Finite phylogenetic complexity and combinatorics of tables
We prove that the phylogenetic complexity -- an invariant introduced by Sturmfels and Sullivant -- of any finite abelian group is finite.
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arXiv subjects
Mateusz Michałek
Publications and source records attributed to Mateusz Michałek.
Abelian Tensors
We analyze tensors in the tensor product of three m-dimensional vector spaces satisfying Strassen's equations for border rank m. Results include: two purely geometric characterizations of the Coppersmith-Winograd tensor, a reduction to the study of symmetric tensors under a mild genericity hypothesis, and numerous additional equations and examples. This study is closely connected to the study of the variety of m-dimensional abelian subspaces of the space of endomorphisms of an m-dimensional vector space, and the subvariety consisting of the Zariski closure of the variety of maximal tori, called the variety of reductions.
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A $2n^2-log(n)-1$ lower bound for the border rank of matrix multiplication
Let M_n denote the matrix multiplication tensor for nxn matrices. We use the border substitution method combined with Koszul flattenings to prove the border rank lower bound of 2n^2-log(n)-1 for M_n.
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Finite phylogenetic complexity of $Z_p$ and invariants for $Z_3$
We study phylogenetic complexity of finite abelian groups - an invariant introduced by Sturmfels and Sullivant. The invariant is hard to compute - so far it was only known for $Z_2$, in which case it equals $2$. We prove that phylogenetic complexity of any group $Z_p$, where $p$ is prime, is finite. We also show, as conjectured by Sturmfels and Sullivant, that the phylogenetic complexity of $Z_3$ equals $3$.
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Real Rank Geometry of Ternary Forms
We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for quadrics and cubics. For quintics we determine the real rank boundary: it is a hypersurface of degree 168. For quartics, sextics and septics we identify some of the components of the real rank boundary. The real varieties of sums of powers are stratified by discriminants that are derived from hyperdeterminants.
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Smooth monomial Togliatti systems of cubics
The goal of this paper is to solve the conjecture stated in a paper of Mezzetti, Miró-Roig, Ottaviani and classify all smooth minimal monomial Togliatti systems of cubics. More precisely, we classify all minimal monomial artinian ideals generated by cubics, failing the weak Lefschetz property and whose apolar cubic system defines a smooth toric variety or, equivalently, we classify all minimal monomial artinian ideals generated by cubics whose apolar cubic system defines a smooth toric variety satisfying at least a Laplace equation of order 2.
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Quantum jumps of normal polytopes
We introduce a partial order on the set of all normal polytopes in R^d. This poset NPol(d) is a natural discrete counterpart of the continuum of convex compact sets in R^d, ordered by inclusion, and exhibits a remarkably rich combinatorial structure. We derive various arithmetic bounds on elementary relations in NPol(d), called "quantum jumps". The existence of extremal objects in NPol(d) is a challenge of number theoretical flavor, leading to interesting classes of normal polytopes: minimal, maximal, spherical. Minimal elements in NPol(5) have played a critical role in disproving various covering conjectures for normal polytopes in the 1990s. Here we report on the first examples of maximal elements in NPol(4) and NPol(5), found by a combination of the developed theory, random generation, and extensive computer search.
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On the geometry of border rank algorithms for matrix multiplication and other tensors with symmetry
We establish basic information about border rank algorithms for the matrix multiplication tensor and other tensors with symmetry. We prove that border rank algorithms for tensors with symmetry (such as matrix multiplication and the determinant polynomial) come in families that include representatives with normal forms. These normal forms will be useful both to develop new efficient algorithms and to prove lower complexity bounds. We derive a border rank version of the substitution method used in proving lower bounds for tensor rank. We use this border-substitution method and a normal form to improve the lower bound on the border rank of matrix multiplication by one, to 2n^2- n+1. We also point out difficulties that will be formidable obstacles to future progress on lower complexity bounds for tensors because of the "wild" structure of the Hilbert scheme of points.
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Examples of $k$-regular maps and interpolation spaces
A continous map $f: \mathbb{C}^n \rightarrow \mathbb{C}^N$ is $k$-regular if the image of any $k$ points spans a $k$-dimensional subspace. It is an important problem in topology and interpolation theory, going back to Borsuk and Chebyshev, to construct $k$-regular maps with small $N$ and only a few nontrivial examples are known so far. Applying tools from algebraic geometry we construct a 4-regular polynomial map $\mathbb{C}^3\rightarrow \mathbb{C}^{11}$ and a 5-regular polynomial map $\mathbb{C}^3\rightarrow \mathbb{C}^{14}$.
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Exponential Varieties
Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Among them are varieties of inverses of symmetric matrices satisfying linear constraints. This class includes Gaussian graphical models. We develop a general theory of exponential varieties. These are derived from hyperbolic polynomials and their integral representations. We compare the multidegrees and ML degrees of the gradient map for hyperbolic polynomials.
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Hackbusch Conjecture on tensor formats
We prove a conjecture of W. Hackbusch about tensor network states related to a perfect binary tree and train track tree. Tensor network states are used to present seemingly complicated tensors in a relatively simple and efficient manner. Each such presentation is described by a binary tree and a collection of vector spaces, one for each vertex of the tree. A problem suggested by Wolfgang Hackbusch and Joseph Landsberg is to compare the complexities of encodings, if one presents the same tensor with respect to two different trees. We answer this question when the two trees are extremal cases: the most "spread" tree (perfect binary tree), and the "deepest" binary tree (train track tree). The corresponding tensor formats are called hierarchical formats (HF) and tensor train (TT) formats, respectively.
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Splitting necklaces and measurable colorings of the real line
A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A well-known application of the Borsuk-Ulam theorem asserts that every $k$-colored necklace can be fairly split by at most $k$ cuts (from the resulting pieces one can form two collections, each capturing the same measure of every color). Here we prove that for every $k\geq 1$ there is a measurable $(k+3)$-coloring of the real line such that no interval can be fairly split using at most $k$ cuts. In particular, there is a measurable $4$-coloring of the real line in which no two adjacent intervals have the same measure of every color. An analogous problem for the integers was posed by Erdős in 1961 and solved in the affirmative in 1991 by Keränen. Curiously, in the discrete case the desired coloring also uses four colors.
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Very ample and Koszul segmental fibrations
In the hierarchy of structural sophistication for lattice polytopes, normal polytopes mark a point of origin; very ample and Koszul polytopes occupy bottom and top spots in this hierarchy, respectively. In this paper we explore a simple construction for lattice polytopes with a twofold aim. On the one hand, we derive an explicit series of very ample 3-dimensional polytopes with arbitrarily large deviation from the normality property, measured via the highest discrepancy degree between the corresponding Hilbert functions and Hilbert polynomials. On the other hand, we describe a large class of Koszul polytopes of arbitrary dimensions, containing many smooth polytopes and extending the previously known class of Nakajima polytopes.
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Non-normal very ample polytopes - constructions and examples
We present a method of constructing non-normal very ample polytopes as a segmental fibration of unimodular graph polytopes. In many cases we explicitly compute their invariants - Hilbert function, Ehrhart polynomial, gap vector. In particular, we answer several questions posed by Beck, Cox, Delgado, Gubeladze, Haase, Hibi, Higashitani and Maclagan.
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On the toric ideal of a matroid
Describing minimal generating set of a toric ideal is a well-studied and difficult problem. In 1980 White conjectured that the toric ideal associated to a matroid is equal to the ideal generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture up to saturation, that is that the saturations of both ideals are equal. In the language of algebraic geometry this means that both ideals define the same projective scheme. Additionally we prove the full conjecture for strongly base orderable matroids.
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Local description of phylogenetic group-based models
Motivated by phylogenetics, our aim is to obtain a system of equations that define a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties defined via group-based models. For any finite abelian group $G$, we provide an explicit construction of $codim X$ phylogenetic invariants (polynomial equations) of degree at most $|G|$ that define the variety $X$ on a Zariski open set $U$. The set $U$ contains all biologically meaningful points when $G$ is the group of the Kimura 3-parameter model. In particular, our main result confirms a conjecture by the third author and, on the set $U$, a couple of conjectures by Bernd Sturmfels and Seth Sullivant.
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Secants of minuscule and cominuscule minimal orbits
We study the geometry of the secant and tangential variety of a cominuscule and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods inspired by statistics we provide an explicit local isomorphism with a product of an affine space with a variety which is the Zariski closure of the image of a map defined by generalized determinants. In particular, equations of the secant or tangential variety correspond to relations among generalized determinants. We also provide a representation theoretic decomposition of cubics in the ideal of the secant variety of any Grassmannian.
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On the graph labellings arising from phylogenetics
We study semigroups of labellings associated to a graph. These generalize the Jukes-Cantor model and phylogenetic toric varieties defined by Buczyńska. Our main theorem bounds the degree of the generators of the semigroup by g+1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.
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