From independent sets and vertex colorings to isotropic spaces and isotropic decompositions: Another bridge between graphs and alternating matrix spaces

In the 1970’s, Lovász built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings (FCT 1979). A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, FOCS 2016; Ivanyos-Qiao-Subrahmanyam, ITCS 2017). In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory. We first show that the maximum independent set problem and the vertex c-coloring problem reduce to the maximum isotropic space problem and the isotropic c-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration

Tur\'an and Ramsey problems for alternating multilinear maps

Guided by the connections between hypergraphs and exterior algebras, we study Tur\'an and Ramsey type problems for alternating multilinear maps. This study lies at the intersection of combinatorics, group theory, and algebraic geometry, and has origins in the works of Lov\'asz (Proc. Sixth British Combinatorial Conf., 1977), Buhler, Gupta, and Harris (J. Algebra, 1987), and Feldman and Propp (Adv. Math., 1992). Our main result is a Ramsey theorem for alternating bilinear maps. Given $s, t\in \mathbb{N}$, $s, t\geq 2$, and an alternating bilinear map $f:V\times V\to U$ with $\dim(V)=s\cdot t^4$, we show that there exists either a dimension-$s$ subspace $W\leq V$ such that $\dim(f(W, W))=0$, or a dimension-$t$ subspace $W\leq V$ such that $\dim(f(W, W))=\binom{t}{2}$. This result has natural group-theoretic (for finite $p$-groups) and geometric (for Grassmannians) implications, and leads to new Ramsey-type questions for varieties of groups and Grassmannians.Comment: 20 pages. v3: rewrite introductio

Networks, (K)nots, Nucleotides, and Nanostructures

Designing self-assembling DNA nanostructures often requires the identification of a route for a scaffolding strand of DNA through the target structure. When the target structure is modeled as a graph, these scaffolding routes correspond to Eulerian circuits subject to turning restrictions imposed by physical constraints on the strands of DNA. Existence of such Eulerian circuits is an NP-hard problem, which can be approached by adapting solutions to a version of the Traveling Salesperson Problem. However, the author and collaborators have demonstrated that even Eulerian circuits obeying these turning restrictions are not necessarily feasible as scaffolding routes by giving examples of nontrivially knotted circuits which cannot be traced by the unknotted scaffolding strand. Often, targets of DNA nanostructure self-assembly are modeled as graphs embedded on surfaces in space. In this case, Eulerian circuits obeying the turning restrictions correspond to A-trails, circuits which turn immediately left or right at each vertex. In any graph embedded on the sphere, all A-trails are unknotted regardless of the embedding of the sphere in space. We show that this does not hold in general for graphs on the torus. However, we show this property does hold for checkerboard-colorable graphs on the torus, that is, those graphs whose faces can be properly 2-colored, and provide a partial converse to this result. As a consequence, we characterize (with one exceptional family) regular triangulations of the torus containing unknotted A-trails. By developing a theory of sums of A-trails, we lift constructions from the torus to arbitrary n-tori, and by generalizing our work on A-trails to smooth circuit decompositions, we construct all torus links and certain sums of torus links from circuit decompositions of rectangular torus grids. Graphs embedded on surfaces are equivalent to ribbon graphs, which are particularly well-suited to modeling DNA nanostructures, as their boundary components correspond to strands of DNA and their twisted ribbons correspond to double-helices. Every ribbon graph has a corresponding delta-matroid, a combinatorial object encoding the structure of the ribbon-graph\u27s spanning quasi-trees (substructures having exactly one boundary component). We show that interlacement with respect to quasi-trees can be generalized to delta-matroids, and use the resulting structure on delta-matroids to provide feasible-set expansions for a family of delta-matroid polynomials, both recovering well-known expansions of this type (such as the spanning-tree expansion of the Tutte polynnomial) as well as providing several previously unknown expansions. Among these are expansions for the transition polynomial, a version of which has been used to study DNA nanostructure self-assembly, and the interlace polynomial, which solves a problem in DNA recombination

Loop Quantum Cosmology: A Status Report

The goal of this article is to provide an overview of the current state of the art in loop quantum cosmology for three sets of audiences: young researchers interested in entering this area; the quantum gravity community in general; and, cosmologists who wish to apply loop quantum cosmology to probe modifications in the standard paradigm of the early universe. An effort has been made to streamline the material so that, as described at the end of section I, each of these communities can read only the sections they are most interested in, without a loss of continuity.Comment: 138 pages, 15 figures. Invited Topical Review, To appear in Classical and Quantum Gravity. Typos corrected, clarifications and references adde