___________________________________________________________________ Hal Schenck

Hal Schenck

Mathematics Department, 1409 W. Green St
University of Illinois
Urbana, IL 61801
tel: 217-333-2229
fax: 217-333-9576



Who am I?

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:

On the more theoretical side, I enjoy working on problems at the interface of discrete geometry and algebra; for example I've also worked on questions involving free resolutions and syzygies, coding theory, rational surfaces and postulation, stability and jump loci of vector bundles, and geometric complexity theory. I did my graduate work at Cornell, and postdoctoral work at Cornell, Harvard, and Northeastern; prior to coming to Illinois I was a professor at Texas A&M. Here's my vita html and pdf.

Information for students

In Spring 2017 I'm teaching Math 231: Calculus II and Math 511: Algebraic Geometry. I also recently taught a minicourse for talented high school students on the mathematics of Google. For students interested in commutative algebra and algebraic geometry, here are a few thoughts on what to read and how to choose an advisor. Below are webpages of students and postdocs:
Here are notes from a lunch talk I've given several times on the Job Hunt.



Upcoming/Recent talks


Conference and Workshop Organization



o Logarithmic vector fields for curve configurations in P^2 with quasihomogeneous singularities, (with Hiroaki Terao and Masahiko Yoshinaga), Mathematical Research Letters, to appear.
o Polynomial interpolation in higher dimension: from simplicial complexes to GC sets (with Nathan Fieldsteel), SIAM Journal on Numerical Analysis, to appear.
o Subdivision and spline spaces (with Tatyana Sorokina), Constructive Approximation, to appear.
o Algebraic methods in Approximation theory, Computer Aided Geometric Design, 45 (2016), 14-31.
o Tensor product surfaces and linear syzygies (with Eliana Duarte), Proceedings of the A.M.S., 144 (2016), 65-72.
o Chen ranks and resonance, (with Dan Cohen), Advances in Mathematics, 285 (2015), 1-27.
o 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.
o Geometry of Wachspress surfaces, (with Corey Irving), Algebra & Number Theory, 8 (2014), 369-396.
o Splines on the Alfeld split of a simplex and type A root systems, Journal of Approximation Theory, 182 (2014), 1-6.
o Syzygies and singularities of tensor product surfaces of bidgree (2,1), (with Alexandra Seceleanu and Javid Validashti), Mathematics of Computation, 83 (2014), 1337-1372.
o Local cohomology of logarithmic forms, (with Graham Denham, Matthias Schulze, Max Wakefield, Uli Walther), Annales de l'Institut Fourier, 63 (2013), 1177-1203.
o Commutative algebra of subspace and hyperplane arrangements, (with Jessica Sidman), in Commutative Algebra, Springer Verlag (2013), 639-665.
o Toric Hirzebruch-Riemann-Roch via Ishida's theorem on the Todd genus, Proceedings of the A.M.S., 141 (2013), 1215-1217.
o High rank linear syzygies on low rank quadrics, (with Mike Stillman), American Journal of Mathematics, 134 (2012), 561-579.
o Equivariant Chow cohomology of nonsimplicial toric varieties, in Transactions of the A.M.S., 364 (2012) 4041-4051.
o Hyperplane Arrangements: Computations and Conjectures, in Advanced Studies in Pure Mathematics, 62, (2012) 323-358.
o 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.
o Resonance varieties via blowups of P^2 and scrolls, in International Mathematics Research Notices, 20, (2011) 4756-4778.
o 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.
o Euler characteristic of coherent sheaves on simplicial torics via the Stanley-Reisner ring, in Journal of Mathematical Physics, 51, (2010) doi 112304.
o 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.
o Freeness of Conic-Line arrangements in P^2, (joint with Stefan Tohaneanu), in Commentarii Mathematici Helvetici, 84 (2009) 235-258.
o 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.
o Piecewise polynomials on polyhedral complexes, (joint with Terry Mcdonald), Advances in Applied Mathematics, 42 (2009), 82-93.
o The Orlik-Terao algebra and 2-formality, (joint with Stefan Tohaneanu), Mathematical Research Letters, 16 (2009), 171-182.
o Efficient computation of resonance varieties via Grassmannians, (joint with Paulo Lima-Filho), Journal of Pure and Applied Algebra, 213 (2009), 1606-1611.
o 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.
o 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.
o Syzygies, multigraded regularity and toric varieties, (joint with Milena Hering and Gregory G. Smith), Compositio Mathematica, 142, (2006), 1499-1506.
o 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.
o Toric surface codes and Minkowski sums, (joint with John Little), SIAM Journal on Discrete Mathematics, 20, (2006), 999-1014.
o Derivation modules of orthogonal duals of hyperplane arrangements, (joint with Joseph P.S. Kung), in Journal of Algebraic Combinatorics, 24, (2006), 253-262.
o 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.
o Linear systems on a special rational surface, in Mathematical Research Letters, 11, (2004) p. 697-714.
o Lattice polygons and Green's theorem, in Proceedings of the A.M.S., 132, (2004) p. 3509-3512.
o Elementary modifications and line configurations in P^2, in Commentarii Mathematici Helvetici, 78, (2003) p. 447-462.
o 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.
o 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.
o Cohomology vanishing and a problem in approximation theory, (joint with Peter Stiller), in Manuscripta Mathematica, 107, (2002) p. 43-58.
o The module of logarithmic p-forms of a locally free arrangement, (joint with Mircea Mustata), in Journal of Algebra, 241, (2001) p. 699-719.
o On a conjecture of Rose, (joint with John Dalbec), in Journal of Pure and Applied Algebra, 165, (2001) p. 151-154.
o A rank two vector bundle associated to a three arrangement, and its Chern polynomial, in Advances in Mathematics, 149, (2000) p. 214-229.
o Subalgebras of the Stanley-Reisner ring, in Discrete and Computational Geometry, 21, (1999) p. 551-556.
o Fat points, inverse systems, and piecewise polynomial functions, (joint with Anthony Geramita), in Journal of Algebra, 204, (1998) p. 116-128.
o A spectral sequence for splines, in Advances in Applied Mathematics, 19, (1997) p. 183-199.
o Local cohomology of bivariate splines, (joint with Mike Stillman), in Journal of Pure and Applied Algebra, 117-118, (1997) p. 535-548.
o 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.


o On minimal free resolutions of sub-permanents and other ideals arising in complexity theory (joint with Klim Efremenko, J.M. Landsberg, Jerzy Weyman).
o The method of shifted partial derivatives cannot separate the permanent from the determinant (joint with Klim Efremenko, J.M. Landsberg, Jerzy Weyman). ___________________________________________________________________

Computational mathematics

oMacaulay2: A program for computing in commutative algebra and algebraic geometry.
oCoCoA: Another option for computing in commutative algebra and algebraic geometry.
oPorta (and others): Programs for polyhedral geometry.
oSingular: A program for computations in local rings and singularities.
oSchubert: A program for enumerative geometry.
oSplinetool: Macaulay2 script which builds the chain complexes and allows computation of various homology modules associated to splines.
oArrangetool: Macaulay2 script for computations involving three arrangements.

Useful math sites

o ams web page
o siam web page
o Algebraic Geometry preprints
oMSRI (Berkeley).