Related papers: On geometric complexity theory: Multiplicity obstr…
While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and…
The classical theory of $G$-structures, which include almost-complex structures, explains the relationship between the curvature of compatible connections and integrability. This note is an effort to understand how the curvature of…
The goal of this paper is to set up an obstruction theory in the context of algebras over an operad and in the framework of differential graded modules over a field. Precisely, the problem we consider is the following: Suppose given two…
We use algebraic geometry to study matrix rigidity, and more generally, the complexity of computing a matrix-vector product, continuing a study initiated by Kumar, et. al. We (i) exhibit many non-obvious equations testing for (border)…
We study the eigenforms of the action of A. Baker's Hecke operators on the holomorphic elliptic homology of various topological spaces. We prove a multiplicity one theorem (i.e., one-dimensionality of the space of these "topological Hecke…
We provide proofs of two theorems stated by Massey in 1961, concerning the obstructions to finding complex structures on real vector bundles. In addition, we determine the second obstruction to a complex structure on a rank six orientable…
The second author previously discussed how classical complexity separation conjectures, we call them "axioms", have implications in three manifold topology: polynomial length stings of operations which preserve certain Jones polynomial…
We provide examples of homogeneous spaces which are neither symmetric spaces nor real cohomology spheres, yet have the property that every invariant metric is geometrically formal. We also extend the known obstructions to geometric…
In 2003, Cohn and Umans proposed a group-theoretic approach to bounding the exponent of matrix multiplication. Previous work within this approach ruled out certain families of groups as a route to obtaining $\omega = 2$, while other…
The subject of this paper is the problem of arrangement of real algebraic curves on real algebraic surfaces. In this paper we extend Rokhlin, Kharlamov-Gudkov-Krakhnov and Kharlamov-Marin congruences for curves on surfaces and give some…
We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky-MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity…
We calculate a certain homological obstruction introduced by De Concini, Procesi and Salvetti in their study of the Schwartz genus of the fibration from the space of ordered configuration of points in the plane to the space of the unordered…
Let U be a smooth quasi-projective variety over a field k that is finite, the algebraic closure of a finite field or algebraically closed of characteristic 0. Let X be a suitable projective compactification of U, and D an effective divisor…
Each Gr-functor of the type $(\varphi,f)$ of a Gr-category of the type $(\Pi,\C)$ has the obstruction be an element $\overline{k}\in H^3(\Pi,\C).$ When this obstruction vanishes, there exists a bijection between congruence classes of…
In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of…
In this article we introduce the notion of a 'good model' in order to study the higher obstructions of complex supermanifolds. We identify necessary and sufficient conditions for such models to exist. Illustrations over Riemann surfaces are…
We generalise the notion of separable equivalence, originally presented by Linckelmann (2011), to an equivalence relation on additive categories. We use this generalisation to show that from an initial equivalence between two algebras we…
Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact,…
Let $R$ be a smooth affine algebra over an infinite perfect field $k$. Let $I\subset R$ be an ideal, $\omega_I:(R/I)^n\to I/I^2$ a surjective homomorphism and $Q_{2n}\subset \mathbb{A}^{2n+1}$ be the smooth quadric defined by the equation…
As it follows from G\"odel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually, this fundamental principle can be efficiently applicable in…