algebras with many surprising properties." Archiv der Mathematik, Volume 115 (2020), no. 5, 509-521.  (with Alexandra Seceleanu) [arXiv] [journal link]
  9. "Asymptotically good homological error correcting codes." Journal of Algebra Combinatorics Discrete Structures and Applications, Volume 6 , Issue 3, (2019) 135 - 145. (with Heather Newman) [pdf] [journal link]
  10. "Stillman's question for exterior algebras and Herzog's conjecture on Betti numbers of syzygy modules." Journal of Pure and Applied Algebra, Volume 223, Issue 2, February 2019, Pages 634-640 . [arXiv] [journal link]
  11. "The projective dimension of three cubics is at most 5." Journal of Pure and Applied Algebra, Volume 223, Issue 4, April 2019, Pages 1383-1410. (with Paolo Mantero) [arXiv] [journal link]
  12. "Regularity of prime ideals." Mathematische Zeitschrift, February 2019, Volume 291, Issue 1–2, pp 421–435. (with Giulio CavigliaMarc ChardinIrena PeevaMatteo Varbaro) [pdf] [journal link]
  13. "Counterexamples to the Eisenbud-Goto regularity conjecture." (with Irena Peeva) Journal of the American Mathematical Society, Volume 31, Number 2 (2018), 473–496. [pdf] [journal link]
  14. "A finite classification of (x,y)-primary ideals of low multiplicity."(with Paolo Mantero) Collectanea Mathematica, 69 Issue 1 (2018), 107–130. [pdf] [journal link]
  15. "A tight bound on the projective dimension of four quadrics." (with Craig HunekePaolo Mantero and Alexandra Seceleanu) Journal of Pure and Applied Algebra, Volume 222, Issue 9, (2018), 2524-2551 [arXiv] [journal link]
  16. "Three themes on syzygies." (with Gunnar Fløystad and Irena Peeva) Bulletin of the American Mathematical Society 53 no. 3, July 2016, 415-435 [pdf] [journal link]
  17. "Multiple structures with arbitrarily large projective dimension on linear subspaces." (with Craig HunekePaolo Mantero and Alexandra Seceleanu) Journal of Algebra 447 (2016), 183–205. [arXiv] [journal link]
  18. "A multiplicity bound for graded rings and a criterion for the Cohen-Macaulay property." (with Craig HunekePaolo Mantero and Alexandra Seceleanu) Proceedings of the American Mathematical Society 143 no. 6 (2015), 2365-2377. [arXiv] [journal link]
  19. "Infinite graded free resolutions." (with Irena Peeva) Commutative Algebra and Noncommutative Algebraic Geometry Volume 1: Expository Articles (Eisenbud, Iyengar, Singh, Stafford, Van den Bergh eds.), Math. Sci. Res. Inst. Publ. No. 67, Cambridge University Press, New York (2015), 215–258. [pdf] [MSRI link
    (The pdf version here is updated from the print version and includes references to recent developments in the theory of Golod rings.)
  20. "Hypergraphs and regularity of square-free monomial ideals." (with Kuei-Nuan Lin) International Journal of Algebra and Computation 23 no. 7 (2013), 1573–1590. [arXiv] [journal link]
  21. "Projective dimension of codimension two algebras presented by quadrics." (with Craig HunekePaolo Mantero and Alexandra Seceleanu) Journal of Algebra 393 (2013), 170-186. [arXiv] [journal link]
  22. "Bounding projective dimension." (with Alexandra Seceleanu ) Commutative Algebra. Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday (I. Peeva, Ed.), Springer-Verlag, London (2013), 551-576. [pdf]
  23. "A polynomial bound on the regularity of an ideal in terms of half of the syzygies." Mathematics Research Letters. 19 (2012), no 3, 555-565.[arXiv] [journal link]
  24. "Ideals with larger projective dimension and regularity." (with Jesse Beder, Luis Nunez-BetancourtAlexandra SeceleanuBart Snapp and Branden Stone) Journal of Symbolic Computation 46 (2011), no. 10, 1105-1113. [arXiv] [journal link]
  25. "A family of ideals with few generators in low degree and large projective dimension." Proceedings of the American Mathematical Society. Volume 139, No. 6. pp. 2017-2023. [arXiv] [journal link]
  26. "A note on the strong direct summand conjecture." Proceedings of the American Mathematical Society. Volume 127, No. 9. 2009. pp. 2857-2864. [pdf] [journal link]
  27. "SRPT optimally uses faster machines to minimize flow time." (with Eric Torng) ACM Transactions on Algorithms. Volume 5, No. 1. November 2008. pp. 1-25. [pdf] [journal link]
  28. "A Generalized Floor Bound on the Minimum Distance of Geometric Goppa Codes." (with Benjamin Lundell) Journal of Pure and Applied Algebra. Volume 207, Issue 1. September 2006. pp. 155-164. [pdf] [journal link]
  29. "SRPT optimally uses faster machines to minimize flow time." (with Eric Torng) SODA '04: Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, January 2004. pp. 350–358. [pdf] [proceedings link]

Slides and Video of Selected Talks

  1. "The Eisenbud-Goto Conjecture" and "Rees-like Algebras" - IIT Bombay Virtual Commutative Algebra Seminar, 2021.  [Video and notes]
  2. "Subadditivity of Syzygies and Related Problems" Fellowship of the Ring - A worldwide online commutative algebra seminar, 2020. [Video & notes]
  3. "Green-Lazarsfeld Index for Toric Edge Ideals of Bipartite Graphs" JMM Denver, CO. 2020. [Slides]
  4. "Degrees of Projective Varieties" Colloquium at the University of Louisiana Lafayette, 2019. [Slides]
  5. "On the Maximal Graded Shifts of Modules over a Polynomial Ring" AMS Sectional Meeting. Special Session on Commutative Algebra. Hunter College. New York City, NY. 2017. [Slides]
  6. "A Tight Bound on the Projective Dimension of Four Quadrics" AMS Sectional Meeting. Special Session on Combinatorial and Computational Commutative Algebra. University of Georgia. Athens, GA. 2016. [Slides]
  7. "Graded Free Resolutions of Ideals" Department Colloquium. University of Nebraska. Lincoln, NE. 2015. [Slides]
  8. "Graded Free Resolutions of Ideals: A UCR Story" Department Colloquium. UC Riverside. Riverside, CA. 2014. [Slides]
  9. "Multiple Structures With Arbitrarily Large Projective Dimension Supported on Linear Spaces." AMS Sectional Meeting. Special Session on Recent Advances in Commutative Algebra. University of Louisville. Louisville, KY. 2013. [Slides]
  10. "Bounds on the Projective Dimension and Regularity of Ideals." MSRI workshop on "Representation Theory, Homological Algebra, and Free Resolutions" Berkeley, CA. 2013. [Video]
  11. "Hypergraphs and Regularity of Square-free Monomial Ideals." Joint AMS-MAA Math Meetings. Special Session on Commutative Algebra and Algebraic Geometry. San Diego, CA. 2013. [Slides]
  12. "Syzygy Bounds on the Regularity of Ideals." AMS Sectional Meeting. Special Session on Commutative Algebra. University of Arizona. Tuscon, AZ. 2012. [Slides]
  13. "MSRI Tutorial Session on Infinite Free Resolutions." Joint Workshop on Commutative Algebra and Cluster Algebras. Berkeley, CA. 2012 [M2File]
  14. "Ideals with Large(r) Projective Dimension and Stillman's Question." AMS Sectional Meeting, Special on Commutative Algebra. University of Utah. Salt Lake City, UT. 2011 [Slides]

Macaulay2 Code and Packages

  1. BigIdeal.m2 - This package generates the ideals defined in "Ideals with Larger Projective Dimension and Regularity" by Beder, McCullough, Nunez, Seceleanu, Snapp and Stone. These ideals have very large projective dimension and regularity relative to the degree and number of generators.
  2. PowerSeries.m2 - This package allows for computation of and manipulation of power series in which more series terms may always be computed later on. Support for rational functions and generating functions is built in.