Related papers: The Sticky Matroid Conjecture

Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the "characteristic polynomial" of a matroid $M$, polynomial whose coefficients enumerate…

We settle three problems from the literature on stable and real zero polynomials and their connection to matroid theory. We disprove the weak real zero amalgamation conjecture by Schweighofer and the second author. We disprove a conjecture…

We give a new proof of Vassiliev's planarity criterion for framed four-valent graphs (and more generally, *-graphs), which is based on Pontryagin-Kuratowski theorem.

The following hypothesis was put forward by Goreinov, Tyrtyshnikov and Zamarashkin in \cite{GTZ1997}. For arbitrary real $n \times k$ matrix with orthonormal columns a sufficiently "good" $k \times k$ submatrix exists. "Good" in the sense…

We prove a conjecture about the concordance invariant $\vartheta$, defined in a recent paper by Lewark and Zibrowius. This result simplifies the relation between $\vartheta$ and Rasmussen's $s$-invariant. The proof relies on Bar-Natan's…

In 1991, Kahn made the following conjecture. For any $n$-dimensional vector space $V$ and any $n\times n$ array of $n^2$ bases of $V$, it is possible to choose a representative vector from each of these bases in such a way that the…

The Gasca-Maeztu conjecture for the case $n=4$ was proved for the first time in [J. R. Busch, A note on Lagrange interpolation in $\mathbb{R}^2$, Rev. Un. Mat. Argentina, 36 (1990) 33--38]. Here we bring a short and simple proof of it.

Ahmadi-Shparlinski conjectured that every ordinary, geometrically simple Jacobian over a finite field has maximal angle rank. Using the L-Functions and Modular Forms Database, we provide two counterexamples to this conjecture in dimension…

As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and…

We give a counterexample to a conjecture made by Cigler, Jerman and Wojciechowski stating that all posets are conclusive. We also provide combinatorial characterizations for conclusiveness of finite posets and the existence of outer…

We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.

For a ribbon knot, it is a folk conjecture that the rank of its knot Floer homology must be 1 modulo 8, and another folk conjecture says the same about reduced Khovanov homology. We give the first counter-examples to both of these folk…

We study the existence and the number of $k$-neighborly reorientations of an oriented matroid. This leads to $k$-variants of McMullen's problem and Roudneff's conjecture, the case $k=1$ being the original statements on complete cells in…

Given two finite matroids on the same ground set, a celebrated result of Edmonds says that the ground set can be partitioned into two disjoint subsets in a manner that there is a common independent set in both matroids whose intersection…

The Jacobian conjecture in dimension $n$ asserts that any polynomial endomorphism of $n$-dimensional affine space over a field of zero characteristic, with the Jacobian equal 1, is invertible. The Dixmier conjecture in rank $n$ asserts that…

White's conjecture asserts that any two tuples of matroid bases that have the same multi-set union can be transformed from one to another by symmetric exchanges; it also implies that the toric ideals of matroids are generated by the…

In 2008, Schmidt and Tuller stated a conjecture concerning optimal packing and covering of integers by translates of a given three-point set. In this note, we confirm their conjecture and relate it to several other problems in…

This is the text of my Bourbaki seminar on the proof of the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic.

WITHDRAWN: The proof contains an uncorrectable gap in the proof of theorem 7 on page 11. A proof of the Krzyz conjecture is presented, based on the application of the variational method, as well as on the use of two classical results and…

We resolve a 25 year old problem by showing that The Paving Conjecture is equivalent to The Paving Conjecture for Triangular Matrices.