Louis Theran

Generic combinatorial rigidity of periodic frameworks

Abstract We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks . The characterization is a true analogue of the Maxwell–Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorithms. To prove our rigidity theorem we introduce and develop periodic direction networks and Z 2 - graded-sparse colored graphs .

The algebraic combinatorial approach for low-rank matrix completion

We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the treatment of single entries in a closed theoretical and practical framework. More specifically, apart from introducing an algebraic combinatorial theory of low-rank matrix completion, we present probability-one algorithms to decide whether a particular entry of the matrix can be completed. We also describe methods to complete that entry from a few others, and to estimate the error which is incurred by any method completing that entry. Furthermore, we show how known results on matrix completion and their sampling assumptions can be related to our new perspective and interpreted in terms of a completability phase transition.

Algebraic Matroids with Graph Symmetry

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. nFor (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal poly- nomial in classical Galois theory, and studying the matroid structure with multivariate methods. nFor (b) matroids with symmetries we introduce combinatorial invariants capturing structural properties of the rank function and its limit behavior, and obtain proofs which are purely combinatorial and do not assume algebraicity of the matroid; these imply and generalize known results in some specific cases where the matroid is also algebraic. These results are motivated by, and readily applicable to framework rigidity, low-rank matrix completion and determinantal varieties, which lie in the intersection of (a) and (b) where additional results can be derived. We study the corresponding matroids and their associated invariants, and for selected cases, we characterize the matroidal structure and the circuit polynomials completely.

Generic Rigidity with Forced Symmetry and Sparse Colored Graphs

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.

Universality Theorems for Inscribed Polytopes and Delaunay Triangulations

We prove that every primary basic semi-algebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semi-algebraic set is, moreover, open, it is, additionally, (up to homotopy) the retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of

Analytic analysis of auxetic metamaterials through analogy with rigid link systems

{mathbb {Q}}

Anchored boundary conditions for locally isostatic networks.

Q are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mnëv universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.

Algorithms for detecting dependencies and rigid subsystems for CAD

We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a reflection. To establish these results, we develop a new technique for deriving linear representations of sparsity matroids on colored graphs and extend the direction network method of proving rigidity characterizations to handle reflections.