Mathematics Department,
1409 W. Green St
University of Illinois
Urbana, IL 61801
tel: 217-333-2229
fax: 217-333-9576
I am a Professor in the Mathematics Department at University of Illinois. My research interests are interdisciplinary: my background is in commutative algebra and algebraic geometry. I'm especially interested in problems which can be studied from a computational standpoint, and in interactions with problems in applied mathematics. My current applied work is on two themes:
The method of shifted partial derivatives cannot separate the permanent from the determinant (with Klim Efremenko, J.M. Landsberg, Jerzy Weyman), Mathematics of Computation, to appear.
Logarithmic vector fields for curve configurations in P^2 with quasihomogeneous singularities, (with Hiroaki Terao and Masahiko Yoshinaga), Mathematical Research Letters, to appear.
Polynomial interpolation in higher dimension: from simplicial complexes to GC sets (with Nathan Fieldsteel), SIAM Journal on Numerical Analysis, 55 (2017), 131-143.
Subdivision and spline spaces (with Tatyana Sorokina), Constructive Approximation, to appear.
Algebraic methods in Approximation theory, Computer Aided Geometric Design, 45 (2016), 14-31.
Tensor product surfaces and linear syzygies (with Eliana Duarte), Proceedings of the A.M.S., 144 (2016), 65-72.
Chen ranks and resonance, (with
Dan Cohen), Advances in Mathematics, 285 (2015), 1-27.
Finitely many smooth d-polytopes with n lattice points, (with
Bogart, Haase, Hering, Lorenz, Nill, Paffenholz, Rote, Santos), Israel Journal of Mathematics, 207, (2015), 301-330.
Geometry of Wachspress surfaces, (with Corey Irving), Algebra & Number Theory, 8 (2014), 369-396.
Splines on the Alfeld split of a simplex and type A root systems, Journal of Approximation Theory, 182 (2014), 1-6.
Syzygies and singularities of tensor product surfaces of bidgree (2,1), (with Alexandra Seceleanu and Javid Validashti), Mathematics of Computation, 83 (2014), 1337-1372.
Local cohomology of logarithmic forms, (with Graham Denham, Matthias Schulze, Max Wakefield, Uli Walther), Annales de l'Institut Fourier, 63 (2013), 1177-1203.
Commutative algebra of subspace and hyperplane arrangements, (with Jessica Sidman), in Commutative Algebra, Springer Verlag (2013), 639-665.
Toric Hirzebruch-Riemann-Roch via
Ishida's theorem on the Todd genus, Proceedings of the A.M.S., 141 (2013), 1215-1217.
High rank linear syzygies on low rank quadrics, (with
Mike Stillman), American Journal of Mathematics, 134 (2012), 561-579.
Equivariant Chow cohomology of nonsimplicial toric varieties, in Transactions of the A.M.S., 364 (2012) 4041-4051.
Hyperplane Arrangements: Computations and Conjectures, in Advanced Studies in Pure Mathematics, 62, (2012) 323-358.
Recent developments and open problems in linear series, (joint with the MFO miniworkshop crew), Contributions to Algebraic Geometry (P. Pragacz, ed.) EMS publishing (2012) 93-140.
Resonance varieties via blowups of P^2 and scrolls, in International Mathematics Research Notices, 20, (2011) 4756-4778.
Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property, (with Brian Harbourne, Alexandra Seceleanu), in Journal of the London Mathematical Society, 84, (2011) 712-730.
Euler characteristic of coherent sheaves on simplicial torics via the Stanley-Reisner ring, in Journal of Mathematical Physics, 51, (2010) doi 112304.
The weak Lefschetz property and powers of linear forms in K[x,y,z],
(joint with
Alexandra Seceleanu), in Proceedings of the A.M.S., 138 (2010) 2335-2339.
Freeness of Conic-Line arrangements in P^2,
(joint with
Stefan Tohaneanu), in Commentarii
Mathematici Helvetici, 84 (2009) 235-258.
Holonomy Lie algebras and the LCS formula for
subarrangements of A_n,
(joint with
Paulo Lima-Filho), International Mathematics Research Notices, 8 (2009) 1421-1432.
Piecewise polynomials on polyhedral complexes, (joint with
Terry Mcdonald), Advances in Applied Mathematics, 42 (2009), 82-93.
The Orlik-Terao algebra and 2-formality,
(joint with
Stefan Tohaneanu), Mathematical Research Letters, 16 (2009), 171-182.
Efficient computation of resonance varieties via Grassmannians,
(joint with
Paulo Lima-Filho), Journal of Pure and Applied Algebra, 213 (2009), 1606-1611.
A case study in bigraded commutative algebra,
(joint with
David Cox and
Alicia Dickenstein), in "Syzygies and Hilbert Functions", edited
by
Irena Peeva, Lecture notes in Pure and Applied Mathematics 254,
(2007), 67--112.
Betti numbers and
degree bounds for some linked zero schemes,
(joint with
Leah Gold and
Hema Srinivasan),
Journal of Pure and Applied Algebra, 210, (2007), 481-491.
Syzygies, multigraded regularity and toric varieties,
(joint with
Milena Hering and
Gregory G. Smith), Compositio Mathematica, 142, (2006), 1499-1506.
Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence,
(joint with
Alex Suciu), in Transactions of the A.M.S., 358, (2006), 2269-2289.
Toric surface codes and
Minkowski sums,
(joint with
John Little), SIAM
Journal on Discrete Mathematics, 20, (2006), 999-1014.
Derivation modules of orthogonal duals of hyperplane arrangements,
(joint with
Joseph P.S. Kung), in Journal of Algebraic Combinatorics, 24, (2006), 253-262.
Cayley-Bacharach and
evaluation codes on complete intersections,
(joint with
Leah Gold and
John Little), in Journal of
Pure and Applied Algebra, 196, (2005) p. 91-99.
Linear systems on a
special rational surface, in Mathematical Research Letters, 11, (2004) p. 697-714.
Lattice polygons and
Green's theorem, in Proceedings of the A.M.S., 132, (2004) p. 3509-3512.
Elementary modifications and line configurations in P^2,
in Commentarii
Mathematici Helvetici, 78, (2003) p. 447-462.
Local complete intersections in P^2 and Koszul syzygies,
(joint with
David Cox), in Proceedings of the A.M.S., 131, (2003) p. 2007-2014.
Lower central series and free resolutions of hyperplane arrangements,
(joint with
Alex Suciu), in Transactions of the A.M.S., 354, (2002) p. 3409-3433.
Cohomology vanishing and a problem in approximation theory,
(joint with
Peter Stiller), in Manuscripta Mathematica, 107,
(2002) p. 43-58.
The module of logarithmic p-forms of a locally free arrangement,
(joint with Mircea Mustata), in Journal of Algebra, 241,
(2001) p. 699-719.
On a conjecture of Rose,
(joint with John Dalbec),
in
Journal of Pure and Applied Algebra, 165, (2001) p. 151-154.
A rank two vector bundle associated to a three arrangement, and its Chern
polynomial, in
Advances in Mathematics, 149,
(2000) p. 214-229.
Subalgebras of the Stanley-Reisner ring, in Discrete and
Computational Geometry, 21, (1999) p. 551-556.
Fat points, inverse systems, and piecewise polynomial functions,
(joint with
Anthony Geramita), in Journal of Algebra, 204,
(1998) p. 116-128.
A spectral sequence for splines, in Advances in Applied
Mathematics, 19, (1997) p. 183-199.
Local cohomology of bivariate splines,
(joint with
Mike Stillman), in Journal of Pure and Applied Algebra, 117-118, (1997) p. 535-548.
A family of ideals of minimal regularity and the Hilbert series of C^r(\Delta),
(joint with
Mike Stillman), in Advances in Applied
Mathematics, 19, (1997) p. 169-182.
On minimal free resolutions of sub-permanents and other ideals arising in complexity theory (joint with Klim Efremenko, J.M. Landsberg, Jerzy Weyman).
The Weak Lefschetz property for quotients by Quadratic Monomials (joint with Juan Migliore and Uwe Nagel).
Macaulay2: A program for computing in commutative algebra and
algebraic geometry.
CoCoA: Another option for computing in commutative algebra and
algebraic geometry.
Porta (and others): Programs for
polyhedral geometry.
Singular: A program for computations
in local rings and singularities.
Schubert: A program for enumerative
geometry.
Splinetool:
Macaulay2 script which builds the chain complexes and allows computation of
various homology modules
associated to splines. Arrangetool:
Macaulay2 script for computations involving three arrangements.
ams web page
siam web page
Algebraic
Geometry preprints MSRI (Berkeley).
