February 10 - 14, 2014
Consider the restricted Cech complex for evenly-spaced points around a circle. The restricted Cech complex is built not from balls in the plane but instead from balls in the circle, i.e. circular arcs. Since the intersection of two such circular arcs need not be contractible, the nerve lemma need not apply. The following two families of homotopy types are obtainable as the restricted Cech complex for evenly-spaced points on a circle. First, for any two nonnegative integers t and n, one can obtain a t-fold wedge sum of copies of the 2n-dimensional sphere. Second, for any n, one can obtain a single copy of the (2n+1)-dimensional sphere. These homotopy types are closely related to the Vietoris-Rips complexes for evenly-spaced points on a circle, which are studied by Michal Adamaszek in "Clique complexes and graph powers." This is joint work with Christopher Peterson and Corrine Previte.
Keywords of the presentation: Integral inequalities; extended Chebyshev functional, Pathway Fractional operators; Curiel and Galu fractional integral operator
Abstract. A remarkably large number of inequalities involving the fractional inte-
gral operators have been investigated in the literature by many authors. Very recently,
Dumitru et al. [Chinese J. Math. (2013)], gave certain interesting fractional integral
inequalities involving the Gauss hypergeometric functions. Using the same technique,
in this paper, we present some(presumably) new fractional integral inequalities involv-
ing Pathway type fractional operators, whose special cases are shown to yield corre-
sponding inequalities associated with Saigo, Erdelyi-Kober and Riemann-Liouville type
fractional integral operators. Relevant connections of the results presented here with
those earlier ones are also pointed out.
Keywords of the presentation: Curve Shortening, Self Similar solutions, Morse Decomposition
Curves that evolve under Curve Shortening by a combination of rescaling and Euclidean motions are solutions to a system of ordinary differential equations on the unit tangent bundle of Rn.
The flow, which has no fixed points, does admit an interesting Morse decomposition, especially after compactifying the phase space.
In this talk I will present the many different solutions to Curve Shortening that arise in this way.
We present some applications of persistent homology to the physics of granular media, which are materials made of inert particles that interact only by dissipative contacts. More concretely we study tapped granular media in a 2D vertical container. The graph of contacts between particles has been used as a mean to study properties of these materials in numerical simulations. However in experimental settings it is difficult to obtain the exact graph, since the contacts are not precisely defined. Starting with noisy data (the position of the particles) we construct a parametrized Vietoris-Rips complex for an appropriate range of the filtration parameter. The corresponding first Betti numbers are then used to characterize different physical states that previous approaches could not distinguish.
Dense granular materials provide a rich setting to explore networks and topology. Of particular interest here are states near the jamming transition. The idea of jamming is simple: if a collection of grains is too loose, then it will not be mechanically stable. If it is very dense, there will be more than enough force-bearing contacts to make the system stable or jammed. The concept of states that are marginally stable is then a natural one. Until recently, it was thought that density alone controls the jamming transition. We have shown that this is not the case: both the density and stress anisotropy control jamming of frictional grains. This means that characterizing the networks of contacts of force-bearing grains is crucial to understanding jamming. In this talk, I will explore the use of computational homology and network methods to characterize force networks near jamming.
Keywords of the presentation: computational topology, dynamical systems, time-series analysis
Most of the traditional time-series analysis techniques that are used
to study trajectories from nonlinear dynamical systems involve
state-space reconstructions and clever approximations of asymptotic
quantities, all in the context of finite and often noisy data. Few of
these techniques work well in the face of nonstationarity. Embedding
a time series that samples different dynamical systems at different
times, for instance---and then calculating a long-term Lyapunov
exponent---does not make sense.
Computational topology offers some important advantages in situations
like this. Topological descriptions of structure are inherently more
qualitative, and thus more robust, than more-rigid geometrical
characterizations. Even when the data contain finite amounts of noise, computational
topology can produce provable results [Day et al. 2008, Mischaikow et
al. 1999], and it is naturally immune to changes of scale and
orientation that skew the data. Faced with a time
series that samples a number of different dynamical systems, one can detect regime changes by looking for shifts in the
topological structure of the reconstructed dynamics: e.g., nearby
points whose immediate future paths are significantly different. This
segmentation strategy works in some situations where others do
not---if the different components of the signal overlap in state
space, for instance. Once a signal has been segmented into
components, one can compute topological "signatures" of those
components---e.g., with witness complexes and Conley index theory.
The notion of persistence can be leveraged to make appropriate choices
for the different free parameters in these algorithms. It may also be
possible to take advantage of persistence in order to handle the
additional challenges that arise when the data arrive in a stream and
must be analyzed 'on the fly.' In this situation, one does not have
the luxury of post facto analysis of the full data set. Rather, one
must detect shifts immediately--and then immediately start building up
a new model from the incoming stream.
This is joint work with Jim Meiss, Vanessa Robins, and Zach Alexander.
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In this poster we explain two numerical algorithms for the
computation of normally hyperbolic invariant tori (NHIT) in families of
discrete dynamical systems. The application of the parameterization method
leads to solving invariance equations for which we use a Newton-like
method adapted to the dynamics and the geometry of the invariant torus and
its invariant bundles.
The first method computes the NHIT and its internal dynamics, which is a
priori unknown.
The second method computes NHIT in which the internal dynamics is a fixed
quasi periodic rotation, by adjusting parameters of the family.
We apply these methods to continue NHIT w.r.t. parameters, and to explore
different mechanisms of breakdown of NHIT.
This is a join work with Alex Haro.
Keywords of the presentation: outer approximation, Conley Index, topological entropy
Outer approximations of continuous, discrete-time systems, or maps, incorporate bounded error and allow for the rigorous extraction of dynamics via Conley Index theory and other tools. Recent advances, including joint work with W. Kalies, M. D. LaMar, R. Trevi'no and R. Frongillo, have extended the class of maps to which these methods may be applied and improved the types of results that one may obtain, specifically in terms of computed lower bounds on topological entropy. I will describe some of this recent work and show sample results for systems ranging from the two-dimensional Henon map to the infinite-dimensional Kot-Schaffer integrodifference operator from ecology.
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This poster details the implementation of a numerical algorithm to compute codimension-one tori in three-dimensional, volume-preserving maps. A torus is defined by its conjugacy to rigid rotation, which is in turn given by its Fourier series. The algorithm employs a quasi-Newton scheme to find the Fourier coefficients of a truncation of the series. It is guaranteed to converge assuming the torus exists, the initial estimate is suitably close, and the map satisfies certain nondegeneracy conditions. We demonstrate that the growth of the largest singular value of the derivative of the conjugacy predicts the threshold for the destruction of the torus. We use these singular values to examine the mechanics of the breakup of the tori, making comparisons to Aubry-Mather and anti-integrability theory when possible.
We present a new approach to nonlinear dynamics based on topological methods that allows for a priori choices in the resolution of the model, both with respect to the variables and the parameters. We apply this approach to switching networks to investigate a transcriptional network driving the cell cycle.
Joint work with B. Cummins, B. Fan, T. Gedeon, S. Harker, A. Goullet and K. Mischaikow, Dept. of Mathematical Sciences, Montana State University and Dept. of Mathematics, Rutgers University.
Keywords of the presentation: Gene regulatory networks, switching models, database for dynamics
Experimental data on gene regulation and protein interaction is often very qualitative, with the only information available about pairwise interactions is the presence of either up-or down- regulation. Since majority of the parameters for any model in such a situation are not constrained by data, it is important to understand how different choices of parameters affect the dynamics and, therefore, the predictions of such a model.
Continuous time Boolean networks, or switching networks, represent an attractive platform for qualitative studies of gene regulation, since the dynamics at fixed parameters is relatively easily to compute. However, it is quite difficult to analytically understand how changes of parameters affect dynamics.
Database for Dynamics is an excellent tool for studying these models, as it rigorously approximates global dynamics over a parameter space. The results obtained by this method provably capture the dynamics a predetermined spatial scale.
We combine these two approaches to present a method to study switching networks over a parameter spaces. We apply our method to experimental data for cell cycle dynamics.
Keywords of the presentation: categorification, homology, euler characteristic
this talk will survey examples of categorification and decategorification in applications to the engineering sciences.
Keywords of the presentation: Arnold diffusion; the three-body problem; shadowing orbits
We present two models for Arnold diffusion in the three-body problem. The first model is the spatial circular restricted three-body problem. We show that, on some fixed energy level, there exist trajectories near one of the libration points whose out-of-plane amplitude of motion changes from nearly zero to nearly the maximum value for that energy level. The second model is the planar elliptic restricted three-body problem, with the eccentricity of the binary regarded as a perturbation parameter. We show that there exist trajectories whose energy changes between two given levels, for all sufficiently small eccentricities. Equivalently, there exist trajectories that start near the Lyapunov orbit of the unperturbed problem for some given energy, and end up near the Lyapunov orbit of the unperturbed problem for some other given energy. In both models we use the existence of a normally hyperbolic invariant manifold, the `inner dynamics' given by the restriction of the flow to this manifold, and the `outer dynamics' given by the heteroclinic connections to this manifold. A key ingredient is a topological shadowing lemma, which allows one to find true orbits near pseudo-orbits given by alternately applying the inner dynamics and the outer dynamics. This approach can be applied in both analytical arguments and rigorous numerical experiments. This is based on joint works with M. Capinski, A. Delshams, R. de la Llave, and P. Roldan.
Evans function analysis has become a standard method of calculating the stability of non-linear waves for PDE's, and building on the machinery of the Evans function, we have proven a related but alternative form of analysis. The Hopf bundle has an embedding in complex space, and locally is represented by the cross product of a circle and complex projective space - the dynamical system associated with the linearized operator for a PDE can thus induce a winding number through parallel transport in the fibre. Our method uses parallel transport to count the multiplicity of eigenvalues contained within a loop in the spectral plane.
When studying a differentiable function, it is difficult to determine the topology of sublevel sets corresponding to points near critical values of the function. Nevertheless, by studying the persistent homology of these sublevel sets, the importance of any one sublevel set is dwarfed by the global behavior of the function. We generalize the algorithm for rigorously computing the homology of two dimensional nodal domains [Day, Kalies, and Wanner 2009] to handle higher dimensional sublevel sets. We then describe how to rigorously compute persistent homology and present computational results.
We model pacemaker effects in a 1 dimensional array of oscillators with nonlocal coupling via an algebraically localized heterogeneity. We assume the oscillators obey simple phase dynamics and that the array is large enough so that it can be approximated by a continuous nonlocal evolution equation. We concentrate on the case of heterogeneities with negative average and show that steady solutions to the nonlocal problem exist. In particular, we show that these heterogeneities act as a wave source, sending out waves in the far field. This effect is not possible in 3 dimensional systems, such as the complex Ginzburg-Landau equation, where the wavenumber of weak sources decays at infinity. To obtain our results we use a series of isomorphisms to relate the nonlocal problem to the viscous eikonal equation. The linearization about the constant solution results in an operator, L, which is not Fredholm in regular Sobolev spaces. We show that when viewed in the setting of Kondratiev spaces the operator, L, is Fredholm. These spaces can be described as Sobolev spaces with algebraic weights that increase in degree with each derivative.
Keywords of the presentation: Maslov Index, Morse Index Theorem, Multi-dimensional domains, dynamical systems
The Maslov Index plays a central role in the geometric view (proof) of the Morse Index Theorem. The theorem is about geodesics which are curves and therefore dependent on a single (real) variable and the Maslov Index is a topological count of the conjugate points. The Maslov Index is extended to the case of elliptic PDEs in multi-dimensional domains and generalized Morse Index Theorems will be discussed. This is joint work with J. Deng, J.Marzuola and G.Cox.
This poster describes an exploratory study of the possibility of using techniques from topological data analysis for studying datasets generated from dynamical systems described by stochastic delay equations. The dataset is generated using Euler-Maryuama simulation for two first order systems with stochastic parameters drawn from a normal distribution. The first system contains additive noise whereas the second one contains parametric or multiplicative noise. Using Taken’s embedding, the dataset is converted into a point cloud in a high-dimensional space. Persistent homology is then employed to analyze the structure of the point cloud in order to study equilibria and periodic solutions of the underlying system. Our results show that the persistent homology successfully differentiates between different types of equilibria. Therefore, we believe this approach will prove useful for automatic data analysis of vibration measurements.
Keywords of the presentation: hierarchical clustering, dynamics in configuration space, dynamics in tree-space, control of robot swarms, active sensing
Hierarchical clustering is a well-known and widely used method of unsupervised data mining and pattern analysis. Less attention has been paid its potential role in specifying and controlling the coordination of swarms of actively controlled particles. Nevertheless, the near ubiquity of hierarchical command structure in human organizations suggests the potential value of formalizing this “relaxed” but “organized “ mode of large group coordination and control. This talk will present a centralized hybrid dynamical controller guaranteed to bring a collection of n distinct, completely actuated particles in Euclidean d-space to a configuration that satisfies an arbitrarily specified binary proximity hierarchy from almost any initial configuration. The insights from that construction suggest approaches to variants that might be implemented in a distributed fashion without loss of theoretical guarantees. The talk will conclude with speculative remarks that return to the more familiar setting of hierarchical clustering for pattern analysis. Specifically, it is intriguing to consider the potential for re-interpreting this control framework as offering a “reactive data mining” approach to online active sensor management, for example, applied to problems of real-time anomaly detection or situational awareness.
Acknowledgements: This research is supported in part by the US Air Force Office of Scientific Research under the CHASE MURI, AFOSR MURI FA9550-10-1-0567. The work described results from a collaborative effort with my co-authors, Omur Arslan, Yuliy Baryshnikov, and Dan Guralnik.
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Keywords of the presentation: time series, basin of attraction, locomotion, Morse decomposition, meteorological data
I show some attempts in my research group for understanding global phase space structures of dynamics from experimental data. I will discuss the following problems as case studies.
One problem is to understand the basin of attraction of the human bipedal locomotion, where we study simple mathematical models of passive bipedal locomotion and locomotion driven by a phase oscillator, and show that there are several common features of the basin of attraction in both models. The analysis is done by numerical time series data from the models, and we explain the common features by using the hyperbolicity of the saddle equilibrium corresponding to the upright standing.
Another problem is to study transition of winter weather patterns from observed meteorological data. Here we are interested in the trend of weather pattern change. We use the idea of Morse decomposition obtained from time series analysis for observed data, as well as for numerical simulation data of General Circulation Model. The results seem to agree with common knowledge among meteorologists, and we discuss the possibility and limitation of the method.
Keywords of the presentation: persistence diagrams
Persistence diagrams are a relatively new topological tool for describing and quantifying complicated patterns in a simple but meaningful way. We will demonstrate this technique on patterns appearing in Rayleigh-Benard convection dense granular media. This procedure allows us to transform experimental or numerical data from experiment or simulation into a point cloud in the space of persistence diagrams. There are a variety of metrics that can be imposed on the space of persistence diagrams. By choosing different metrics one can interrogate the pattern locally or globally, which provides deeper insight into the dynamics of the process of pattern formation. Because the quantification is being done in the space of persistence diagrams this technique allows us to compare directly numerical simulations with experimental data.
Understanding connecting orbits between equilibria and periodic orbits of differential equations is a fundamental step toward understanding the global dynamics of differential equations. Connecting orbits organize transport between different regions of phase space, force the existence of complicated dynamics, and form the basic objects of classical tools in nonlinear analysis such as Morse/Floer homology. By first studying local stable and unstable manifolds it is possible to express connecting orbits as solutions of certain finite time boundary value problems. One theme in my research is the development of mathematically rigorous computational tools for solving these boundary value problems.
Keywords of the presentation: algorithm, homology, fundamental group, CW complex, Discrete Morse Theory
Efficient algorithmic computation of homology groups and the fundamental
group of a subset of Euclidean space is among the basic problems
in present-day Applied Topology. Particular interest is in algorithms
taking finite CW complexes on input. At the end of 20th century
Robin Forman proposed a version of the classical
Morse theory for a CW complex embedded with a combinatorial counterpart of
a vector field. In the talk we will demonstrate how this theory may be
fruitfully used in the construction of homology algorithms.
We will also show a recent extension of this approach to the algorithmic
computation of a presentation of the fundamental group.
This is a joint work with B. Fiedler (Berlin), A. Mochizuki & G. Kurosawa (RIKEN), H. Kokubu (Kyoto)
The outline is as follows:
(1) I will introduce the notion determining node, feedback vertex set (FVS) for Regulatory network, which is given by
B. Fiedler, A. Mochizuki, G. Kurosawa, D. Saito, J. Dynam. Diff Eqns, 25 (2013)
(2) We will generalize these idea to dynamical time series analysis for Morse decomposition
(3) Example: Mirsky’s model of mammalian circadian rhythm
Analysis of point cloud data using simplicial complexes is a relatively new concept.
This has been developed and implemented mainly by the Stanford Computational topology group.
We analyzed the synthesized PCD of a hollow torus using the three simplicial complexes,
namely Vietoris-Rips, Witness and Lazy witness complexes with the help of the open source
software javaplex. We compared the three complexes for their computational time complexity
and their efficiency for identifying the real feature. According to our analysis the witness
complex is more efficient both in terms of computation time and feature identification.
Dynamical systems with a coupled cell network structure arise in applications that range from statistical mechanics and electrical circuits to neural networks, systems biology, power grids and the world wide web.
A network structure can have a strong impact on the behaviour of a dynamical system. For example, it has been observed that networks can robustly exhibit (partial) synchronisation, multiple eigenvalues and degenerate bifurcations. In this talk I will explain how semigroups and their representations can be used to understand and predict these phenomena. As an application of our theory, I will discuss how a simple feed-forward motif can act as an amplifier.
This is joint work with Jan Sanders.
Keywords of the presentation: discrete Morse theory, persistent homology, experimental data
Our work with x-ray micro-CT images of complex porous materials has required the development of topologically valid and efficient algorithms for studying and quantifying their intricate structure. As an example, simulations of two-phase fluid displacements in a porous rock depend on network models that accurately reflect the connectivity and geometry of the pore space. These network models are usually derived from curve skeletons and watershed basins. Existing algorithms compute these separately and may give inconsistent results. We have recently shown that Forman's discrete Morse theory provides a unifying framework for producing both topologically faithful skeletons and compatible pore space partitions. Although this work is grounded in image analysis, the theory and algorithms may be extended to simplicial and more general complexes, and our formulation of discrete unstable and stable sets may also be relevant to general combinatorial vector fields of interest in dynamical systems.
This talk is based on joint work with Olaf Delgado-Friedrichs and Adrian Sheppard.
There is perhaps no simpler mathematical object that unifies forces of the nature better than the globo-toroid. The dynamics of this object can be modeled by, also, a very simple 3-dimensional ODE, which in an abstract sense may be viewed as the nature’s dynamo that packages and releases energy in our universe. Its signatures are found in the natural processes of creation, life and destruction. In this poster presentation the aim is to introduce the globo-toroid concept, and to lay the ground for future work. In doing so the simulated big data is used to show how the well-known mathematical notions of the periodic attractors, Poincaré sections, Lypunov exponents, Riemann sphere, slow and fast manifolds, as well as physics concept of the wormhole, are used to explain and illustrate the extraordinary behavior of the globo-toroid dynamics.
Keywords of the presentation: Computational homology, fluid dynamics, turbulence, convection
Mathematical tools based on algebraic topology (homology) provide new ways to
describe complex patterns of fluid flow observed in laboratory experiments.
First, we will discuss how homology can be used to quantify
non-Oberbeck-Boussinesq (NOB) effects in weakly turbulent Rayleigh-Benard convection patterns.
We then will describe homology-based methods to measure dynamical
finite-size effects in spatiotemporally-chaotic convective flows. Finally, we will
outline work in progress in which topological characteristics of fluid flows from both lab experiments and
corresponding numerical simulations are encoded in persistence diagrams.
Keywords of the presentation: vector field topology, computational geometry
Vector fields naturally arise in numerous areas of science and engineering,
such as fluid dynamics, aerodynamics, electromagnetism, computer vision and
biology. A popular approach to vector field analysis is vector field topology,
whose goal has traditionally been to describe vector field data in terms of
features such as stationary points, periodic orbits and separatrices.
Designing robust algorithms for finding such features and estimating their
importance is one of the important fundamental problems in scientific
visualization. Classical approaches to this problem are built upon the foundation
provided by classical dynamical systems theory. Roughly speaking, numerical
methods are used to approximate trajectories of the vector field and features
are a result of analysis of the structure of these approximations.
While these approaches have been used to obtain beautiful visualizations,
they have been hindered by high complexity of the output, sensitivity to
numerical error, high computational cost and consistency issues.
This talk will provide an overview of an alternative, purely computational
geometric framework based on discontinuous (piecewise constant) vector fields.
Our approach guarantees consistency of the output, can be used to estimate
stability of features with respect to perturbation of the input and supports
multi-scale analysis of vector field topology.
Keywords of the presentation: topology, fluid dynamics, mapping class groups, pseudo-Anosov maps
Topological chaos is a type of chaotic behavior that is forced by the motion of obstacles in some domain. I will review two approaches to topological chaos, with applications in particular to stirring and mixing in fluid dynamics. The first approach involves constructing devices where the fluid motion is topologically complex, usually by imposing a specific motion of stirring rods. I will then discuss optimization strategies that can be implemented. The second approach is diagnostic, where flow characteristics are deduced from observations of periodic or random orbits and their topological properties. Many tools and concepts from topological surface dynamics have direct applications: mapping class groups, braids, the Thurston-Nielsen classification theorem, topological entropy, coordinates for equivalence classes of loops, and the Bestvina-Handel algorithm for train tracks.
Keywords of the presentation: braids, Morse-Conley-Floer homology, computer assistence
Pieces of string or curves in three dimensional space may be knotted or
braided. This topological tool can be used to study certain types of nonlinear
differential equations. In particular, such an approach leads to forcing
theorems in the spirit of the famous "period three implies chaos" for interval
maps. We have identified three types of differential equations that respect the braid structure, and for these we can construct Morse-Conley-Floer type (homological) invariants. I will also illustrate how the topological arguments can be combined with a computer-assisted approach.
We present our study on the topology of the space of the coverings. The configuration space Cov(n,r) of coverings is defined as the space of collections of n-element subsets of a metric space X forming an r-net in X. Here we consider the covering spaces over 2D grid domains, or some metric trees. We focus on the excess one covering (meaning that one needs at least n-1 metric r-balls to cover the whole space), and determine the homotopy type of the space. As an application of our study, the feedback control algorithms for repeated coverage are considered.
Keywords of the presentation: Homology, nodal domain, pattern characterization, non-uniform rectangular grid
Homology has long been accepted as an important computational tool for
quantifying complex structures. In many applications these structures arise as
nodal domains or excursion sets of real-valued functions, and are therefore
amenable to a numerical study based on suitable discretizations. Such an
approach immediately raises the question of how accurately the resulting
homology can be computed. In this talk we present a probabilistic algorithm for
correctly determining the topology of two-dimensional excursion sets. The
approach relies on constructing an appropriate cubical approximation for the
nodal domains based on the behavior of the defining function at the vertices of
an adaptively generated grid. The algorithm is probabilistic in nature in order
to alleviate grid alignment issues, and the homology of the resulting nonuniform
cubical complex is determined using coreductions. We illustrate this approach
with applications to time-dependent patterns generated by models for phase
separation and to Conley index theory.
The intrinsic volumes generalize both Euler characteristic and volume, quantifying the “size” of a set in various ways. Lifting the intrinsic volumes from sets to functions over sets, we obtain the Hadwiger Integrals, a family of integrals that generalize both the Euler integral and the Lebesgue integral. The classic Hadwiger Theorem says that the intrinsic volumes form a basis for the space of all valuations on sets. An analogous result holds for valuations on functions: with certain assumptions, any valuation on functions can be expressed in terms of Hadwiger integrals. These integrals provide various notions of the size of a function, which are potentially useful for analyzing data arising from sensor networks, cell dynamics, image processing, and other areas. This poster provides an overview of the intrinsic volumes, Hadwiger integrals, and possible applications.
Tensor fields appear in a wide range of science, engineering, and medical applications. The topology of a tensor field is complicated and intriguing. We show recent advances in understanding the topology of 3D symmetric tensor fields, with applications in scientific visualization and computer graphics.