News
Spring 2023: I'm organising the SMG Spring Meeting. Please attend!
Fall 2022: I'll be on sabbatical at the IAS as a participant in the Special Year on Dynamics, Additive Number Theory and Algebraic Geometry.

January 2022: I have succeeded Bernd Sturmfels as Editor-in-Chief of the SIAM Journal on Applied Algebra and Geometry.

Dec 2020: my former Ph.D. student Arthur Bik as won an SNF Postdoc.Mobility grant, with which he will spend a year at MPI MIS (Leipzig) and a year at Texas A&M (College Station). Congrats to Arthur!

Nov 2020: my Ph.D. student Alejandro Vargas won an SNF Early Postdoc.Mobility grant, with which he will spend time in Frankfurt and Nantes. Congrats to Alejandro!

Feb 2020: We have created a movie visualising parts of my joint paper with Alejandro Vargas on the gonality of metric graphs.

Feb 2020: Shreehari Bodas, a Bachelor's student from IIT Bombay, has successfully applied for a ThinkSwiss 2020 Research Scholarship to visit my group for two months this summer---unfortunately, due to the Covid-19 pandemic, his visit did not take place.

Apr 2019: The current, online version of the programme for SIAM AG 19 is online.

Mar 2019: My paper on topological Noetherianity of polynomial functors was accepted for publication in the Journal of the AMS.

July 2018: My former student Rob Eggermont has won a Veni grant.

June 2018: The website for MEGA 2019 is up and running.

Apr 2018: The website for SIAM AG 19 is up and running.

Dec 2017: My student Arthur Bik won the DIAMANT prize for best Ph.D. talk at the DIAMANT symposium; see this picture.

Nov 2017: MEGA (Effective Methods in Algebraic Geometry) now has a permanent website collecting information about past and future MEGAs!

May 2017: Christandl, Jensen, and Zuiddam give very simple and elegant examples of strict submultiplicativity of tensor rank under the tensor product in this preprint!

Jan 2017: Paul Breiding has succeeded at evaluating our formula for the average ED degree of the cone over Veronese embeddings. See Paul's impressive preprint and our paper for the formula.

Jan 2017: Andries Brouwer and Dan Christensen refuted the Gale-Neyman conjecture that in chomp on the n-dimensional hypercube {0,1}^n with poisoned block at the origin, the all-one vector is a winning first move. They show this is not true for n=7. See this page.

Aug 2016: I have moved to the Bern math institute, succeeding Christine Riedtmann.

Mar 2016: Anna Seigal wrote a blog about the pictures on the cover of our new SIAM journal.

Feb 2016: My Vici proposal entitled Stabilisation in Algebra and Geometry was successful!

Feb 2016: The website for the journal is up, and paper submission will open on 23 March.

Jan 2016: Here is the cover art for the new SIAM Journal on Applied Algebra and Geometry---the website will be up soon!


Dec 2015: Our proposal for a new SIAM Journal on Applied Algebra and Geometry was approved by SIAM. We aim to attract the best papers in this broad area. Submission will open soon, but please contact me if you have fantastic work that could fit!

June 2015: the Mittag-Leffler 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 last-year'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 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.

Preprints

Christopher Chiu, Jan Draisma, Rob Eggermont, Tim Seynnaeve, and Nafie Tairi: Topological Noetherianity of the infinite half-spin representations, 2024. Preprint.
We show that each of the two infinite half-spin representations of the infinite spin group is Noetherian up to the action of that group. This implies, for instance, that the k-th secant variety to the maximal isotropic Grassmann in its spinor embedding is defined set-theoretically by finitely many types of equations, independently of the dimension of the underlying orthogonal space.

Benjamin Biaggi, Chia-Yu Chang, Jan Draisma, and Filip Rupniewski: Border subrank via a generalised Hilbert-Mumford criterion, 2024. Preprint, submitted.
[We show that the border subrank of a generic nxnxn-tensor is Θ(sqrt(n)), and establish a similar result for order-d tensors. To this end, we establish a generalisation of the Hilbert-Mumford criterion that we believe might be useful in other contexts, as well.]

Arthur Bik, Jan Draisma, and Andrew Snowden: Two improvements in Brauer's theorem on forms, 2024. Preprint.
[Brauer's theorem on forms says that over fields with the property that diagonal forms in sufficiently many variables have nontrivial rational roots, also every system of a fixed number of forms of fixed degrees in sufficiently many variables has a nontrivial common root. We establish two improvements. One says that the codimension of the Zariski closure of the set of rational roots can be bounded independently of the number of variables, and the other says that if the system has high enough strength, then the rational roots are in fact dense in the set of all roots.]

Arthur Bik, Jan Draisma, Rob Eggermont, and Andrew Snowden: Uniformity for limits of tensors, 2023. Preprint.
[We show that, for many notions of tensor rank, tensors in the closure of rank at most r can be written as limits of a one-parameter family T(ε) of tensors of rank at most r in which the negative exponent of ε is bounded from below independently of the dimension of the underlying vector space. We derive this result from a more general result about the geometry of GL-varieties.]

Jan Draisma and Thomas Karam: On subtensors of high partition rank, 2023. Preprint, submitted.
[We prove a tensor variant of the statement that a rank-r matrix has an (r+1)x(r+1)-submatrix of rank r.]

Jan Draisma, Rob H. Eggermont, Azhar Farooq, and Leandro Meier: Image closure of symmetric wide-matrix varieties, 2022. Preprint, submitted.
[We prove that the image closure of a wide matrix variety (roughly, the spectrum of a Sym(natural numbers)-polynomial ring where the variables have a single index) is defined by finitely many (Sym(natural numbers)-orbits of) equations. This greatly generalises my 2010 Advances paper on finiteness for the Gaussian k-factor model.]

Christopher H. Chiu, Alessandro Danelon, Jan Draisma, Rob H. Eggermont, and Azhar Farooq: Sym-Noetherianity for powers of GL-varieties, 2022. Preprint, submitted.
[We prove that, for a variety X on which the infinite general linear group GL acts in a suitable manner so that X is GL-Noetherian by earlier work, the infinite power X^N, where N={natural numbers}, is Noetherian under the product of the infinite symmetric group Sym(N) and GL.]

Andreas Blatter, Jan Draisma, and Filip Rupniewski: A tensor restriction theorem over finite fields, 2022. Preprint, submitted.
[We prove that tensors over a finite field are well-quasi-ordered under restriction, and study the coarse structure of restriction-closed tensor properties.]

Jan Draisma, Rob H. Eggermont, Tim Seynnaeve, Nafie Tairi, and Emanuele Ventura: Quasihomomorphisms from the integers into Hamming metrics, 2022. Preprint, submitted.
[A conjecture by Kazhdan-Ziegler says that if f is a map from the integers to nxn-matrices over the complex numbers such that the rank of f(x+y)-f(x)-f(y) has rank at most some bound c for all x,y, then for some matrix A, f(x)-x A has rank at most some C=C(c) for all x. We prove this conjecture in the case of diagonal matrices, where the rank metric reduces to the Hamming metric.]

To appear

Diego Cifuentes, Jan Draisma, Oskar Henriksson, Annachiara Korchmaros, and Kaie Kubjas: 3D genome reconstruction from partially phased Hi-C data . Bull. Math. Biology 86, article number 33 (2024).
[We study the problem of reconstructing the 3D structure of chromosomes from Hi-C data of diploid organisms, which poses additional challenges compared to the better-studied haploid setting. With the help of techniques from algebraic geometry, we prove that a small amount of phased data is sufficient to ensure finite identifiability, both for noiseless and noisy data.]

Andries E. Brouwer, Jan Draisma, and Çiçek Güven: The unique coclique extension property for apartments of buildings, Innov. Inc. Geom. 20(2-3), 209–221 (2023).
[We prove that any inclusion-wise maximal coclique in an ordinary Kneser graph on k-sets in {1,...,n} extends to a unique coclique in the corresponding q-Kneser graph on k-subspaces of F_q^n. We further establish similar results for other buildings.]

Arthur Bik, Alessandro Danelon, and Jan Draisma: Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum, Math. Ann. 385, 1879–1921 (2023).
[In earlier work, the third author proved that polynomial functors over infinite fields are topologically Noetherian. In this paper, we establish the generalisation to arbitrary base rings whose spectrum is a Noetherian topological space. In particular, over the integers, this implies the existence of uniform degree bounds for equations that set-theoretically cut out varieties such as the r-th secant variety of the d-th Veronese embedding, independent of the ground field and of the dimension of the underlying vector space.]

Jan Draisma, Thomas Kahle, and Finn Wiersig: No short polynomials vanish on bounded rank matrices, Bull. Lond. Math. Soc. 55(4), 1791–1807 (2023).
[We prove that any nonzero polynomial that vanishes on the variety of rank-r matrices has at least (r+1)! terms, and that equality holds essentially only for the r-minors. We also establish an analogue for skew-symmetric matrices and Pfaffians. Both results build on the auxilliary result that on a sufficiently general r-dimensional linear space in K^n, no nonzero polynomials with fewer than r terms vanish.]

Benjamin Biaggi, Jan Draisma, and Tim Seynnaeve: On the quadratic equations for odeco tensors, Rend. Ist. Mat. Univ. Trieste 54, paper 17 (2022) (special issue on the occasion of Giorgio Ottaviani's 60th birthday).
[We study the locus defined by Robeva's quadratic equations for orthogonally decomposable tensors, and its connection to the Gorenstein locus in Hilbert schemes of points in affine space.]

Jan Draisma, Rob H. Eggermont, and Azhar Farooq: Components of symmetric wide-matrix varieties, J. Reine Angew. Math. 793, 143-184 (2022).
[In this paper we prove that if X_n is a variety of c-times-n matrices that is preserved the group S_n acting by column permutations, and if the map forgetting the last column maps X_{n+1} into X_n, then for n>>0 the number of irreducible components of X_n is a quasipolynomial in n. To this end, the paper develops the theory of FI^op-schemes of width one, and develops a subtle combinatorial machinery, in which S_n-orbits on components of X_n correspond to orbits of certain finite groupoids on integral points in rational polyhedra. Our theorem also has interesting purely combinatorial consequences, such as this one: for any k, the number of n-times-n symmetric zero/one matrices of rank k, counted modulo S_n acting by simultaneous row and column permutations, is, for n>>0, a quasipolynomial in n.]

Arthur Bik, Alessandro Danelon, Jan Draisma, and Rob H. Eggermont: Universality of high-strength tensors, Vietnam J. Math. 50(2), 557-580 (2022). Special issue on the occasion of Bernd Sturmfels's 60th birthday.
[A theorem by Kazhdan and Ziegler says that any sufficiently high-strength polynomial specialises to all polynomials in a prescribed number of variables. We generalise that theorem to arbitrary polynomial functors, and use this to show that among points with dense orbits in the limit space of a homogeneous polynomial functor there exists a unique minimal class under the partial order given by specialisation.]

Guus P. Bollen, Dustin Cartwright, and Jan Draisma: Matroids over one-dimensional groups, Int. Math. Res. Not. 3, 2298-2336 (2022).
[We develop the theory of matroids over one-dimensional algebraic groups, with special emphasis on positive characteristic. In particular, we compute the Lindström valuations and Frobenius flocks of such matroids. Building on work by Evans and Hrushovski, we show that the class of algebraic matroids, paired with their Lindström valuations, is not closed under duality of valuated matroids.]

Jan Draisma: The irreducible control property in matrix groups, Linear Algebra Appl. 634, 15-29 (2022).
[We study the problem of decomposing an element in a matrix group as a product of matrices in prescribed subvarieties, in a prescribed order, and show that often, the solution space to this problem is irreducible for each target matrix. As an example, adding a factor U in the well-known LU decomposition leads to the statement that each irreducible variety of invertible matrices lifts to an irreducible variety of ULU-decompositions.]

Jan Draisma and Alejandro Vargas: On the gonality of metric graphs, Notices Am. Math. Soc. 68(5), 687-695 (2021).
[This is an expository paper on the gonality of metric graphs, with an emphasis on our joint work below, as well as independent follow-up work by Alejandro Vargas.]

Jan Draisma and Felipe Rincón: Tropical ideals do not realise all Bergman fans, Res. Math. Sci. 8(3), article 44, 2021.
[Using a result of Las Vergnas on the non-existence of tensor products of matroids, we show that the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank 2 on three elements, equipped with weight 1, is not the tropical variety of any tropical ideal in the sense of Maclagan-Rincón.]

Jan Draisma and Alejandro Vargas: Catalan-many tropical morphisms to trees; Part I: Constructions, J. Symb. Comp. 104, 580-629, 2021.
[We give a purely combinatorial proof of the result, proved first by Matt Baker using specialisation of divisors from algebraic curves to metric graphs, that every genus-g metric graph has gonality at most ceiling(g/2)+1. We do so by studying morphisms that move in a family of the right dimension. See also this movie. In part II we will globalise our proof and construct a space of morphisms that maps onto the moduli space of genus-g metric graphs with fibres of cardinality a Catalan number.]

Arthur Bik, Jan Draisma, Alessandro Oneto, and Emanuele Ventura: The monic rank, Math. Comp. 89(325), 2481-2505, 2020.
[We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone X; and we describe an algorithmic technique based on classical invariant theory to determine, in certain concrete situations, the maximal monic rank. Using this, we prove that each univariate complex polynomial of degree 6,9,12 is the sum of 3 cubes of polynomials of degrees 2,3,4, respectively, and similarly that each univariate octic is a sum of 4 fourth powers of quadrics---special cases of a question by Boris Shapiro.]

Jan Draisma, Johannes Rau, and Chi Ho Yuen: The dimension of an amoeba, Bull. Lond. Math. Soc. 52(1), 16-23, 2020.
[Answering a question by Nisse and Sottile, we derive a formula for the dimension of the amoeba of an irreducible algebraic variety.]

Arthur Bik, Jan Draisma, and Rob H. Eggermont: Polynomials and tensors of bounded strength, Commun. Contemp. Math. 21(7), paper number 1850062 (24 pages), 2019.
[The strength of a homogeneous polynomial is the minimal number of terms in any expression as a sum of products of lower-degree polynomials. We prove that any nontrivial Zariski-closed property of polynomials (and tensors) that is functorial in the underlying vector space implies bounded strength.]

Jan Draisma, Michał Lasoń, and Anton Leykin: Stillman's conjecture via generic initial ideals, Commun. Algebra 47(6) (special issue dedicated to Gennady Lyubeznik), 2384-2395, 2019.
[We prove that an ideal generated by homogeneous polynomials of prescribed degrees d₁,...,d_k can have only finitely many generic initial grevlex ideals, independent of the number of variables. This yields a fourth, constructive proof of Stillman's conjecture, after Ananyan-Hochster's proof and two proofs by Erman-Sam-Snowden.]

Jan Draisma: Partial correlation hypersurfaces in Gaussian graphical models, Algebraic Combinatorics 2(3), 439-446 (2019).
[I give a positive answer to the question by Lin-Uhler-Sturmels-Bühlmann whether partial correlation hypersurfaces for complete DAGs are always nonsingular.]

Jan Draisma: Topological Noetherianity of polynomial functors, J. Am. Math. Soc. 32(3), 691-707 (2019).
[I prove that finite-degree polynomial functors from the category of finite-dimensional vector spaces to itself are topologically Noetherian. This implies the existence of uniform bounds on degrees of equations needed to characterise Zariski-closed tensor properties that are preserved under base changes and extending the dimension of the underlying vector space.]

Arthur Bik and Jan Draisma: A note on ED degrees of stable subvarieties in polar representations, Isr. J. Math. 228(1), 353-377 (2018).
[In this paper, Drusvyatskiy, Lee, Ottaviani, and Thomas establish a "transfer principle" by means of which the Euclidean distance degree of an orthogonally invariant matrix variety can be computed from the Euclidean distance degree of its intersection with a linear subspace. We simplify the proof and generalise this principle to varieties in orthogonal polar representations.]

Jan Draisma and Florian M. Oosterhof: Markov random fields and iterated toric fibre products, Adv. App. Math., 97, 64-79 (2018).
[We prove Conjecture 56 in this paper by Rauh and Sullivant, to the effect that the Markov degree of a Markov random field based on a large graph obtained by gluing copies from a finite collection of small graphs along a common subgraph is bounded, independently of the number of copies.]

Jan Draisma and Rob H. Eggermont: Plücker varieties and higher secants of Sato's Grassmannian, J. Reine Angew. Math. 737, 189-215 (2018).
[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.]

Jan Draisma, Giorgio Ottaviani, and Alicia Tocino: Best rank-k approximations for tensors: generalizing Eckart-Young, Res. Math. Sci. 5(2), article 27, 2018.
[We prove that the best rank-k approximation to a sufficiently general tensor lies in the linear span of its complex critical rank-1 approximations---a variant of the Eckart-Young theorem.]

Filip Cools and Jan Draisma: On metric graphs with prescribed gonality, J. Comb. Th. A 156, 1-21 (2018).
[We prove that in the moduli space of genus-g metric graphs the locus of graphs with gonality at most d has the classical dimension min{3g-3,2g+2d-5}.]

Guus Bollen, Jan Draisma, and Rudi Pendavingh: Algebraic matroids and Frobenius Flocks, Adv. Math. (323), 688-719 (2018).
[To each algebraic representation of a matroid M in characteristic p>0, we associate a Frobenius flock: an infinite lattice worth of vector spaces related by two simple axioms. From this flock we construct a valuation on M. One of the consequences is that if M is rigid, then it is algebraic in characteristic p if and only if it is linear in that characteristic.]
Note: the published version contains typos in the formulation of Lemma 18 and Theorem 38: B^ν \ B^ν_β should read B^ν and "matroid flock" should read "Frobenius flock".

Ada Boralevi, Jan Draisma, Emil Horobeţ, and Elina Robeva: Orthogonal and unitary tensor decomposition from an algebraic perspective, Israel J. Math. 222, 223-260 (2017).
[We prove that the set of orthogonally decomposable tensors is a real algebraic variety defined by polynomials of degrees at most four.]

Ada Boralevi, Jasper van Doornmalen, Michiel E. Hochstenbach, and Bor Plestenjak: Uniform determinantal representations SIAM Journal on Applied Algebra and Geometry 1(1), 415-441 (2017).
[We study determinantal representations of degree-d polynomials that depend bi-affine-linearly on the n variables and on the coefficients of the polynomial, respectively. Our main result is that if n is fixed and d varies, the minimal size of such a representation is Theta(d^{n/2}).]

Jan Draisma and Piotr Zwiernik: Automorphism groups of Gaussian chain graph models, Bernoulli, 23(2), 1102-1129, 2017.
[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 so-called 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 rank-one approximations to a tensor, Linear Multilinear Algebra, 64(12), 2498-2518 (2016).
[For a real-valued 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 rank-one tensors. We treat both the case of symmetric tensors and the case of general tensors.]

Jasmijn A. Baaijens and Jan Draisma: On the existence of identifiable reparametrizations for linear compartment models, SIAM J. Appl. Math. 76(4), 1577–1605 (2016).
[Following work by Nicolette Meshkat and Seth Sullivant and using algebraic geometry, combinatorics, and Lie theory, we derive simple necessary and sufficient conditions for a linear compartment model to be as identifiable as can be.]

Jan Draisma and Elisa Postinghel: Torus actions and faithful tropicalisation, Manuscr. Math. 149(3-4), 315-338 (2016).
[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, Emil Horobeţ, Giorgio Ottaviani, Bernd Sturmfels, and Rekha R. Thomas: The Euclidean distance degree of an algebraic variety, Found. Comput. Math. 16(1), 99-149 (2016).
[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 20(4), 953-967 (2015).
[We study the possible degrees of hsops of invariant rings of binary forms.]

Jan Draisma, Rob H. Eggermont, Robert Krone, and Anton Leykin: Noetherianity for infinite-dimensional toric varieties, Algebra & Number Theory 9(8), 1857-1880 (2015).
[We prove that many families of toric ideals stabilize up to symmetry.]

Andries E. Brouwer, Jan Draisma, and Bart J. Frenk: Lossy gossip and the composition of metrics, Discrete Comput. Geom. 53(4), 890-913 (2015).
[We prove that the monoid generated by n-by-n 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.]

Guus P. Bollen and Jan Draisma: An online version of Rota's basis conjecture, J. Alg. Comb. 41(4), 1001-1012 (2015).
[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 and Rob H. Eggermont: Finiteness results for Abelian tree models, J. Eur. Math. Soc. 17, 711-738, 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, 174-187, 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, 5648-5666, 2014.
[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.]

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), 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 Transform. 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 Transform. 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