Related papers: Toric Ideals of Flow Polytopes

We show that if a hyperbolic knot manifold $M$ contains an essential twice-punctured torus $F$ with boundary slope $\beta$ and admits a filling with slope $\alpha$ producing a Seifert fibred space, then the distance between the slopes…

Results of Koebe (1936), Schramm (1992), and Springborn (2005) yield realizations of $3$-polytopes with edges tangent to the unit sphere. Here we study the algebraic degrees of such realizations. This initiates the research on constrained…

We introduce a very natural topology on the set of total orderings of monomials of any algebra having a countable basis over a field. This topological space and some notable subspaces are compact. This topological framework allows us to…

We consider the topologically constrained random walk model for topological polymers. In this model, the polymer forms an arbitrary graph whose edges are selected from an appropriate multivariate Gaussian which takes into account the…

We show that any nonzero polynomial in the ideal generated by the $r \times r$ minors of an $n \times n$ matrix $X$ can be used to efficiently approximate the determinant. For any nonzero polynomial $f$ in this ideal, we construct a small…

Let $Y_{3,2}$ be the $3$-uniform hypergraph with two edges intersecting in two vertices. Our main result is that any $n$-vertex 3-uniform hypergraph with at least $\binom{n}{3} - \binom{n-m+1}{3} + o(n^3)$ edges contains a collection of $m$…

This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and…

We investigate products J of ideals of "row initial" minors in the polynomial ring K[X] defined by a generic m-by-n matrix. Such ideals are shown to be generated by a certain set of standard bitableaux that we call superstandard. These…

Part I of this paper presented a systematic derivation of the Stokes Dirac structure underlying the port-Hamiltonian model of ideal fluid flow on Riemannian manifolds. Starting from the group of diffeomorphisms as a configuration space for…

In this article we analyze totally periodic pseudo-Anosov flows in graph three manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a…

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound for the projective dimension of all homogeneous ideals, in polynomial rings over a field, generated by n forms of degree at most d.…

If a graph is in bridge position in a 3-manifold so that the graph complement is irreducible and boundary irreducible, we generalize a result of Bachman and Schleimer to prove that the complexity of a surface properly embedded in the…

Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads…

Let C be an irreducible projective curve of degree d in Pn(K), where K is an algebraically closed field, and let I be the associated homogeneous prime ideal. We wish to compute generators for I, assuming we are given sufficiently many…

Let $I = ( f_1, \dots, f_n )$ be a homogeneous ideal in the polynomial ring $K[x_1, \dots,x_n]$ over a field $K$ generated by generic polynomials. Using an incremental approach based on a method by Gao, Guan and Volny, and properties of the…

We present a higher-dimensional version of the Poincar\'e-Birkhoff theorem which applies to Poincar\'e time maps of Hamiltonian systems. The maps under consideration are neither required to be close to the identity nor to have a monotone…

We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces…

In Boij-Soderberg theory, it is known that for any degree sequence $\mathbf{d}$, there exists a finitely generated module that has a pure resolution of type $\mathbf{d}$. On the other hand, in the case of ideal, there are two necessary…

We prove that the geodesic flow on the unit tangent bundle to every hyperbolic 2-orbifold that is a sphere with 3 or 4 singular points admits explicit genus one Birkhoff sections, and we determine the associated first return maps.

Recent progress on flow polytopes indicates many interesting families with product formulas for their volume. These product formulas are all proved using analytic techniques. Our work breaks from this pattern. We define a family of closely…