DANIEL ERMAN: RESEARCH


My research is in commutative algebra and algebraic geometry, with applications to algorithms and computation.

Papers

  1. 48. A short proof of the Hanlon-Hicks-Lazarev Theorem
    (w/ Michael K. Brown).
  2. 47. Linear syzygies of curves in weighted projective space
    (w/ Michael K. Brown).
  3. 46. Matrix factorizations of generic polynomials.

  4. 45. Linear strands of multigraded free resolutions
    (w/ Michael K. Brown).
    Mathematisches Annalen (to appear).
  5. 44. Positivity and nonstandard graded Betti numbers
    (w/ Michael K. Brown).
  6. 43. Tate resolutions on toric varieties
    (w/ Michael K. Brown).
    JEMS, to appear.
  7. 42. Syzygies of P^1 x P^1: data and conjectures
    (w/ Juliette Bruce, Daniel Corey, Steve Goldstein, Robert P. Laudone, and Jay Yang).
    Journal of Algebra, to appear.
  8. 41. Minimal free resolutions of differential modules
    (w/ Michael K. Brown),
    Transactions of the AMS, to appear.
  9. 40. Characteristic Dependence of Syzygies of Random Monomial Ideals
    (w/ Caitlyn Booms and Jay Yang),
    SIAM Journal on Discrete Math 36 (2022), 682--701.
  10. 39. The SchurVeronese Package in Macaulay2
    (w/ Juliette Bruce, Steve Goldstein, and Jay Yang),
    Journal for Software in Algebra and Geometry, 11 (2022), 83--87.
  11. 38. Small projective spaces and Stillman Uniformity for sheaves,
    (w/ Steven V Sam and Andrew Snowden),
    Algebraic Geometry, 8 (2021), 374--388.
  12. 37. Tate Resolutions on Products of Projective Spaces: Cohomology and PushForward Complexes
    (w/ David Eisenbud and Frank-Olaf Schreyer).
  13. 36. Big polynomial rings with imperfect coefficient fields
    (w/ Steven V Sam and Andrew Snowden),
    Michigan Math Journal, 70 (2021), 649--672.
  14. 35. Strength and Hartshorne's Conjecture in high degree
    (w/ Steven V Sam and Andrew Snowden),
    Mathematisches Zeitschrift, 297 (2020), 1467--1471.
  15. 34. Interpolation over ZZ and torsion in class groups
    (w/ John D. Berman.)
    Journal of Commutative Algebra, to appear
  16. 33. An equivariant Hilbert Basis Theorem
    (w/ Steven V Sam and Andrew Snowden),
    Math. Res. Lett., 27 67-77, (2020).
  17. 32. Virtual Resolutions for a Product of Projective Spaces
    (w/ Christine Berkesch and Gregory G. Smith).
    Algebraic Geometry, 7, 460-481, (2020).
  18. 31. Big Polynomial rings and Stillman's conjecture
    (w/ Steven V Sam and Andrew Snowden).
    Inventiones, 218, 413-439 (2019).
  19. 30. A probabilistic approach to systems of parameters and Noether normalization
    (w/ Juliette Bruce)
    20. Algebra Number Theory 13, 2081-2102, (2019).
  20. 29. Generalizations of Stillman's conjecture via twisted commutative algebras
    (w/ Steven V Sam and Andrew Snowden). .
    IMRN, (2019), 12281--12304.
  21. 28. Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials
    (w/ Steven V Sam and Andrew Snowden).
    Bulletin of the AMS, 56, 87-114, (2019).
  22. 27. Random Flag Complexes and Asymptotic Syzygies
    (w/ Jay Yang).
    Algebra Number Theory, 12, 2151-2166 (2018).
  23. 26. Conjectures and computations about Veronese syzygies
    (w/ Juliette Bruce, Steve Goldstein, and Jay Yang).
    Experimental Mathematics, 29 (2018), 398-413.
  24. 25. Divergent series and Serre's intersection formula for graded rings,
    Advances in Mathematics 314, 573-582 (2017).
  25. 24. Categorified duality in Boij--Söderberg Theory and invariants of free complexes
    (w/ David Eisenbud.)
    J. Eur. Math. Soc. (JEMS), 19, 2657-2695, (2017).
  26. 23. Questions about Boij-Söderberg Theory
    (w/ Steven V Sam)
    Surveys on recent developments in algebraic geometry, eds Coskun, de Fernex, Gibney, (2017).
  27. 22. Gauss Composition for P^1, and the universal Jacobian of the Hurwitz space of double covers
    (w/ Melanie Matchett Wood.)
    Journal of Algebra, 470, 320-352 (2017).
  28. 21. Supernatural analogues of Beilinson Monads
    (w/ Steven V Sam.)
    Compositio Mahematica, 152 , 2545-2562 (2016).
  29. 20. Furstenberg sets and Furstenberg schemes over finite fields
    (w/ Jordan S. Ellenberg.)
    Algebra and Number Theory, 10, 1415-1436 (2016).
  30. 19. A quick proof of nonvanishing for asymptotic syzygies
    (w/ Lawrence Ein and Rob Lazarsfeld.)
    Algebraic Geometry, 3, 211-22 (2016).
  31. 18. Semiample Bertini Theorems over finite fields
    (w/ Melanie Matchett Wood.)
    Duke Math. J. 164, 1-38 (2015).
  32. 17. A heuristic for the distribution of point counts for random curves over a finite field
    (w/ Jeffrey D. Achter, Kiran S. Kedlaya, Melanie Matchett Wood, and David Zureick-Brown.)
    Philosophical Transactions A, 373 (2015).
  33. 16. Tate resolutions for products of projective spaces
    (w/ David Eisenbud and Frank-Olaf Schreyer.)
    Acta Math. Vietnam. 40, 5-36 (2015).
  34. 15. The Betti table of a high degree curve is asymptotically pure.
    Recent advances in algebraic geometry: A volume in honor of Rob Lazarsfeld's 60th birthday (ed. Christopher Hacon, Mircea Mustata, and Mihnea Popa)
    pp. 200-206, London Mathematical Society Lecture Notes (No. 417), Cambridge University Press, Cambridge, 2015.
  35. 14. Asymptotics of random Betti tables
    (w/ Lawrence Ein and Rob Lazarsfeld.)
    Journal für die reine und angewandte Mathematik (Crelle's Journal), 702, (2015) 55-75.
  36. 13. Tensor complexes: Multilinear free resolutions constructed from higher tensors
    (w/ Christine Berkesch, Manoj Kummini, and Steven V Sam.)
    J. Eur. Math. Soc. (JEMS) 15, 2257-2295 (2013).
  37. 12. Three flavors of extremal Betti tables
    (w/ Christine Berkesch and Manoj Kummini.)
    Commutative algebra: expository papers dedicated to David Eisenbud on the occasion of his 65th birthday (ed. Irena Peeva), 99-121, Springer, New York, 2013.
  38. 11. Filtering free resolutions (formerly titled Beyond numerics: The existence of pure filtrations)
    (w/ David Eisenbud and Frank-Olaf Schreyer.)
    Compositio Mathematica, 149, 754-772 (2013).
  39. 10. Murphy's Law for Hilbert function strata in the Hilbert scheme of points
    Mathematical Research Letters, 19, 1277-1281 (2012).
  40. 9. Shapes of free resolutions over a local ring
    (w/ Christine Berkesch, Manoj Kummini, and Steven V Sam.)
    Mathematische Annalen, 354, 939-9545 (2012).
  41. 8. Poset structures in Boij-Söderberg theory
    (w/ Christine Berkesch, Manoj Kummini, and Steven V Sam.)
    International Mathematics Research Notices, 2012, 5132-5160 (2012).
  42. 7. The cone of Betti diagrams over a hypersurface ring of low embedding dimension
    (w/ Christine Berkesch, Jesse Burke and Courtney Gibbons.)
    Journal of Pure and Applied Algebra, 216 (2012), 2256-2268.
  43. 6. Secant varieties of P^2 x P^n embedded by O(1,2)
    (w/ Dustin A. Cartwright and Luke Oeding.)
    Journal of the London Mathematical Society, 85 (2012), 121-141.
  44. 5. Laurent polynomials and Eulerian numbers
    (w/ Gregory G. Smith and Anthony Várilly-Alvarado.)
    Journal of Combinatorial Theory, Series A, 118 (2011), 396-402.
  45. 4. A syzygetic approach to the smoothability of zero dimensional schemes
    (w/ Mauricio Velasco.)
    Advances in Mathematics, 224 (2010), 1143-1166.
  46. 3. A special case of the Buchsbaum-Eisenbud-Horrocks rank conjecture
    Mathematical Research Letters, 17 (2010), 1079-1089.
  47. 2. The Semigroup of Betti Diagrams
    Algebra and Number Theory, 3 (2009), 341-365.
  48. 1. Hilbert schemes of 8 points
    (w/ Dustin A. Cartwright, Mauricio Velasco and Bianca Viray.)
    Algebra and Number Theory, 3 (2009), 763-795.