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SFB Colloquium


During the semester, the SFB TRR109 organizes a colloquium which takes place every four weeks. The organization of the colloquium alternates between the TU Berlin and the TU Munich. The presentations are broadcast live from the hosting university to the partner university.

  • Last Occurrence: 12.07.2022, 14:15 - 16:30
  • Type: Seminar
  • Location: TU Berlin and TU Munich

Contact: Fraeya Whiffin (TUB) and Diane Clayton-Winter (TUM)

Room:
TU Berlin: MA 875, Mathematics Building, Strasse des 17. Juni 136, Berlin
TU Munich: Room 02.06.011, Boltzmannstr. 3, Garching
 
Time: Tuesdays 14:15-16:30
 
 
 
Talks

12.07.2022

  • 14:15 - 15:15 A canonical discrete analogue of the classical circular cross sections of ellipsoids and their isometric deformation, Wolfgang Schief (University of New South Wales)
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  • Two classical but perhaps little known facts of "elementary" geometry are that an ellipsoid may be sliced into two one-parameter families of circles and that ellipsoids may be deformed into each other in such a manner that these circles are preserved. In fact, as an illustration of these remarkable properties of ellipsoids, Hilbert and Cohn-Vossen present pictures of deformable wooden models of sliced ellipsoids in their classical monograph Geometry and the Imagination. The purpose of this talk is to demonstrate that recently introduced concepts in (integrable) Discrete Differential Geometry allow one to discretise ellipsoids in such a manner that a variety of classical properties such as those mentioned above survive. Connections with (discrete) confocal coordinate systems are explored.
  • 15:30 - 16:30 Grid Peeling and the Affine Curve-Shortening Flow, Günter Rote (FU Berlin)
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  • Grid Peeling is the process of taking the integer grid points inside a convex region and repeatedly removing the convex hull vertices. It has been observed by Eppstein, Har-Peled, and Nivasch, that, as the grid is refined, this process converges to the Affine Curve-Shortening Flow (ACSF), which is defined as a deformation of a smooth curve. As part of the M.Ed. thesis of Moritz Rüber, we have investigated the grid peeling process for special parabolas, and we could observe some striking phenomena. This has lead to a conjecture for the value of the constant that relates the two processes.

29.03.2022

  • 14:15 - 15:15 (@TUB) Harmonic Morphisms, Proper Harmonic Functions - The Method of Eigenfamilies, Sigmundur Gudmundsson (Lund University)
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  • Harmonic morphisms are maps $F: (M^m,g) \rightarrow (N^n,h)$ between Riemannian manifolds which preserve harmonic functions on the manifolds involved. In the late 1970s they were characterised, independently by B. Fuglede and T. Ishihara, as those harmonic maps which are horizontally conformal. This means that they are solutions to an over-determined non-linear system of partial differential equations. For this reason their existence theory becomes non-trivial. From the geometric point of view, the case when the codomain is a surface ($n=2$) is particularly interesting. Then each regular fibre of $F: (M,g) \rightarrow (N^2,h)$ is a minimal submanifold of $(M,g)$ of codimension 2. This means that harmonic morphisms can be seen as a useful tool for constructing such interesting geometric objects. In the first part of this talk, we will describe connections with classical complex analysis and a certain duality to the theory of minimal surfaces in Euclidean 3-space. We will then introduce our notion of eigenfamilies of complex-valued functions on $(M,g)$ and show how these can be employed to construct a wide collection of solutions to our over-determined non-linear problem on Riemannian Lie groups $(G,g)$ and symmetric spaces $(G/K,g)$. The literature on biharmonic functions $f: (M,g) \rightarrow C$ is vast, but with only very few exceptions the domains are either surfaces or open subsets of flat Euclidean space. The development of the last few years has changed this. We will describe how the above mentioned method of eigenfamilies gives $p$-harmonic functions on Lie groups and symmetric spaces for any $p>1$.
  • 15:30 - 16:30 (@TUM) Smoothing of transport plans and Density Functional Theory, Ugo Bindini (TU Munich)
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  • In Optimal Transport problems, the optimal couplings tend to concentrate on low dimensional sets (e.g. Monge maps), while for applications to physical problems it is often needed to have diffuse probability measures with good regularity properties. In 2017, in collaboration with Luigi De Pascale, we introduced a new technique for regularizing transport plans while keeping the marginal constraint. I will present this technique and how it led to new results in the field of Density Functional Theory.

08.02.2022

  • 14:15 - 15:15 (@zoom) Regge symmetry, Ivan Izmestiev (TU Vienna)
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  • The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here we give an elementary geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry. The talk is based on joint work with Arseniy Akopyan.

30.11.2021

  • 14:15 - 15:15 (@TUB) Complete families of embedded high genus CMC surfaces in the 3-sphere, Sebastian Heller (University of Hannover)
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  • In this talk I will report on recent joint work with L. Heller and M. Traizet about the existence of 1-dimensional families of Constant Mean Curvature surfaces passing through the Lawson minimal surfaces ξg,1, of high genus. I will discuss the main geometric properties of the surfaces like embeddedness, conformal type, mean curvature as well as their area and Willmore energy. I will show that these families of CMC surfaces are complete, i.e., the surfaces in each family limit to a doubly covered sphere while the conformal type degenerates. Additionally, I will discuss the relationship of the area of the Lawson surfaces with special values of multiple polylogarithms and the Riemann ζ-function.
  • 15:30 - 16:30 (@TUB) From geometry to dynamics. Integrable bilinear discretizations of quadratic vector fields., Yuri Suris (TU Berlin)
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  • We discuss dynamics of birational maps which appear as bilinear discretizations of quadratic vector fields. The corresponding dynamical systems turn out to be integrable much more often than could be expected a priori. Recently, geometric constructions behind integrable discretizations of this kind have been discovered. We will discuss these constructions, along with various aspects of integrability of birational dynamical systems in general.

04.02.2020

  • 14:15 - 15:15 (@TUM) Algebraic Systems Biology, Heather Harrington (University of Oxford)
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  • Signalling pathways can be modelled as a biochemical reaction network. When the kinetics are to follow mass-action kinetics, the resulting mathematical model is a polynomial dynamical system. I will overview approaches to analyse these models using computational algebraic geometry and statistics. Then I will present how to analyse such models with time-course data using differential algebra and geometry for model identifiability, specifically focusing on parameter identifiability for parameter inference. I will briefly mention how topological data analysis can help distinguish models and data.

03.12.2019

  • 14:15 - 15:15 (@ TUM) Gromov's notion of 'partial differential relation' and applications: a tutorial, Gero Friesecke (TU München)
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  • In 1986, Gromov introduced the notion of 'partial differential relation' (PDR) as well as a very interesting notion of 'ellipticity' which is loosely related to the traditional meaning of this word in PDE but also applies to fully nonlinear problems. My plan for the tutorial is as follows:
    1. I begin with a hiker's-guide introduction to these notions, guided by beautiful examples each of which can be recast as a solution set to a PDR:
    (i) holomorphic functions
    (ii) rigid motions
    (iii) microstructures in the Ball-James model of solid-solid phase transitions
    (iv) Euler equations of fluid dynamics.
    2. I then give a brief introduction to how 'wild' solutions to non-elliptic PDR can be constructed (for (iii) this is elementary and for (iv) this yields a celebrated theorem DeLellis and Szekelyhidi on the existence of nontrivial weak solutions to the Euler equations which are compactly supported in spacetime).
    3. I close by discussing the deep question whether elliptic PDRs admit approximate solutions not close to exact solutions. The intuitive answer (No) is correct, e.g., for example (ii) (this is a theorem by Friesecke, James and Mueller) but in general wrong, as I will explain.
  • 15:30 - 16:30 (@TUM) Optimal Lattice Quantizers and Best Approximation in the Wasserstein Metric, David Bourne (Heriot-Watt University, Edinburgh)
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  • In this talk I will discuss the problem of the best approximation of the three-dimensional Lebesgue measure by a discrete measure supported on a Bravais lattice. Here 'best approximation' means best approximation with respect to the Wasserstein metric W_p, p \in [1,\infty). This problem is known as the quantization problem and it arises in numerical integration, electrical engineering, discrete geometry, and statistics.

05.11.2019

  • 14:15 - 15:15 (@TUM) Semi-discrete unbalanced optimal transport and quantization, Bernhard Schmitzer (TUM)
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  • Semi-discrete optimal transport between a discrete source and a continuous target has intriguing geometric properties and applications in modelling and numerical methods. Unbalanced transport, which allows the comparison of measures with unequal mass, has recently been studied in great detail by various authors. In this talk we consider the combination of both concepts. The tessellation structure of semi-discrete transport survives and there is an interplay between the length scales of the discrete source and unbalanced transport which leads to qualitatively new regimes in the crystallization limit. Based on joint work with David P. Bourne and Benedikt Wirth.
  • 15:30 - 16:30 (@TUM) Gromov-Wasserstein distances and distributional invariants of datasets, Facundo Memoli (Ohio State University)
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  • The Gromov-Wasserstein (GW) distance is a generalization of the standard Wasserstein distance between two probability measures on a given ambient metric space. The GW distance assumes that these two probability measures might live on different ambient spaces and therefore implements an actual comparison of pairs of metric measure spaces. Metric-measure spaces are triples (X,dX,muX) where (X,dX) is a metric space and muX is a Borel probability measure over X and serve as a model for datasets. In practical applications, this distance is estimated either directly via gradient based optimization approaches, or through the computation of lower bounds which arise from distributional invariants of metric-measure spaces. One particular such invariant is the so called ‘global distance distribution’ of pairwise distances. This talk will overview the construction of the GW distance, the stability of distribution based invariants, and will discuss some recent results regarding the injectivity of the global distribution of distances for smooth planar curves, hypersurfaces, and metric trees.

02.07.2019

  • 14:15 - 15:15 (@TUB) Attractors of Trees of Maps and of Sequences of Maps between Spaces with Applications to Subdivision, David Levin (Tel-Aviv University, Israel)
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  • Both in subdivision methods and in fractal analysis, geometrical objects are generated by iterative mapping procedures. Motivated by both subjects we consider more general frameworks of mappings, namely, trees of maps and maps between spaces. We define iterated procedures on such sequences of maps and investigate their convergence. The main tool here is the analysis of backward trajectories of maps, leading to non-stationary versions of Banach fixed-point theory. Trees of maps can represent different kinds of subdivision processes, from stationary uniform subdivision, to non-stationary subdivision, and also non-uniform and non-linear subdivision. In the fractals' perspective, such trees of maps suggest new types of fractals generated by code-dependent function systems.

04.06.2019

  • 15:00 - 16:00 (@TUB) The Z-Dirac and massive Laplacian operators in the Z-invariant Ising model, Béatrice de Tilière (University Paris Dauphine)
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  • The topic of this talk is Baxter's Z-invariant Ising model defined on isoradial graphs. We prove that certain key quantities of the model – the partition function, and edge-probabilities in the contour representation – can be explicitly expressed using the Z-massive Laplacian and its inverse, the massive Green function, introduced by Boutillier, dT, Raschel. This establishes a deep relation between classical 2d-models of statistical mechanics: the Ising model, spanning forests and random walk. In order to prove these results, we introduce the Z-Dirac operator; we relate it to the Z-massive Laplacian, extending to the full Z-invariant regime results obtained by Kenyon at the critical point; we then relate the Z-Dirac operator to the Ising model. The proof consists in establishing matrix relations allowing to compare matrix inverses and also, using additional arguments, determinants.
  • 16:15 - 17:15 (@TUM) Physics-based Deep Learning for Fluids, Nils Thuerey (Technische Universität München)
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  • In this talk I will focus on the possibilities that arise from recent advances in the area of deep learning for accelerating and improving physics simulations. I will focus on fluids, which encompass a large class of materials we encounter in our everyday lives. In addition to being ubiquitous, the underlying physical model, the Navier-Stokes equations, at the same time represent a challenging, non-linear advection-diffusion PDE that poses interesting challenges for deep learning methods. I will explain and discuss several recent research projects from that focus on temporal predictions of physical functions, temporally coherent adversarial training, and predictions of steady-state turbulence solutions. Among other things, it turns out to be useful to make the learning process aware of the underlying physical principles. E.g., the transport component of the Navier-Stokes equations play a crucial role. I will also give an outlook about open challenges in the area of deep learning for general physics problems. Among others, trained models could server as priors for a variety of inverse problems and control tasks.

07.05.2019

  • 14:15 - 15:15 (@TUB) Perfectly matched layers with discrete complex analysis, Albert Chern (Technische Universität Berlin)
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  • Seeking reflectionless boundary conditions for wave equations has been one of the most classical non-trivial problems in numerical PDEs. The method of perfectly matched layer (PML) is regarded as the state-of-the-art approach. A PML is a wave-absorbing layer attached to the boundary designed so ideally that the interface produces no reflection wave. Mathematically, PML equations are viewed as alternate realizations of the analytic continuation of the wave equation. However, it had been believed that numerical reflections are inevitable in discretized PML equations due to discretization error. In this work we adopt the linear discrete complex analysis and replicate the PML theory in the discrete setup. The discovered discrete PML becomes the first truly reflectionless boundary treatment for the discrete wave equation.

29.01.2019

  • 14:15 - 15:15 (@TUM) Introduction to pde2path (and some recent additions), Hannes Uecker (Carl von Ossietzky Universität Oldenburg)
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  • pde2path is a Matlab package for numerical continuation and bifurcation analysis of partial differential equations. Its main design goals are flexibility (to be able to treat large classes of problems), easy usage, and hackability (easy customization). It is based on the finite element method and arclength continuation. Currently, it can handle branches of steady solutions and time periodic orbits for various classes of systems of PDEs, and the associated bifurcations. We first briefly review the basic setup using the Allen-Cahn equation as a simple example, and then explain some more advanced features such as bifurcations of higher multiplicity, including some connections to geometry by considering problems of pattern formation on curved surfaces.
  • 15:30 - 16:30 (@TUB) News from Celestial Mechanics, Andreas Knauf (Friedrich-Alexander-Universität Erlangen-Nürnberg)
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  • For the oldest differential equation, the one of the n-body problem, it is unknown whether for almost all initial conditions a global solution exists. We present a general method, developed together with St. Fleischer, that has be used to show that collisions and, for up to four particles, non-collision singularities and initial conditions violating asymptotic completeness are of measure zero.

08.01.2019

  • 14:15 - 15:15 (@TUM) Finite-Difference Discretization of the Ambrosio-Tortorelli Functionals, Annika Bach (TU München)
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  • Motivated by applications to image reconstruction, in this talk we present some results on the asymptotic behavior of a finite-differences discretization of the Ambrosio-Tortorelli functionals. These functionals provide an approximation via Gamma-convergence of the Mumford-Shah functional. Denoted by epsilon the approximation parameter of the Ambrosio-Tortorelli functionals and by delta the discretization step-size, we fully describe the relative impact of epsilon and delta in terms of Gamma-limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when varepsilon and delta are of the same order, the underlying lattice structure affects the Gamma-limit, which turns out to be an anisotropic free-discontinuity functional. This is joint work with Caterina Ida Zeppieri (Münster) and Andrea Braides (Rome).
  • 15:30 - 16:30 (@TUM) Variational Convergence of Discrete Euler Elastica, Henrik Schumacher (RWTH Aachen)
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  • In this talk, I am going to give a brief exposition of my recent work on discrete Euler elastica (joint work with Sebastian Scholtes and Max Wardetzky). Discrete Euler elastica are critical points of the Euler-Bernoulli bending energy subject to inextensibility constraints and clamped boundary conditions. A very popular discrete bending energy for polygonal lines has been proposed by Hencky; it basically consists of a weighted sum of squared turning angles. We show that, under mesh refinement, a certain subset of almost minimizers of this energy (subject to inextensibility constraints and clamped boundary conditions) converges to the set of smooth minimizers: When using piecewise-linear interpolation as reconstruction operator, we obtain Hausdorff-convergence in the space of Lipschitz curves. However, the proof hinges on a reconstruction operator that embeds discrete curves into $BV^3$ (the space of curves whose third derivative is a measure of bounded variation). Modulo this operator, we even obtain Hausdorff-convergence in the Sobolev space $W^{2,p}$, $p <\infty$. The main observation needed for the proof is that the intersections of a compactum with a decreasing sequence of sublevel sets of a lower semi-continuous function converges in Hausdorff distance to the set of minimizers. In order to exploit this observation, we have to take the following steps: (i) a~priori analysis of the minimizers in \emph{both} the smooth and discrete setting, (ii) estimation of the energies' consistency errors and of the constraint violations due to discretization/smoothing, and (iii) repairing these constraint violations by means of the Newton-Kantorovich theorem. If time allows, I will briefly discuss modifications to this machinery that are needed for treating the convergence analysis of discrete minimal surfaces.

11.12.2018

  • 14:15 - 15:15 (@TUM) Approximate curvatures of a varifold, Gian Paolo Leonardi (University of Modena and Reggio Emilia)
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  • Varifolds, i.e. Radon measures on the Grassmannian bundle of unoriented tangent d-planes of a Riemannian n-manifold M, represent a variational generalization of unoriented, d-dimensional submanifolds of M. By a suitable extension of classical variation operators, we introduce a notion of approximate second fundamental form that is well-defined for a generic varifold. Rectifiability, compactness, and convergence results are proved, showing in particular the consistency and stability of approximate curvatures with respect to varifold convergence. If restricted to the case of "discrete varifolds", this theory provides a general framework for extracting key features from discrete geometric data. Some numerical tests on point clouds (evaluation of curvatures and geometric flows, also in presence of noise and singularities) will be shown. We shall finally discuss some future perspectives and open problems. This is a joint research with Blanche Buet (Univ. Paris XI - Orsay) and Simon Masnou (Univ. Lyon 1).
  • 15:30 - 16:30 (@TUM) A review on some mathematical results about crystallization, Xavier Blanc (Université Paris-Diderot)
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  • In this talk, I will review some results on the crystallization conjecture, that is, the mathematical proof of the fact that, under appropriate conditions, interacting particles place themselves into periodic configurations. I will first review classical models at zero temperature, in which few results have been proved. Apart from the question of the emergence of periodicity, determination of the optimal lattice, in link with special functions, will be addressed. Then, some problems regarding the quantum case, in which the notion of crystalline order needs to be defined in a different way. Similarities with positive temperature classical models will be outlined. This is a joint work with M. Lewin (Univ. Paris Dauphine)

27.11.2018

  • 14:15 - 15:15 (@TUB) Holomorphic dynamics connecting geometry, root-finding, algebra, and the Mandelbrot set, Dierk Schleicher (Jakobs University Bremen)
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  • We will discuss the interplay between various aspects of holomorphic dynamical systems that are investigated in my ERC grant. We present some of the fundamental questions in the field and how they relate to geometric aspects (such as Thurston’s geometrization theory), algebra, and numerical tasks, especially root finding.

03.07.2018

  • 14:15 - 15:15 (@TU Berlin) D'Alembert type representation for discrete surfaces, Shimpei Kobayashi (Hokkaido University, Japan & DGD Guest Professor)
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  • It is known that smooth constant negative Gaussian curvature surfaces can be constructed by a pair of ordinary differential equations (Krichever, Toda), which is an analogue of d'Alembert formula for wave equation. The heart of the construction is based on a method of integrable systems, especially infinite dimensional Lie groups, the loop groups. Recently by using the method of integrable systems, discretization of surfaces have been intensively considered. I will talk about d'Alembert type representation for discrete constant negative Gaussian curvature surfaces and discrete indefinite affine spheres.

12.06.2018

  • 14:15 - 15:15 (@TUB) Conformal geometry and random surfaces, Steffen Rohde (University of Washington & DGD Guest Professor)
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  • What do you typically get when you glue n equilateral triangles along their edges and constrain the resulting surface to be a topological sphere? The (conjectural) limit as n tends to infinity is well-known to physicists but poses many challenges to mathematicians. I will give precise formulations of this problem, will describe work of probabilists (Browninan map, Peano sphere), physicists (Liouville quantum gravity), geometers and complex analysts (uniformization), and will discuss open problems.
  • 15:30 - 16:30 (@TUM) Optimizing over convex functions: motivations and challenges, Guillaume Carlier (Université Paris Dauphine & John von Neumann Professor@TUM)
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  • Optimization problems subject to a convexity constraint on the admissible profiles arise in a variety of contexts in aerodynamics, economics, geometry, shape optimization, mass transport.... They are challenging because discretizing convex functions or shapes is in general very costly and naive algorithms maybe inconsistent. In this survey talk, after describing some applications and theoretical results I will review some tractable numerical approaches to these problems.

15.05.2018

  • 14:15 - 15:15 (@TUM) Solving x’=?, Konstantin Mischaikow (Rutgers University, USA & DGD Guest Professor)
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  • With the advent of every improving information technologies, science and engineering is being being evermore guided by data-driven models and large-scale computations. In this setting, one often is forced to work with models for which the nonlinearities are not derived from first principles and quantitative values for parameters are not known. With this in mind, I will describe an alternative approach formulated in the language of combinatorics and algebraic topology that is inherently multiscale, amenable to mathematically rigorous results based on discrete descriptions of dynamics, computable, and capable of recovering robust dynamic structures. To keep the talk grounded, I will discuss the ideas in the context of modeling of gene regulatory networks.

17.04.2018

  • 14:15 - 15:15 (@TUM) Wulff shape emergence in graphene, Elisa Davoli (Universität Wien)
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  • Graphene samples are identified as minimizers of configurational energies featuring both two- and three-body atomic-interaction terms. This variational viewpoint allows for a detailed description of ground-state geometries as connected subsets of a regular hexagonal lattice. In this talk we will discuss how these geometries evolve as the number n of carbon atoms in the graphene sample increases. By means of an equivalent characterization of minimality via a discrete isoperimetric inequality, we will prove that ground states converge to the ideal hexagonal Wulff shape as n tends to infinity. In particular, we will show that ground states deviate from such hexagonal Wulff shape by at most Kn^{3/4}+o(n^{3/4}) atoms, where both the constant K and the rate n^{3/4} are sharp.
  • 15:30 - 16:30 (@TUM) Numerical Dynamics of Integrodifference Equations, Christian Pötzsche (Alpen-Adria-Universität, Klagenfurt)
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  • Integrodifference equations (IDEs for short) are a popular tool in theoretical ecology to describe the spatial dispersal of populations with nonoverlapping generations. From a mathematical perspective, IDEs are recursions on ambient spaces of continuous or integrable functions and thus generate infinite-dimensional dynamical systems. Hence, for simulation purposes an appropriate numerical approximation yielding a finite-dimensional state space is due. Our goal is to study dynamical properties of IDEs (e.g. existence of reference solutions, attractors, invariant manifolds) which are preserved under corresponding numerical methods and to establish convergence for increasingly more accurate schemes.

06.02.2018

  • 14:15 - 15:15 (@TUM) Rattling in spatially discrete reaction-diffusion equations with hysteresis, Sergey Tikhomirov (Saint-Petersburg State University and John v. Neumann Professor @ TUM)
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  • Hysteresis naturally appears as a mechanism of self-organization and is often used in control theory. Important features of hysteresis are discontinuity and memory. We consider reaction-diffusion equations with hysteresis. Such equations describe processes in which diffusive and non-diffusive instances interacts according to a hysteresis law. Due to the discontinuity of hysteresis, these equations are not always well-posed. We consider a spatial discretization of the problem and present a new mechanism of pattern formation, which we call rattling. The profile of the solution forms two hills propagating with non-constant velocity. The profile of hysteresis forms a highly oscillating quasiperiodic pattern, which explains mechanism of illposedness of the original problem and suggests a possible regularization. Rattling is very robust and persists in arbitrary dimension and in systems acting on different time scales. The talk is based on joint works with P. Gurevich.
  • 15:30 - 16:30 (@TUB) Matrix KP solitons in a tropical limit, Yang-Baxter equation and related structures, Folkert Müller-Hoissen (Max Planck Institute for Dynamics and Self-Organization)
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  • The KP equation generalizes the famous Korteweg-deVries (KdV) equation to a "completely integrable" PDE in three dimensions. In physics it models, in particular, certain wave structures on a fluid surface, but it also plays a role in pure mathematics. The KP equation possesses multi-soliton solutions forming network-like structures on a plane, evolving in time. In a "tropical limit", they are given, at fixed time, by a graph, with a number (wave height) attached to each edge. We consider a matrix version of the KP equation and a (binary Darboux) method to generate exact solutions. In the tropical limit we now have vector or matrix data attached to the edges of the corresponding graph. For a subclass of soliton solutions ("pure solitons"), the distribution of matrix data over the tropical limit graph turns out to be ruled by the famous Yang-Baxter equation, which is a familiar structure of quantum, rather than classical, integrable systems. For other classes of matrix KP solitons different rules exist, and there is much more to discover from an analysis of matrix KP solitons in the tropical limit. In particular, we also meet the pentagon equation.

09.01.2018

  • 14:15 - 15:15 (@ TUM) Exact periodic stripes for minimizers of a local/non-local interaction functional in general dimension, Eris Runa (MPI Leipzig)
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  • In this talk we will consider a functional consisting of a perimeter term and a non-local term which are in competition. In the discrete setting such functional was introduced by Giuliani, Lebowitz, Lieb and Seiringer. We show that the minimizers of such functional are optimal periodic stripes for both the discrete and continuous setting. In the discrete setting, such behaviour has been shown by Giuliani and Seiringer using different techniques for a smaller range of exponents. One striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one dimensional. This model has many similarities with the celebrated Ohta-Kawasaki functional. In particular for Ohta-Kawasaki functional, the minimality of periodic stripes is conjectured. This work is in collaboration with Sara Daneri.
  • 15:30 - 16:30 (@TUM) Local phenomena in random dynamical systems: bifurcations, synchronisation, and quasi-stationary dynamics, Maximilian Engel (TU München)
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  • We consider several related topics in the bifurcation theory of random dynamical systems: synchronisation by noise, noise-induced chaos and qualitative changes of fi nite-time behaviour. We study these phenomena with reference to a Hopf bifurcation subject to white noise and the model of a stochastically driven limit cycle on the cylinder. Furthermore, we investigate the bifurcation behaviour of unbounded noise systems in bounded domains, exhibiting the local character of random bifurcations which are usually hidden in the global analysis.

12.12.2017

  • 14:15 - 15:15 (@ TU München) Combinatorial Solutions to Elastic Shape Matching, Daniel Cremers (TU München)
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  • In my presentation, I will focus on four different shape matching problems, namely the matching between two planar shapes, the matching between two 3D shapes, the matching between a shape and an image and the matching between a planar and a 3D shape. In all cases, I will discuss combinatorial formulations for elastic shape matching and show how optimal or near-optimal solutions can be computed using dynamic programming or integer linear programming.

21.11.2017

  • 14:15 - 15:15 (@TUB) Complexity of 3-Manifolds, Stephan Tillmann  (The University of Sydney)
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  • The complexity of a 3-manifold is the minimum number of tetrahedra in a triangulation of the manifold. It was defined and first studied by Matveev in 1990. The complexity is generally difficult to compute, and various upper and lower bounds have been derived during the last decade using fundamental group, homology or hyperbolic volume.
    Effective bounds have only been found in joint work with Jaco, Rubinstein and, more recently, Spreer. Our bounds not only allowed us to determine the first infinite classes of minimal triangulations of closed 3-manifolds, but they also lead to a structure theory of minimal triangulations.
    In this talk, I will give a general introduction to complexity of 3-manifolds and explain its ramifications for algorithmic problems in the study of 3-manifolds.
  • 15:30 - 16:30 (@TUB) The story of infinitesimals: a conspiracy of silence ?, Daryl Cooper (University of California at Santa Barbara)
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  • The hyperreals are an ordered field containing the real numbers as well as infinitesimals and infinitely large numbers. In some sense they have ALL the properties of the real numbers. They have languished for 60 years, spurned by most professionals. The situation recalls the slow acceptance of other extensions of the concept of number. I will construct the hyperreals, and say a few words about doing geometry and topology with these guys.

18.07.2017

  • 14:15 - 15:15 (@TUB) Deformations of Spectral curves, Martin Schmidt (Universität Mannheim)
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  • The talk reviews some results on spectral curves of integrable systems. First we show, how the theory of the Whitham equations for spectral curves fits into the theory of deformations of algebraic curves. In the context of the algebraic geometric correspondence between spectral data and integrable systems with Lax operators there exists different notions of spectral curves. They are all one-sheeted coverings of the multiplier curve and have the same normalization. If the spectral curves are chosen to be locally planar, then the theory of Whitham deformations yields a smooth universal deformation. As an application we consider spectral curves of closed curves and CMC tori in three-dimensional Euclidean space. There are two different density results related to these two examples: Firstly, the closed curves of finite type in three dimensional Euclidean space are dense in all closed curves. Secondly, for fixed spectral genus the closure of the spectral curves of CMC tori in three dimensional Euclidean space inside the spectral curves of CMC planes can be determined. These results are the first steps toward the soul conjecture of Ulrich Pinkall.
  • 15:30 - 16:30 (@TUB) The geometric discretisation of the Suslov problem: a case study of consistency for nonholonomic integrators, Luis García Naranjo (IIMAS-UNAM, Mexico)
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  • In mechanics, constraints that restrict the possible configurations of the system are termed holonomic. A simple example is the fixed length of the rod of a pendulum. Mechanical systems with constraints on the velocities that are not consequences of constraints on positions are called nonholonomic. These often arise in rolling systems, like a sphere rotating without slipping on a table.
    The geometric discretisation of nonholonomic systems is a an open research subject that goes back to a paper by Cortés and Martinez from 2001. In their approach, it is necessary to propose a discrete version of the constraints and of the Lagrangian of the system.
    In this talk I will first present a general introduction to nonholonomic systems and their discretisation. Next, I will present recent results on the geometric discretisation of the Suslov problem, which is one of the simplest examples of a nonholonomic system that exhibits some of the main features that distinguish nonholonomic from Hamiltonian systems. Our results indicate that the so-called consistency condition on the choice of the discrete constraints and the discrete Lagrangian, is not relevant in the design of nonholonomic geometric integrators with optimal properties.

20.06.2017

  • 14:15 - 15:15 (@ TUM) Iterative refinement processes for manifold-valued data, Caroline Moosmüller (University of Passau)
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  • We consider iterative refinement processes which operate on discrete data and produce a smooth curve or surface in the limit. Most results on such processes, also called subdivision schemes, concern data in vector spaces and rules which are linear. We are interested in studying manifold-valued data and refinement rules which are solely defined by the intrinsic geometry of the underlying manifold. In this talk we focus on a particular class of subdivision schemes, called Hermite schemes. These algorithms successively refine discrete point-vector data and, via a limit process, produce a curve and its derivatives. We give an introduction to linear Hermite subdivision schemes and present adaptations to manifolds using geodesics and parallel transport. Furthermore, we analyse the resulting nonlinear algorithms with respect to convergence and C1 smoothness.
  • 15:30 - 16:30 (@ TUM) Near-optimal quantization and encoding under various measurement models, Rayan Saab (University of California, San Diego)
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  • In the era of digital computation, data acquisition consists of a series of steps. A sampling or measurement process is typically followed by quantization, or digitization, which allows digital storage and transmission of data. In turn, quantization is often followed by encoding, or compression, to efficiently represent the quantized data. In this talk, we discuss quantization and encoding schemes for a variety of measurement processes, along with their associated reconstruction algorithms. We show results for classically oversampled band-limited functions, oversampled linear measurements of finite dimensional signals, and compressed sensing measurements of sparse and compressible signals. The encoding methods we discuss are practical, rely on noise-shaping quantization schemes such as Sigma-Delta quantization, and also work in the extreme case of 1-bit quantization. Moreover, they yield near-optimal approximation accuracy as a function of the bit-rate.

23.05.2017

  • 15:30 - 16:30 (@TUM) A theoretical framework for the analysis of Mapper, Mathieu Carrière (INRIA France)
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  • Mapper is probably the most widely used TDA (Topological Data Analysis) tool in the applied sciences and industry. Its main application is in exploratory analysis, where it provides novel data representations that allow for a higher-level understanding of the geometric structures underlying the data. The output of Mapper takes the form of a graph, whose vertices represent homogeneous subpopulations of the data, and whose edges represent certain types of proximity relations. Nevertheless, the inherent instability of the output and the difficult parameter tuning make the method rather difficult to use in practice. This talk will focus on the study of the structural properties of the graphs produced by Mapper, together with their partial stability properties, with a view towards the design of new tools to help users set up the parameters and interpret the outputs.

02.05.2017

  • 14:15 - 15:15 (@TUB) Computing symplectic capacities combinatorially, Michael Hutchings (UC Berkeley)
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  • A basic question in symplectic geometry is to determine when one symplectic manifold with boundary (such as a domain in $\mathbb{R}^{2n}$) can be symplectically embedded into another. Another basic question is to understand the periodic orbits of Hamiltonian vector fields (more precisely Reeb vector fields) on the boundaries of such domains. It turns out that these two questions are closely related: the periodic orbits of the Reeb vector field give rise to obstructions to symplectic embeddings. In particular, the periodic orbits can be used to define numerical invariants of symplectic manifolds with boundary, called “symplectic capacities”, which are monotone under symplectic embeddings. In the talk, we will first review the above story. We will then discuss some recent work on defining and computing symplectic capacities combinatorially for “convex toric domains” in $\mathbb{R}^{2n}$ (joint work with Jean Gutt). We will also mention some work in progress on numerically computing perodic orbits and symplectic capacities of convex domains in $\mathbb{R}^4$ by approximating them by convex polyhedra (joint with Julian Chaidez).
  • 15:30 - 16:30 (@TUB) Abelian Higgs Vortices and Discrete Conformal Maps, Ananth Sridhar (TU Berlin)
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  • We explain a connection between recent developments in the study of vortices in the abelian Higgs models, and in the theory of structure-preserving discrete conformal maps. We explain how both are related via conformal mapping problems involving prescribed linear combinations of the curvature and volume form, and show how the discrete conformal theory can be used to construct discrete vortex solutions.

07.02.2017

  • 14:15 - 15:15 (@TUM) Orientational order on surfaces - the coupling of topology, geometry and dynamics, Axel Voigt (TU Dresden)
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  • We consider the numerical investigation of surface bound orientational order using unit tangential vector fields by means of a gradient-flow equation of a weak surface Frank-Oseen energy. The energy is composed of intrinsic and extrinsic contributions, as well as a penalization term to enforce the unity of the vector field. Four different numerical discretizations, namely a discrete exterior calculus approach, a method based on vector spherical harmonics, a surface finite-element method, and an approach utilizing an implicit surface description, the diffuse interface method, are described and compared with each other for surfaces with Euler characteristic 2. We demonstrate the influence of geometric properties on realizations of the Poincare-Hopf theorem and show examples where the energy is decreased by introducing additional orientational defects.
  • 15:30 - 16:30 (@TUM) Persistent homology for data: stability and statistical properties, Frederic Chazal (INRIA France)
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  • Computational topology has recently seen an important development toward data analysis, giving birth to Topological Data Analysis. Persistent homology appears as a fundamental tool in this field. It is usually computed from filtrations built on top of data sets sampled from some unknown (metric) space, providing "topological signatures" revealing the structure of the underlying space. When the size of the sample is large, direct computation of persistent homology often suffers two issues. First, it becomes prohibitive due to the combinatorial size of the considered filtrations and, second, it appears to be very sensitive to noise and outliers. The goal of the talk is twofold. First, we will briefly introduce the notion of persistent homology and show how it can be used to infer relevant topological information from metric data through stability properties. Second, we will present a method to overcome the above mentioned computational and noise issues by computing persistent diagrams from several subsamples and combining them in order to efficiently infer robust and relevant topological information.

10.01.2017

  • 14:15 - 15:15 (@TUB) The hyperbolic geometry of Markov’s theorem on Diophantine approximation and quadratic forms, Boris Springborn (TU Berlin)
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  • 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?
  • 15:30 - 16:30 (@TUM) On the connection between A/D conversion and the roots of Chebyshev polynomials, Felix Krahmer (TU München)
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  • 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.

13.12.2016

  • 14:15 - 15:15 (@TUM) Atomistically inspired origami, Richard D. James (University of Minnesota)
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  • “Objective Structures” are structures generated as orbits of discrete groups of isometries. We comment on their unexpected prevalence in nanoscience, materials science and biology and also explain why they arise in a natural way as distinguished structures in quantum mechanics, molecular dynamics and continuum mechanics. The underlying mathematical idea is that the isometry group that generates the structure matches the invariance group of the differential equations. Their characteristic features in molecular science imply highly desirable features for macroscopic structures, particularly 4D structures that deform. We illustrate the latter by constructing some “objective origami” structures.

15.11.2016

  • 14:15 - 15:15 (@TUM) Salem numbers and discrete groups of automorphisms of algebraic surfaces, Igor Dolgachev (University of Michigan at Ann Arbor & John von Neumann Guest Professor @ Research Group Algebra, TUM)
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  • A Salem number is a real algebraic integer greater than one whose all conjugates have absolute value at most one and at least one of them has absolute value one. In particular, a Salem number is a root of a reciprocal monic polynomial with integer coefficients. In complex dynamics the logarithms of Salem numbers are realized as topological entropy of an automorphism of an algebraic surface. In my talk I will explain when such an automorphism exists and which Salem numbers occur in this way.
  • 15:30 - 16:30 (@TUB) Constant mean curvature surfaces in $\mathbb{R}^3$ with Delaunay ends and Hilbert's 21st problem, Franz Pedit (University of Massachusetts, Amherst)
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  • We will discuss the classification problem for constant mean curvature surfaces with Delaunay ends, its relation to surface group representations of punctured compact Riemann surfaces into a loop group, and Fuchsian connections (differential equations) with coefficients in a loop algebra.

05.07.2016

  • 14:15 - 15:15 (@TUB) Schrödinger's Smoke, Peter Schröder (Caltech)
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  • We describe a new approach for the purely Eulerian simulation of incompressible fluids. In it, the fluid state is represented by a $C^2$-valued wave function evolving under the Schrödinger equation subject to incompressibility constraints. The underlying dynamical system is Hamiltonian and governed by the kinetic energy of the fluid together with an energy of Landau-Lifshitz type. The latter ensures that dynamics due to thin vortical structures, all important for visual simulation, are faithfully reproduced. This enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments, even on modestly sized grids. Our implementation uses a simple splitting method for time integration, employing the FFT for Schrödinger evolution as well as constraint projection. Using a standard penalty method we also allow arbitrary obstacles. The resulting algorithm is simple, unconditionally stable, and efficient. In particular it does not require any Lagrangian techniques for advection or to counteract the loss of vorticity. We demonstrate its use in a variety of scenarios, compare it with experiments, and evaluate it against benchmark tests.
    Joint work with Albert Chern, Felix Knöppel, Ulrich Pinkall and Steffen Weißmann. For more details search "Schrödinger's Smoke" on YouTube.
  • 15:30 - 16:30 (@TUB) News about confocal quadrics, Yuri Suris (TU Berlin)
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  • I will report about some recent developments, both in geometry and dynamics, related to confocal quadrics. First, I will provide a new approach to billiards in confocal quadrics and their integrability, based on the pluri-Lagrangian structure. Second, I will describe a novel construction of discrete confocal coordinate systems, based on a joint work with Bobenko, Schief and Techter.

31.05.2016

  • 14:15 - 15:15 (@TUB) Tire track geometry and the filament equation: results and conjectures, Sergei Tabachnikov (Pennsylvania State University)
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  • The simplest model of a bicycle is a segment of fixed length that can move, in n-dimensional Euclidean space, so that the velocity of the rear end is always aligned with the segment (the rear wheel is fixed on the frame). The rear wheel track and a choice of direction uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. The two track are related by the bicycle (Darboux) transformation which defines a discrete time dynamical system on the space of curves. I shall discuss the symplectic, and in dimension 3, bi-symplectic, nature of this transformation and, in dimension 3, its relation with the filament equation.
    An interesting problem is to describe the curves that are in the bicycle correspondence with themselves (in this case, given the front and rear tracks, one cannot tell which way the bicycle went). In dimension two, such curves yield solutions to Ulam's problem: is the round ball the only body that floats in equilibrium in all positions? I shall discuss F. Wegner's results on this problem and relate them with the planar filament equation.
    Open problems and conjectures will be emphasized.
  • 15:30 - 16:30 (@TUM) Dynamic isoperimetry and Lagrangian coherent structures, Gary Froyland (UNSW, Australia)
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  • The study of transport and mixing processes in dynamical systems is important for the analysis of mathematical models of physical systems. I will describe a novel, direct geometric method to identify subsets of phase space that remain strongly coherent over a finite time duration. The method is based on a dynamic extension of classical (static) isoperimetric problems; the latter are concerned with identifying submanifolds with the smallest boundary size relative to their volume. I will introduce dynamic isoperimetric problems; the study of sets with small boundary size relative to volume as they are evolved by a general dynamical system. I will state dynamic versions of the fundamental (static) isoperimetric (in)equalities; a dynamic Federer-Fleming theorem and a dynamic Cheeger inequality. I will also introduce a dynamic Laplace operator and describe a computational method to identify coherent sets based on eigenfunctions of the dynamic Laplacian. Our results include formal mathematical statements concerning geometric properties of finite-time coherent sets, whose boundaries can be regarded as Lagrangian coherent structures. The computational advantages of this approach are a well-separated spectrum for the dynamic Laplacian, and flexibility in appropriate numerical approximation methods. Finally, we demonstrate that the dynamic Laplace operator can be realised as a zero-diffusion limit of a recent probabilistic transfer operator method for finding coherent sets, based on small diffusion.

24.05.2016

  • 14:15 - 15:15 (@TUM) An Introduction to Distance Preserving Projections of Smooth Manifolds, Mark Iwen (Michigan State University)
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  • Manifold-based image models are assumed in many engineering applications involving imaging and image classification. In the setting of image classification, in particular, proposed designs for small and cheap cameras motivate compressive imaging applications involving manifolds. Interesting mathematics results when one considers that the problem one needs to solve in this setting ultimately involves questions concerning how well one can embed a low-dimensional smooth sub-manifold of high-dimensional Euclidean space into a much lower dimensional space without knowing any of its detailed structure. We will motivate this problem and discuss how one might accomplish this seemingly difficult task using random projections. Little if any prerequisites will be assumed.
  • 15:30 - 16:30 (@TUB) Numerics of nonlinear Schrödinger equations on long time intervals, Ludwig Gauckler (TU Berlin)
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  • In the talk, the long-time behaviour of numerical methods for Hamiltonian differential equations is discussed, in particular the near-conservation of energy by symplectic numerical methods on long time intervals. In the case of Hamiltonian ordinary differential equations, this can be rigorously shown by a backward error analysis. After an introduction to this classical result, the difficulties in extending such a result to Hamiltonian partial differential equations like the nonlinear Schrödinger equation are described. Finally, a recent result on long-time near-conservation of energy by the (symplectic) split-step Fourier method applied to the (Hamiltonian) nonlinear Schrödinger equation is presented.

19.04.2016

  • 14:15 - 15:15 (@TUM) On a discretization of Conley theory for flows in the setting of discrete Morse theory, Marian Mrozek (Jageillonian University, Kraków)
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  • Conley theory studies qualitative features of dynamical systems by means of topological invariants of isolated invariant sets. Since the invariants are stable under perturbations and computable from a finite sample, they provide a tool for rigorous, qualitative numerical analysis of dynamical systems. In this talk, after a brief introduction to Conley theory, I will present its recent extension to combinatorial multivector fields, a generalization of the concept introduced by R. Forman in his discrete (combinatorial) Morse theory. I will also show some numerical examples indicating potential applications in the study of sampled dynamical systems.
  • 15:30 - 16:30 (@TUM) Stability and Approximations in Topological Data Analysis, Magnus Bakke Botnan  (TU München)
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  • Persistent homology assigns a topological descriptor to a real-valued function defined on a topological space. This descriptor is called a persistence diagram and comprises a collection of intervals summarizing the dimension and 'size' of the homological features of the associated sublevel filtration. Two immediate questions arise: 1) is the descriptor stable with respect to perturbation of the function? 2) can we approximate the descriptor? By introducing the language of interleavings we shall see how positive answers have been given to both these questions. From here we shall discuss similar questions related to other, albeit related, topological descriptors.

02.02.2016

  • 14:15 - 15:15 (@TUM) Polyhedral Patterns, Helmut Pottmann (TU Wien, John von Neumann Gastprofessor @ TUM)
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  • 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 ap- proximations of smooth surfaces by polyhedral surfaces and present some initial results in this direction.
  • 15:30 - 16:30 (@TUM) The flux integral revisited: the Lagrangian perspective, Daniel Karrasch (TU München)
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  • Advective transport of scalar quantities through surfaces is of fundamental importance in many scientific applications. From the Eulerian perspective of the surface it can be quantified by the well-known flux integral. The recent development of highly accurate semi-Lagrangian methods for solving scalar conservation laws and of Lagrangian approaches to coherent structures in turbulent (geophysical) fluid flows necessitate a new approach to transport, i.e. accumulated (over time) flux, from the (Lagrangian) material perspective. In my talk, I present a Lagrangian framework for calculating transport of conserved quantities through a given (hyper-)surface in n-dimensional, fully aperiodic, volume-preserving flows.

08.12.2015

  • 14:15 - 15:15 (@TUB) On limit shapes and their integrability, Nicolai Reshetikhin (UC Berkeley)
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  • The talk will start with the introduction into the limit shape phenomenon which is the emergence of a deterministic structure from a random one for large systems. Then the focus will be on the 6-vertex model and on dimer models and it will be argued that corresponding limit shapes are given by integrable PDE's.
  • 15:30 - 16:30 (@TUB) Surfaces containing two circles through each point, Mikhail Skopenkov (NRU HSE, IITP RAS)
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  • Motivated by potential applications in architecture, we find all analytic surfaces in space $\mathbb{R}^3$ such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface. The problem of finding such surfaces traces back to the works of Darboux from XIXth century. The proof uses a new factorization technique for quaternionic polynomials. A substantial part of the talk is elementary and is accessible even for high school students. Several open problems are stated.

10.11.2015

  • 14:15 - 15:15 (TU Berlin) Geometric Yang-Baxter maps in non-commuting variables, and their semi-commutative reductions,  Adam Doliwa (University of Warmia and Mazury)
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  • We discuss a family of solutions of the functional Yang-Baxter equation obtained from periodic reductions of the non-commutative Hirota (discrete KP) system. Such maps involve 2N totally non-commuting variables, and in the commutative case reduce to maps discussed by Etingof and by Noumi and Yamada. We study then the maps upon additional assumption of commutativity of certain products of the variables. In the simplest case we obtain this way non-commutative Hirota (discrete sin-Gordon or the lattice mKdV) equation, and the corresponding Yang-Baxter map. Finally, we show how the relaxed periodicity condition gives the non-isospectral extensions of the equations. In particular, we present how self-similarity reduction of the non-isospectral lattice mKdV equation leads to a non-commutative q-P-VI equation.
  • 15:30 - 16:30 (TU Berlin) The Ribbonlength of Knot Diagrams, John Sullivan (TU Berlin)
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  • The ropelength problem asks to minimize the length of a knotted space curve such that a unit tube around the curve remains embedded. A two-dimenaional analog has a much more combinatorial flavor: we require a unit-width ribbon around a knot diagram to be immersed with consistent crossing information. Attempting to characterize critical points for ribbonlength leads us to new results about the medial axis of an immersed disk in the plane, including a certain topological stability for thin disks. This is joint work with Elizabeth Denne and Nancy Wrinkle.

22.09.2015

  • 14:15 - 15:15 (@TUM, live broadcast to TUB) Domain Filling Circle Packings - Existence and Uniqueness, David Krieg (TU Freiberg)
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  • Circle packings are discrete models of analytic functions which mimic and approximate their classical counterparts. For example, William Thurston conjectured and Rodin and Sullivan proved that discrete conformal mappings modelled by circle packings converge to the classical mappings if the packings are appropriately refined. In the standard approach to discrete conformal mapping the domain packing is constructed by "cookie cutting", which results in poor approximation of the domain near its boundary. To overcome this deficiency, different algorithms have been proposed which allow one to construct domain filling packings, with boundary circles touching the boundary of the given domain. Though these algorithms work well in practice, some questions concerning existence and uniqueness of domain filling packings remained open. Oded Schramm proved very general results for domain filling (not only circle) packings, but in particular his uniqueness statements require assumptions on the boundary (being ``decent''), which seem not to be completely natural. For more general domains Schramm predicted the appearance of degenerate circles with radius zero; but he gives no explicit criteria when this may happen. In the talk I present existence and uniqueness statements for circle packings filling arbitrary bounded, simply connected domains. As normalization three boundary points (prime ends if the domain is not Jordan) are associated with three boundary circles, which have to be touched in a generalized sense. This leads to a discrete version of Carathéodory's Theorem, which provides the existence and uniqueness of discrete conformal mappings under weak assumptions. Afterwards a generalization of circle packings is introduced (circle agglomerations), degeneracy is taken into account, and other normalizations are discussed."
  • 15:30 - 16:15 (@TUM, live broadcast to TUB) Molecular structures generated by discrete symmetries, X-ray diffraction, and structure identification., Gero Friesecke (TU München)
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  • The scattering of incoming plane waves at crystals, i.e., periodic structures, results in discrete structure- identifying diffraction patterns. Much of what is known from experiment about the atomic-scale structure of matter is found using this method. Mathematically, the remarkable diffraction patterns have been explained by the multi-dimensional Poisson summation formula. But why - really - does all this work, i.e. how do you get from some continuous object with discrete symmetry (the electron density of a crystal, a continuous periodic function on $\R^3$) to a discrete object on some other space (the X-ray diffraction pattern) and back? A closer examination (joint work with Dominik Juestel, TUM, and Richard James, University of Minnesota) reveals that, in fact, the hypothesis of periodicity is not fundamentally what is being used, but rather a group structure. The key point is that the structure is invariant under a DISCRETE subgroup of the Euclidean group of rotations and translations, and the incoming radiation is equivariant under a related CONTINUOUS subgroup. This suggests the possiblity of novel X-ray methods, via the replacement of the pair (plane waves, crystals) by (other solutions to Maxwell's equations, 'objective structures'). The details have been worked out for helical structures and lead, surprisingly, to the recently discovered beams with orbital angular momentum (OAM). Objective structures are defined mathematically as orbits of a finite number of points under a discrete Euclidean group. Many important structures in biology and nanoscience are of this form, including buckyballs and many fullerenes, the parts of many viruses, actin, carbon nanotubes (all chiralities). The talk will informally present a mathematical picture of all the objects involved, not assuming any background in the underlying physics.

14.07.2015

  • 14:15 - 15:15 (@TUM) Interactive Visualization of 2-D Persistent Homology, Michael Lesnick (IMA, University of Minneapolis & Columbia University, NY)
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  • In topological data analysis, we study data by associating to the data a filtered topological space, whose structure we can then examine using persistent homology. However, in many settings, such as in the study of point cloud data with noise or non-uniformities in density, a single filtered space is not a rich enough invariant to encode the interesting structure of our data. This motivates the study of multidimensional persistence, which associates to the data a topological space simultaneously equipped with two or more filtrations. The homological invariants of these “multifiltered spaces,” while much richer than their 1-D counterparts, are also far more complicated. As such, adapting the usual 1-D persistent homology methodology for data analysis to the multi-D setting requires new ideas. In this talk, I’ll introduce multi-D persistent homology and discuss joint work with Matthew Wright on the development of a practical tool for the interactive visualization of 2-D persistence.
  • 15:30 - 16:30 (@TUM) Cheeger constants and frustration indices for magnetic Laplacians on graphs and manifolds, Norbert Peyerimhoff (University of Durham, UK)
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  • Due to Mark Kac' famous question "Can one hear the shape of a drum?" in 1966, the study of spectra of Laplacians became a very fashionable research area with many beautiful results. A fundamental fact in this area is Cheeger's inequality, which relates an isoperimetric constant to the first non-trivial eigenvalue of the Laplace operator. Remarkably, this result is not only fundamental in the setting of Riemannian manifolds: its combinatorial counterpart is of central importance for the development of efficient and well connected networks or, more theoretically formulated, in the construction of expander graph families. It is natural to try to extend spectral results from classical Laplacians to more general operators like magnetic Laplacians. The magnetic potential leads to a specific first order modification of the classical Laplacian and requires to change from a real to a complex-valued Hilbert space. However, self-adjointness guarantees that the spectrum of a magnetic Laplacian is still real. In this talk we introduce these operators on graphs and on Riemannian manifolds, and we will discuss in detail how the classical Cheeger inequality needs to be modified. The classical isoperimetric constant has to be replaced by a new invariant, balancing between isoperimetric properties of subsets and an contribution of the additional magnetic potential. The contribution of the magnetic potential is measured by the so-called frustration index. If time permits, I will also explain how our results can be applied for the construction of a particular spectral clustering algorithm for partially oriented graphs. This is joint work with Carsten Lange (TU Munich), Shiping Liu (Durham) and Olaf Post (Uni Trier).

16.06.2015

  • 14:15 - 15:15 (TU Berlin) Algebraic entropy, a measure of complexity for rational discrete systems, Claude Viallet (Université Pierre et Marie Curie, Paris)
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  • We will present the definition of algebraic entropy, a global index of complexity of birational discrete dynamical systems. Initially constructed to detect integrability it has become an object of interest by itself. We will describe ways to calculate it, explain its relation to the singularity structure of the systems, and motivate a fundamental conjecture on the values it can assume (log of an algebraic integer).
  • 15:30 - 16:30 (TU Berlin) Tiling of triply-periodic minimal surfaces, Myfanwy Evans (TU Berlin)
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  • High symmetry dense packings of trees and lines in the two-dimensional hyperbolic plane can be projected to triply-periodic minimal surfaces. The resulting three-dimensional structures are space-filling, symmetric and entangled structures composed of multiple networks or filaments. In this talk, I will discuss the construction and characterisation of these complex entangled structures alongside applications from star terpolymer self-assembly to skin swelling.

19.05.2015

  • 14:15 - 15:15 (@ TUM) Confined elastic curves, Matteo Novaga (Universitá di Pisa, John von Neumann Gastprofessor)
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  • I will consider curves confined in a prescribed container, minimizing their bending or elastic energy. In particular, I will discuss the existence and regularity of the optimal curves, and describe their shape in some particular cases.
  • 15:30 - 16:30 (@TUM) Dynamics of Discrete Screw Dislocations in SC, BCC, FCC and HCP crystals, Lucia de Luca (TU München)
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  • Dislocations are line defects in the periodic structure of crystals and their motion represents the main mechanism of plastic deformation in metals. We discuss a variational approach for modeling the dynamics of screw dislocations in some specific crystal structures, i.e., Simple Cubic (SC), Body Centered Cubic (BCC), Face Centered Cubic (FCC), and Hexagonal Close Packed (HCP). More precisely, using a discrete (in space and in time) variational scheme, we study the motion of a configuration of dislocations toward low energy con gurations. We deduce an effective fully overdamped dynamics that follows the maximal dissipation criterion introduced by Cermelli & Gurtin and predicts motion along the glide directions of the crystal. The results are fruit of a joint work with Roberto Alicandro (University of Cassino), Adriana Garroni and Marcello Ponsiglione ("La Sapienza", University of Rome).

21.04.2015

  • 14:15 - 15:15 (TU Berlin) Navigating the space of symmetric CMC surfaces, Sebastian Heller (Universität Tübingen)
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  • We consider compact surfaces of constant mean curvature (CMC) in 3-dimensional space forms. While the only CMC spheres are round spheres and CMC tori can be explicitly parametrized via integrable systems methods, only very little is known about higher genus CMC surfaces.
    In this talk we first give a brief introduction to the spectral curve approach to CMC tori due to Hitchin. In general the constant mean curvature condition of a surface can be translated into the flatness condition of an associated family of $SL(2,\mathbb{C})$-connections. For tori, the abelian fundamental group allows to reduce flat $SL(2,\mathbb{C})$-connections to flat line bundle connections, and the associated family can be parametrized in terms of certain algebraic geometric objects - the spectral data - from which the conformal immersion can be reconstructed.
    Under the assumption of certain discrete symmetries, irreducible connections on higher genus surfaces can also be parametrized by flat line bundle connections. This enables us to define a generalization of the spectral curve theory for higher genus CMC surfaces. Due to the non-abelian nature, it is hard to construct spectral data for higher genus CMC surfaces explicitly.
    In a recent preprint (joint work with L. Heller and N. Schmitt) we have introduced a flow on the spectral data from CMC tori towards higher genus CMC surfaces. We explain how this flow can be used to construct spectral data for higher genus CMC surfaces and to study the moduli space of symmetric CMC surfaces of higher genus.
  • 15:30 - 16:30 (TU Berlin) Tropical $(p,q)$-homology and algebraic cycles, Kristin Shaw (TU Berlin)
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  • I will define tropical $(p,q)$-homology as introduced by Itenberg, Katzarkov, Mikhalkin and Zharkov (IKMZ). In fact, despite their name these homology groups can be defined for any polyhedral space not necessarily having a tropical structure. However, in the tropical setting these groups are particularly interesting since it follows from (IKMZ) that these homology groups can give Hodge numbers of complex projective varieties.
    The second part of this talk I will describe how one can use these groups to study "tropical algebraic cycles". Here I will focus on two main examples. Firstly tropical hypersurfaces in $\mathbb{R}^3$, which are polyhedral complexes dual to subdivisions of lattice polytopes. Secondly, tropical Abelian varieties, which are real tori equipped with an integral affine structure.

10.02.2015

  • 14:15 - 15:15 (TU Berlin) Discrete variational mechanics in structure-preserving integration and optimal control, Sina Ober-Blöbaum (Uni Paderborn/FU Berlin)
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  • Discrete variational mechanics plays a fundamental role in constructing and analyzing structure-preserving numerical methods for the simulation and optimization of mechanical systems. Based on a discrete variational principle that approximates the continuous one, time-stepping schemes denoted as variational integrators can be derived that are structure preserving, i.e. they are symplectic-momentum conserving and exhibit good energy behavior, meaning that no artificial dissipation is present and the energy error stays bounded over longterm simulations. In recent years, much effort has been put into the analysis and the further development of variational integrators to make them applicable to a broad class of mechanical systems.
    After a brief introduction to variational integrators and an overview of their different fields of applications, we derive two different kinds of high order variational integrators based on different dimensions of the underlying approximation space. While the first well-known integrator is equivalent to a symplectic partitioned Runge-Kutta method, the second integrator, denoted as symplectic Galerkin integrator, yields a method which in general, cannot be written as a standard symplectic Runge-Kutta scheme.
    In the second part of the talk, we use these integrators for the discretization of optimal control problems. For the numerical solution of optimal control problems, direct methods are based on a discretization of the underlying differential equations which serve as equality constraints for the resulting finite dimensional nonlinear optimization problem. By analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that for these particular integrators dualization and discretization commute. This property guarantees that the convergence rates are preserved for the adjoint system which is also referred to as the Covector Mapping Principle.
  • 15:30 - 16:30 (TU Berlin) The Chemical Master Equation as a discretization of the Fokker-Planck and Liouville equation for chemical reactions, Alexander Mielke (WIAS Berlin)
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  • We discuss the reaction kinetics according to a finite set of mass-action type reactions under the additional assumption of detailed balance. This nonlinear ODE system has an entropic gradient structure. The associated Chemical Master Equation is the Markov chain counting the number of particles for each species, where a reaction corresponds to a jump in the discrete state space $\mathbb N_0^I$. This Markov process has again a gradient structure on the set of probability measures.
    Scaling the number of particles by the total volume $V$, we show evolutionary $\Gamma$-convergence for $V\to \infty$ of the discrete Markov process to the continuous Liouville equation which is a gradient flow with respect to a Wasserstein-type metric. We discuss the role of the Fokker-Planck equation as a singularly perturbed Liouville equation with better approximation properties.
    This is joint work with Jan Maas, IST Wien.

27.01.2015

  • 14:15 - 15:15 (@ TUM) What is the semiclassical limit of non-Hermitian time evolution? , Roman Schubert (Bristol University, UK)
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  • It is well known that in the semiclassical limit (or the high frequency limit) solutions to many PDE's in physics and other sciences are driven by a Hamiltonian flow, i.e., by solutions to a system of ODE's. Examples include Maxwell's equations and the Schroedinger equation. In this setting the propagation of waves can be described geometrically as the propagation of Lagrangian submanifolds of a symplectic manifold by the Hamiltonian flow. We will consider the case that the Hamiltonian flow is generated by a complex valued Hamilton function, which corresponds to a system with loss or gain and a Schroedinger equation where the Hamilton operator is non-Hermitian. We derive a new class of ODE's which govern the semiclassical limit, and which combine a Hamiltonian vectorfield, generated by the real part of the Hamilton function, with a gradient vectorfield, generated by the imaginary part. It turns out that the propagation can again be described in terms of symplectic geometry, but this time it includes a complex structure which is generated by the dynamics along the rays. We will give an overview of these new geometric structures emerging from the semiclassical limit. This is joint work with Eva Maria Graefe.
  • 15:30 - 16:30 (@ TUM) Combinatorial Models for Random Matrices with Gaussian Entries, Michael La Croix (MIT Massachusetts)
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  • Random matrix theory studies the statistics of functions defined on matrices with random entries. Typical questions involve understanding how the spectrum of such a matrix depends on the size of the matrix. Universality principles show that it is often sufficient to consider matrices constructed from independent Gaussian entries. For several such models, expectations of symmetric functions of eigenvalues are polynomial in the size of the matrix. With appropriate scaling and choice of basis, these polynomials appear with non-negative integer coefficients, and can be given combinatorial interpretations as generating series for combinatorial maps, equivalence classes of graphs embedded in surfaces, considered up to homeomorphism. The analytic problem of evaluating high-dimensional integrals is thus discretized as a counting problem, which in this case can be studied using algebraic combinatorics, by encoding the same maps in terms of permutation factorizations. The equivalence between the discreet and continuous approaches is a consequence of the duality relating the representation theories of the finite groups and Lie groups acting on the permutation and matrix models, but in some ways the equivalence is more general than either of the models. Parallel theories for matrices with complex entries (combinatorialized by orientable maps) and matrices with real entries (combinatorialized by non-oriented maps) can be combined into a single parametrized theory that continues to make sense even when the parameter is evaluated outside of its natural domain.

13.01.2015

  • 14:15 - 15:15 (TU München) Signed graphs, nested set complexes and spines, Carsten Lange (TU München)
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  • Fulton and MacPherson used nested set structures to describe compactifications of configuration spaces in 1994 and examples for related nested set complexes can be constructed for every finite graph. These examples, called graph associahedra, have a simple combinatorial description, can be realized by convex polyhedra and yield instances of generalized permutahedra. All necessary objects will be introduced in the first part of my presentation. In the second part, I present a generalized construction of nested set complexes for finite graphs with an addditional vertex labeling by + and - which exhibits a larger class of generalized permutahedra. A key ingredient of this construction are spines which are used to describe a generalization of nested sets.
  • 15:30 - 16:30 (TU München) Approximation and Convergence of the Intrinsic Volume, Herbert Edelsbrunner (IST Austria)
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  • We study the computation of intrinsic volumes of a solid body from a sequence of binary images of progressively finer resolution. While the intrinsic volumes of the binary images do not necessarily converge to the correct value, we show that the formula can be rigged to give the correct limit for the first intrinsic volume, which in R^3 relates to the total mean curvature of the boundary of the body. Work with Florian Pausinger.

16.12.2014

  • 14:15 - 15:15 (TU Berlin) A portrait of Arnold Diffusion, Vassili Gelfreich (University of Warwick)
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  • "Arnold diffusion" is a name for the long time instability in nearly integrable Hamiltonian systems. Arnold diffusion requires the Hamiltonian system to have at least 3 degrees of freedom or, in the case of the discrete time, a symplectic map to be in dimension 4 or higher.
    In this talk we informally discuss mechanisms which lead to Arnold diffusion and illustrate the phenomenon with visualisations of the dynamics of a 4d symplectic map.
  • 15:30 - 16:30 (TU Berlin) Hausdorff dimension in transcendental dynamics, Dierk Schleicher (Jacobs University Bremen)
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  • Transcendental dynamics, the iteration of transcendental entire functions from C to C, has become a very active field of research especially in the last decade (even though foundations were laid by Pierre Fatou about one hundred years ago). We describe some of the basic questions and properties, including the interesting dynamics near essential singularities. The dynamics also has some interesting and surprising dynamical properties: for instance, in a most natural way we obtain a decomposition of the complex plane into two disjoint sets E and R such that every point in E is connected to infinity by its own curve in R so that all curves are disjoint from each other and from E — so that one might think that R is much bigger than E. However, R has Hausdorff dimension 1, while E is the complement of R and thus has full planar measure.

18.11.2014

  • 14:15 - 15:15 (TU München) The Morse theory of Čech and Delaunay filtrations, Ulrich Bauer (TU München)
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  • Given a finite set of points in ℝⁿ and a positive radius, we consider the Čech, Delaunay–Čech, Delaunay (alpha shape), and wrap complexes as examples of a generalized discrete Morse theory. We prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, and the same is true for their weighted versions. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data. Joint work with Herbert Edelsbrunner.
  • 15:30 - 16:30 (TU München) From Shapes to Shape Spaces, Max Wardetzky (Universität Göttingen)
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  • Over the past decades, a large amount of research in geometry processing has been dedicated to computational tools for processing single geometries. From a more global perspective, one might ask to study surfaces as points in a space of shapes. Over the past years, concepts from Riemannian manifolds have been applied to design and investigate shape spaces, with applications to shape morphing and modeling, computational anatomy, as well as shape statistics. Studying shape space from the point of view of Riemannian geometry enables transfer of important geometric concepts, such as geodesics or curvatures, from classical geometry to these (usually) infinite-dimensional spaces of shapes. In my talk I will highlight some of the recent developments and open questions in understanding shape space from a Riemannian perspective, with a particular down-to-earth focus on how one can relate Riemannian metrics to elastic energies.

21.10.2014

  • 14:15 - 15:15 (TU Munich) The symplectic camel and the uncertainty principle, Maurice de Gosson (Universität Wien)
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  • Gromov's discovery in 1985 of the symplectic non-squeezing theorem, dubbed "the principle of the symplectic camel" by Arnol'd, can be viewed as a classical version of the uncertainty principle of quantum mechanics. We will show that a derived notion, the symplectic capacity of subsets of phase space, allows a topological reformulation of the uncertainty principle. Using recent results by Artstein-Avidan, R. Karasev, and Y. Ostrover we propose a new notion of indeterminacy. We also discuss the relationship between the notion of symplectic capacity and Hardy's uncertainty principle about the localization of a function and its Fourier transform, which can be reformulated in terms of the symplectic capacity of the covariance ellipsoid of the Wigner transform of that function.
  • 15:30 - 16:30 (TU Berlin) Integer partitions from a geometric viewpoint, Matthias Beck (San Francisco State University)
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  • The study of partitions and compositions (i.e., ordered partitions) of integers goes back centuries and has applications in various areas within and outside of mathematics. Partition analysis is full of beautiful--and sometimes surprising--identities, starting with Euler's classic theorem that the number of partitions of an integer k into odd parts equals the number of partitions of k into distinct parts.
    Motivated by work of G. Andrews et al from the last 1 1/2 decades, we will show how one can shed light on certain classes of partition identities by interpreting partitions as integer points in polyhedra. Our approach yields both "short" proofs of known results and new theorems.
    This is joint work with Ben Braun, Ira Gessel, Nguyen Le, Sunyoung Lee, and Carla Savage.

24.06.2014

  • 14:15 - 15:15 (TU Berlin) Universal aspects of geometric and algebraic integrability in Pluecker and Lie geometry, Wolfgang Schief (University of New South Wales, Australia)
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  • We discuss a novel geometric interpretation of a "universal" system of difference equations in terms of line complexes in a three-dimensional complex projective space. This system of algebraic equations is integrable in the sense of multi-dimensional consistency and admits a variety of canonical reductions. These are associated with particular classes of integrable line complexes such as linear and quadric line complexes. In the former case, one may reinterpret the line complexes in terms of Lie circle geometry. In fact, another reduction of the universal system leads to integrability in Lie sphere geometry and a generalisation to quaternionic line complexes is available which corresponds to Lie 4-sphere geometry. Connections with "master" equations such as the discrete BKP, CKP and Darboux equations are presented.
  • 15:30 - 16:30 (TU Munich) The Geometry and Dynamics of Semiclassical Wave Packets, Tomoki Ohsawa (University of Michigan-Dearborn, USA)
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  • I will talk about the geometry and dynamics of semiclassical wave packets, which provide a description of the transition regime between quantum and classical mechanics. It is well known that both classical and quantum mechanical systems are described as Hamiltonian systems: finite-dimensional one for the former, and infinite-dimensional for the latter with respect to the corresponding symplectic geometric structures. I will show how to exploit such geometric structures to formulate semiclassical dynamics from the Hamiltonian/symplectic point of view. The Hamiltonian/symplectic formulation reveals the role of symmetry, conservation laws, and reduction in semiclassical wave packet dynamics.

27.05.2014

  • 14:15 - 15:15 (TU Munich) Solving problems in algebraic geometry using applied mathematics, Hartmut Monien (Universität Bonn)
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  • Algebraic geometry is concerned with finding the locus of zeros of sets of polynomial equations in a commutative ring. In my talk I will show how some hard questions in that field can in fact be answered by explicitly solving a 2D partial differential equation on a Riemann surface. The monodromy group of the Riemann surface is closely related to the "dessin d'enfant" of Grothendiek which allows to gain insight into the absolute Galois group. We will present some interesting non-congruence subgroups with interesting Galois groups.
  • 15:30 - 16:30 (TU Berlin) Smoke Rings from Smoke, Peter Schröder (Caltech)
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  • We give an algorithm which extracts vortex filaments ("smoke rings") from a given 3D velocity field. Given a filament strength h>0, an optimal number of vortex filaments, together with their extent and placement, is given by the zero set of a complex valued function over the domain. This function is the global minimizer of a quadratic energy based on a Schr{ödinger operator. Computationally this amounts to finding the eigenvector belonging to the smallest eigenvalue of a Laplacian type sparse matrix.

    Turning traditional vector field representations of flows, for example, on a regular grid, into a corresponding set of vortex filaments is useful for visualization, analysis of measured flows, hybrid simulation methods, and sparse representations. To demonstrate our method we give examples from each of these.

    Joint work with Ulrich Pinkall and Steffen Weißmann

    Bio: Peter Schröder is the Shaler Arthur Hanisch Professor of Computer Science and Applied and Computational Mathematics at Caltech where he has been on the faculty for the past 19 years. His research is focused on applications of discrete differential geometry in computer graphics. He got his start as an undergraduate in mathematics and computer science at TU Berlin.

29.04.2014

  • 14:15 - 15:15 (TU Munich) The multivariate Hermite-Laguerre connection, Caroline Lasser (TU München)
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  • 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.
  • 15:30 - 16:30 (TU Munich) Geometric Methods for the Approximation of High-dimensional Dynamical Systems, Mauro Maggioni (Duke University, Durham)
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  • 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.

04.02.2014

  • 14:15 - 15:15 (TU Berlin) The Riemann-Hilbert Method, Alexander Its (Indianapolis)
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  • In its original setting, the Riemann-Hilbert problem is the question of surjectivity of the monodromy map in the theory of Fuchsian systems, and it was included by Hilbert in his famous list as problem number twenty-one. Subsequent developments put the Riemann-Hilbert problem into the context of analytic factorization of matrix- valued functions and brought to the area the methods of singular integral equations (Plemelj, 1908) and holomorphic vector bundles (R\"ohrl. 1957). This resulted eventually in a (negative) solution, due to Bolibruch (1989) of the Riemann-Hilbert problem in its original setting and to a number of deep results (Bolibruch, Kostov) concerning a thorough analysis of relevant solvability conditions.

    Simultaneously, and to a great extent independently of the solution of the Riemann-Hilbert problem {\it per se}, a powerful analytic apparatus - the Riemann-Hilbert method - was developed for solving a vast variety of problems in pure and applied mathematics. A classical example of the use of analytic factorization techniques is the Wiener-Hopf method in linear elasticity, hydrodynamics, and diffraction.

    Another array of problems that have fallen under the Riemann-Hilbert formalism over the last twenty - twenty five years came from modern theory of integrable systems. In this new area, the Riemann-Hilbert approach exploits ideas which go far beyond both the usual Wiener-Hopf scheme and the theory of singular integral equations, and they have their roots in the inverse scattering method of soliton theory and in the theory of isomonodromy deformations. The main beneficiary of this, latest version of the Riemann-Hilbert method, is the global asymptotic analysis of nonlinear systems. Indeed, many long-standing asymptotic problems in the diverse areas of pure and applied math have been recently solved with the help of the Riemann-Hilbert technique.

    In this talk a general overview of the Riemann-Hilbert method will be given. The most recent applications of the Riemann-Hilbert approach to asymptotic problems arising in the theory of matrix models, orthogonal polynomials, and statistical mechanics will be outlined. The talk is based on the works of many authors spanned over a number of years.
  • 15:30 - 16:30 (TU Berlin) Lattice Polygons and Real Roots, Michael Joswig (TU Berlin)
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  • It is known from theorems of Bernstein, Kushnirenko and Khovanskii from the 1970s that the number of complex solutions of a system of multivariate polynomial equations can be expressed in terms of subdivisions of the Newton polytopes of the polynomials. For very special systems of polynomials Soprunova and Sottile (2006) found an analogue for the number of real solutions. In joint work with Günter M. Ziegler we could give a simple combinatorial formula and an elementary proof for the signature of foldable triangulation of a lattice polygon. Via the Soprunova-Sottile result this translates into lower bounds for the number of real roots of certain bivariate polynomial systems.

07.01.2014

  • 14:15 - 15:15 (TU Munich) Variational methods for lattice systems, Marco Cicalese (TU München)
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  • 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.
  • 15:30 - 16:30 (TU Munich) Mean Field Sparse Optimal Control, Massimo Fornasier (TU München)
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  • 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.

10.12.2013

  • 14:15 - 15:15 (TU Berlin) The Euclidean Distance Degree, Bernd Sturmfels (UC Berkeley and MPI Bonn)
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  • 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.
  • 15:30 - 16:30 (TU Munich) Cell Packing Structures, Johannes Wallner (University Graz)
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  • 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.

12.11.2013

  • 14:15 - 15:15 (TU Munich) On a surface theory for quadrilateral nets, Tim Hoffmann (TU München)
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  • I 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 and Max Wardetzky.

15.10.2013

  • 14:15 - 15:15 (TU Berlin) Bianchi (hyper-)cubes and a geometric unification of the Hirota and Miwa equations, Wolfgang Schief (University of New South Wales, Australia)
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  • We present a geometric and algebraic way of unifying two discrete master equations of integrable 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 solutions 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.
  • 15:30 - 16:30 (TU Berlin) The critical temperature for the Ising model on biperiodic graphs, David Cimasoni (University of Geneva)
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  • 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 introduction 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,…)

    This is joint work with Hugo Duminil-Copin.

02.07.2013

  • 16:15 - 17:15 (TU Munich) Rigidity of origami surfaces, Ileana Streinu (Smith College, Northampton, USA)
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  • 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.
  • 17:30 - 18:30 (TU Munich) Rigidity and flexibility of periodic frameworks”, Ciprian S. Borcea (Rider University, Northampton, USA)
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  • 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.

04.06.2013

  • 16:15 - 17:15 (TU Berlin) Minimum-cardinality triangulations of polytopes and manifolds, Julian Pfeifle (Universitat Politècnica de Catalunya)
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  • 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.

  • 17:30 - 18:30 (TU Berlin) Variational Time Discretization of Geodesic Calculus in Shape Space, Martin Rumpf (Universität Bonn)
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  • 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.

07.05.2013

  • 16:15 - 17:15 (TU Munich) Linearizing Hilbert Nullstellensatz and the Orientability of Matroids, Jesus de Loera (Univ. of California, Davis & John Neumann Gastprofessor, TU Munich)
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  • 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.

  • 17:30 - 18:30 (TU Munich) Complex matroids: rigidity and syzygies, Jürgen Richter-Gebert (TU Munich)
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  • 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. 

16.04.2013

  • 16:15 - 17:15 (TU Berlin) Discrete and continuous integrable systems on cluster varieties, Vladimir Fock (Univ. Strasbourg)
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  • A.B. Goncharov and R. Kenyon defined a class of integrable systems on cluster varieties enumerated by convex polygons on a plane with integral vertices. Every such system has a family of commuting continuous flows enumerated by integral points inside the polygons and discrete flows enumerated by integral points on the boundary and represented by algebraic transformations. We will present our study of their construction using elementary diagramatic technique introduced by Dylan Thurston. We will also show the relation of this construction with affine Lie groups and show that the integrable systems coincide with well known ones on such groups (for example the relativistic Toda chain). However, we will try to show that construction of Goncharov and Kenyon gives a rather new point of view on integrable systems simplifying the constructions and admitting several generalisations.
  • 17:30 - 18:30 (TU Berlin) Variational formulation of commuting Hamiltonian flows, Yuri B. Suris (TU Berlin)
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  • We give a complete solution of the following problem: one looks for a function of several variables (interpreted as multi-time) which delivers critical points to the action functionals obtained by integrating a Lagrangian 1-form along any smooth curve in the multi-time. We derive the corresponding multi-time Euler-Lagrange equations and show that, under the multi-time Legendre transformation, they are equivalent to a system of commuting Hamiltonian flows. Involutivity of the Hamilton functions turns out to be equivalent to closeness of the Lagrangian 1-form on solutions of the multi-time Euler-Lagrange equations. In the discrete time context, the analogous extremal property turns out to be characteristic for systems of commuting symplectic maps.