Related papers: Rigidity, Residues and Duality: Overview and Recen…
We introduce the notion of mixed Hodge complex on an algebraic variety, improving Du Bois' filtered complex, and relate Deligne's theory of mixed Hodge structure with the theory of mixed Hodge module. This was supposed to be true, but is…
We prove that rigid representations of the fundamental group of a surface into the group of oreintation-preserving homeomorphisms of the circle are geometric, thereby establishing a converse statement of a theorem by the first author.
A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle…
In this paper, we study residues of differential 2-forms on a smooth algebraic surface over an arbitrary field and give several statements about sums of residues. Afterwards, using these results we construct algebraic-geometric codes which…
This work focuses on the bearing rigidity theory, namely the branch of knowledge investigating the structural properties necessary for multi-element systems to preserve the inter-units bearings when exposed to deformations. The original…
Fix a prime number $p$. We report on some recent developments in algebraic geometry (broadly construed) over $p$-adically complete commutative rings. These developments include foundational advances within the subject as well as external…
We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear…
A discrete duality is a relationship between classes of algebras and classes of relational systems (frames) resulting in two representation theorems building on the early work of J\'onsson and Tarski, Kripke, and van Benthem. In this…
We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of…
In this paper, we give the rigidity theorem for a log morphism as an extension of a fixed scheme morphism. We also give several applications of the rigidity theorem.
Relativistic thick ring models are constructed using previously found analytical Newtonian potential-density pairs for flat rings and toroidal structures obtained from Kuzmin-Toomre family of discs. In particular, we present systems with…
We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the…
We study dg-manifolds which are R[2]-bundles over R[1]-bundles over manifolds, we calculate its symmetries, its derived symmetries and we introduce the concept of T-dual dg-manifolds. Within this framework we construct the T-duality map as…
These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. Recent results are surveyed, along with their classical predecessors. The notes…
We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent…
The algebraic stability theorem for $\mathbb{R}$-persistence modules is a fundamental result in topological data analysis. We present a stability theorem for $n$-dimensional rectangle decomposable persistence modules up to a constant…
Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large class of cluster varieties associated to compact oriented…
Generalized Chinese Remainder Theorem (CRT) is a well-known approach to solve ambiguity resolution related problems. In this paper, we study the robust CRT reconstruction for multiple numbers from a view of statistics. To the best of our…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
An important, if relatively less well known aspect of the singularity theorems in Lorentzian Geometry is to understand how their conclusions fare upon weakening or suppression of one or more of their hypotheses. Then, theorems with modified…