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We investigate the liberation question for the compact Lie groups, by using various "soft" and "hard" methods, based respectively on joint generation with a free quantum group, and joint generation with a free torus. The soft methods extend…
We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of,…
In this paper we present a new characterization of free group actions (in classical differential geometry), involving dynamical systems and representations of the corresponding transformation groups. In fact, given a dynamical system, we…
It is shown that, for any pair of cardinals with infinite sum, there exist a group and an equation over this group such that the first cardinal is the number of solutions to this equation and the second cardinal is the number of…
In this paper we study simplicial complexes as higher dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen-Macaulay quotients, and a…
We construct a generalized version for the free product of unital C*-algebras over a family of unital C*-subalgebras, starting from the group-analogue. When all the subalgebras are the same, we recover the free product with amalgamation…
Consider a group $G$ acting nicely on a simply-connected simplicial complex $X$. Numerous classical methods exist for using this group action to produce a presentation for $G$. For the case that $X/G$ is 2-connected, we give a new method…
Higher-order notions of Kreweras complementation have appeared in the literature in the works of Krawczyk, Speicher, Mastnak, Nica, Arizmendi, Vargas, and others. While the theory has been developed primarily for specific applications in…
The conformal crossing equation puts very stringent constraints on the conformal data. We formulate it in way that makes the conformal symmetry more transparent. This allows for generalization of the crossing equation to arbitrary Lie group…
We give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity…
We derive a formula for expressing free cumulants whose entries are products of random variables in terms of the lattice structure of non-crossing partitions. We show the usefulness of that result by giving direct and conceptually simple…
We introduce the notions of mixed resolutions and simplicial sections, and prove a theorem relating them. This result is used (in another paper) to study deformation quantization in algebraic geometry.
We extend the notion of (smooth) stable generalized complex structures to allow for an anticanonical section with normal self-crossing singularities. This weakening not only allows for a number of natural examples in higher dimensions but…
We prove that a connected simplicial complex is uniquely determined by its complex of discrete Morse functions. This settles a question raised by Chari and Joswig. In the 1-dimensional case, this implies that the complex of rooted forests…
In this paper, we prove classical affine W-algebras associated to Lie superalgebras (W-superalgebras) can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasi-classical limits of quantum…
In this paper, we extend the classical theory of crossed $G$-sets and the crossed Burnside ring from a finite group $G$ to a finite groupoid $\mathcal{G}$. We introduce a natural monoidal structure on the category of crossed…
The aim of this paper is to provide an explicit basis of the miniversal deformation of a monomial curve defined by a free semigroup -- these curves make up a notable family of complete intersection monomial curves. First, we dispense a…
We prove that the simplicial complex whose simplices are the nonempty partial bases of $\mathbb{F}_n$ is homotopy equivalent to a wedge of $(n-1)$-spheres. Moreover, we show that it is Cohen-Macaulay.
Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d=2), and strictly contains the class of…
We refine Brink's theorem, that the non-reflection part of a reflection centralizer in a Coxeter group W is a free group. We give an explicit set of generators for centralizer, which is finitely generated when W is. And we give a method for…