"Deformations of spectral curves"

"The geometric discretisation of the Suslov problem: a case study of consistency for nonholonomic integrators"

"Iterative refinement processes for manifold-valued data"

"Near-optimal quantization and encoding under various measurement models"

"A theoretical framework for the analysis of Mapper"

"Orientational order on surfaces - the coupling of topology, geometry and dynamics"

"Persistent homology for data: stability and statistical properties"

"The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms" Coffee break

"On the connection between A/D conversion and the roots of Chebyshev polynomials"

Boris Springborn: The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic form Markov's theorem classifies the worst irrational numbers (with respect to rational approximation) and the binary quadratic forms whose values stay far away from zero (for nonzero integer arguments). I will present a proof of Markov's theorem that uses hyperbolic geometry. The main ingredients are a dictionary to translate between algebra/arithmetic and hyperbolic geometry, and some basic ideas from decorated Teichmüller theory. This will explain what Diophantine approximation has to do with simple closed geodesics, and also with the question: How far can a straight line crossing a triangle stay away from the vertices? Felix Krahmer: On the connection between A/D conversion and the roots of Chebyshev polynomials Sigma-Delta modulation is a popular approach for coarse quantization of audio signals. That is, rather than taking a minimal amount of samples and representing them with high resolution, one considers redundant representations and works with a low resolution. The underlying idea is to employ a feedback loop, incorporating the prior evolution of the sampling error. In this way, the representation of a sample can partially compensate for errors made in previous steps. The design of the filter at the core of the feedback loop is crucial for stability and hence for performance guarantees. Building on work of Güntürk (2003) who proposed to use sparse filters, we optimize the sparsity pattern, showing that a distribution mimicking the roots of Chebyshev polynomials of the second kind is optimal. The focus of this talk will be on the interplay between complex variables, orthogonal polynomials, and signal processing in the proof. This is joint work with Percy Deift and Sinan Güntürk (Courant Institute of Mathematical Sciences, NYU), derived as a part of my doctoral dissertation.

"Atomistically inspired origami"

"Salem numbers and discrete groups of automorphisms of algebraic surfaces"

"Constant mean curvature surfaces in $\mathbb{R}^3$ with Delaunay ends and Hilbert's 21st problem"

"Schrödinger's Smoke"

"News about confocal quadrics"

"Non-Negative Dimensionality Reduction in Signal Separation"

"Tire track geometry and the filament equation: results and conjectures"

"Dynamic isoperimetry and Lagrangian coherent structures"

"An Introduction to Distance Preserving Projections of Smooth Manifolds"

"Numerical long-time energy conservation for the nonlinear Schrödinger equation"

"On a discretization of Conley theory for flows in the setting of discrete Morse theory"

"Stability and Approximations in Topological Data Analysis"

"Polyhedral Patterns"

"The flux integral revisited: the Lagrangian perspective"

"The flux integral revisited: the Lagrangian perspective"

"On limit shapes and their integrability"

"Domain Filling Circle Packings - Existence and Uniqueness"

"Molecular structures generated by discrete symmetries, X-ray diffraction, and structure identification"

"Interactive Visualization of 2-D Persistent Homology"

"Cheeger constants and frustration indices for magnetic Laplacians on graphs and manifolds"

"Confined elastic curves"

"Dynamics of Discrete Screw Dislocations in SC, BCC, FCC and HCP crystals"

"Navigating the space of symmetric CMC surfaces"

"Tropical (p,q)-homology and algebraic cycles"

Workshop aus Berlin mit Live-Schaltung nach München

"Discrete variational mechanics in structure-preserving integration and optimal control"

"The Chemical Master Equation as a discretization of the Fokker-Planck and Liouville equation for chemical reactions"

"What is the semiclassical limit of non-Hermitian time evolution?"

"Combinatorial Models for Random Matrices with Gaussian Entries"

"Signed graphs, nested set complexes and spines"

"Approximation and Convergence of the Intrinsic Volume"

"Hausdorff dimension in transcendental dynamics"

"A portrait of Arnold Diffusion"

"The Morse theory of Čech and Delaunay filtrations"

"From Shapes to Shape Spaces"

"The Symplectic Camel and the Uncertainty Principle"

"Integer partitions from a geometric viewpoint"

"Universal aspects of geometric and algebraic integrability in Pluecker and Lie geometry"

"The Geometry and Dynamics of Semiclassical Wave Packets"

"Solving problems in algebraic geometry using applied mathematics"

"Smoke Rings from Smoke"

"A tour through modern methods in multiple time scale dynamics"

An overview of multiple time scale systems and singular perturbation problems will be given. It is the goal of this talk to show the breadth of this field and outline some of its major techniques and applications developed within the last 15 years. First, I am going to briefly introduce the background for the geometric viewpoint for normally hyperbolic systems covering Fenichel theory and the notion of canards. Next, switching between fast and slow systems will be considered using the Exchange Lemma. When normal hyperbolicity is lost the blow-up method for desingularization is employed. To conclude we briefly illustrate the general problem of multiscale dynamics near instability in the context of stochastic fast-slow systems. Throughout the talk, specialized numerical methods will be used to illustrate the dynamics.

"The multivariate Hermite-Laguerre connection"

"Geometric Methods for the Approximation of High-dimensional Dynamical Systems"

The Hermite polynomials have been born in the 19th century and live in many mathematical fields: in numerical analysis, in quantum theory, in combinatorics, in probability. The same can be said about Laguerre polynomials. Our talk will review the connection between the two and lift this connection to the multivariate situation such that the symplectic geometry of classical phase space comes into play. Our results are joint work with S. Troppmann.

We discuss a novel statistical learning framework for performing model reduction and modeling of stochastic high-dimensional dynamical systems. We consider two complementary settings. In the first one, we are given long trajectories of the system and we discuss new techniques for estimating, in a robust fashion, an effective number of degrees of freedom of the system, which may vary in the state space of the system, and a local scale where the dynamics is well-approximated by a reduced dynamics with a small number of degrees of freedom. We then use these ideas to produce an approximation to the generator of the system and obtain reaction coordinates for the system that capture the large time behavior of the dynamics. We present various examples from molecular dynamics illustrating these ideas. This is joint work with C. Clementi, M. Rohrdanz and W. Zheng. In the second setting we only have access to a (large number of expensive) simulators that can return short simulations of high-dimensional stochastic system, and introduce a novel statistical learning framework for learning automatically a family of local approximations to the system, that can be (automatically) pieced together to form a fast global reduced model for the system, called ATLAS. ATLAS is guaranteed to be accurate (in the sense of producing stochastic paths whose distribution is close to that of paths generated by the original system) not only at small time scales, but also at large time scales, under suitable assumptions on the dynamics. We discuss applications to homogenization of rough diffusions in low and high dimensions, as well as relatively simple systems with separations of time scales, and deterministic chaotic systems in high-dimensions, that are well-approximated by stochastic differential equations. This is joint work with M. Crosskey.

"Variational Integrators in Plasma Physics"

Variational integrators provide a systematic way to derive geometric numerical methods for Lagrangian dynamical systems, which preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether’s theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered dynamical system. Even though a large class of systems fulfills this requirement, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type like they are often found in fluid dynamics or plasma physics.
We propose the application of the variational integrator method to so called adjoint Lagrangians, which formally allow us to embed any dynamical system into a Lagrangian system by doubling the number of variables. Thereby we are able to derive variational integrators for arbitrary systems, extending the applicability of the method significantly. A discrete version of the Noether theorem for adjoint Lagrangians yields the discrete momenta preserved by the resulting numerical schemes. The presented method therefore provides a systematic way to construct numerical schemes which respect certain conservation laws of a given system.
The basics of variational integrators for field theories are presented including the discrete Noether theorem. The theory is then applied to several prototypical systems from plasma physics like the Vlasov-Poisson system and ideal magnetohydrodynamics. Numerical examples confirm the good theoretical properties.

"Formfinding with statics for polyhedral meshes"

*Abstract*:

We report on recent progress in the efficient modeling and computation of polyhedral meshes or otherwise constrained meshes, in particular meshes to be used in architectural and industrial design. As it turns out, in many cases the constraint equations can be rewritten to allow almost-standard numerical methods to converge quickly, with appropriate regularization taking care of constraints which are both redundant and under-determined. We also demonstrate how equilibrium forces, with or without compression-only constraints, are part of the formfinding process. This is joint work with C.-C. Tang, X. Sun, A. Gomes and Helmut Pottmann.

"Variational methods for lattice systems"

"Mean Field Sparse Optimal Control"

I will give a concise introduction to the variational analysis of the micro-to-macro limits of energy driven lattice systems. To address the problem I will review a general scheme based on the notion of Gamma-convergence. Then I will present several examples from materials science devoting special attention to the continuum limit of some simple network models entailing multiple scales where new effects of microscopic origin add up to the usual macroscopic description and give rise to complex energies.

We present the concept of mean-field optimal control which is the rigorous limit process connecting finite dimensional optimal control problems with ODE constraints modeling multi-agent interactions to an infinite dimensional optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of interacting agents. While in the classical mean-field theory one studies the behavior of a large number of small individuals freely interacting with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect, we address the situation where the individuals are actually influenced also by an external policy maker, and we propagate its effect for the number N of individuals going to infinity. On the one hand, from a modeling point of view, we take into account also that the policy maker is constrained to act according to optimal strategies promoting its most parsimonious interaction with the group of individuals. This is realized by considering cost functionals including L^1-norm terms penalizing a broadly distributed control of the group, while promoting its sparsity. On the other hand, from the analysis point of view, and for the sake of generality, we consider broader classes of convex control penalizations. In order to develop this new concept of limit rigorously, we need to carefully combine the classical concept of mean-field limit, connecting the finite dimensional system of ODE describing the dynamics of each individual of the group to the PDE describing the dynamics of the respective probability distribution, with the well-known concept of Gamma-convergence to show that optimal strategies for the finite dimensional problems converge to optimal strategies of the infinite dimensional problem.

"The Euclidean Distance Degree"

"Cell Packing Structures"

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. The Euclidean distance degree is the number of critical points of this optimization problem. We focus on varieties seen in engineering applications, and we discuss exact computational methods. Our running example is the Eckart-Young Theorem which states that the nearest point map for low rank matrices is given by the singular value decomposition. This is joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani, Rekha Thomas.

We give an overview of architectural structures which are either composed of polyhedral cells or closely related to them. In particular we discuss so-called support structures of polyhedral cell packings, which are mostly relevant if they are derived from quadrilateral or hexagonal meshes. There are interesting connections between discrete differential geometry on the one hand, and applications on the other hand. Such applications range from load-bearing structures to shading and lighting systems. On a higher level, we illustrate the interplay between geometry, optimization, statics, and manufacturing, with the overall aim of combining form, function and fabrication into novel integrated design tools. This is joint work with H. Pottmann et al.

"Optimal topological simplification of discrete functions on surfaces"

Given a function f on a surface and a tolerance d>0, a fundamental problem
is the construction of a perturbed function g such that N(f-g) is at most d with
respect to some norm N and g has the minimum number of critical points.
I will present a solution to this problem (with respect to the supremum norm) and
describe how homological noise of persistence at most 2d can be completely
removed from an input function on a discrete surface. The solution obtained is
not unique and a convex polyhedron of possible solutions is identified. As a
consequence, the method can be complemented to construct a solution that
satisfies an additional constraint.
The construction relies on a connection between discrete Morse theory and
persistence homology. A brief introduction to both subjects will be included.

"Transfer operator based numerical analysis of time-dependent transport"

"Topological chaos, braiding and bifurcation of almost-cyclic sets"

Numerical methods involving transfer operators have only recently been recognized as powerful tools for analyzing and quantifying transport processes in time-dependent systems. This talk discusses several different constructions that allow us to extract coherent structures and dynamic transport barriers in nonautonomous dynamical systems. Moreover, we will explore in example systems how diffusion as well as the finite-time duration of the computations influence the structures of interest.

In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem. Periodic orbits generated by the dynamics can behave as physical obstructions that 'stir' the surrounding domain and serve as the basis for this topological analysis. We provide evidence that, even for the case where periodic orbits are absent, almost-cyclic sets can be used. These are individual components of almost-invariant sets identified using a transfer operator approach which act as stirrers or 'ghost rods' around which the surrounding fluid appears to be stretched and folded. We discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes’ flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. Hence, we develop a connection between set-oriented statistical methods and topological methods, which provides an additional analysis tool in the study of complex flows.

"On a surface theory for quadrilateral nets"

will report on recent work on a discrete version of surface theory for quadrilateral nets. Our approach aims to generalize the known integrable cases into a more general framework. There are many well working examples of integrable discretizations of special surface classes as well as well working discrete definitions of fundamental forms, curvatures, shape operator and similar fundamental objects of surface theory for but so far little effort has been made to formulate a general framework that covers the integrable cases with their fundamental properties and still works on a broader class of nets. This is joint work with Andrew O. Sageman-Furnas (Furnas) and Max Wardetzky.

"Bianchi (hyper-)cubes and a geometric unification of the Hirota and Miwa equations"

"The critical temperature for the Ising model on biperiodic graphs"

We present a geometric and algebraic way of unifying two discrete master equations of inte- grable system theory, namely the dKP (Hirota) and dBKP (Miwa) equations. We demonstrate that so-called Cox lattices encapsulate Bianchi (hyper-)cubes associated with either simultaneous so- lutions of a novel 14-point and the dBKP equations or solutions of the dKP equation, depending on whether the Cox lattices are generic or degenerate.

The Ising model is one of the most studied models in statistical physics. It is one of the simplest models to exhibit a ”phase transition”, that is, a sharp change of behavior when some parameter (here, the temperature) crosses some critical value. In this talk, I will start with a gentle introduc- tion to the Ising model and its phase transition. Then, I will explain how to determine the critical temperature of the Ising model on any biperiodic planar weighted graph (or equivalently, on any finite weighted graph embedded in the torus). Although this result lies in the realm of statistical physics, the statement is formulated in homological terms, and the proof uses several geometric tools (Kramers-Wannier duality on surfaces, Harnack curves, . . . )

Steinitz's theorem is a classical but very remarkable result characterizing graphs of convex polytopes in R^3. In this talk, I will first survey several known proofs, and present one that is especially simple. I will then discuss the quantitative version and recent advances in this direction. Joint work with Stedman Wilson.

Bieberbach constructed in 1933 domains in $\mathbb{C}^2$ which were biholomorphic to $\mathbb{C}^2$ but omitted an open set. The existence of such domains was unexpected, because the analogous statement for the one-dimensional complex plane is false. The special domains Bieberbach considered are given as basins of attraction of a cubic Henon map. This classical method of the construction is one of the first applications of dynamical systems to complex analysis. In this talk the boundaries of the real sections of Bieberbach’s domains will be shown to be smooth. The real Julia sets of Bieberbach’s map will be calculated explicitly and illustrated with computer generated graphics.

“Rigidity of origami surfaces” 17.15 Kaffeepause 17.30 Dr. Ciprian S. Borcea (Rider University, New Jersey, USA)

"Rigidity and flexibility of periodic frameworks"

Cauchy's famous rigidity theorem for 3D convex polyhedra has been extended in various directions by Dehn, Weyl, A.D.Alexandrov, Gluck and Connelly. These results imply that a disk-like polyhedral surface with simplicial faces is, generically, flexible, if the boundary has at least 4 vertices. What about surfaces with rigid but not necessarily simplicial faces? A natural, albeit extreme family is given by flat-faced origamis.

Around 1995, Robert Lang, a well-known origamist, proposed a method for designing a crease pattern on a flat piece of paper such that it has an isometric flat-folded realization with an underlying, predetermined metric tree structure. Important mathematical properties of this algorithm remain elusive to this day.

In this talk I will show that Lang's beautiful method leads, often but not always, to a crease pattern that cannot be continuously deformed to the desired flat-folded shape if its faces are to be kept rigid. Most surprisingly, sometimes the initial crease pattern is simply rigid: the (real) configuration space of such a structure may be disconnected, with one of the components being an isolated point.

Joint work with my PhD student John Bowers, who has also implemented a computer program to visualize the research.

A d-periodic bar-and-joint framework is an abstraction (and generalization to arbitrary dimension d) of an atom-and-bond crystal structure. We present a general introduction to the deformation theory of this type of frameworks. Questions of generic rigidity highlight the role of sparsity conditions on the underlying quotient graph.

Joint work with Dr. Ileana Streinu, Smith College.

"Minimum-cardinality triangulations of polytopes and manifolds" 17.15 Kaffeepause 17.30 Prof. Dr. Martin Rumpf (Univerität Bonn)

"Variational Time Discretization of Geodesic Calculus in Shape Space"

"Minimum-cardinality triangulations of polytopes and manifolds"

Triangulations are important in both discrete and numerical mathematics, but different properties are studied in each of these areas. On the discrete side, attention tends to focus on structural and combinatorial properties, such as the ``shape'' of the set of all triangulations of a fixed object, or the (minimal) number of simplices in any one of them. In this talk, I will (briefly) survey some of the principal results in this area, and report on recent progress in finding triangulations of minimal cardinality of some interesting polytopes and topological manifolds derived from them. Some of these results exploit the new capabilities of the software package ``polymake'' for exact and efficient calculations in quadratic extension rings of the rationals.

"Variational Time Discretization of Geodesic Calculus in Shape Space"

The talk will introduce a time discrete geometric calculus on the space of shapes with applications in geometry processing and computer vision. The discretization is based on a suitable local approximation of the squared distance, which can be efficiently computed. The approach covers shape morphing and the robust distance evaluation between shapes based on the computation of discrete geodesic paths, shape extrapolation via a discrete exponential map, and natural transfer of geometric details along shape paths using discrete parallel transport. Furthermore, it can be used for the statistical analysis of time indexed shape data in terms of discrete geodesic regression.

The talk will describe how concepts from Riemannian manifold theory are combined with application dependent models of physical dissipation. Furthermore, a rigorous consistency and convergence analysis will be outlined. Applications will be presented in the shape space of viscous fluidic objects and the space of viscous thin shells.

"Linearizing Hilbert Nullstellensatz and the Orientability of Matroids" 17.15 Kaffeepause 17.30 Prof. Jürgen Richter-Gebert, TU München

"Complex matroids: rigidity and syzygies"

"Linearizing Hilbert Nullstellensatz and the Orientability of Matroids"

Systems of multivariate polynomial equations can be used to model the combinatorial problems. In this way, a problem is feasible (e.g. a graph is 3-colorable, Hamiltonian, etc.) if and only if a certain system of polynomial equations has a solution over an algebraically closed field. Such modeling has being used to prove non-trivial combinatorial results via polynomials (e.g. work by Alon, Tarsi, Karolyi,etc). But the polynomial method is not just for proving theorems but a rather exciting method to compute with combinatorial objects. In this talk we introduce the audience to this new idea. We show that for combinatorial feasibility problems, Hilbert's Nullstellensatz gives a sequence of linear algebra problems, over an algebraically closed field, that eventually decides feasibility. We call this method the Nullstellensatz-Linear Algebra approach or NulLA method for short. Matroids and oriented matroids play an important role in discrete geometry and questions about orientability and realizability of matroids give rise to highly structured systems of polynomial equations with connections to classical mathematics. In the second part of the talk we connect the study of matroids to the NulLA method. We present systems of polynomial equations that correspond to a matroid M and each of these systems has a zero solution if and only if M is orientable. In this case Hilbert's Nullstellensatz gives us that M is non-orientable if and only if certain certificate to the given polynomials system exists. Since Richter-Gebert showed that determining if a matroid is orientable is NP-complete, thus determining if these systems have solutions is also NP-complete. However, we also show that these systems of equations and the corresponding linear-algebra relaxations have rather rich structure. For example, it turns out the associated polynomial systems corresponding to M is linear if M is a binary matroid and thus one can determined if binary matroids are orientable much more easily. This talk is based on joint work with J. Lee, J. Miller, and S. Margulies.

„Complex matroids: rigidity and syzygies“

The talk focuses on the interrelation of phirotopes and chirotopes -- the latter forming an abstraction of the signature information of a real vector configuration, the first forming an abstraction of phase information of a complex vector configuration. We will see that in contrast to the real case, the complex phirotope in general already encodes the geometric location of the vectors of the configuration. Thus phirotopes are by far more rigid than chirotopes. Within the realm of phirotopes those related to real chirotopes form in a sense a singular situation. This singularity is the reason that chirotopes have a by far richer realization theory than phirotopes. As a consequence of the rigidity of phirotopes, explicit algebraic relations must exist among the data of a phirotope. We will extract these relations and interpret them as (slightly surprising) results in elementary geometry.

"Discrete and continuous integrable systems on cluster varieties" 17.15 Kaffeepause 17.30 Yuri B. Suris (TU Berlin)

"Variational formulation of commuting Hamiltonian flows"

During the last years it became quite popular to visualize complex (analytic) functions as images. The talk gives an introduction to “phase plots” (or “phase portraits”), which depict a function f directly on its domain by color-coding the argument of f.

By changing the combinatorics of the underlying symmetry group, images of Euclidean ornaments can be transported into the hyperbolic plane. It can be seen that conformality is an important requirement when mapping the actual artistic content. Discrete conformal maps make such computations feasible, leading to new pictures derived from existing works of art.

Since the seminal work of Jordan, Kinderlehrer and Otto, it is known that the heat flow on $R^n$ can be regarded as the gradient flow of the entropy in the Wasserstein space of probability measures. Meanwhile this interpretation has been extended to very general classes of metric measure spaces, but it seems to break down if the underlying space is discrete. In this talk we shall present a new metric on the space of probability measures on a discrete space, based on a discrete Benamou-Brenier formula. This metric defines a Riemannian structure on the space of probability measures and it allows to prove a discrete version of the JKO-theorem. This naturally leads to a notion of Ricci curvature based on convexity of the entropy in the spirit of Lott-Sturm-Villani. We shall discuss how this is related to functional inequalities and present discrete analogues of results from Bakry-Emery and Otto-Villani. This is joint work with Matthias Erbar (Bonn).

Minimal surfaces in projective differential geometry may be characterised in various different ways. Based on discrete notions of Lie quadrics and their envelopes, we propose a canonical definition of (integrable) discrete projective-minimal surfaces. We discuss various algebraic and geometric properties of these surfaces. In particular, we present a classification of discrete projective-minimal surfaces in terms of the number of envelopes of the associated Lie quadrics. It turns out that this classification is richer than the classical analogue and sheds new light on the latter.

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