Related papers: Slider-pinning Rigidity: a Maxwell-Laman-type Theo…

In this paper Moussouris' algorithm for the decomposition of spin networks is reviewed and the implicit assumptions made in the Decomposition Theorem relating a spin network with its state sum are examined. It is found that the theorem in…

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's…

In this text we introduce the torsion of spinor connections. In terms of the torsion we give conditions on a spinor connection to produce Killing vector fields. We relate the Bianchi type identities for the torsion of spinor connections…

This paper proposes methods for identification of large-scale networked systems with guarantees that the resulting model will be contracting -- a strong form of nonlinear stability -- and/or monotone, i.e. order relations between states are…

This is a concise overview of the definitions and properties of the linking number and its higher-order generalization, Milnor invariants.

We introduce mixed twistor $D$-modules, and establish the fundamental functorial property. We also prove that they are described as the gluing of admissible variations of mixed twistor structure. In a sense, mixed twistor $D$-modules could…

We give a survey of various rigidity results involving scalar curvature. Many of these results are inspired by the positive mass theorem in general relativity. In particular, we discuss the recent solution of Min-Oo's Conjecture for the…

In this paper, we study Legendrian realizations of cable links of knot types that are uniformly thick but not Legendrian simple, extending prior work of Dalton, the second author, and Traynor. This leads to new phenomena, such as stabilized…

These are lectures given at the 2022 Arizona Winter School. It gives an introduction to the rigidity method for constructing automorphic forms for semisimple groups over function fields. The rigidity method leads to explicit constructions…

This paper is concerned with the concept of linear repetitivity in the theory of tilings. We prove a general uniform subadditive ergodic theorem for linearly repetitive tilings. This theorem unifies and extends various known (sub)additive…

Consider a compact Riemannian manifold with boundary endowed with a magnetic field. A path taken by a particle of unit charge, mass, and energy is called a magnetic geodesic. It is shown that if everything is real-analytic, the topology,…

Ladder polymers, known for their rigid, ladder-like structures, exhibit exceptional thermal stability and mechanical strength, positioning them as candidates for advanced applications. However, accurately determining their structure from…

The Reidemeister theorem states that any link in $3$-space can be encoded by a diagram (a suitably decorated projection) on a plane, and provides a finite set of combinatorial moves relating two diagrams of the same link up to isotopy. In…

We describe our recent results concerning the rigidity/unlockability properties of clusters of rigid bodies sliding over the unit sphere.

Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems of metrics in a same conformal class that fixes the boundary and satisfy certain integral conditions on the the scalar curvatures and the mean…

We derive a model for the finite motion of a magneto-elastic rod reinforced with isotropic (spherical) or anisotropic (ellipsoidal) inclusions. The particles are assumed weakly and uniformly magnetised, rigid and firmly embedded into the…

We prove large-data local stability theorems for several spin models in two dimensions.

The square lattice with central forces between nearest neighbors is isostatic with a subextensive number of floppy modes. It can be made rigid by the random addition of next-nearest neighbor bonds. This constitutes a rigidity percolation…

We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and…

In this paper, we see several basic properties of graded linear series. We firstly see that, if a graded linear series contains an ample series, then so are the pullbacks of the system under birational morphisms. Using this proposition, we…