David Hilbert

My Research Interest:
Rings..&..Things

Emmy Noether


My primary research interest is commutative and computational algebra, with related interests in homological algebra and algebraic geometry.
In essence, these fields find their roots in the study of algebraic equations in relation to the geometry of their solutions. Such a line of investigation goes back at least to Descartes and the idea of coordinatizing the plane.
Nowadays, though, commutative algebra and algebraic geometry study the solutions of those equations by forming an algebraic object, called a ring, which consists of the `generic' solutions. The algebraic properties of these generic solutions then give insight into the geometric and algebraic nature of the equations.
The methods in use are no longer from algebra alone, but also from analysis and topology. Conversely, they have been extensively used in those fields as well, and have proven useful in other fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.

Eeyore

Here is the list of the volumes that I have edited:
[30] Commutative Algebra and its Interactions to Geometry (A. Corso and C. Polini, Eds.), Contemp. Math. 555, Amer. Math. Soc., Providence, RI, 2011.
[29] Algebra, Geometry and Their Interactions (A. Corso, J. Migliore and C. Polini, Eds.), Contemp. Math. 448, Amer. Math. Soc., Providence, RI, 2007.
[28] Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects (A. Corso, P. Gimenez, M. Vaz Pinto and S. Zarzuela, Eds.), Lecture Notes in Pure and Applied Mathematics 244, Chapman & Hall/CRC, Boca Raton, FL, 2006

Here is the updated list of my publications:
[27] On generalized Loewy lengths (with C. Huneke, C. Polini, and B. Ulrich), in progress.
[26] Distance in resolutions with applications to integral dependence, Loewy lengths, and blowup algebras (with C. Huneke, C. Polini, and B. Ulrich), in preparation.
[24] Iterated socles and integral dependence in regular rings (with S. Goto, C. Huneke, C. Polini, and B. Ulrich), arXiv:1409.5481, Trans. Amer. Math. Soc., to appear.
[25] Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas (with U. Nagel, S. Petrovic, and C. Yuen), Forum Math 29 (2017), 799-830.
[23] Bounds on the normal Hilbert coefficients (with C. Polini and M.E. Rossi), Proc. Amer. Math. Soc. 144 (2016), 1919-1930.
[22] Sally modules of m-primary ideals in local rings, Comm. Algebra 37 (2009), 4503-4515.
[21] Monomial and toric ideals associated to Ferrers graphs (with U. Nagel), Trans. Amer. Math. Soc. 361 (2009), 1371-1395.
[20] Specializations of Ferrers ideals (with U. Nagel), J. Algebraic Comb. 28 (2008), 425-437.
[19] Hilbert coefficients of ideals with a view toward blowup algebras (with C. Polini), in Syzygies and Hilbert functions, Lecture Notes in Pure and Applied Mathematics, 254 (2007), 41-65.
[18] Multiplicity of the special fiber ring of blowups (with C. Polini and W.V. Vasconcelos), Math. Proc. Camb. Phil. Soc. 140 (2006), 207-219.
[17] Integral closure of ideals and annihilators of homology (with C. Huneke, D. Katz and W.V. Vasconcelos), Lecture Notes in Pure and Applied Mathematics 244 (2006), 33-48.
[16] Depth of associated graded rings via Hilbert coefficients of ideals (with M.E. Rossi and C. Polini), J. Pure Appl. Algebra 201 (2005), 126-141.
[15] Core of projective dimension one modules (with C. Polini and B. Ulrich), manuscripta math. 111 (2003), 427-433.
[14] Cohen-Macaulayness of special fiber rings (with L. Ghezzi, C. Polini and B. Ulrich), Comm. Algebra 31 (special issue in honor of S. Kleiman) (2003), 3713-3734.
[13] Core and residual intersections of ideals (with C. Polini and B. Ulrich), Trans. Amer. Math. Soc. 354 (2002), 2579-2594.
[12] The structure of the core of ideals (with C. Polini and B. Ulrich), Math. Ann. 321 (2001), 89-105.
[11] A class of strongly Cohen-Macaulay ideals (with C. Polini), Bull. London Math. Soc. 33 (2001), 662-668.
[10] Gaussian ideals and the Dedekind-Mertens lemma (with S. Glaz), Lecture Notes in Pure and Applied Mathematics 217 (2001), 131-144.
[9] On residually S2 ideals and projective dimension one modules (with C. Polini), Proc. Amer. Math. Soc. 129 (2001), 1309-1315.
[8] A generalized Dedekind-Mertens lemma and its converse (with W. Heinzer and C. Huneke), Trans. Amer. Math. Soc. 350 (1998), 5095-5109.
[7] Sally modules and associated graded rings (with C. Polini and M. Vaz Pinto), Comm. Algebra, 26 (1998), 2689-2708.
[6] On the integral closure of ideals (with C. Huneke and W.V. Vasconcelos), manuscripta math. 95 (1998), 331-347.
[5] Generic Gaussian ideals (with W.V. Vasconcelos and R.H. Villarreal), J. Pure Appl. Algebra 125 (1998), 117-127.
[4] Reduction number of links of irreducible varieties (with C. Polini), J. Pure Appl. Algebra 121 (1997), 29-43.
[3] Links of prime ideals and their Rees Algebra (with C. Polini), J. Algebra 178 (1995), 224-238.
[2] Blowups of links of irreducible varieties, Ph.D. Thesis, Rutgers University, 1995.
[1] Links of prime ideals (with C. Polini and W.V. Vasconcelos), Math. Proc. Camb. Phil. Soc. 115 (1994), 431-436.