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ZACHARIAH C. TEITLER
My area of research is algebraic geometry and commutative algebra (MSC 14 and 13). I am interested in a range of problems involving algebraic geometry and commutative algebra, usually with a combinatorial or computational flavor.
1. Computation in algebraic geometry While massive computations have become common in applied mathematics, they are still rare in areas of pure mathematics such as algebraic geometry, where “computation” often means the calculation of a small number of examples. In collaboration with Hillar, Sottile, et al., I have developed and deployed a massive parallel computation to explore the behavior of Schubert calculus over the real numbers [HGPdC+ 09]. We plan more massive computations to explore phenomena such as monodromy of Schubert problems, building on [LS09]. I am also interested in computing resolutions of singularities, multiplier ideals, and associated invariants (see below). For these efforts we will develop new mathematical software, particularly for the resolution of singularities, to extend the capabilities of the Singular [GPS09] libraries desing.lib [BS00] and resolve.lib [FKP04]. This new software will be highly useful to the mathematical community. In addition it takes a certain amount of expertise and experience to organize and run a massive computation on hundreds of computers simultaneously; my experience with Hillar and Sottile has prepared me for this. I will prepare and run one or more large computational experiments focusing on efforts such as monodromy of Schubert problems and resolution of singularities. These projects will be good opportunities for undergraduate and graduate students to learn about these areas of mathematics and also gain experience with a wide variety of software development tools, ranging from mathematical software such as Singular, Macaulay2 [GS], and Sage [Ste08] to programming tools such as Perl [Per] and database tools such as MySQL [Sun] (all of which I am using currently). This would follow a common model of research in the physical sciences, in which a senior scientist organizes and mentors a lab of postdocs, graduate students, and undergraduate students working on one or more related projects. Our current project uses hundreds of instructional computers at Texas A&M University during non-class times. If these are available, I will develop them into a highly cost-effective resource for research computing.
September 16, 2009
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2. Problems with a combinatorial flavor I am interested in many problems with a combinatorial flavor. Here I describe three. Rank problems. The rank of a homogeneous polynomial is the least number of terms needed to write it as a sum of powers of linear forms. The rank problem of determining rank has been well-studied in algebraic geometry since at least the mid-19th century; it has applications to a number of areas of engineering such as antenna array processing and data analysis, see for example [BCMT09] for a detailed discussion and many references to applications. JM Landsberg and I found new bounds for rank [LT09] in terms of singularities of the hypersurface defined by the polynomial. This highly geometric result arises by appealingly elementary methods, and appears to be unprecedented in over 100 years of study of the rank problem. These bounds are not sharp and it would be extremely interesting to improve them. I am also interested in generalizing. The above description of the rank problem refers to secant varieties of Veronese varieties; the well-known rank problem for tensors can be regarded as the analogous problem for secant varieties of Segre or Segre-Veronese varieties. It is not clear whether the same technique will give interesting bounds in these other cases. Whether by the same or a new technique, any bound for ranks of tensors would be of great interest in many areas of mathematics and engineering. Combinatorics enters here in analyzing the arrangement of the points corresponding to the linear forms appearing in a power-sum expression, in that we consider the matroid-like collection of subsets of these points which are collinear, or more generally lie on a hypersurface of low degree. An interesting inverse problem is to use this combinatorial understanding to produce polynomials of unexpectedly high rank. Hilbert functions of fat point schemes. The problem of characterizing and classifying Hilbert functions of geometric and algebraic objects is central in algebraic geometry and commutative algebra. In particular, there is great interest in studying the Hilbert functions of fat point schemes, among other reasons because of the Segre–Harbourne–Gimigliano– Hirschowitz conjecture1, see for example [Har02]. Let p1 , . . . , pr ∈ Pn be distinct points, and let m1 , . . . , mr be nonnegative integers. The scheme Z defined by the ideal r I(pi )mi , i=1 where I(pi ) is the homogeneous ideal of pi , is a fat point scheme. The Hilbert function hZ describes how many conditions Z imposes on hypersurfaces of each degree. Brian Harbourne, Susan Cooper, and I have developed an approach to studying Hilbert functions of fat points in the plane based on being given only partial information about the fat point scheme, in particular, a list of some (or all or none) of the collinear sets of points of Z. We give upper and lower bounds for the Hilbert function, and a necessary and sufficient condition for the bounds to coincide; when the condition is met, we give bounds for the graded Betti numbers and the condition for these bounds to coincide. This generalizes results of [GMS06], although the methods are closer to those of [FHL06]. Our results should
1Names
listed in chronological order.
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lead to large steps forward in the effort to classify Hilbert functions of zero-dimensional schemes. We expect to finish this work before the end of 2009 [CHT09]. Preliminary exploration suggests that while the same method does give bounds on Hilbert functions of fat point schemes in higher dimensions, the bounds are rather loose; and while the same method can be applied with hypersurfaces of higher degree (rather than only lines), the bounds, while sharper, are very difficult to work with. We will study this further and also explore connections to the “basic double links” of [GMS06]. Arithmetic toric varieties. Toric varieties over closed fields have been well-studied in algebraic geometry for several reasons including: they have a simple combinatorial classification; in the words of W. Fulton [Ful93, pg. ix] they “have provided a remarkably fertile testing ground for general theories”; and many of the varieties appearing in applications such as algebraic statistics, geometric modeling, and algebraic phylogenetics are in fact toric varieties. In these applications, however, one is interested in working over the real numbers. Toric varieties over non-closed fields appear also in number theory, as in [BT95] for example. In joint work in progress with Javier Elizondo, Paulo Lima-Filho, and Frank Sottile, we study the classification of arithmetic toric varieties [ELFST], greatly extending work of Claire Delaunay [Del04] on the classification of real forms of toric surfaces and threefolds. We describe the classification of forms of toric varieties in terms of Galois cohomology. This is worked out in detail for classes of examples including projective spaces, affine toric varieties and toric ideals, and toric surfaces. Subsequent work will address the topology of real forms of toric varieties following [Del04], including equivariant cohomology with respect to the semidirect product of the torus and the Galois group. 3. Resolution of singularities and multiplier ideals To an ideal I ⊂ C[x1 , . . . , xn ] (or more generally on a normal variety [dFH08]) and a real number t ≥ 0 one may associate the multiplier ideal J (I t ) defined in terms of a log resolution, a strong embedded resolution of singularities of the variety defined by I. See [BL04, Laz04]. Multiplier ideals are hard to compute and very few examples are known, mainly due to the difficulty of computing resolution of singularities. The Singular libraries desing.lib [BS00] and resolve.lib [FKP04] each compute resolution of singularities, but it remains a nontrivial programming task to implement the remaining steps to compute multiplier ideals. (Specifically, desing.lib computes the strong resolution of singularities but provides no mechanism to access the numerical data of the resolution; resolve.lib provides such a mechanism but computes a weak resolution of singularities.) Similarly, Macaulay2 together with Normaliz [BK09] can do most of the work to compute multiplier ideals of monomial ideals, but not all, so again more programming is needed. (Specifically, Normaliz can handle affine semigroups defined by weak inequalities, but Howald’s
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theorem [How01] for multiplier ideals of monomial ideals describes them in terms of strict inequalities, i.e., by the interior of a polyhedron.) Once these software programs are written the mathematical community will have valuable new tools for exploring and testing conjectures about multiplier ideals, including questions related to syzygies [LL07, LLS08, Lee07] and jumping numbers [ELSV04, BS09, Nai09, Tuc08]. Note that resolution of singularities is somewhat parallelizable, by simply entering each new coordinate chart in the resolution process into a queue from which a pool of computing nodes draw tasks. So this approach to studying multiplier ideals would be greatly enhanced by access to large-scale computing resources; but such resources would certainly not be required for this research to have an impact. I am particularly interested in studying subspace arrangements, including jumping numbers of hyperplane arrangements. A real number t > 0 is a jumping number if J (I t ) = J (I t− ); the jumping numbers are related to the Bernstein–Sato polynomial and other local invariants of singularities, see [ELSV04]. The jumping numbers of hyperplane arrangement are characterized in [BS09], but one would like to find a simpler characterization. I would also like to treat some special classes such as Coxeter arrangements. The answer should have combinatorial and representation-theoretic significance. An exceptional divisor E is relevant for (the multiplier ideals of) an ideal I if, roughly speaking, deleting E from the formula for the multiplier ideals J (I t ) changes the result. (More precisely, if µ : Y → Cn is a log resolution of I on which E appears, the total transform IOY is OY (−F ), and the relative canonical divisor is denoted KY /Cn , then J (I t ) = µ∗ OY (KY /Cn − tF ); we may say that E is relevant at t if µ∗ OY (KY /Cn − tF +E) = J (I t ), and E is relevant if it is relevant at some t.) A conjecture of Smith–Thompson [ST07] relates the set of relevant exceptional divisors to a log minimal model. This is verified in [ST07] for plane curve singularities, where it is shown that relevant exceptional divisors are characterized by the combinatorics of the resolution tree: they are the exceptional divisors corresponding to nodes of degree at least 3. This was applied in [Nai09] to study jumping numbers and extended to curve singularities on surfaces with rational singularities in [Tuc08]. For the cases studied in my paper on multiplier ideals of line arrangements [Tei07], the dual complex (analogue of resolution tree) of the exceptional divisors is a path, and it is shown that only the exceptional divisors corresponding to the first and last nodes on the path are relevant. I am interested in studying other line arrangements for which the exceptional divisors have more intricate combinatorics to test the extent to which relevance is determined combinatorially. More generally I would like to study the relevant exceptional divisors for other ideals, such as monomial ideals and hyperplane arrangements (where I expect they are precisely the exceptional divisors corresponding to the minimal building set, which is indeed combinatorially determined). References
[BCMT09] Jerome Brachat, Pierre Comon, Bernard Mourrain, and Elias Tsigaridas, Symmetric tensor decomposition, arXiv:0901.3706, Jan 2009.
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Winfried Bruns and Gesa Kaempf, A macaulay 2 interface for normaliz, arXiv:0908.1308, Aug 2009. [BL04] Manuel Blickle and Robert Lazarsfeld, An informal introduction to multiplier ideals, Trends in commutative algebra, Math. Sci. Res. Inst. Publ., vol. 51, Cambridge Univ. Press, Cambridge, 2004, arXiv:math.AG/0302.5351, pp. 87–114. MR MR2132649 (2007h:14003) [BS00] G´bor Bodn´r and Josef Schicho, A computer program for the resolution of singularities, a a Resolution of singularities (Obergurgl, 1997), Progr. Math., vol. 181, Birkh¨user, Basel, 2000, a pp. 231–238. MR MR1748621 (2001e:14001) [BS09] Nero Budur and Morihiko Saito, Jumping coefficients and spectrum of a hyperplane arrangement, arXiv:0903.3839, Mar 2009. [BT95] Victor V. Batyrev and Yuri Tschinkel, Rational points of bounded height on compactifications of anistropic tori, Int. Math. Res. Not. 12 (1995), 591–635. [CHT09] Susan Cooper, Brian Harbourne, and Zach Teitler, Bounding hilbert functions of fat points in the plane via matroids and residuation, In preparation, 2009. [Del04] Claire Delaunay, Real structures on compact toric varieties, Pr´publication de l’Institut de e Recherche Math´matique Avanc´e [Prepublication of the Institute of Advanced Mathematie e cal Research], 2004/18, Universit´ Louis Pasteur D´partement de Math´matique Institut de e e e Recherche Math´matique Avanc´e, Strasbourg, 2004, Th`se, Universit´ Louis Pasteur, Strase e e e bourg, 2004. MR MR2097589 (2005f:14099) [dFH08] Tommaso de Fernex and Christopher D. Hacon, Singularities on normal varieties, To appear in Compositio Mathematica, May 2008. [ELFST] Javier Elizondo, Paulo Lima-Filho, Frank Sottile, and Zach Teitler, Arithmetic toric varieties, In progress. [ELSV04] Lawrence Ein, Robert Lazarsfeld, Karen E. Smith, and Dror Varolin, Jumping coefficients of multiplier ideals, Duke Math. J. 123 (2004), no. 3, 469–506. MR MR2068967 (2005k:14004) [FHL06] Giuliana Fatabbi, Brian Harbourne, and Anna Lorenzini, Resolutions of ideals of fat points with support in a hyperplane, Proc. Amer. Math. Soc. 134 (2006), no. 12, 3475–3483 (electronic). MR MR2240658 (2007d:13018) [FKP04] Anne Fr¨hbis-Kr¨ger and Gerhard Pfister, Practical Aspects of Algorithmic Resolution of u u Singularities, Reports On Computer Algebra, no. 33, Centre for Computer Algebra, University of Kaiserslautern, October 2004, Online available at http://www.mathematik.uni-kl.de/ ~zca. [Ful93] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry. MR MR1234037 (94g:14028) [GMS06] A. V. Geramita, J. Migliore, and L. Sabourin, On the first infinitesimal neighborhood of a linear configuration of points in P2 , J. Algebra 298 (2006), no. 2, 563–611. MR MR2217628 (2007a:13016) [GPS09] G.-M. Greuel, G. Pfister, and H. Sch¨nemann, Singular 3.1.0 — A computer algebra system o for polynomial computations, 2009, http://www.singular.uni-kl.de. [GS] Daniel R. Grayson and Michael E. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/. [Har02] Brian Harbourne, Problems and progress: a survey on fat points in P2 , Zero-dimensional schemes and applications (Naples, 2000), Queen’s Papers in Pure and Appl. Math., vol. 123, Queen’s Univ., Kingston, ON, 2002, pp. 85–132. MR MR1898832 (2003f:13032) [HGPdC+ 09] Christopher Hillar, Luis Garcia-Puente, Abraham Martin del Campo, James Ruffo, Zach Teitler, Stephen L. Johnson, and Frank Sottile, Experimentation at the frontiers of reality in schubert calculus, arXiv:0906.2497, Jun 2009.
[BK09]
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[How01] [Laz04]
[Lee07] [LL07] [LLS08]
[LS09] [LT09] [Nai09] [Per] [ST07]
[Ste08] [Sun] [Tei07] [Tuc08]
J. A. Howald, Multiplier ideals of monomial ideals, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2665–2671 (electronic). MR MR1828466 (2002b:14061) Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik., vol. 49, Springer-Verlag, Berlin, 2004, Positivity for vector bundles, and multiplier ideals. MR MR2095472 Seunghun Lee, Filtrations and local syzygies of multiplier ideals, J. Algebra 315 (2007), no. 2, 629–639. MR MR2351883 (2008h:14001) Robert Lazarsfeld and Kyungyong Lee, Local syzygies of multiplier ideals, Invent. Math. 167 (2007), no. 2, 409–418. MR MR2270459 (2007h:13021) Robert Lazarsfeld, Kyungyong Lee, and Karen E. Smith, Syzygies of multiplier ideals on singular varieties, Michigan Math. J. 57 (2008), 511–521, Special volume in honor of Melvin Hochster. MR MR2492466 Anton Leykin and Frank Sottile, Galois groups of Schubert problems via homotopy computation, Math. Comp. 78 (2009), no. 267, 1749–1765. MR MR2501073 J.M. Landsberg and Zach Teitler, On the ranks and border ranks of symmetric tensors, arXiv: 0901.0487, Jan 2009. Daniel Naie, Jumping numbers of a unibranch curve on a smooth surface, Manuscripta Math. 128 (2009), no. 1, 33–49. MR MR2470185 (2009j:14034) Perl Foundation, The, Perl, http://www.perl.org. Karen E. Smith and Howard M. Thompson, Irrelevant exceptional divisors for curves on a smooth surface, Algebra, geometry and their interactions, Contemp. Math., vol. 448, Amer. Math. Soc., Providence, RI, 2007, pp. 245–254. MR MR2389246 William Stein, Sage: Open Source Mathematical Software, The Sage Group, 2008, http://www.sagemath.org. Sun Microsystems, Inc., MySQL, http://www.mysql.com. Zachariah C. Teitler, Multiplier ideals of general line arrangements in C3 , Comm. Algebra 35 (2007), no. 6, 1902–1913. Kevin Tucker, Jumping numbers on algebraic surfaces with rational singularities, arXiv:0801. 0734 [math.AG], Jan 2008.