TABLE OF CONTENTS

In the first part, we provide a tool for detecting material coherence from a set of spatially sparse particle trajectories via the study of a homology-induced map by the braid corresponding to particle motion. We apply our tools to the analysis of high-dimensional geospatial sensor data and provide a metric for quantifying climate anomalies.

PREFACE

Carlsson's success in applying topological methods to studying microarray data inspired my collaboration with Samuel Volchenboum and Stephen Skapek at the University of Chicago, where we applied topological methods to analyze gene expression profiles of rhabdomyosarcoma tumor samples to predict how a patient might respond on current treatment protocols. and identify opportunities for future research into therapies that are more effective and less toxic. John Harer (Duke University), Konstantin Mischaikow (Rutgers University), Amit Patel (Institute for Advanced Study), Jean-Luc Thiffeault (University of Wisconsin-Madison), and many others at the Institute for Mathematics and its Applications at the University of Minnesota; and David Cohen-Steiner at Inria Sophia Antipolis.

INTRODUCTION

In the second part, we formalize the local-to-global structure captured by topology in the setting of point clouds. Finally, we introduce the hierarchical complex, which facilitates the study of topological features captured by mapping constructs over a range of resolutions.

BRAIDS AND MATERIAL COHERENCE

Introduction

Fried [34] and Kolev [35] showed that the topological entropy of a braid, which is related to the Lyapunov flow exponent, is bounded below by the logarithm of the spectral radius of the Burau matrix of the braid. Thiffeault shows experimentally that the magnitude of the braiding exponent is proportional to the Lyapunov exponent for a pulsating eddy current.

Contributions

The accuracy is limited by the length of the historical trajectories, and the analysis requires the identification of a projection line on which the different trajectories do not coincide. This braid is equivalent to a product of tubular braids with a trivial inner braid, followed by a product of inner braids with a trivial braid between the tubular braids.

Figure 2.1: Illustration of a tubular braid with non-trivial braiding within a tube. This braid is equivalent to a product of tubular braids with trivial inte-rior braiding, followed by a product of inteinte-rior braids with trivial braiding between tubul

Braid groups

• Braided strands
• Mapping class groups
• Automorphisms of a free group

The essential reduction system for the mapping class f ∈Mod(S) is sometimes called the acanonical reduction system defined by Farb and Margalit [40]. The essential reduction system ERS(β) is nonempty if and only if β is reducible and nonperiodic [48].

Figure 2.2: Illustration of σ i , the braid consisting of a single crossing given by passing the ith strand behind x i the ( i + 1 ) st strand x i + 1

Application to the analysis of flows

• Numerical implementation

In our experience, the infidelity of the Burau representation has not been an obstacle to deriving coherent sets from physical systems. The entries of the Burau representation matrix can also be interpreted as algebraic intersection numbers [50, 55]. Then there exists δ > 0 such that for all t satisfying |t−1| < δ, we have that the Burau matrix B(α)(t) has an eigenvector that is partially constant in (Dn,C).

Informally, since the Burau matrix B(γ)(1) is a permutation matrix, then the Burau matrix B(γ)(t) for t ≈ 1 is approximately a permutation matrix. In the next section, we calculate the Burau matrix for two different dynamical systems and visualize corresponding eigenvectors whose level sets correspond to components of the Nielsen-Thurston decomposition.

Figure 2.3: Illustration of µ j in D e n .

Examples

• Blinking vortex flow
• Modified Du ffi ng oscillator

Once the algebraic convolution is computed, we map each generator σi of the algebraic convolution to its corresponding Burau matrix, given in equation (2.2), and we instantiate the Burau matrices with a fixed real value such that t≈1. Corollary 2.27 can be used to choose the value of t.) The product QB(σb`)(t≈1), whereβ=Q`σb`, is the Burau matrix corresponding to the concatenation of trajectories. The parameter µ is the flow strength, and its value controls the behavior of the system. As µ increases, the size of the chaotic regions grows, destroying limiting KAM surfaces as the chaotic regions merge.

Trajectories of the modified Duffing oscillator belong to one of three types, each illustrated in Figure 2.12 with its corresponding colors. Allshouse and Thiffeault [36] argue that the ability of the braid-theoretic approach to detect coherent sets, even in an incompressible flow, is evidence of the wide applicability of the method.

Figure 2.8: Illustrations of Poincar´e sections for Aref’s blinking vortex flow;

Discussion and future directions

• A faithful representation of the braid group
• Parallelism

We note that since the modified Duffing oscillator is a compressible system, particle positions may in fact coincide. To form a well-defined braid, we choose a sufficiently sparse sample of the domain and a time window such that the dynamics are reasonably well captured, but no two particles coincide at any time. The dots (numbered from left to right) are the initial conditions for the trajectories further studied by Allshouse and Thiffeault as representative trajectories for the two types of initial conditions.

This effectively splits the sequence of matrix multiplications into subsequences each consisting of (mostly) sparse matrix multiplications, thereby reducing the computational requirements of our analysis.

Figure 2.14: Allshouse and Thiffeault detect two types of initial conditions for the modified Duffing oscillator [36]

TOPOLOGICAL DATA ANALYSIS

Introduction

We also provide an overview of previous work on Reeb graphs and their higher-dimensional analogues, Reeb spaces, which allow us to summarize the structure of a topological space X with respect to level sets of a continuous function f : X→Rd,d≥ 1. Reeb graphs and Reeb spaces can also be adapted for constructing point clouds using a construct called mapper, introduced by Singh, M´emoli and Carlsson [5]. In their preprint, Carri`ere and Oudot relate 1-dimensional mapping constructions of a topological space X and a continuous real-valued function f : X → R to the corresponding Reeb graph and show that 1-dimensional mapping constructions are stable under disturbances.

Munch and Wang use category theory to provide a more general framework and relate mapping constructions for a topological space X and a continuous multivariate function f : X → Rd, d ≥ 1, to the corresponding Reeb space. They show that categorical Reeb spaces and categorical map constructions converge under an interleaving distance between their categorical representations.

Contributions

To compare mapper constructions over a range of clustering resolutions, we introduce the hierarchical abstract mapper, which is given by a family of abstract mappers. Now each functor ˙CK,δf : Sel(K)up → Set can be pushed into the category SetOpen(Rd) and seen as a functor ˚CK,δf : Open(Rd)→Set, which enables us to define an interpage distance between hierarchical abstract mappers C˚f. Furthermore, we show that hierarchical abstract mappers correspond to dendrograms of single-link hierarchical clustering when we take the filter function f : X → R as a constant function.

In these cases, we show that the interleaving distance between hierarchical abstract mappers over two different sets of XandY is bounded above by the Gromov-Hausdorff distance between the dendrograms over XandY. Finally, we provide an algorithm for hierarchical mapping constructions that allows the study of topological features of mapping constructions over a range of cluster resolutions, enabling the analysis of topological features using statistical methods.

Simplicial complexes

The nerve of A, denoted N(A), is the abstract simplicial complex with vertices consisting of the elements of One and simplicity given by the finite subset n. Persistent homology is motivated by the desire to distinguish noise from features in data and the need to examine data over a range of scales. Note that the persistent homology groupsHi,pj consist of homology classes of Kit that still exist inKj, andHi,ip =Hp(Ki).

The homology classα∈Hp(Ki) is bornKi if it is not in the image of the mapping induced by the inclusion Ki−1 ⊂ Ki. Then the p-th persistent filtration diagram (K, f), denoted dgmp(f), consists of points (ai,aj), each with multiplicity µi,pj.

Reeb graphs and Reeb spaces .1 Reeb graphs.1Reeb graphs

20] show that the intersection distance in the space of Reeb graphs is an extended pseudometric. Munch and Wang define Reeb spaces using categorical means and show that categorical representations of Reeb spaces and categorical representations of map constructions converge on the intersection distance [19]. An ε-intersection between two abstract Reeb spaces F,G : Open(Rd) → Set is a pair of map families.

Munch and Wang [19] showed that the interlacing distance on the space of abstract Reeb spaces is an extended pseudometric. The ideas used to prove the stability of Reeb graphs carry over to the setting of Reeb spaces.

Figure 3.1: Reeb graph of height function.

Mapper constructions

• Classical mapper construction for topological spaces
• Classical mapper construction for point clouds

The construction of the mapping was introduced as a simplicial complex given by the neural withdrawal of the open lid with a federal function [5]. Given a continuous mapping f : X→Y and a finite open covering UofY, the construction of the mapping over X is the back-retraction nerve f∗(U), where we write f∗(U) for the covering X given by the collection of path components f −1( U),U∈U. Moreover, the set of clusters given by ∼δ is exactly the set of path components of the Vietoris-Rips complex VR(X,δ) [76], which we use to define the mapping construction for point clouds.

Given a point cloudX, a map f : X→Y, and a finite open envelope UorY, let V denote the set of path components of the Vietoris-Rips complex V(f−1(U),δ), U∈U. A different filter function can lead to a simple complex with a different shape, highlighting different features of the data.

Abstract mapper and hierarchical abstract mapper

• Abstract mapper for topological spaces
• Convergence of abstract mappers to abstract Reeb spaces
• Abstract mapper and hierarchical abstract mapper for point clouds The abstract mapper stores the data of classical mapper constructions for
• Correspondence between single-linkage hierarchical clustering and hierarchical mapper constructions

We consider the opposite category, Cell(K)op, whose objects consist of the simplexes of K and whose morphismsτ →σ are given by the plane relationσ≤τ. Instead, the classical mapper construction for point clouds is given by the nerve of the set of path components of a Vietoris-Rips complex (definition 3.24). This allows us to compare mapper constructions resulting from a sampling or approximation of a topological space with the mapper construction, or Reeb space, of the topological space.

As before, we let f :X →Rdbe a mapping into the ad-dimensional parameter space and Ube a bounded open covering of the off image. For any finite metric space (X, dX), we can associate a stable pair (X, θX), where for every r ≥ 0, the partition blocks θX(r) consist of equivalence classes of ∼r.

diagram of inclusions Ω δ f − 1 S

Application to the analysis of geospatial sensor data

• Algorithm for mapper constructions
• Algorithm for hierarchical mapper constructions
• Study of sea surface temperatures using persistent homology In this example we examine sea surface temperatures as an example of a

The hierarchical mapper construction starts much like the mapper construction, but instead of choosing a single clustering threshold, we consider a finite series of clustering thresholdsδ1 <· · ·< δj <· · ·< δm. We let X be the set of data points given by (t,xt), where xt is a grid of global sea surface temperatures for time. We let the filter function f : X → S1 be given by projecting the time series onto a circle (representing the annual cycle).

There exist resolution parameters δ1,δm > 0 such that the mapper construction given by ¯fX∗(U× {δm}) is a circle, and the mapper construction given by ¯fX∗(U× {δ1}) is a spiral. For a fixed δ such that δ1 < δ < δm, the mapper construction given by f¯X∗(U× {δ}) can contain loops (non-trivial 1-dimensional homology classes).

Figure 3.2: (left) Simplices within clustering thresholds. (center) 1-simplices between clustering thresholds

Discussion and future directions

We find that the most significant signals occur during June-July-August (JJA) and December-January-February (DJF).

Figure 3.4: Mapper constructions for global sea surface temperatures. Loops correspond to non-trivial 1-dimensional persistence classes.

NOMENCLATURE

BIBLIOGRAPHY

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