Conference: Discretization in Geometry and Dynamics

Implicit solvent models are empirical models which can easily be coupled to quantum or classical molecular models to take solvation effects into account. In this talk, I will present a new discretization method for simulating these models based on domain decomposition and boundary integral methods. This approach is extremely effective to simulate large biomolecules composed of thousands of atoms, and the potential energy surfaces computed with this discretization procedure are smooth, which is a critical property to perform geometry optimization and molecular dynamics simulations. 

Affiliation: Ecole des Ponts, Paris Tech

The general pipeline for analyzing point data with persistent homology uses a notion of distance to convert the points into a filtration of complexes, it uses the homology functor to get a tower of vector spaces, and it finally converts the tower into bars or, equivalently, points in the birth-death plane. In this talk, we will focus on the first step, arguing that the pipeline also works for Bregman divergences, which are generally not symmetric and do not satisfy the triangle inequality.  Examples are the Kullback-Leibler divergence commonly used for text and images, and the Itakura-Saito divergence preferred for sound data.

Affiliation: IST Austria

A flexible polyhedron in an $n$-dimensional space of constant curvature is an $(n-1)$-dimensional closed polyhedral surface that can be deformed continuously so that every its face remains congruent to itself during the deformation, but the deformation is not induced by an ambient rotation of the space. Intuitively, one may think of a flexible polyhedron as of a polyhedral surface with faces made of some rigid material and with hinges at codimension~$2$ faces that allow dihedral angles to change continuously.

The bellows conjecture stated by Connelly in 1978 asserts that the volume of any flexible polyhedron in dimensions greater than or equal to~$3$ is constant during the flexion. (Originally this conjecture was stated for the three-dimensional Euclidean space.)

The bellows conjecture was proved in Euclidean spaces of all dimensions (Sabitov for $n=3$, and the author for $n\ge 4$).

In the talk, we shall present the author's recent results on the bellows conjecture in non-Euclidean spaces. Namely, it turns out that the bellows conjecture is false in spheres of all dimensions and is true in odd-dimensional Lobachevsky spaces.

Affiliation: Steklov Mathematical Institute & Kharkevich Institute for Information Transmission Problems, Moscow

Many gauge field theories can be described using a multisymplectic Lagrangian formulation, where the Lagrangian density involves space-time differential forms. While there has been prior work on finite-element exterior calculus for spatial and tensor product space-time domains, less has been done from the perspective of space-time simplicial complexes. One critical aspect is that the Hodge star is now taken with respect to a pseudo-Riemannian metric, and this is most naturally expressed in space-time adapted coordinates, as opposed to the barycentric coordinates that Whitney forms are typically expressed in terms of.

We introduce a novel characterization of Whitney forms and their Hodge dual with respect to a pseudo-Riemannian metric that is independent of the choice of coordinates, and then apply it to a variational discretization of the covariant formulation of Maxwell's equations. Since the Lagrangian density for this is expressed in terms of the exterior derivative of the four-potential, the use of finite-dimensional function spaces that respects the de Rham cohomology results in a discretization that inherits the gauge symmetries of the continuous problem. This yields a variational discretization that exhibits a discrete Noether's theorem.

Affiliation: University of California, San Diego

This talk reviews differential equations on manifolds of matrices or tensors of low rank. They serve to approximate, in a low-rank format, large time-dependent matrices and tensors that are either given explicitly via their increments or are unknown solutions of high-dimensional differential equations, such as multi-particle time-dependent Schr\"odinger equations. Suitable numerical integrators are based on splitting the projector onto the tangent space of the low-rank manifold at the current approximation. In contrast to all standard integrators, these projector-splitting methods are robust with respect to the presence of small singular values in the low-rank approximation. This robustness relies on geometric properties of the low-rank manifolds.

The talk is based on work done intermittently over the last decade with Othmar Koch, Achim Nonnenmacher, Ivan Oseledets, Bart Vandereycken, Emil Kieri and Hanna Walach.

Affiliation: Universität Tübingen

We describe a framework for applying linear algebra routines directly to infinite-dimensional linear equations without discretization, which can be used to solve linear ordinary differential equations with general boundary condition and singular integral equations.  The algorithm achieves O(n) complexity, where n is the number of degrees of freedom required to achieve a desired accuracy, which is determined adaptively.   We include an example arising from Riemann–Hilbert problems, where direct  discretization of the operator naturally leads to singular or extremely badly conditioned systems, whereas solving in infinite-dimensions without discretization is successful.  

Affiliation: University of Sydney

This talk presents a framework for the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid surface or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and  discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In presence of symmetry, a discrete version of Noether's theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

Affiliation: EPFL Lausanne

A lattice 3-polytope is a polytope $P$ with integer vertices. We call size of $P$ the number of lattice points it contains, and width of P the minimum, over all integer linear functionals $f$, of the length of the interval $f(P)$. For example, $P$ has width one if it is contained between two consecutive parallel lattice planes.

We are interested in the classification of lattice 3-polytopes modulo affine integer isomorphism. Lattice 3-polytopes of size four (that is, “empty lattice tetrahedra”) are classified since long, and they are known to have width one. However, there are infinitely many of them, which implies that there are also infinitely many 3-polytopes of any given size greater than four. However, we show that all but finitely many of them have width one, which suggests the possibility of completely classifying those of width larger than one. In this talk we report on an algorithm to do this.

For sizes five and six our methods are quite ad-hoc, combining geometric and oriented matroid techniques with a detailed case study of the possible configurations that arise. Starting with size seven we use a more structured approach, based on the proof that every lattice 3-polytope of width larger than one falls into one of the following three categories:

a) It projects orthogonally to one of a list of five particular $2$-polytopes of sizes four to six. We call $3$-polytopes of this type spiked, and they can be described explicitly, for each size.

b) Except for three of its lattice points, $P$ is contained in a rational parallelepiped of width one with respect to every facet. We call these polytopes boxed. They have size at most 10 and we have completely enumerated them with the help of computer (there are 23 of size 7, seven of size 8, and only one of sizes 9 and 10).

c) It has (at least) two vertices $u$ and $v$ whose removal still has width larger than one. Polytopes of this type can be all obtained ``gluing'' smaller polytopes of width larger than one.

The complete classification of boxed and spiked $3$-polytopes allows for a computational enumeration of all lattice $3$-polytopes of width larger than one up to any given size. We have completed this enumeration up to size nine. In particular, this implies the complete classification of ``distinct-pair-sum'' (or dps, for short) $3$-polytopes, which have size at most eight, and allows us to answer certain questions  by  Reznick on dps polytopes.

This is joint work with Monica Blanco.

Affiliation: Universidad de Cantabria

The class of symplectic Runge-Kutta methods, which has attracted much attention in the last twenty five years, coincides with the class of Runge-Kutta methods that exactly preserve quadratic integrals of motion. In this talk I shall survey a number of applications where symplectic Runge-Kutta schemes are useful due to the quadratic conservation property while the preservation of the symplectic structure plays no role. In particular I shall discuss the integration of adjoint problems, and of optimal control problems. 

Affiliation: Universidad Carlos III de Madrid

We study heat equation and optimal transports on time-dependent metric measure spaces with particular emphasis on mm-spaces which evolve as super-Ricci flows. A time-dependent family of Riemannian manifolds is a super-Ricci flow if  $2 \mathrm{Ric} + \partial_t g \ge 0$. This includes all static manifolds of nonnegative Ricci curvature as well as all solutions to the Ricci flow equation. 

We characterize super-Ricci flows of metric measure spaces in terms of coupling properties of (backward) Brownian motions, gradient estimates for the (forward) heat equation, as well as dynamical convexity of the Boltzmann entropy on the Wasserstein space. And we prove stability and compactness of super-Ricci flows under measured Gromov-Hausdorff limits.

Affiliation: Universität Bonn

The evolute of a plane curve is the envelope of its normals; the involute is a converse construction: the evolute of an involute is the given curve. These classical notions have been studied since 17th century.

In this talk I shall discuss iterations of these two constructions, in the continuous and the discrete settings. In the latter case, one deals with polygons, and one considers two discrete analogs of the normals: the bisectors of the angles, and the perpendicular bisectors of the sides. The dynamics of these two systems are very different. Our study relies heavily on computer experiments, and some observations still remain mysterious.

This is a joint work in progress with M. Arnold, D. Fuchs, I. Izmestiev, and E. Tsukerman.

Affiliation: Pennsylvania State University

In this talk we discuss several problems from convex geometry, like Tverberg type coincidence problems, Grünbaum--Hadwiger--Ramos hyperplane mass partition problem and Nandakumar \& Ramana-Rao problem with a view towards embeddability and periodic billiard trajectories problems. Using the configuration space / test map ansatz we construct a bridge that connects the convex geometry problems with particular problems from (equivariant) topology. Advanced machinery of equivariant obstruction theory is used efficiently in addressing these problems.

This talk is based on the joint work with Florian Frick, Albert Haase, and Günter M. Ziegler

Two triangle meshes are conformally equivalent if their edge lengths are related by scale factors associated to the vertices. Such a pair can be considered as preimage and image of a discrete conformal map.

In this talk we consider the approximation of a given smooth conformal map $f$ by such discrete conformal maps $f^\eps$ defined on triangular lattices. In particular, let $T$ be an infinite triangulation of the plane with congruent strictly acute triangles. We scale this triangular lattice by $\eps>0$ and approximate a compact subset of the domain of $f$ with a portion of it. For $\eps$ small enough we prove that there exists a conformally equivalent triangle mesh whose scale factors are given by $\log|f'|$ on the boundary. Furthermore we show that the corresponding discrete conformal (piecewise linear) maps $f^\eps$ converge to  $f$ uniformly in $C^1$ with error of order $\eps$.

Surfaces with constant mean curvature or constant Gauss curvature have been of interest in differential geometry for a long time. The modern viewpoint is to associate these special geometries with the theory of integrable systems, in particular discrete ones. An essential feature of this approach is that surfaces with constant curvature come with one-parameter associated families.

In this talk we investigate semidiscrete surfaces with constant mean curvature, where the notion of curvature comes from a recently developed curvature theory for quadrilateral meshes equipped with unit normal vectors. In case of vanishing mean curvature, associated families are defined via a Weierstrass representation, otherwise by a Lax pair representation.

In this talk I will present a discrete version of the classical uniformization theorem based on the theory of hyper-ideal circle patterns. It applies to surfaces represented as finite branched covers over the Riemann sphere as well as to compact polyhedral surfaces with non-positive cone singularities. The former include all Riemann surfaces represented as smooth algebraic curves, and more generally, any closed Riemann surface with a choice of a meromorphic function on it. The latter include any closed Riemann surface with a choice of a quadratic differential on it. We show that for such surfaces discrete uniformization via hyper-ideal circle patterns always exists and is unique (up to isometry). We also propose a numerical algorithm, utilizing convex optimization, that constructs the desired discrete uniformization.

The classification of polytopes has been studied since antiquity. Since the 1980s, no significant progress has been made in the classification of simplicial polytopes with few vertices in low dimensions. Recently we were able to enumerate all simplicial $4$-polytopes with $10$ vertices and neighbourly simplicial $d$-polytopes with $n$ vertices for the pairs $(d,k)$=$(4,11),(5,10),(6,11)$ and $(7,11)$. We also decided for almost all enumerated polytopes, whether they can be realized with all vertices on the unit sphere. We will indicate how these results were obtained using optimization techniques and outline possible future applications.

We present a novel constructive approach to define time evolution of critical points of an energy functional. Differently from other more established approaches based on viscosity approximations in infinite dimension, our procedure is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite dimensional. Nevertheless, in the infinite dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as limit of evolutions of critical points of fi nite dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we show several numerical experiments. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.

I discuss the role of continuous and discrete symmetries in classical and novel (G.F., James, Juestel, arXiv:1506.04240)
X-ray methods for atomic structure determination.

So-called spin holomorphic functions on rhombic quad-graphs played a key-role in Chelkak's and Smirnov's proof of the universality of the Ising model. These functions are in one-to-one correspondence to real spinors on a double cover of the medial graph, that satisfy the propagation equation introduced by Mercat. It turns out that spin holomorphic functions cannot be defined on a larger class of quad-graphs, but discrete spinors and the discrete propagation equation can be defined on general discrete Riemann surfaces. Still, decompositions into parallelograms play a special role, as did rhombic decompositions in the work of Mercat. The aim of this talk is to motivate the propagation equation as a discrete holomorphicity condition and to sketch how this discrete holomorphicity on the medial graph is related to the established linear theory of discrete Riemann surfaces. Some possible directions for future research will be also mentioned.

In last years we successfully applied line bundle theory to problems in computer graphics. This talk is about the discrete theory behind it. This includes classification theorems, a discrete Poincaré-Hopf index theorem and a generalization of the well-known cotan-Laplace operator to discrete hermitian line bundles with curvature.

We consider a discrete surface as an arbitrary cell decomposition of a surface. We define two types of discrete minimal surfaces and show that they are conjugate to each other. By triangulating their Gauss maps, these discrete minimal surfaces correspond to harmonic functions on planar meshes with respect to the cotangent Laplacian. They generalize earlier notions of discrete minimal surfaces as quad meshes: circular minimal and conical minimal.

In addition, the associated family of certain minimal surfaces, including those from discrete isothermic nets and Schramm's orthogonal circle patterns, are shown to be critical points of the area functional.

Separation in graphs is related to graphs being expanders. A conjecture by Kalai generalizes the planar separation theorem to simple polytopes. It states that the graph of a simple d-polytope can be separated to two roughly equal parts by removing O(n^((d-2)/(d-1))) vertices. We provide a counterexample to this conjecture. This is joint work with Günter Ziegler.

Mapping every simplex in the Delaunay triangulation of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Assuming the points are chosen by a Poisson point process in R^2 or in R^3, we determine the expected number of critical simplices as well as of intervals in the generalized discrete gradient.

Driven by an overwhelming amount of applications numerical approximation of partial differential equations was established as one of the core areas in applied mathematics.
During the last decades a trend for the solution of PDEs emerged, that focuses on employing systems from applied harmonic analysis for the adaptive solution of these equations. Most notably wavelet systems have been used and lead for instance to provably optimal solvers for elliptic PDEs. Inspired by this success story also other systems with various advantages in different directions should be employed in various discretization problems. For instance, ridgelets where recently successfully used in the discretization of linear transport equations.

Another famous system is that of shearlets, which admits optimal representations of functions that have singularities along smooth curves. However, in order to apply these methods in adaptive discretization algorithms it is necessary to have a system on a bounded domain or even a manifold. To apply standard algorithms this system should yields a frame, should be capable of incorporating boundary conditions, and should characterize Sobolev spaces.
Although there have been first approaches to construct shearlet systems for the solution of PDEs on bounded domains they fail to satisfy all the desiderata above.
In this talk we will introduce a novel shearlet system that meets all the requirements mentioned above and admits optimal approximation rates for functions with discontinuities along curves.

I will present an overview of the results achieved in the framework of the project B07. In particular, I will focus on the notion of discrete pluri-Lagrangian systems, thus promoting them as integrable discrete variational systems. Concrete results will be presented for systems in dimension 1,2 and 3. 

We deal with the analysis of a class of evolution equations with time-dependent perturbations in the framework of a general metric space and with the asymptotic behavior of solutions to these evolution equations.

We generalize several results on the metric formulation, which extends the notion of curves of steepest descent for autonomous gradient flows in metric spaces and prove the existence of such solutions for the related non-autonomous initial value problem by means of a variational approximation scheme by time discretization. Additionally, we show the exponential contractivity property of convex gradient flows and the related asymptotic behavior of the corresponding solutions. As a particular application, we use the previous results to show the existence of curves of steepest descent for the Fokker-Planck equation in the Wasserstein space. In the last part, we analyze the non-autonomous Fokker-Planck equation with time-dependent quadratic potential energies. We explicitly compute the solutions for this evolution equation and show that in the long-time behavior solutions behave asymptotically like a shifted and dilated Maxwellian distribution or respectively in the non-linear case like a shifted and dilated Barenblatt solution. At last, we show that in the high-frequency limit the solution curves will converges to a solution curve related to an average.

We study the design and optimization of polyhedral patterns, which are patterns of planar polygonal faces on freeform surfaces. Working with polyhedral patterns is desirable in architectural geometry and industrial design. However, the classical tiling patterns on the plane must take on various shapes in order to faithfully and feasibly approximate curved surfaces. We define and analyze the deformations these tiles must undertake to account for curvature, and discover the symmetries that remain invariant under such deformations. We propose a novel method to regularize polyhedral patterns while maintaining these symmetries into a plethora of aesthetic and feasible patterns. Finally, we raise the question of faithful approx imations of smooth surfaces by polyhedral surfaces.

For a plane graph G with k special "terminal" vertices, we want to embed a specified system of non-crossing paths between terminal vertices. The cyclic order of the paths around each terminal is prescribed, thus specifying the homotopy type of the system of paths. Each path has its own cost function, as a mapping from the edges to the reals, and we want to minimize the overall cost of all paths. This problem arises in the choice of reparametrizations for the numeric solution of Riemann-Hilbert problems.

We solve the problem for k=2 terminals (with many parallel paths between them) in polynomial time by a linear-programming formulation.

In the spirit of Klein's Erlangen Program, we present various aspects of the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix interpreted as Pluecker coordinates. Particular emphasis is put on the restriction to Lie circle geometry which is intimately related to the master dCKP equation of discrete integrable systems theory. We discuss the geometric interpretation, construction and integrability of fundamental line complexes in Moebius, Laguerre and hyperbolic geometry. In the process, we encounter avatars of classical and novel incidence theorems and associated cross- and multi-ratio identities for particular hypercomplex numbers. In the context of Moebius geometry, this leads to novel doubly hexagonal circle patterns.

Discretizations of ordinary differential equations are often studied through their modified equation. This is a differential equation with solutions that interpolate the solutions of the discrete system. The modified equation is usually obtained as a power series in the step size of the discretization. It is well-known that if a symplectic integrator is applied to a Hamiltonian system, then its modified equation is again Hamiltonian. We discuss this property from the variational (Lagrangian) side. We present a technique to construct a Lagrangian for the modified equation directly from the discrete Lagrangian, without passing through the Hamiltonian side.