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We study cobordisms of a class of topological operads called ``manifold operads''. These operads are generalizations of the Fulton-MacPherson operad: an operad built from configurations of points in Euclidean space. Cobordism of manifold…

Let O be a cyclic topological operad with multiplication. In the framework of the cosimplicial machinery by McClure and Smith, we prove that the totalization of the cosimplicial space associated to O has an action of an operad equivalent to…

We consider the decomposition of bounded linear operators on Hilbert spaces in terms of functions forming frames. Similar to the singular-value decomposition, the resulting frame decompositions encode information on the structure and…

A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the…

We prove Macdonald-type deformations of a number of well-known classical branching rules by employing identities for elliptic hypergeometric integrals and series. We also propose some conjectural branching rules and allied conjectures…

In this work we introduce the broken line construction, which is a geometric and combinatorial algorithm that computes periodic Sturmian angles of a given period, yielding the locations of their landing parameters in the Mandelbrot set. An…

First, we give a functorial construction of a group associated to a symmetric operad. Applied to the endomorphism operad it gives the group of formal diffeomorphisms. Second, we associate a symmetric operad to any family of decorated graphs…

An arbitrary linear relation (multivalued operator) acting from one Hilbert space to another Hilbert space is shown to be the sum of a closable operator and a singular relation whose closure is the Cartesian product of closed subspaces.…

Carlsson's construction is a simplicial group whose geometric realization is the loop space of the 1-stunted reduced Borel construction. Our main results are: i) Given a pointed simplicial set acted upon by the discrete cyclic group C_2 of…

The purpose of this paper is to study the derived category of simplicial multicategories with arbitrary sets of objects (also known as, colored operads in simplicial sets). Our main result is a derived Morita theory for operads-where we…

Dorpalen-Barry et al. proved Elser's conjecture about sign of Elser's number by interpreting them as certain sums of reduced Euler characteristics of an abstract simplicial complex known as $U$-nucleus complex. We prove a conjecture posed…

Umbilics are points of a surface embedded in three space where normal curvatures are independent of direction. The (in)famous Carath\'{e}odory Conjecture states that a compact simply connected embedded surface has at least two umbilic…

A new hierarchy of combinatorial operads is introduced, involving families of regular polygons with configurations of arcs, called decorated cliques. This hierarchy contains, among others, operads on noncrossing configurations, Motzkin…

The purpose of this paper is two-fold. In Part 1 we introduce a new theory of operadic categories and their operads. This theory is, in our opinion, of an independent value. In Part 2 we use this new theory together with our previous…

We consider the composition product of symmetric sequences in the case where the underlying symmetric monoidal structure does not commute with coproducts. Even though this composition product is not a monoidal structure on symmetric…

This paper proves that the homotopy type of a pointed, simply-connected, 2-reduced simplicial set is determined by the chain-complex augmented by functorial diagonal and higher diagonal maps (a simple generalization of the ones used to…

This paper develops a complete foundational treatment of simplicial complexes from Euclidean spaces through geometric realizations, emphasizing concrete computations, examples, and practical verification methods. Beginning with finite point…

We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well…

We introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. They are defined as simply connected prism complexes satisfying some local combinatorial conditions. We study…

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a…