Mateusz MichalekUNDER CONSTRUCTION
Some conjectures I worked on (mostely with other people). Do not worry, almost in all cases there is plenty of work left to be done!
Conjecture 1: 6.2 in E. Mezzetti, R. Miró-Roig and G. Ottaviani, Laplace Equations and the Weak lefschetz Property, Canadian Mah. J. 65 (2013) extending and correcting previous conjecture p. 12 in G. Ilardi, Togliatti systems, Osaka J. Math. 43 (2006), 1-12.
This conjecture was on classification of (smooth minimal monomial) Togliatti systems (of cubics). So in down-to-earth terms on some very special sets of monomials of degree three. The word 'special' here means several things. To a set of monomials one associates two ideals/schemes: one is simply the ideal defined by the given monomials. The other one is the image of the map given by monomials that are NOT in our set, but are of the same degree (the apolar) - so a nice toric variety. As you guess there are interplays between the two. The failing of weak Lefschetz property (in some degree) for the first one is related to satisfying Laplace equation (of some order) of the second one. WLP in short means that (multiplication by) a linear form defines a map that is not of maximal rank. Satisfying Laplace equation means the osculating spaces (higher analogs of tangent spaces) are degenerate. For more see our article. The conjecture was proved with Rosa-Maria Miró-Roig, who is an expert. (If you plan to work on the subject, contacting Rosa is a good idea.)
What is left? A lot: as you can see we proved the case of cubics, as there is not even a conjectural classification in higher degree. Currently Hal Schenck is working on these topics.
Conjecture 2: Conjecture 12.7, Wolfgang Hackbusch, Tensor spaces and numerical tensor calculus, volume 42 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2012
How do you represent a tensor? It depends who you are, but if a tensor is in a very big product of vector spaces, you probably have a problem. Unless of course you are lucky and your tensor is special. People dealing with applications often have to work in hudge tensor spaces, but fortunatelly their tensors are often special. One of the ways to represent them are tensor networks - in short, an inductive way to build up your tensor from pieces. Still, there are many ways how you build your representation. The Conjecture of W. Hackbusch compared (in a very precise way!) two such representations: one given by a perfect binary tree (so-called hierarchical format) and one by a caterpillar tree/train track tree. The conjecture was proved with Weronika and Jarek Buczyńscy in here. Many thanks are also due to Joseph Landsberg who gave us lectures on this topic and pointed us toward the conjecture.
What is left? It is quite amazing how well-developped is this approach towards tensors. Dispite hudge amount of applications, it seems quite far from algebraic geometry (and unknow to algebraic geometers). I am sure both areas would profit from interactions - as I hope the example above proves. However, these would require people who understand both stories - and there are not many of them. How deep conections will be found between the algbraic geometry approach and the above one is up to you!
Conjecture 3: 3.6(d) in R. Howe, (GL(n);GL(m))-duality and symmetric plethysm, Proc. Indian Acad. Sci. Math. Sci. 97 (1987)
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 come with a natural group action of GL(V) - simple change of coordinates - and as such form an irreducible representation. Looking back at XIX century, a large part of mathematics (maybe even majority!) was focused on understanding what happens to polynomials if there is an additional group action. The main example are polynomials with variables that are ceoefficients of polynomials, e.g SL(V) invariants. (For d=2 and dim V=2 think about Delta.) Now, in modern language one could say mathematicians were studying plethysm S^k(S^d(V^*)). This is already a reducible representation, however general formulas for the decomposition probably will never be known. (Some mathematicians argue with me about the validity of last statement, but I just claim the formulas are simply too complicated.) Once we cannot compute something, usually mathematicians try to estimate it, very often in the limit. OK let us look at S^k(S^d(V)), where we consider d as a variable. The (hopeless) question we ask: given a partition of \lambda of kd what is the multiplicity m(\lambda) in S^k(S^d(V))? A beautiful miracle is that m is a piecewise quasipolynomial, i.e. one can partition the vector space where all \lambda live into cones and in each cone the fomula is a quasipolynomial (i.e. a polynomial but the coefficients are not constants but depend on parameters, however only modulo a fixed integer). We can make our question about 'estimate' precise: we can ask what is the leading term of m (in each chamber)? It turns out that the leading term is not a quasipolynomial, but an honest polynomial. Here comes a natural conjecture of Howe: among all tensors, more or less 1/k! are symmetric, or more precisely the leading term of m is the same as 1/k! times (the leading term of) the multiplicity of \lambda in S^d(V)^{\otimes k}. The latter multiplicity is the famous Richardson-Littlewood coefficient and in fact can be expressed as a counting problem of lattice points in certain polytopes. In particular, the leading term is related to volumes of the polytopes. We proved this (conjectur was stated not only for outer symmetric power but arbitrary Schur functor - detalis in the paper) together with Thomas Kahle . That is, in each chamger, we determined the leading term confirming the conjecture. Seems the story is over, however as Michele Vergne pointed out to us there is a very interesting question left, see below.
What is left? If we fix a \lambda we can ask about (one variable quasipolynomial) f:s-> m(s\lambda), i.e. we scall \lambda, and ask about the leading term. It seems that: if we determined the leading terms of formulas in general then, restricting to ray through \lambda, the leading term of f will be just the restriction of the leading term of m. Haha! This works for general \lambda (here in fact when all rows of \lambda have different lengths), however for special \lambda the leading term of m may restrict to 0 (and then the leading term of f of course is different). Still one can naturally make an analogue of Howe's conjecture: the leading term of f equals 1/k! times the leading term of the multiplicity of s\lambda in (S^{sd}(V))^{\otimes k}. This remains open!
Conjecture 4: by J. Rhodes: Conjecture 0 in E. Ballico and A. Bernardi, Stratification of the fourth secant variety of Veronese varieties via the symmetric rank, Adv. Pure Appl. Math. 4 (2013), no. 2, 215–250.
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 that a tensor has border rank (at most) k if it may be approximated by tensors of rank k. Still such a tensor may have rank (much) greater than k. Say T\in A\otimes B\otimes C. Is there a bound for the quotient rank(T)/border rank(T)? As you compute some examples you find that the following conjecture stated by John Rhodes is very natural: the quotient before is at most 2. This holds in small dimension, however we provided the first counterexample with JM Landsberg in our article and more counterexamples were presented by Jeroen Zuiddam here.What is left? In general, for T\in A\otimes B\otimes C bounding rank T/border rank of T is open. If you want a BIG challenge try to find a tensor with quotient greater than 3. (You will also solve other open problems in process :) )
Question 5: Question on p.4 and p.5 in A. Iliev and L. Manivel, Varieties of reductions for gl(n), Projective varieties with unexpected properties, Walter de Gruyter GmbH & Co. KG, Berlin, (2005).
Question 6: Question 3.5 (1) and (2), Question 3.6 in D. Cox et al. Integer Decomposition Property of Dilated Polytopes, The Electronic Journal of Combinatorics 21.4 (2014)
Conjecture 3.5(a),(b), Very ample and Koszul segmental fibrations Open question 3 (a),(b) p. 2310, Question p. 2316, C.~Haase, T.~Hibi, D.~Maclagan, Oberwolfach ReportAnswered in Non-Normal Very Ample Polytopes – Constructions and Examples
Conjecture 7: Conjecture 9.3 in A. King Tilting bundles on some rational surfaces (1997).
Conjecture 8: Conjectures 29 and 30 in B. Sturmfels and S. Sullivant, Toric ideals of phylogenetic invariants, Journal of Computational Biology (2005) 12(2): 204-228
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 (now I know I was wrong!) I thought that mathematics is only about proving hard open conjectures. So I tried really hard to prove those and... failed. They are very interesting: in turns out that one can encode the algebraic properties of a finite group by a lattice polytope. Further, according to the conjectures algebraic properties of the defining equations of the toric variety represented by the polytope are closely related to the original group. Precisely the conjecture says that the degree of the generators of the associated ideal (Markov basis) is bounded by the cardinality of the (finite, abelian) group. What is probably most beautiful is that these toric varieties come to us naturally form other sciences: here it is phylogenetics and a special role is played by the group Z2xZ2 (so called 3-Kimura model). After 10 years of fight (maybe even a war) finally with Emanuele Ventura we managed to prove the easier Conjecture 29!
What is left? If you manage to prove the more general Conjecture 30 you have my highest respect! I can offer a prize for this research from my grant (say 3000 euro), I can offer a postdoc, I can... well basically a lot - just name it! (Btw. when I was thinking about ofering prize for proving these conjectures Thomas Kahle told me that maybe I am still too young to do this. But that was over 6 years ago, when I was still a PhD student - hope it is fine now :) ) (I also realized I assumed conjecture is true. Well, if you find a counterexample I probably cannot offer such a prize, but I will figure something out. An invitation for a good dinner is a lower bound.)
Conjecture 9: Conjectures 12 and 13 in N. White, A unique exchange property for bases, Linear Algebra and its Applications (1980) 31: 81-91
Conjecture 10: Conjecture 4.6 in E. Mezzetti, R.M. Miró-Roig, Togliatti systems and Galois coverings, Journal of Algebra (2018) 509
Answered in Circulant matrices and Galois-Togliatti systems
Conjecture 11: D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. Matrix product state representations. Quantum Inf. Comput. 7(5–6):401–430, 2007.
Quantum version of Wielandt’s Inequality revisited
Conjecture 12: