Jan Draisma

Associate Professor

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Spring 2014: three members of the SIAM (AG)^2 were elected SIAM fellows: Jean Lasserre, Peter Olver, and Bernd Sturmfels. Congratulations!

24 February 2014: I wrote a guest post on Rota's basis conjecture for The Matroid Union.

9 January 2014: I won the GEWIS teaching award 2013/2014 for Mathematics. Needless to say, I'm very, very proud of this!

December 2013: I was elected Chair of the SIAM activity group on Algebraic Geometry. This group brings together researchers who use algebraic geometry in industrial and applied mathematics. If you are an algebraic geometer interested in applications, or if you have a maths/statistics/engineering/CS/... problem that you think might benefit from algebraic techniques, please check out this activity group (or contact me). It's this interplay that makes the group such a success!

24 October 2013: my paper with Eggermont on the existence of poly-time membership tests for a wide class of phylogenetic models has just been accepted for J. Eur. Math. Soc.

1 May 2013: my paper with Kuhnt and Zwiernik on groups acting on Gaussian graphical models has just been accepted for Annals of Statistics.

Spring 2013: Robert Krone from GeorgiaTech is visiting for three months. Together with Anton Leykin and Rob Eggermont we aim to prove finiteness-up-to-symmetry results for certain infinite-dimensional toric varieties.

15 October 2012: the website of the CIME/CIRM course Combinatorial Algebraic Geometry, taking place from 10-15 June 2013 in Levico Terme, is up. Check it out!

1 September 2012: Emil Horobeţ from Babes-Bolyai university joins the group on the project Tensors of Bounded Rank.

23 January 2012: Tensors of Bounded Rank, an NWO free competition Ph.D. project proposal together with Monique Laurent and Siep Weiland has been awarded funding!

23 January 2012: Piotr Zwiernik is starting a Post-doc in the Vidi project.

1 September 2011: Rob Eggermont started his Ph.D. in the Vidi project.

2 March 2011: a manuscript with Johan P. de Jong on his Bachelor's project has been accepted for publication in the EMS Newsletter. See this page.

6 October 2010: A Vidi grant! See this page or this page or this page.


































Important: most links below are to the arxiv preprint, which may differ from the published version.
Please contact me if you want a reprint of the latter.

Preprints

Jan Draisma and Elisa Postinghel: Torus actions and faithful tropicalisation, 2014. Preprint.
[We exploit torus actions to construct explicit sections of the map from Berkovich spaces to tropical varieties, and give many concrete examples where our methods apply.]

Jan Draisma and Rob H. Eggermont: Plücker varieties and higher secants of Sato's Grassmannian, 2014. Preprint.
[We introduce the notion of Plücker variety, which encompasses varieties such as Grassmannians and their secant varieties. We prove that each bounded Plücker variety is defined in bounded degree. In particular, this holds for higher secant varieties of Grassmannians.]

Guus P. Bollen and Jan Draisma: An online version of Rota's basis conjecture, 2013. Preprint.
[We formulate an online version of Rota's basis conjecture for vector spaces. We establish the rather surprising dichotomy that this online version follows from the Alon-Tarsi conjecture in even dimensions, and on the other hand is simply wrong in odd dimensions.]

Jan Draisma, Emil Horobet, Giorgio Ottaviani, Bernd Sturmfels, and Rekha R. Thomas: The Euclidean distance degree of an algebraic variety, 2013. Preprint.
[We introduce the notion of Euclidean distance degree, which is an algebraic measure for the complexity of computing the nearest point on an algebraic variety from a given data point. We compute the degree for many examples from applications, and develop general theory for future applications.]

Jan Draisma, Rob H. Eggermont, Robert Krone, and Anton Leykin: Noetherianity for infinite-dimensional toric varieties, 2013. Preprint.
[We prove that many families of toric ideals stabilize up to symmetry.]

To appear

Jan Draisma: Noetherianity up to symmetry, lecture notes for the 2013 CIME/CIRM course on Combinatorial Algebraic Geometry, to appear in the CIME subseries of Springer's Lecture Notes in Mathematics.
[These lecture notes serve as a gentle introduction to exciting developments on the interplay between algebraic geometry, combinatorics, and algebraic statistics: the use of infinite-dimensional methods to prove stabilisation for mathematical models in various applications.]

Jan Draisma and Rob H. Eggermont: Finiteness results for Abelian tree models, 2012. J. Eur. Math. Soc., to appear.
[We show that phylogenetic tree models with additional symmetry constraints on the transition matrices are defined by polynomials of bounded degree, independent of the tree, provided that the symmetry constraints come from an Abelian group action.]

Jan Draisma and Jose Rodriguez: Maximum likelihood duality for determinantal varieties, 2012. Int. Math. Res. Not., to appear.
[We prove that the maximum-likelihood degree of the variety of rank-r matrices equals that of the variety of co-rank (r-1)-matrices; and also establish variants for symmetric and skew-symmetric matrices.]

Appeared

Jan Draisma and Ron Shaw: Some noteworthy alternating trilinear forms, J. Geom. 105(1), 167-176, 2014.
[We study interesting classes of alternating trilinear forms.]

Jan Draisma and Jochen Kuttler: Bounded-rank tensors are defined in bounded degree, Duke Math. J. 163(1), 35-63, 2014.
[We prove that tensors of bounded rank are defined by polynomials of bounded degree, independent of the dimension or size of the tensors. In the process, we construct a space of infinite-dimensional tensors that should be of independent interest.]

Jan Draisma, Sonja Kuhnt, and Piotr Zwiernik: Groups acting on Gaussian graphical models, Annals of Statistics 41(4), 229-250, 2013.
[We describe the group G of matrices stabilising an undirected Gaussian graphical model, and pave the way for G-invariant tests and G-equivariant estimators.]

Jan Draisma, Seth Sullivant, and Kelli Talaska: Positivity for Gaussian graphical models, Adv. Appl. Math. 50(5), 661-674, 2013.
[We give explicit, cancellation-free formulas for the sub-determinants of Gaussian graphical models.]

Niek Bouman, Jan Draisma, and Johan van Leeuwaarden: Energy minimisation of repelling particles on a toric grid, SIAM J. Discrete Math. 27(3), 1295-1312, 2013.
[We prove that the unique lowest-energy configurations of 32 repelling particles on a toric 8-by-8 chessboard are those where all particles sit on squares of the same color; as well as generalisations to larger and/or higher-dimensional toric grids.]

Jan Draisma and Guus Regts: Tensor invariants for certain subgroups of the orthogonal group, J. Alg. Comb. 38(2), 393-405, 2013.
[We associate to every vertex model a "better" model that gives rise to the same partition function and whose orbit under the orthogonal group is closed. This better model can be used to determine the rank of edge connection matrices.]

Jan Draisma and Bart J. Frenk: Tropically unirational varieties, Contemporary Mathematics 589, 109-123, 2013. Algebraic and Combinatorial Aspects of Tropical Geometry. CIEM Workshop Tropical Geometry, December 2011.
[We introduce the notion of tropically unirational varieties, which are unirational varieties with a parameterisation whose tropicalisation is surjective, and we construct many classes of examples.]

Jan Draisma, Tyrrell B. McAllister, and Benjamin Nill: Lattice width directions and Minkowski's 3^d-theorem, SIAM Journal on Discrete Mathematics 26(3), 1104-1107, 2012.
[A characterisation of the cross-polytope as the unique lattice polytope whose lattice width is attained in the maximal possible number of directions.]

Rina Foygel, Jan Draisma, and Mathias Drton: Half-trek criterion for generic identifiability of linear structural equation models, Annals of Statistics 40(3), 1682-1713, 2012.
[New, combinatorial criteria for generic (non-)identifiability of mixed Gaussian graphical models.]

Jan Draisma: Transitive Lie algebras of vector fields---an overview, Qualitative Theory of Dynamical Systems 11(1), 39-60, 2012.
[A more elaborate intro to the theory of (transitive) Lie algebras of vector fields.]

Filip Cools, Jan Draisma, Sam Payne, and Elina Robeva: A tropical proof of the Brill-Noether Theorem, Advances in Mathematics 230(2), 759-776, 2012.
[We give a new, tropical proof of Griffiths and Harris's theorem on the non-existence of special divisors on a general curve of given genus.]

Jan Draisma, Dion Gijswijt, László Lovász, Guus Regts, and Alexander Schrijver: Characterizing partition functions of the vertex model, Journal of Algebra 350, 197-206, 2012.
[Using invariant theory, we characterise graph parameters that are partition functions of a vertex model over an algebraically closed field of characteristic 0.]

Jan Draisma: On Lie algebras of vector fields, Oberwolfach report 57(4), 3327-3329, 2010, on the Oberwolfach mini-workshop Algebraic and Analytic Techniques for Polynomial Vector Fields.
[A quick intro to the theory of (transitive) Lie algebras of vector fields.]

Jan Draisma, Eyal Kushilevitz, and Enav Weinreb: Partition Arguments in Multiparty Communication Complexity, Theoretical Computer Science 412(24), 2611-2622, 2011.
[The journal version of our ICALP paper below.]

Jan Draisma and Johan P. de Jong: On the Casas-Alvero conjecture, Eur. Math. Soc. Newsl. 80: 29-33, 2011.
[The outcome of a Bachelor's project on the Casas-Alvero conjecture. And an erratum thanks to Christiaan van de Woestijne.]

Andries E. Brouwer and Jan Draisma: Equivariant Gröbner bases and the Gaussian two-factor model, Math. Comp. 80: 1123-1133 (2011).
[Infinite-dimensional Gröbner bases up to symmetry, and a computational proof that the ideal of the Gaussian two-factor model is generated by pentads and three-by-three minors. There is also an appendix containing the equivariant Gröbner basis of the two-factor model and a zipped tar-file containing it in Singular format.]

Seth Sullivant, Kelli Talaska, and Jan Draisma: Trek separation for Gaussian graphical models, 2010. Annals of Statistics 38(3): 1665-1685, 2010.
[We give a complete combinatorial characterisation of which subdeterminants vanish identically on a Gaussian graphical model.]

Jan Draisma and Ron Shaw: Singular lines of trilinear forms, 2010. Linear Algebra Appl.433(3): 690-697, 2010.
[We prove the conjecture by the second author that, over any quasi-algebraically closed field, any trilinear form has a singular line.]

Karin Baur and Jan Draisma: Secant dimensions of low-dimensional homogeneous varieties, Advances in Geometry 10:1-29, 2010.
[Using tropical lower bounds, we compute the higher secant dimensions of all homogeneous varieties of dimension at most three, in all equivariant embeddings.]

Jan Draisma: Finiteness for the k-factor model and chirality varieties, Advances in Mathematics 223:243-256, 2010.
[I prove conjectures by Dress and by Drton, Sturmfels, and Sullivant, that certain varieties arising from chemistry and statistics are defined by finitely many equations up to symmetry.]

Jan Draisma, Eyal Kushilevitz, and Enav Weinreb: Partition Arguments in Multiparty Communication Complexity, in Automata, Languages and Programming, proceedings of ICALP 2009, pages 390-402. Lecture Notes in Computer Science 5555, Springer, 2009.
[Partition arguments are a common technique for showing that the multiparty communication complexity of some multi-argument function is high. We show that they are, in a sense, the only technique if one assumes a generalisation of the log-rank conjecture.]

Jan Draisma and Jochen Kuttler: On the ideals of equivariant tree models, Mathematische Annalen 344(3):619-644, 2009.
[A proof that the ideal of a phylogenetic tree model can be derived from the ideals of its flattenings at vertices.]

Thomas Decker, Jan Draisma, and Pawel Wocjan: Efficient Quantum Algorithm for Identifying Hidden Polynomials, Quantum Information and Computation 9:0215-0230, 2009.
[An algorithm for a natural polynomial variant of the Abelian hidden subgroup problem in quantum computing, whose complexity analysis uses interesting commutative algebra.]

Jan Draisma: Some tropical geometry of algebraic groups, minimal orbits, and secant varieties, Oberwolfach report 4(4),57 Tropical Geometry December 2007, 3292--3294.
[Some links between tropical geometry, algebraic groups, and varieties in their representations.]

Jan Draisma and Jos in 't panhuis: Constructing simply laced Lie algebras from extremal elements, Algebra and Number Theory 2(5):551-572, 2008.
[A new construction of the simple Lie algebras of simply laced Dynkin type as the generic Lie algebras generated by extremal elements with prescribed commutation relations.]

Jan Draisma: A tropical approach to secant dimensions, J. Pure Appl. Algebra 212(2):349--363, 2008.
[A lower bound on dimensions of higher secant varieties using polyhedral tools from tropical geometry.]

Jan Draisma, Gregor Kemper, and David Wehlau: Polarization of Separating Invariants, Canad. J. Math. 60(3): 556--571, 2008.
[A characteristic-independent analogue, for separating invariants, of Weyl's theorem that polynomial invariants of large tuples are generated by polarisations from smaller tuples.]

Karin Baur, Jan Draisma, and Willem de Graaf: Secant dimensions of minimal orbits: computations and conjectures Experimental Mathematics 16(2):239--250, 2007.
[A probabilistic algorithm for computing dimensions of secant varieties, and a number of conjectures based on experiments.]

Jan Draisma: Counting components of the null-cone on tuples Transformation groups 11(4):609--624, 2006. (Presented at MEGA 2005.)
[We prove that the number of components of the null-cone on tuples stabilises as the size of the tuple grows, and give a combinatorial formula for this limit.]

Matthias Bürgin and Jan Draisma: The Hilbert null-cone on tuples of matrices and bilinear forms Math. Z. 254(5):785--809, 2006.
[We explicitly determine the components of the null-cone of certain classical representations; among our tools is the max-flow-min-cut theorem.]

Jan Draisma: Small maximal spaces of non-invertible matrices Bull. Lond. Math. Soc. 38(5):764--776, 2006.
[I construct eight-dimensional, non-extendible vector spaces of non-invertible n-by-n matrices for infinitely many n, thus settling the old question whether such a space can have dimension less than n.]

Jan Draisma, Hanspeter Kraft and Jochen Kuttler: Nilpotent subspaces of maximal dimension in semisimple Lie algebras Compositio Math. 142:464--761, 2006.
[Gerstenhaber proved that the maximal dimension of a vector space of nilpotent n-by-n matrices is n(n-1)/2, and that equality holds iff the space can be conjugated into the upper triangular matrices. We extend this to general semisimple Lie algebras.]

Jan Draisma: Representation theory on the open Bruhat cell J. Symb. Comp. 39:279--303, 2005.
[Differentiating the action of a simple algebraic group G on a G/P gives an action of Lie(G) by polynomial vector fields on the open Bruhat cell of G/P. I describe explicitly the Lie(G)-module structure on polynomials and on polynomial vector fields on that cell, in the case where the unipotent radical is abelian.]

Karin Baur and Jan Draisma: Higher secant varieties of the minimal adjoint orbit J. Algebra 280:743--761, 2004.
[We determine the higher secant varieties of the orbit of long-root vectors in classical simple Lie algebras.]

Jan Draisma: Constructing Lie Algebras of First Order Differential Operators J. Symb. Comp. 36(5):685--698, 2003.
[An algorithm for constructing (truncated) Lie algebras of first order differential operators from given Lie-algebraic data, along with a number of constructions leading to polynomial Lie algebras.]

Arjeh M. Cohen and Jan Draisma: From Lie Algebras of Vector Fields to Algebraic Group Actions Transformation Groups 8(1):51--68, 2003.
[Necessary and sufficient (though not very easily verifiable) conditions for a Lie algebra of vector fields to come from an algebraic group action.]

Jan Draisma: Lie Algebras of Vector Fields, 2002. Ph. D. thesis. (TU/e, Eindhoven, the Netherlands.)

Jan Draisma: On a Conjecture of Sophus Lie, 2002. In Differential Equations and the Stokes Phenomenon, World Scientific Publishing Company, Incorporated, Singapore. (Proceedings of a workshop held at Groningen University from May 8--30, 2001.)
[Lie conjectured that any finite-dimensional transitive Lie algebra of formal vector fields can be realised using coefficients built up from polynomials and simple exponentials. I prove this conjecture in a number of cases.]

Jan Draisma: Recognizing the Symmetry Type of O.D.E.'s J. Pure Appl. Algebra 164(1--2):109--128, 2001.
[An algorithm for determining the Lie algebra of point symmetries of an o.d.e., provided that it is transitive, up to conjugation with an automorphism of formal power series.]

Miscellaneous

Marko Boon, Alessandro Di Buchianico, Jan Draisma, Remco van der Hofstad, Adrian Muntean, Mark Peletier, and Jan-Jaap Oosterwijk, editors. Proceedings of the 84th European Study Group Mathematics with Industry, SWI 2012 , Eindhoven, Netherlands, January 30--February 3, 2012.

Jan Draisma: Het Handelsreizigerprobleem, De Groene Amsterdammer, mei 2012.

Jan Draisma: Jan en Differentiaalmeetkunde Contribution to the Liber Amicorum for Jan de Graaf, 2007.

Jan Draisma: Leonhard Euler: ein Mann, mit dem man rechnen kann Nieuw Arch. Wisk., vijfde serie 8(3):192--193, 2007. Book review.

Magdalena Caubergh, Jan Draisma, Geert-Jan Franx, Geertje Hek, Georg Prokert, Sjoerd Rienstra, and Arie Verhoeven: Measure under Pressure, Calibration of pressure measurement Proceedings of the Studygroup Mathematics in Industry, 2006.

Jan Draisma, Hanspeter Kraft: Proceedings of the Rhine Workshop on Computer algebra 2006 on 16 and 17 March 2006.

Jan Draisma: An Algebraic Approach to Lie Point Symmetries of Ordinary Differential Equations, 1998. Master's thesis. (TU/e, Eindhoven, the Netherlands.)