Jan Draisma
Professor of Mathematics
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Organisational
Recreational Maths
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News
June 2015: the MittagLeffler proposal by Anders Jensen, Hannah Markwig, Benjamin Nill, and myself for a semester on Tropical geometry, amoebas, and polytopes was accepted for Spring 2018!
May 2015: Rob Eggermont got his Ph.D. with the distinction cum laude; congratulations! See this news item.
April 2015: My article on Plücker varieties with Rob Eggermont was accepted by Crelle's journal.
April 2015: I have joined the editorial boards of Linear and Multilinear Algebra, Nieuw Archief voor Wiskunde and, last year, of Experimental Mathematics. Where appropriate, please consider submitting your work to one of these journals.
March 2015: I wrote a SIAM News article loosely based on lectures by Peter Bürgisser and others during the Simons Institute semester on Algorithms and Complexity in Algebraic Geometry.
January 2015: I was appointed parttime full professor at the VU Amsterdam; see
this news item.
October 2014: Rekha
Thomas wrote a beautiful article for SIAM News on the Euclidean distance degree.
Fall 2014: The MEGA 2015 website is in place, and submission of papers, extended abstracts, computations, and posters is open!
October 2014: both my lastyear's Master students have started Ph.D.'s: Jasmijn Baaijens in the CWI life sciences group, under the supervision of Alexander Schönhuth; and Guus Bollen in Discrete Maths here at TU/e, under the supervision of Hans Cuypers, Rudi Pendavingh, and myself. The best of luck to both!
September 2014: The conference
website for the SIAM Conference on Applied Algebraic Geometry is up
and running. Check it out, and consider organising a minisymposium! The conference has also been accepted as a ICIAM 2015 satellite.
July 2014: my Department nominated me for a TU/e Education Award, as
their candidate in the category Best Master's Program Lecturer.
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 polytime 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 finitenessuptosymmetry results for
certain infinitedimensional toric varieties.
15 October 2012: the website of the CIME/CIRM course Combinatorial
Algebraic Geometry, taking place from 1015 June 2013 in Levico Terme,
is up. Check it out!
1 September 2012: Emil Horobeţ from BabesBolyai 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 Postdoc 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 Piotr Zwiernik: Automorphism groups of Gaussian chain graph models, 2015. Preprint, submitted
[We determine the automorphism group of a Gaussian graphical model given by a chain graph without flags. This group has a connected component of 1 given by a certain poset, and the factor group is related to the authomorphism group of the socalled essential graph. These results extend earlier work with Kuhnt on undirected graphs, and comprise the case of directed acyclic graphs as another special case.]
Jan Draisma and Emil Horobeţ: The average number of critical rankone approximations to a tensor, 2014. Preprint, submitted
[For a realvalued tensor v whose entries are independent, standard normal variables, we compute the expected number of critical points of the distance function to v on the manifold of real rankone tensors. We treat both the case of symmetric tensors and the case of general tensors.]
Jan Draisma and Elisa Postinghel: Torus actions and faithful tropicalisation, 2014. Preprint, submitted.
[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.]
To appear
Jan Draisma, Rob H. Eggermont, Robert Krone, and Anton Leykin:
Noetherianity for
infinitedimensional toric varieties, 2013. Algebra & Number Theory, to appear.
[We prove that many families of toric ideals stabilize up to symmetry.]
Jan Draisma and Rob H. Eggermont: Plücker varieties and higher secants of Sato's Grassmannian, J. Reine Angew. Math., to appear.
[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.]
Andries E. Brouwer, Jan Draisma, and Bart J. Frenk: Lossy gossip and the composition of metrics,
Discrete Comput. Geom., to appear.
[We prove that the monoid generated by nbyn distance matrices is a polyhedral complex of dimension n choose 2, compute the structure of this complex for n at most 5, and relate the elements of this monoid to gossip over lossy phone lines.]
Jan Draisma, Emil Horobeţ, Giorgio Ottaviani, Bernd Sturmfels, and
Rekha R. Thomas:
The Euclidean distance degree of an algebraic variety, Found. Comput. Math., to appear.
[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.]
Andries E. Brouwer, Jan Draisma, and Mihaela Popoviciu: The degrees of a system of parameters of the ring of invariants of a binary form, Transform. Groups, to appear.
[We study the possible degrees of hsops of invariant rings of binary forms.]
Guus P. Bollen and Jan Draisma: An online version of Rota's basis conjecture, J. Alg. Comb., to appear.
[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 AlonTarsi conjecture in even dimensions, and on the other hand is simply wrong in odd dimensions.]
Appeared
Jan Draisma and Rob H. Eggermont: Finiteness results for Abelian tree models, J. Eur. Math. Soc. 17, 711738, 2015.
[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: Geometry, Invariants, and the Search for Elusive Complexity Lower Bounds, SIAM News 48(2), March 2015.
[In this article, based on a Simons Open Lecture by Peter
Bürgisser with almost the same title and on various lectures by
others, I discuss some recent developments on geometric and representation
theoric methods in complexity theory. As Al Hales pointed,
there is some overkill early on in the article: p_n(T) can be computed
in O(n^2) by expanding brackets, without interpolation :) ]
Jasmijn A. Baaijens and Jan Draisma:
Euclidean distance degrees of real algebraic groups, Linear Algebra Appl 467, 174187, 2015.
[We study the problem of finding the matrix in a matrix group closest to a given matrix.]
Jan Draisma and Jose Rodriguez: Maximum likelihood duality for determinantal varieties, Int. Math. Res. Not. 20, 56485666, 2014.
[We prove that the maximumlikelihood degree of the variety of rankr matrices equals that of the variety of corank (r1)matrices; and also establish variants for symmetric and skewsymmetric matrices.]
Aldo Conca, Sandra Di Rocco, Jan Draisma, June Huh, Bernd Sturmfels, and Filippo Viviani: Combinatorial Algebraic Geometry, Lecture Notes in Mathematics 2108, Springer, 2014.
[These are lecture notes for a CIME/CIRM summer school in Levico Terme,
2013. The common theme is the study of algebraic varieties equipped
with a rich combinatorial structure. My own notes on Noetherianity up
to Symmetry can be found
here.]
Jan Draisma and Ron Shaw: Some noteworthy alternating trilinear forms, J. Geom. 105(1), 167176,
2014.
[We study interesting classes of alternating trilinear forms.]
Jan Draisma and Jochen Kuttler: Boundedrank tensors are defined in bounded degree, Duke Math. J. 163(1), 3563, 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 infinitedimensional 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), 229250, 2013.
[We describe the group G of matrices stabilising an undirected
Gaussian graphical model, and pave the way for Ginvariant tests and
Gequivariant estimators.]
Jan Draisma, Seth Sullivant, and Kelli Talaska: Positivity for Gaussian graphical models, Adv. Appl. Math. 50(5), 661674, 2013.
[We give explicit, cancellationfree formulas for the subdeterminants 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), 12951312, 2013.
[We prove that the unique lowestenergy configurations of 32 repelling
particles on a toric 8by8 chessboard are those where all particles sit on squares of the same color; as well as generalisations to larger and/or higherdimensional toric grids.]
Jan Draisma and Guus Regts: Tensor invariants for certain subgroups of the orthogonal group,
J. Alg. Comb. 38(2), 393405, 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, 109123,
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^dtheorem, SIAM Journal on Discrete Mathematics 26(3), 11041107, 2012.
[A characterisation of the crosspolytope as the unique lattice polytope whose lattice width is attained in the maximal possible number of directions.]
Rina Foygel, Jan Draisma, and Mathias Drton: Halftrek criterion for generic identifiability of linear structural
equation models, Annals of Statistics 40(3), 16821713, 2012.
[New, combinatorial criteria for generic (non)identifiability of mixed Gaussian graphical models.]
Jan Draisma:
Transitive Lie algebras of vector fieldsan overview, Qualitative Theory of Dynamical Systems 11(1), 3960, 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 BrillNoether Theorem, Advances in Mathematics 230(2), 759776, 2012.
[We give a new, tropical proof of Griffiths and Harris's theorem on the
nonexistence 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, 197206, 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), 33273329, 2010, on the Oberwolfach miniworkshop 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), 26112622, 2011.
[The journal version of our ICALP paper below.]
Jan Draisma and Johan P. de Jong:
On the CasasAlvero conjecture, Eur. Math. Soc. Newsl. 80: 2933, 2011.
[The outcome of a Bachelor's project on the CasasAlvero conjecture.
And an erratum thanks to Christiaan van de Woestijne.]
Andries E. Brouwer and Jan Draisma: Equivariant
Gröbner bases and the Gaussian twofactor model,
Math. Comp. 80: 11231133 (2011).
[Infinitedimensional Gröbner bases up to symmetry, and a
computational proof that the ideal of the Gaussian twofactor model
is generated by pentads and threebythree minors. There is also an
appendix containing
the equivariant Gröbner basis of the twofactor model and a zipped tarfile containing
it in Singular format.]
Seth Sullivant, Kelli Talaska, and Jan Draisma:
Trek separation for Gaussian graphical models, 2010. Annals of Statistics 38(3): 16651685, 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): 690697, 2010.
[We prove the conjecture by the second author that, over any
quasialgebraically closed field, any trilinear form has a singular line.]
Karin Baur and Jan Draisma:
Secant dimensions of lowdimensional homogeneous varieties, Advances in Geometry 10:129, 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 kfactor model and chirality varieties, Advances in Mathematics 223:243256, 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 390402. Lecture Notes in Computer Science 5555, Springer, 2009.
[Partition arguments are a common technique for showing that the
multiparty communication complexity of some multiargument function is
high. We show that they are, in a sense, the only technique if one assumes
a generalisation of the logrank conjecture.]
Jan Draisma and Jochen Kuttler: On the ideals of equivariant
tree models, Mathematische Annalen 344(3):619644, 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 9:02150230, 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, 32923294.
[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):551572, 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):349363, 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): 556571, 2008.
[A characteristicindependent 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):239250, 2007.
[A probabilistic algorithm for computing dimensions of secant varieties, and a number of conjectures based on experiments.]
Jan Draisma:
Counting components of the nullcone on
tuples Transform. groups 11(4):609624, 2006.
(Presented at MEGA 2005.)
[We prove that the number of components of the nullcone 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 nullcone on tuples of
matrices and bilinear forms
Math. Z. 254(5):785809, 2006.
[We explicitly determine the components of the nullcone of certain
classical representations; among our tools is the maxflowmincut theorem.]
Jan Draisma:
Small maximal spaces of noninvertible matrices
Bull. Lond. Math. Soc. 38(5):764776, 2006.
[I construct eightdimensional, nonextendible vector spaces of
noninvertible nbyn 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. 142:464761, 2006.
[Gerstenhaber proved that the maximal dimension of a vector space of nilpotent nbyn matrices is n(n1)/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. 39:279303, 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 280:743761, 2004.
[We determine the higher secant varieties of the orbit of longroot vectors in classical simple Lie algebras.]
Jan Draisma:
Constructing Lie Algebras of First Order Differential Operators
J. Symb. Comp. 36(5):685698, 2003.
[An algorithm for constructing (truncated) Lie algebras of first order differential operators from given Liealgebraic 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
Transform. Groups 8(1):5168, 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 830, 2001.)
[Lie conjectured that any finitedimensional 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(12):109128, 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 JanJaap Oosterwijk, editors.
Proceedings of the 84th European Study Group Mathematics with Industry, SWI 2012 , Eindhoven, Netherlands, January 30February 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):192193, 2007. Book review.
Magdalena Caubergh, Jan Draisma, GeertJan 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.)
