相关论文: The rigidity of Doyle circle packings on the infin…

Discrete conformal mappings based on circle packing, vertex scaling, and related structures has had significant activity since Thurston proposed circle packing as a way to approximate conformal maps in the 1980s. The first convergence…

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are…

Circles of a single size can pack together densely in a hexagonal lattice, but adding in size variety disrupts the order of those packings. We conduct simulations which generate dense random packings of circles with specified size…

We show that for certain triangulations of surfaces, circle packings realising the triangulation can be found by solving a system of polynomial equations. We also present a similar system of equations for unbranched circle packings. The…

We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the…

The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincar\'e complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more…

We present a symmetry result regarding stationary solutions of the 2D Euler equations in a disk. We prove that in a disk, a steady flow with only one stagnation point and tangential boundary conditions is a circular flow, which confirms a…

If $\Gamma$ is the fundamental group of a complete finite volume hyperbolic $3$-manifold, Guilloux conjectured that the Borel function on the $\text{PSL}(n,\mathbb{C})$-character variety of $\Gamma$ should be rigid at infinity, that is it…

Brjuno and R\"ussmann proved that every irrationally indifferent fixed point of an analytic function with a Brjuno rotation number is linearizable, and Yoccoz proved that this is sharp for quadratic polynomials. Douady conjectured that this…

Liouville conformal field theory is a prototypical example of an exactly solvable quantum field theory, in the sense that the correlation functions in an arbitrary background can be determined exactly using only the constraints of unitarity…

We establish quantitative homogenization, large-scale regularity and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is…

A saddle loop is a germ of a holomorphic foliation near a homoclinic saddle connection. We prove that they are classied by their Poincar{\'e} rst-return map. We also prove that they are formally rigid when the Poincar{\'e} map is…

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and proper…

In this paper we generalize the classical theorem of Thue about the optimal circular disc packing in the plane. We are given a family of circular discs, not necessarily of equal radii, with the property that the inflation of every disc by a…

In this paper, we establish a Liouville type rigidity result for a class of asymptotically hyperbolic non-compact Einstein metrics defined on manifolds of dimension $d\ge 5$ extending the earlier result in dimension $d=4$.

We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is…

Liouville's theorem says that in dimension greater than two, all conformal maps are M\"obius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally…

This paper encompasses the mathematical derivations of the analytic and generalized formula and recurrence relations to find out the radii of n umber of circles inscribed or packed in the plane region bounded by circular arcs (including…

We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete…

Four types of discrete transforms of Weyl orbit functions on the finite point sets are developed. The point sets are formed by intersections of the dual-root lattices with the fundamental domains of the affine Weyl groups. The finite sets…