Vincent Coll | Lehigh University (original) (raw)

Papers by Vincent Coll

arXiv (Cornell University), Oct 26, 2020

arXiv (Cornell University), Oct 31, 2022

This paper is a continuation of earlier work on the construction of contact forms on seaweed alge... more This paper is a continuation of earlier work on the construction of contact forms on seaweed algebras. In the prequel to this paper, we show that every index-one seaweed subalgebra of An−1 = sl(n) is contact by identifying contact forms that arise from Dougherty's framework. We extend this result to include index-one seaweed subalgebras of Cn = sp(2n). Our methods are graph-theoretic and combinatorial.

Discussiones Mathematicae Graph Theory, 2020

A proper edge-coloring of a graph is a coloring in which adjacent edges receive distinct colors. ... more A proper edge-coloring of a graph is a coloring in which adjacent edges receive distinct colors. A path is properly colored if consecutive edges have distinct colors, and an edge-colored graph is properly connected if there exists a properly colored path between every pair of vertices. In such a graph, we introduce the notion of the graph's proper diameter-which is a function of both the graph and the coloring-and define it to be the maximum length of a shortest properly colored path between any two vertices in the graph. We consider various families of graphs to find bounds on the gap between the diameter and possible proper diameters, paying singular attention to 2-colorings.

Electronic Journal of Combinatorics, Dec 6, 2012

For a fixed integer m, we consider edge colorings of complete graphs which contain no properly ed... more For a fixed integer m, we consider edge colorings of complete graphs which contain no properly edge colored cycle C m as a subgraph. Within colorings free of these subgraphs, we establish a global structure by bounding the number of colors that can induce a spanning and connected subgraph. In the case of small cycles, namely C 4 , C 5 , and C 6 , we show that our bounds are sharp.

Journal of Algebraic Combinatorics, May 10, 2021

We introduce a family of posets which generate Lie poset subalgebras of An−1 = sl(n) whose index ... more We introduce a family of posets which generate Lie poset subalgebras of An−1 = sl(n) whose index can be realized topologically. In particular, if P is such a toral poset, then it has a simplicial realization which is homotopic to a wedge sum of d one-spheres, where d is the index of the corresponding type-A Lie poset algebra gA(P). Moreover, when gA(P) is Frobenius, its spectrum is binary; that is, consists of an equal number of 0's and 1's. We also find that all Frobenius, type-A Lie poset algebras corresponding to a poset whose largest totally ordered subset is of cardinality at most three have a binary spectrum.

Journal of Algebraic Combinatorics, Nov 6, 2017

If g is a Frobenius Lie algebra, then for certain F ∈ g * the natural map g −→ g * given by x −→ ... more If g is a Frobenius Lie algebra, then for certain F ∈ g * the natural map g −→ g * given by x −→ F [x, −] is an isomorphism. The inverse image of F under this isomorphism is called a principal element. We show that if g is a Frobenius seaweed subalgebra of An−1 = sl(n) then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicites have a symmetric distribution. Our proof methods are constructive and combinatorial in nature.

Journal of Generalized Lie Theory and Applications, 2015

Mathematics Magazine, Oct 1, 2014

Summary We generalize hypersurfaces of revolution by allowing the profile curve to be dependent o... more Summary We generalize hypersurfaces of revolution by allowing the profile curve to be dependent on more than one variable. We call these more general objects spherical arrays and we use them to introduce the reader to volume calculations, which are normally reserved for a more advanced course in differential geometry. The rich symmetry of the spherical arrays allow us to build objects with properties reminiscent of those for which Gabriel's Horn became a cause célèbre.

American Mathematical Monthly, 2014

Advances in Geometry, Jul 1, 2013

arXiv (Cornell University), Aug 27, 2020

The category of Frobenius Lie algebras is stable under deformation, and here we examine explicit ... more The category of Frobenius Lie algebras is stable under deformation, and here we examine explicit infinitesimal deformations of four and six dimensional Frobenius Lie algebras with the goal of understanding if the spectrum of a Frobenius Lie algebra can evolve under deformation. It can.

arXiv (Cornell University), Aug 2, 2014

In 2000, Enomoto and Ota conjectured that if a graph G satisfies σ2(G) ≥ n + k − 1, then for any ... more In 2000, Enomoto and Ota conjectured that if a graph G satisfies σ2(G) ≥ n + k − 1, then for any set of k vertices v1,. .. , v k and for any positive integers n1,. .. , n k with ni = |G|, there exists a partition of V (G) into k paths P1,. .. , P k such that vi is an end of Pi and |Pi| = ni for all i. We prove this conjecture when |G| is large. Our proof uses the Regularity Lemma along with several extremal lemmas, concluding with an absorbing argument to retrieve misbehaving vertices.

arXiv (Cornell University), Jun 13, 2012

We provide a recursive classification of meander graphs, showing that each meander is identified ... more We provide a recursive classification of meander graphs, showing that each meander is identified by a unique sequence of fundamental graph-theoretic moves. This sequence is called the meander's signature. The signature not only provides a fast algorithm for the computation of the index of a Lie algebra associated with the meander, but also allows for the speedy determination of the graph's plane homotopy type-a finer invariant than the index. The signature can be used to construct arbitrarily large sets of meanders, Frobenius and otherwise, of any given size and configuration. Making use of a more refined signature, we are able to prove an important conjecture of Gerstenhaber & Giaquinto: The spectrum of the adjoint of a principal element in a Frobenius seaweed Lie algebra consists of an unbroken chain of integers. Additionally, we show the dimensions of the associated eigenspaces to be symmetric about 0.5. In certain special cases, the signature is used to produce an explicit formula for the index of a seaweed Lie subalgebra of sl(n) in terms of elementary functions.

Given a multigraph H, a graph G is H-linked if every injective map f : V (H) ! V (G) can be exten... more Given a multigraph H, a graph G is H-linked if every injective map f : V (H) ! V (G) can be extended to an H-subdivision (f , g) in G for some g. Given a multigraph H and an integer sequence w = {w e | e 2 E(H), w e 2}, a graph G is (H, w , m)-linked if every injective map f : V (H) ! V (G) can be extended to an H-subdivision (f , g) in G such that each path g(e) has length w e ,. .. , or w e + m. If m = 0, then we say G is (H, w)-linked. We show that the sharp minimum degree condition for a graph to be H-linked is the same as the sharp minimum degree condition for a large graph to be (H, w , m)-linked for m 1 and all sets w with each value w e 2 w at least 14. Additionally, we establish a sharp minimum degree condition for a large graph to be (H, w)-linked.

arXiv (Cornell University), Dec 13, 2020

We provide a combinatorial recipe for constructing all posets of height at most two for which the... more We provide a combinatorial recipe for constructing all posets of height at most two for which the corresponding type-A Lie poset algebra is contact. In the case that such posets are connected, a discrete Morse theory argument establishes that the posets' simplicial realizations are contractible. It follows from a cohomological result of Coll and Gerstenhaber on Lie semi-direct products that the corresponding contact Lie algebras are absolutely rigid.

arXiv (Cornell University), Apr 12, 2015

Let H be a hypersurface in R n and let π be an orthogonal projection in R n restricted to H. We s... more Let H be a hypersurface in R n and let π be an orthogonal projection in R n restricted to H. We say that H satisfies the Archimedean projection property corresponding to π if there exists a constant C such that Vol(π −1 (U)) = C • Vol(U) for every measurable U in the range of π. It is well-known that the (n − 1)-dimensional sphere, as a hypersurface in R n , satisfies the Archimedean projection property corresponding to any codimension 2 orthogonal projection in R n , the range of any such projection being an (n − 2)-dimensional ball. Here we construct new hypersurfaces that satisfy Archimedean projection properties. Our construction works for any projection codimension k, 2 ≤ k ≤ n − 1, and it allows us to specify a wide variety of desired projection ranges Ω n−k ⊂ R n−k. Letting Ω n−k be an (n − k)-dimensional ball for each k, it produces a new family of smooth, compact hypersurfaces in R n satisfying codimension k Archimedean projection properties that includes, in the special case k = 2, the (n − 1)-dimensional spheres.

The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine ... more The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine this graph in several ways with a goal of determining families of Frobenius (index zero) seaweed algebras. Our analysis gives two new families of Frobenius seaweed algebras as well as elementary proofs of known families of such Lie algebras.

arXiv (Cornell University), Apr 13, 2022

The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult t... more The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult to compute. However, for the suggestively-named seaweed algebras, the computation of the index can be reduced to a combinatorial formula based on the connected components of a "meander": a planar graph associated with the algebra. Our index analysis on seaweed algebras requires only basic linear and abstract algebra. Indeed, the main goal of this survey-type article is to introduce a broader audience to seaweed algebras with minimal appeal to specialized language and notation from Lie theory. This said, we present several results that do not appear elsewhere and do appeal to more advanced language in the Introduction to provide added context.