Conjectures

Polynomiality of the maximum likelihood degree

For a generic d-dimensional space of symmetric matrices, is the ML-degree φ(n,d) a polynomial in n of degree d−1 for each fixed d?

Proved with Laurent Manivel, Leonid Monin, Tim Seynnaeve and Martin Vodička in Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming, J. Eur. Math. Soc. 26 (2024), 3091–3135.

Hierarchical versus train-track tensor formats

How do you represent a tensor? It depends who you are, but if your tensor lives in a very big product of vector spaces you probably have a problem — unless you are lucky and your tensor is special. People working on applications often have to work in huge tensor spaces, but fortunately their tensors are often special. One way to represent them is by tensor networks: in short, an inductive way to build your tensor up from pieces. Still, there are many ways to build such a representation.

Hackbusch's conjecture compared, in a very precise way, two of them: the one given by a perfect binary tree, the so-called hierarchical format, and the one given by a caterpillar or train-track tree. It was proved with Weronika Buczyńska and Jarosław Buczyński in The Hackbusch conjecture on tensor formats, J. Math. Pures Appl. 104 (2015), 749–761. Many thanks are also due to Joseph Landsberg, who gave us lectures on this topic and pointed us towards the conjecture.

What is left: It is quite amazing how well developed this approach to tensors is. Despite a huge number of applications it seems quite far from algebraic geometry, and largely unknown to algebraic geometers. I am sure both areas would profit from interaction — as I hope the example above shows. But that requires people who understand both stories, and there are not many of them. How deep the connections run is up to you.

Mulmuley's positivity hypothesis in Geometric Complexity Theory

Mulmuley's programme needed plethysm stretching functions to be Ehrhart quasi-polynomials of rational polytopes, with the constraint matrix independent of the partitions. That would give a positive combinatorial rule for plethysm.

Disproved with Thomas Kahle in Obstructions to combinatorial formulas for plethysm, Electron. J. Combin. 25 (2018). An explicit multiplicity function violates Ehrhart–Macdonald reciprocity, so it is not the Ehrhart function of any rational polyhedron.

What is left: The weaker asymptotic hypothesis is constrained but not settled, and Stanley's Problem 9 on a positive rule for plethysm remains open.

The algebraic degree of semidefinite programming

Nie, Ranestad and Sturmfels gave a formula for the algebraic degree δ(m,n,r) of semidefinite programming, but could prove it only in the range where the relevant determinantal variety is smooth. They conjectured it holds for all m.

Proved in the same complete quadrics paper, by relating the Lascoux coefficients to Q-Schur polynomials evaluated at n copies of ½.

Toric ideals of phylogenetic invariants

These conjectures hold a special place in my heart. I think the paper of Sturmfels and Sullivant was the first one my advisor, Jarosław Wiśniewski, gave me for my PhD thesis. At that point — and now I know I was wrong — I thought mathematics was only about proving hard open conjectures. So I tried really hard to prove those, and failed.

They are very interesting. It turns out that one can encode the algebraic properties of a finite group by a lattice polytope. Further, according to the conjectures, the algebraic properties of the defining equations of the toric variety represented by that polytope are closely related to the original group. Precisely, the conjecture says that the degree of the generators of the associated ideal — the Markov basis — is bounded by the cardinality of the finite abelian group.

What is perhaps most beautiful is that these toric varieties come to us naturally from other sciences: here it is phylogenetics, and a special role is played by the group ℤ₂×ℤ₂, the so-called 3-Kimura model. That is exactly Conjecture 30, and after ten years of fight — maybe even war — we finally managed to prove it with Emanuele Ventura, in Phylogenetic complexity of the Kimura 3-parameter model, Adv. Math. 343 (2019), 640–680. Conjecture 29 asks the same for an arbitrary finite abelian group.

Is rank at most twice border rank?

It is a well-known fact that, contrary to the case of matrices, tensors of rank at most k do not form a closed set. We say a tensor has border rank at most k if it can be approximated by tensors of rank k — but such a tensor may have rank much greater than k. For T in A⊗B⊗C, is there a bound on rank(T)/border rank(T)? Computing examples, the conjecture stated by John Rhodes looks very natural: the quotient is at most 2.

This holds in small dimension. With J. M. Landsberg we provided the first counterexample, in Abelian tensors, J. Math. Pures Appl. 108 (2017), 333–371; independently and at the same time, by different methods, so did Jeroen Zuiddam in A note on the gap between rank and border rank, Linear Algebra Appl. 525 (2017), 33–44.

What is left: In general, bounding rank(T)/border rank(T) for T in A⊗B⊗C is open. If you want a big challenge, try to find a tensor with quotient greater than 3 — you will solve several other open problems along the way.

Howe's conjecture on symmetric plethysm

Algebraic geometry is all about polynomials. Homogeneous polynomials of degree d on a vector space V are naturally identified with elements of the symmetric power S^d(V*). They carry a natural action of GL(V) — a simple change of coordinates — and as such form an irreducible representation. Looking back at the nineteenth century, a large part of mathematics, maybe even the majority, was focused on understanding what happens to polynomials when there is an additional group action.

In modern language one could say that mathematicians were studying the plethysm S^k(S^d(V*)). This is already a reducible representation, and general formulas for its decomposition will probably never be known — some mathematicians argue with me about that, but I claim the formulas are simply too complicated. Once we cannot compute something, we try to estimate it, very often in the limit.

Given a partition λ of kd, what is the multiplicity m(λ) in S^k(S^d(V))? A beautiful miracle is that m is a piecewise quasipolynomial: one can partition the space where all λ live into cones, and in each cone the formula is a quasipolynomial. We can then ask for the leading term of m in each chamber. It turns out the leading term is not a quasipolynomial but an honest polynomial, and Howe's conjecture is natural: among all tensors, roughly 1/k! are symmetric. Precisely, the leading term of m equals 1/k! times the leading term of the multiplicity of λ in S^d(V)^⊗k. That latter multiplicity is the famous Littlewood–Richardson coefficient, which can be expressed as a lattice-point count in certain polytopes — so the leading term is related to volumes of polytopes.

We proved this with Thomas Kahle in Plethysm and lattice point counting, Found. Comput. Math. 16 (2016), 1241–1261; the conjecture was stated not only for the outer symmetric power but for an arbitrary Schur functor, details in the paper. That is, in each chamber we determined the leading term, confirming the conjecture. The story seems over, but as Michèle Vergne pointed out to us, a very interesting question remains.

What is left: Fix λ and consider the one-variable quasipolynomial f: s ↦ m(sλ); that is, scale λ and ask for the leading term. It seems that if we know the leading terms in general then, restricting to the ray through λ, the leading term of f is just the restriction of the leading term of m. This works for general λ — in fact when all rows of λ have different lengths — but for special λ the leading term of m may restrict to 0, and then the leading term of f is different. One can still make the natural analogue of Howe's conjecture: the leading term of f equals 1/k! times the leading term of the multiplicity of sλ in (S^sd(V))^⊗k.

White's conjecture on toric ideals of matroids

White's conjecture predicts quadratic generators for the toric ideal of a matroid base polytope. In that form it is still open.

With Michał Lasoń we proved it scheme-theoretically — the two ideals define the same scheme — in On the toric ideal of a matroid, Adv. Math. 259 (2014), 1–12. With Kangjin Han and Julian Weigert we later showed the conjecture propagates between matroids differing by a single basis, in White's conjecture for matroids and inner projections, arXiv:2501.17738.

The stronger forms, however, have fallen. Asking for a quadratic Gröbner basis — a strengthening due to Herzog and Hibi rather than to White — is too much: the toric ideal of the Fano matroid has none, as we showed with Spencer Backman, Nathan Cheung, Michał Lasoń and Gaku Liu in The Gröbner version of White's conjecture is false, arXiv:2606.13960 — found independently by De Loera, Ferroni, Morales and Rambau.

What is left: The conjecture itself — quadratic generators — is wide open.

The degree of the Gaussian graphical model of the cycle

Sturmfels and Uhler computed the degree of the variety attached to the Gaussian graphical model of the m-cycle for m ≤ 8 and guessed the pattern: (m+2)/4 · C(2m,m) − 3·2^2m−3.

Proved with Rodica Dinu and Martin Vodička in Geometry of the Gaussian graphical model of the cycle, arXiv:2111.02937, using intersection theory on the variety of complete quadrics.

The Sullivant–Talaska ideal of the cyclic Gaussian graphical model

Do the 3×3 minors generating the Sullivant–Talaska ideal already cut out the whole prime ideal of the cyclic model?

Proved with Austin Conner and Kangjin Han in Sullivant–Talaska ideal of the cyclic Gaussian graphical model, Adv. Appl. Math. 179 (2026) — in fact those minors form a squarefree Gröbner basis.

Which matroid invariants are matroidal mixed Eulerian numbers?

Answered with Gaku Liu and Julian Weigert in Mixed Eulerian numbers and beyond, arXiv:2502.04980: two loopless matroids of the same rank on the same ground set have the same matroidal mixed Eulerian numbers precisely when they have the same Derksen G-invariant. So the invariant recorded by these numbers is exactly Derksen's.

Are all algebraic polymatroids Chow?

Answered affirmatively with Lukas Grund, June Huh, Hendrik Süss and Botong Wang in Linear operators preserving volume polynomials, arXiv:2506.22415, which in fact characterises algebraic polymatroids by their sets of bases.

Do determinants generate the ideal of singular matrix tuples?

Answered negatively with Julian Vill and Alexander Taveira Blomenhofer in Ideals of spaces of degenerate matrices, Linear Algebra Appl. 648 (2022), 56–69. For 2×2 matrices the determinants generate only when m ≤ 2; from m ≥ 3 cubics are needed, and for every n > 1 and m ≥ n²−n+1 the ideal is not generated in degree n.

What is left: Determining the generators for n ≥ 3 is still open, as is Makam and Wigderson's Problem 12.7 on Cohen–Macaulayness.

Classification of smooth minimal monomial Togliatti systems of cubics

This conjecture concerns the classification of (smooth minimal monomial) Togliatti systems of cubics. In down-to-earth terms: certain very special sets of monomials of degree three. The word “special” means several things at once. To a set of monomials one associates two ideals or schemes. One is simply the ideal defined by the given monomials. The other is the image of the map given by the monomials that are not in our set but have the same degree — the apolar system — which is a nice toric variety. As you would guess, there is an interplay between the two.

The failure of the weak Lefschetz property in some degree for the first is related to the second satisfying a Laplace equation of some order. Failing the weak Lefschetz property means, in short, that multiplication by a linear form defines a map that is not of maximal rank; satisfying a Laplace equation means the osculating spaces — higher analogues of tangent spaces — are degenerate.

The conjecture was proved with Rosa Maria Miró-Roig, who is an expert, in Smooth monomial Togliatti systems of cubics, J. Combin. Theory Ser. A 143 (2016), 66–87. If you plan to work on the subject, contacting Rosa is a good idea.

What is left: A lot. We proved the case of cubics, and there is not even a conjectural classification in higher degree. Hal Schenck is currently working on these topics.

Classifying Gorenstein matroids

Which matroids have Gorenstein base ring, or Gorenstein independence polytope?

The graphic case was settled with Takayuki Hibi, Michał Lasoń, Kazunori Matsuda and Martin Vodička in Gorenstein graphic matroids, Israel J. Math. 243 (2021), 1–26; the full classification followed with Michał Lasoń in Gorenstein matroids, IMRN 2023, in both the base-polytope and independence-polytope forms.

What is left: Herzog and Hibi asked about discrete polymatroids in general; the classification above is for matroids.

Roots of Ehrhart polynomials of symmetric edge polytopes

Do all roots of the Ehrhart polynomial of the symmetric edge polytope of a complete graph — or of a complete bipartite graph of type (2,n) — lie on the line Re(z) = −½?

Both proved with Akihiro Higashitani and Mario Kummer in Interlacing Ehrhart polynomials of reflexive polytopes, Selecta Math. 23 (2017), 2977–2998, via a stronger interlacing property. The bipartite case comes out for every graph of type (2,n), not only the complete ones.

Identifiability and equations for uniform matrix product states

Both settled with Adam Czapliński and Tim Seynnaeve in Uniform matrix product states from an algebraic geometer's point of view, Adv. Appl. Math. 142 (2023): the trace parametrisation is generically N-to-one for N ≥ 5, and the ideal of uMPS(2,2,5) is generated by the 3 quartics and 27 sextics Critch and Morton exhibited. The ideal for N = 6 comes out too.

Leitner's conditions on conjugacy limits of the diagonal Cartan subgroup

Leitner proved two necessary conditions for an abelian subgroup to be a conjugacy limit of the diagonal Cartan subgroup, and asked whether they are sufficient.

Answered negatively with J. M. Landsberg in Abelian tensors, J. Math. Pures Appl. 108 (2017), 333–371: an explicit 9×9×9 tensor satisfies both conditions but fails a flag condition, so its border rank is too large.

Is the set of cyclic matrix product states closed?

Hackbusch asked whether the site-independent (cyclic) matrix product states of fixed bond dimension form a closed set, expected the answer to be no, and asked for an explicit tensor in the closure but not the set.

Answered negatively with Corey Harris and Emre Can Sertöz in Computing images of polynomial maps, Adv. Comput. Math. 45 (2019), 2845–2865, with exactly such an explicit curve — and the same for infinitely many parameter triples.

Minimality of Galois–Togliatti systems

The Togliatti story above, continued. Galois–Togliatti systems are the monomial systems invariant under a cyclic group action, and the conjecture concerned their minimality.

The route to an answer turned out to run through linear algebra: comparing the coefficients in the expansion of the permanent with those in the expansion of the determinant of a three-line circulant matrix. Proved with Pietro De Poi, Emilia Mezzetti, Rosa Maria Miró-Roig and Eran Nevo in Circulant matrices and Galois–Togliatti systems, J. Pure Appl. Algebra 224 (2020), no. 11.

Tilting bundles on rational surfaces

A family of counterexamples is given in Family of counterexamples to King's conjecture, C. R. Math. Acad. Sci. Paris 349 (2011), no. 1–2, 67–69.

Seminormality of cut polytopes is minor-closed

Proved with Michał Lasoń in A note on seminormality of cut polytopes, SIAM J. Discrete Math. 36 (2022): a cut polytope is seminormal exactly when it is normal, which makes the property minor-closed.

Koley and Römer accepted both conjectures as settled and removed them from the published version of their paper.

What is left: Their Conjecture 4.5 does not disappear so much as change character: it becomes literally the Sturmfels–Sullivant conjecture that cut polytopes are normal exactly for K₅-minor-free graphs, which is still open.

Integer decomposition property of dilated polytopes

Answered in Non-normal very ample polytopes — constructions and examples.

Measurable colourings of the real line

Is there a measurable colouring of ℝ with finitely many colours in which no two distinct intervals contain the same measure of every colour?

Yes, and five colours suffice — with Noga Alon, Jarosław Grytczuk and Michał Lasoń in Splitting necklaces and measurable colorings of the real line, Proc. Amer. Math. Soc. 137 (2009), 1593–1599, by a Baire category argument.

What is left: Whether five is optimal is open; two colours are known not to suffice.

Weintraub's conjecture on even partitions in plethysms

For an even partition λ of 2pq with at most p parts, does the Schur module S_λW appear in the plethysm S^p(S^2qW)?

With Laurent Manivel we gave a short, fully constructive proof in Effective constructions in plethysms and Weintraub's conjecture, Algebr. Represent. Theory 17 (2014), 433–443, producing an explicit highest weight vector.

What is left: Credit where it is due: the conjecture was first proved by Bürgisser, Christandl and Ikenmeyer in 2011.

Varieties of reductions for gl(n)

The quantum Wielandt inequality

Take a linear space L of complex D×D operators, and suppose some power L^k is already all of M_D. How soon does that happen? Pérez-García, Verstraete, Wolf and Cirac conjectured that the sequence L¹, L², … stabilises after O(D²) steps. The question governs how long a chain of a matrix product state must be before the representation becomes injective.

With Yaroslav Shitov we brought the bound down from the previously known O(D⁴) to O(D² log D), in Quantum version of Wielandt's inequality revisited, IEEE Trans. Inform. Theory 65 (2019), 5239–5242. A tensor analogue was developed with Tim Seynnaeve and Frank Verstraete in A tensor version of the quantum Wielandt theorem, SIAM J. Matrix Anal. Appl. 40 (2019), 1125–1130.

What is left: Only a logarithmic factor now separates what is known from what was conjectured — but the O(D²) bound itself is still open.

The maximum likelihood degree of the Gaussian cycle

The conjectured value is (m−3)·2^m−2 + 1. With Carlos Améndola, Rodica Dinu and Martin Vodička we proved the matching lower bound in On the maximum likelihood degree for Gaussian graphical models, arXiv:2410.07007.

What is left: The conjecture has since been proved in full — by Rodica Dinu and Martin Vodička, arXiv:2503.02704, without me.

Complete monotonicity for inverse powers of elementary symmetric polynomials

With Khazhgali Kozhasov and Bernd Sturmfels we proved the necessity direction in full, in Positivity certificates via integral representations (Facets of Algebraic Geometry, Cambridge University Press, 2022): small negative powers are never completely monotone.

What is left: Sufficiency — reaching the sharp threshold (n−m)/2 — is still open.

Gal's conjecture on γ-vectors of flag spheres

With Akihiro Higashitani and Katharina Jochemko we established it for the symmetric edge polytopes of complete bipartite graphs, in Arithmetic aspects of symmetric edge polytopes, Mathematika 65 (2019), 763–784, by giving a closed, real-rooted formula for the h*-polynomial.

What is left: The conjecture in general is wide open.

The Nevo–Petersen strengthening of Gal's conjecture

In the same Mathematika paper we showed the γ-polynomial of any flag unimodular triangulation of these symmetric edge polytopes is the f-polynomial of a flag balanced complex — slightly more than the conjecture asks, for that family. Further families followed with Alessio D'Alì and Emanuele Delucchi in Many faces of symmetric edge polytopes.

What is left: Open in general.

Real rank boundaries as secant-tangential joins

Confirmed for quaternary cubics with Hyunsuk Moon in Spaces of sums of powers and real rank boundaries: the real rank boundary is an irreducible hypersurface of degree 40 in ℙ¹⁹.

What is left: The general statement is open, and so is the quaternary quartic case, where the relevant variety of sums of powers is a fivefold nobody has described.

Nonnegativity of the Riesz kernel of a hyperbolic polynomial

Partially addressed in Positivity certificates via integral representations, with Khazhgali Kozhasov and Bernd Sturmfels.

What is left: Not settled.

Is there an algorithm to decide whether a variety is toric?

Does there exist an algorithm to decide whether a given affine variety is toric? I suspect the answer is no.

And a sharper form of the same worry: is there an algorithm to decide whether a given variety is affine space? Equivalently — is there an algorithm to decide whether a given finitely generated ℂ-algebra is a polynomial ring?

Can every n-dimensional variety be injected into 2n-dimensional space?

Can every (smooth) projective algebraic variety of dimension n be injected into 2n-dimensional projective space by an algebraic map — not necessarily an embedding? And, a good place to start, what about curves?

Paul Görlach studied exactly this in Injection dimensions of projective varieties, arXiv:1905.11306.

Do the lattice points in a ball form a normal polytope?

Take a ball in ℝⁿ and let P be the convex hull of the lattice points inside it. Is P always normal? The question is embarrassingly concrete.

Can a single lattice point destroy normality?

Do there exist normal lattice polytopes so delicately balanced that removing any one lattice point — or adding any one — makes them non-normal?

Gap vectors of very ample polytopes

We conjectured that the gap vector of a very ample lattice polytope has no internal zeros, and that it is unimodal when the facets are normal.

Both parts are false. With Michał Lasoń in Non-normal very ample polytopes — constructions and examples, Experimental Math. 26 (2017), 130–137, we produced very ample polytopes with normal facets whose gap vectors have all internal entries zero, and others that are not unimodal.

What is left: A conjecture is allowed to be wrong; it is less pleasant when you are one of the people who made it.

Degree equals ML-degree for exponential varieties

Proved, and without the hyperbolicity hypothesis, with Leonid Monin and Jarosław Wiśniewski in Maximum likelihood degree, complete quadrics and ℂ*-action, SIAM J. Appl. Algebra Geom. 5 (2021), 60–85. The proof is a corollary of Teissier's theorem on polar varieties — the work was in noticing that it applies.

Is every variety from the 3-Kimura model projectively normal?

The 3-Kimura model of phylogenetics produces toric varieties attached to the group ℤ₂×ℤ₂ — the same family as in the Sturmfels–Sullivant entry above. Are they all projectively normal?

Proved by Martin Vodička in Normality of the Kimura 3-parameter model, arXiv:1902.11057.

Claw trees as intersections of smaller-valency trees

On the torus orbit, is the phylogenetic variety of a claw tree the intersection of the varieties attached to trees with strictly smaller valency?

Proved with Marta Casanellas and Jesús Fernández-Sánchez in Low degree equations for phylogenetic group-based models, Collect. Math. 66 (2015), 203–225 — with a complete intersection description on that open set.

Explicit formulas for the ML-degrees φ(n,6), …, φ(n,12)

Seven closed-form polynomials predicted by our algorithms, conditional on the Sturmfels–Uhler polynomiality conjecture.

Confirmed by the complete quadrics paper, which proves polynomiality and then computes φ(n,d) explicitly for every d ≤ 50.