Related papers: An $E_\infty$ structure on the matroid grassmannia…
We develop a theory of variable elliptic structures on planar domains, in which the imaginary unit $i(x,y)$ is a moving generator of a rank-two real algebra bundle defined by a smoothly varying quadratic relation. Differentiating this…
By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…
We build new algebraic structures, which we call genuine equivariant operads, which can be thought of as a hybrid between equivariant operads and coefficient systems. We then prove an Elmendorf-Piacenza type theorem stating that equivariant…
Cut-and-paste $K$-theory is a new variant of higher algebraic $K$-theory that has proven to be useful in problems involving decompositions of combinatorial and geometric objects, e.g., scissors congruence of polyhedra and reconstruction…
We introduce the notion of an online matroid embedding, which is an algorithm for mapping an unknown matroid that is revealed in an online fashion to a larger-but-known matroid. We establish the existence of such an embedding for binary…
The purpose of this note is to prove the existence of a remarkable structure in an iterated sumset derived from a set $P$ in a Cartesian square $\mathbb{F}_p^n\times\mathbb{F}_p^n$. More precisely, we perform horizontal and vertical sums…
The notion of interchange of two multiplicative structures on a topological space is encoded by the tensor product of the two operads parametrizing these structures. Intuitively one might thus expect that the tensor product of an E_m and an…
It is a classic result that the geometry of the total space of a principal bundle with reference to the action of the bundle's structure group is codified in the bundle's operation, a collection of derivations comprising the de Rham…
We construct a combinatorial moduli space closely related to the KSV-compactification of the moduli space of bordered marked Riemann surfaces. The open part arises from symmetric metric ribbon graphs. The compactification is obtained by…
We introduce the $\mathcal{T}$-construction, an endofunctor on the category of generalized operads as a general mechanism by which various notions of plethystic substitution arise from more ordinary notions of substitution. In the special…
We define the e\~ne product for the multiplicative group of polynomials and formal power series with coefficients on a commutative ring and unitary constant coefficient. This defines a commutative ring structure where multiplication is the…
Let $K$ be a field of characteristic zero complete with respect to a non-trivial, non-Archimedean valuation. We relate the sheaf $\widehat{\mathcal{D}}$ of infinite order differential operators on smooth rigid $K$-analytic spaces to the…
We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an R-module according to some axioms. When R is a field, we recover matroids. When R=$\mathbb{Z}$, and when R is a DVR, we get…
We show the existence of a polynomial-size extended formulation for the base polytope of a $(k,\ell)$-sparsity matroid. For an undirected graph $G=(V,E)$, the size of the formulation is $O(|V||E|)$ when $k \geq \ell$ and $O(|V|^2 |E|)$ when…
We show that various combinatorial invariants of matroids such as Chow rings and Orlik--Solomon algebras may be assembled into "operad-like" structures. Specifically, one obtains several operads over a certain Feynman category which we…
The Grassmannian, which is the manifold of all $k$-dimensional subspaces in the Euclidean space $\mathbb{R}^n$, was decomposed through three equivalent methods connecting combinatorial geometries, Schubert cells and convex polyhedra by…
The equivariant Kazhdan-Lusztig polynomial of a braid matroid may be interpreted as the intersection cohomology of a certain partial compactification of the configuration space of n distinct labeled points in the plane, regarded as a graded…
We prove a decomposition theorem for irreducible components of Grassmannians of submodules, as well as for other schemes arising from representation theory, thus generalising the result of Crawley-Boevey and Schroer for module varieties.…
We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion,…
Equivariant Ehrhart theory enumerates the lattice points in a polytope with respect to a group action. Answering a question of Stapledon, we describe the equivariant Ehrhart theory of the permutahedron, and we prove his Effectiveness…