Hal Schenck
Hal Schenck
Mathematics Department ,
1409 W. Green St
University of Illinois
Urbana, IL 61801
tel: 217-333-2229
fax: 217-333-9576
schenck@math.uiuc.edu
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:
Approximation theory, Interpolation, and Geometric modeling
Topological data analysis and trading networks, Multidimensional persistent homology
On the more theoretical side, I enjoy working on problems at the interface
of discrete geometry and algebra; for example
Hyperplane and hypersurface arrangements
Lefschetz properties
Toric varieties
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 Fall 2015 I taught Math 241: Calculus III and Math 502: Commutative Algebra .
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 .
Books
Computational Algebraic Geometry (Cambridge, 2003). This book grew out of an
undergraduate algebraic geometry class (Math 99 ) I taught at Harvard, and covers basics of commutative algebra and Grobner bases. It also gives a quick taste of homological algebra (Ext and Tor) and a bit of sheaf cohomology.
Toric Varieties
(American Mathematical Society, 2011). This book (joint with
David Cox and
John Little ) covers standard and advanced topics in toric geometry. In the first part we begin each chapter with a section on basic algebraic geometry background. Advanced topics include equivariant cohomology, Hirzebruch-Riemann-Roch, and the toric minimal model program.
Upcoming/Recent talks
Sante Fe Institute, Working group on geometric complexity theory (2016)
Oxford, Industrial and Applied Mathematics Seminar (2016)
KIAS conference on Syzygies, exterior algebra, and coherent sheaf cohomology, Jeju, Korea, (2016)
BIRS-CMO Computational algebra and surface modeling, Oaxaca, (2016)
Mathematical Methods for Curves and Surfaces, Tonsberg, Norway (2016)
INdAM Homological and computational methods in commutative algebra, Cortona (2016)
MFO Toric geometry, Oberwolfach (2016)
BIRS Lefschetz Properties and Artinian Algebras, Banff (2016)
MFO Arrangements of subvarieties and applications, Oberwolfach (2016)
ICERM Computational topology in arrangement theory, Brown University (2015) (stream talk)
Geometry and Data Analysis, University of Chicago (2015)
ICMS conference on Minimal free resolutions, Betti numbers, and combinatorics, Edinburgh (2015)
Oxford, Geometry and Analysis Seminar (2015)
MFO workshop on approximation theory and algebraic geometry, Oberwolfach (2015)
FoCM Computational algebraic geometry, Montevideo (2014)
INdAM conference on Configuration Spaces, Cortona (2014)
BIRS Hyperplane Arrangements, Wonderful Compactifications, and Tropicalization, Banff (2014)
Southwest Local Algebra Meeting, Texas A&M (2014)
Virginia Tech colloquium (2014)
University of Tennessee colloquium (2014)
Texas A&M colloquium (2014)
RIMS conference on Characteristic classes and hyperplane arrangements, Kyoto (2013)
KUMUNU conference on Commutative algebra and algebraic geometry, Missouri (2013)
SIAM algebraic geometry meeting (2013) and toric tutorial (2013)
NSF-DFG Graduate/Postdoc summer school: Syzygies in Berlin (2013)
BIRS Algebraic Geometry and Geometric Modeling, Banff (2013) (stream talk)
MSRI year on commutative algebra, (2012-13)
AMS sectional meeting invited address, New Orleans (2012)
MFO Cohomology rings and fundamental groups of arrangements, Oberwolfach (2012)
BIRS Syzygies in Algebraic Geometry, Banff (2012)
MFO Toric Geometry, Oberwolfach (2012)
PIMS conference: Arrangements and applications, Vancouver (2011)
INdAM-MSRI-SMI workshop on toric varieties, Cortona (2011)
BIRS Topological methods in toric geometry, symplectic geometry, and combinatorics, Banff (2010)
MFO Linear series on algebraic varieties, Oberwolfach (2010)
Configuration spaces: geometry, combinatorics, topology, Centro De Giorgi, Pisa (2010)
AMS special session on Combinatorial algebra, Lexington (2010)
AMS special session on Zonotopal algebra, San Francisco (2010)
Mathematical Society of Japan seasonal institute, Hokkaido (2009)
AIM Combinatorial challenges in toric varieties (2009)
2nd workshop on Algebraic Geometry and Approximation Theory (2009)
Bluegrass Algebra conference (2009)
Fields Institute Conference in honor of Peter Orlik (2008)
Macaulay2 Conference, Cornell (2008)
MFO Surface modelling and syzygies, Oberwolfach (2007)
MFO Toric varieties and projective normality, Oberwolfach (2007)
Conference and Workshop Organization
BIRS-CMO workshop on computational algebra and surface modeling, Oaxaca, (2016)
MFO workshop on toric geometry, Oberwolfach (2016)
ICERM workshop on Computational geometric topology in arrangement theory, Brown University (2015)
MFO workshop on approximation theory and algebraic geometry, Oberwolfach (2015)
AMS special session on toric varieties (2014)
Macaulay2 Research Meeting and School (2014)
IMSE symposium on Applied Geometry, Topology and Networks (2014)
SIAM tutorial on toric varieties: math in the mountains (2013)
NSF-DFG Graduate/Postdoc summer school: Syzygies in Berlin (2013)
AMS special session on approximation theory and geometric modelling, New Orleans (2012)
Illinois-Purdue Algebra Geometry and Combinatorics Day 4/28/2012
Commutative Algebra and Algebraic Geometry Conference - UIUC (2011)
PIMS conference: Arrangements and applications-Vancouver (2011)
INdAM-MSRI-SMI workshop on toric varieties, Cortona (2011)
MSRI Graduate workshop on
toric varieties (2009)
AMS-MRC Workshop on Computational Algebra & Convexity, Snowbird (2008) and Tentative Schedule
MFO workshop on Surface modelling and syzygies, Oberwolfach (2007)
Publications
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 , to appear.
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.
Preprints
On minimal free resolutions of sub-permanents and other ideals arising in complexity theory (joint with Klim Efremenko , J.M. Landsberg , Jerzy Weyman ).
The method of shifted partial derivatives cannot separate the permanent from the determinant (joint with Klim Efremenko , J.M. Landsberg , Jerzy Weyman ).
Computational mathematics
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.
Useful math sites
ams web page
siam web page
Algebraic
Geometry preprints
MSRI (Berkeley).