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We prove the following monotonicity result for the holonomy group: Given a sequence of metric connections converging in $C^0$ such that all its members have holonomy contained in a closed group $H$, also their limit connection needs to have…
Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups:…
The aim of this article is to provide space level maps between configuration spaces of graphs that are predicted by algebraic manipulations of cellular chains. More explicitly, we consider edge contraction and half-edge deletion, and…
We give a complete description and classification of locally homogeneous real hypersurfaces in $\mathbb C^3$. Various groups of mathematicians have been studying this problem in the last 25 years, and several significant classes of…
We describe the deformation cohomology of a symplectic groupoid, and use it to study deformations via Moser path methods, proving a symplectic groupoid version of the Moser Theorem. Our construction uses the deformation cohomologies of Lie…
We give some basics about homological algebra of difference representations. We consider both the difference-discrete and the difference-rational case. We define the corresponding cohomology theories and show the existence of spectral…
In this paper, we mainly focus on formal deformation theory of module homomorphisms. We first introduce the cohomology of module homomorphisms and study formal one-parameter deformation. We obtain some properties about obstructions. Then we…
This is an expository paper. It is well known that a linear transformation can be defined to have any desired action on a basis. From this fact, one can show that every group homomorphism from Z^k to R^d extends to a homomorphism from R^k…
We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc…
Many first-order equational theories, such as the theory of groups or boolean algebras, can be presented by a smaller set of axioms than the original one. Recent studies showed that a homological approach to equational theories gives us…
A result of Ben-Or, Coppersmith, Luby and Rubinfeld on testing whether a map be two groups is close to a homomorphism implies a tight lower bound on the distance between the multiplication tables of two non-isomorphic groups.
We introduce a new cohomology theory related to deformations of Lie algebra morphisms. This notion involves simultaneous deformations of two Lie algebras and a homomorphism between them.
Stratified digraphs are popular models for feedforward neural networks. However, computation of their path homologies has been limited to low dimensions due to high computational complexity. A recursive algorithm is proposed to compute…
We develop a new method for visualizing and refining the invariances of learned representations. Specifically, we test for a general form of invariance, linearization, in which the action of a transformation is confined to a low-dimensional…
We classify the representation type of the descent algebras of type $\A$ in the positive characteristic case. The algebras have finite representation type only for a few small degrees; otherwise, they are wild. Our main reduction method…
We study the topological structure and the topological dynamics of groups of homeomorphisms of scattered spaces. For a large class of them (including the homeomorphism group of any ordinal space or of any locally compact scattered space),…
This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the…
A heterogeneous graph consists of different vertices and edges types. Learning on heterogeneous graphs typically employs meta-paths to deal with the heterogeneity by reducing the graph to a homogeneous network, guide random walks or capture…
We show that contact reductions can be described in terms of symplectic reductions in the traditional Marsden-Weinstein-Meyer as well as the constant rank picture. The point is that we view contact structures as particular (homogeneous)…
Homomorphism indistinguishability is a way of characterising many natural equivalence relations on graphs. Two graphs $G$ and $H$ are called homomorphism indistinguishable over a graph class $\mathcal{F}$ if for each $F \in \mathcal{F}$,…