Research

msri   lie   oberwolfach   banff

RESEARCH INTERESTS: 

Commutative algebra, computational algebra, algebraic geometry.

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

PAPERS

  1. "Properties of LCM Lattices of Monomial Ideals." (with Matthew Dorang) submitted.
  2. "Hilbert functions." (with Juan Migliore, Satoshi Murai, Uwe Nagel, and Irena Peeva) submitted.
  3. "F-purity of binomial edge ideals." (with Adam LaClair) to appear in the Journal of the London Mathematics Society.
  4. "Koszul binomial edge ideals." (with Adam LaClair, Matthew Mastroeni, and Irena Peeva) to appear in Forum of Mathematics, Sigma.
  5. "Koszul graded Mobius algebras and strongly chordal graphs." Selecta Mathematica (N.S.) vol 31 (2025), no. 2, Paper No. 29. (with Adam LaClair, Matthew Mastroeni, and Irena Peeva) [arXiv 2412.18499]
  6. "Depth and singular varieties of exterior edge ideals." to appear in Rocky Mountain Journal of Mathematics. (with Matthew Mastroeni, Andrew Osborne, Joshua Rice, and Cole Willis) [arXiv 2208.03366]
  7. "Prime ideals and three-generated ideals with large regularity." Comptes Rendus Mathematique, Volume 362 (2024), p. 251-255. [arXiv 2305.06532]
  8. "Chow rings of matroids are Koszul." Mathematische Annalen, 387, 1819–1851 (2023). (with Matthew Mastroeni) [arXiv 2111.00393]
  9. "Canonical modules and class groups of Rees-like algebras." Michigan Mathematics Journal 73 (3), 571-591, (July 2023).
     (with Paolo Mantero and Lance Edward Miller) [Project Euclid journal link]
  10. "G-quadratic, LG-quadratic, and Koszul quotients of exterior algebras." Communications in Algebra. Volume 50 (2022), no. 8, 3284–3300.  (with Zach Mere) [arXiv 2105.13457]
  11. "Subadditivity of syzygies of ideals and related problems." Commutative Algebra. Expository Papers Dedicated to David Eisenbud on the Occasion of His 75th Birthday (I. Peeva, Ed.), Springer-Verlag, London (2022). [Springer link: Commutative Algebra]
  12. "Singularities of Rees-like algebras." Mathematische Zeitschrift, Volume 297 (2021), no. 1-2, 535–555. (with Paolo Mantero and Lance Edward Miller) [arXiv 1903.00973]
  13. "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 1801.08194]
  14. "The regularity conjecture for prime ideals in polynomial rings." EMS Surveys in Mathematical Sciences, Volume 7 (2020), no. 1, 173-206. (with Irena Peeva) [EMS Press journal link]
  15. "Green-Lazarsfeld condition for toric edge ideals of bipartite graphs." Journal of Algebra, Volume 562 (2020), 1-27.  (with Zach Greif) [arXiv 1908.02744]
  16. "Quadratic Gorenstein algebras with many surprising properties." Archiv der Mathematik, Volume 115 (2020), no. 5, 509-521.  (with Alexandra Seceleanu) [arXiv 2004.10237]
  17. "Asymptotically good homological error correcting codes." Journal of Algebra Combinatorics Discrete Structures and Applications, Volume 6 , Issue 3, (2019) 135 - 145. (with Heather Newman) [DergiPark journal link]
  18. "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 1307.8162]
  19. "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 1801.08195]
  20. "Regularity of prime ideals." Mathematische Zeitschrift, February 2019, Volume 291, Issue 1–2, pp 421–435. (with Giulio Caviglia, Marc Chardin, Irena Peeva, Matteo Varbaro) [DOI: 10.1007/s00209-018-2089-y]
  21. "Counterexamples to the Eisenbud-Goto regularity conjecture." (with Irena Peeva) Journal of the American Mathematical Society, Volume 31, Number 2 (2018), 473–496.
  22. "A finite classification of (x,y)-primary ideals of low multiplicity."(with Paolo Mantero) Collectanea Mathematica, 69 Issue 1 (2018), 107–130. [DOI: 10.1007/s13348-017-0196-4]
  23. "A tight bound on the projective dimension of four quadrics." (with Craig Huneke, Paolo Mantero and Alexandra Seceleanu) Journal of Pure and Applied Algebra, Volume 222, Issue 9, (2018), 2524-2551 [arXiv 1403.6334]
  24. "Three themes on syzygies." (with Gunnar Fløystad and Irena Peeva) Bulletin of the American Mathematical Society 53 no. 3, July 2016, 415-435
  25. "Multiple structures with arbitrarily large projective dimension on linear subspaces." (with Craig Huneke, Paolo Mantero and Alexandra Seceleanu) Journal of Algebra 447 (2016), 183–205. [arXiv 1301.4147]
  26. "A multiplicity bound for graded rings and a criterion for the Cohen-Macaulay property." (with Craig Huneke, Paolo Mantero and Alexandra Seceleanu) Proceedings of the American Mathematical Society 143 no. 6 (2015), 2365-2377. [arXiv 1401.6216]
  27. "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.
    (The pdf version here is updated from the print version and includes references to recent developments in the theory of Golod rings.)
  28. "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 1211.4301]
  29. "Projective dimension of codimension two algebras presented by quadrics." (with Craig Huneke, Paolo Mantero and Alexandra Seceleanu) Journal of Algebra 393 (2013), 170-186. [arXiv 1304.0745]
  30. "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. [DOI: 10.1007/978-1-4614-5292-8]
  31. "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 1112.0058]
  32. "Ideals with larger projective dimension and regularity." (with Jesse Beder, Luis Nunez-Betancourt, Alexandra Seceleanu, Bart Snapp and Branden Stone) Journal of Symbolic Computation 46 (2011), no. 10, 1105-1113.
  33. "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 1005.3361]
  34. "A note on the strong direct summand conjecture." Proceedings of the American Mathematical Society. Volume 127, No. 9. 2009. pp. 2857-2864.
  35. "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. [DOI: 10.1145/1435375.1435376]
  36. "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.
  37. "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. [DOI: 10.5555/982792.982842]

Slides and Video of Selected Talks

  1. "Regularity of Prime ideals" - SIAM Meeting on Applied Algebraic Geometry, Madison, WI, 2025.
  2. "Koszul Graded Moebius Algebras and Strongly Chordal Graphs" - AMS Sectional Meeting, Lawrence, KS, 2025.
  3. "The Koszul property for algebras associated to matroids" - 44th Annual Japan Symposium on Commutative Algebra, Hayama, Japan, 2023
  4. "Chow rings of matroids are Koszul" - Mathematische Forschungsinstitute Oberwolfach workshop on Resolutions and Singularities, 2023.
  5. "The Eisenbud-Goto Conjecture" and "Rees-like Algebras" - IIT Bombay Virtual Commutative Algebra Seminar, 2021.
  6. "Subadditivity of Syzygies and Related Problems" Fellowship of the Ring - A worldwide online commutative algebra seminar, 2020.
  7. "Green-Lazarsfeld Index for Toric Edge Ideals of Bipartite Graphs" JMM Denver, CO. 2020.
  8. "Degrees of Projective Varieties" Colloquium at the University of Louisiana Lafayette, 2019.
  9. "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.
  10. "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.
  11. "Graded Free Resolutions of Ideals" Department Colloquium. University of Nebraska. Lincoln, NE. 2015.
  12. "Graded Free Resolutions of Ideals: A UCR Story" Department Colloquium. UC Riverside. Riverside, CA. 2014.
  13. "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.
  14. "Bounds on the Projective Dimension and Regularity of Ideals." MSRI workshop on "Representation Theory, Homological Algebra, and Free Resolutions" Berkeley, CA. 2013.
  15. "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.
  16. "Syzygy Bounds on the Regularity of Ideals." AMS Sectional Meeting. Special Session on Commutative Algebra. University of Arizona. Tuscon, AZ. 2012.
  17. "MSRI Tutorial Session on Infinite Free Resolutions." Joint Workshop on Commutative Algebra and Cluster Algebras. Berkeley, CA. 2012 [M2File]
  18. "Ideals with Large(r) Projective Dimension and Stillman's Question." AMS Sectional Meeting, Special on Commutative Algebra. University of Utah. Salt Lake City, UT. 201

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.