Stochastic and Applied Topology Seminar
at the Technion

Abstract: The persistent homology summarizes the topological features, and it can be done using a distance function. But, the empirical distance function is highly non-robust to noise and outliers. In this talk we present two methods to deal with this problem: one is of Chazal et al. (2014) that suggest using the distance-to-a-measure (DTM) as a distance function instead of the usual one, and the second is of Phillips et al (2014) that suggest using the kernel distance. The both methods are smooth functions that are robust to noise and outliers, and provide useful topological information.

Abstract: In this talk, a random surface will be a surface that is obtained by randomly gluing together a finite number of hyperbolic triangles. Brooks and Makover introduced this model to study the geometry of a typical hyperbolic surface of large genus. The length spectrum (the set of lengths of closed geodesics on the surface) provides a lot of information on the geometry of a hyperbolic surface. I will speak about recent joint work with Christoph Thaele on the bottom part of the length spectrum of these random surfaces. I will not assume any familiarity with hyperbolic geometry.

Abstract: A nearly ubiquitous assumption in modern data analysis is that a given high-dimensional data set concentrates around a lower dimensional space. Recently, a great deal of attention has been focused on how to use point samples from a metric measure space to estimate the topological and geometric invariants of the space, and on applying the resulting algorithms to real data sets. Most techniques for studying the topology of data, in particular persistent homology, proceed by considering families of topological spaces which were in some way thicker than the original set. In this talk, we ask instead what sort of homotopy theory one may construct directly on finite sets of points. We construct such a homotopy theory, and show that it may also be applied to a variety of combinatorial objects, and give several computations for homotopy groups on point clouds, graphs, and simplicial complexes.

Abstract: We establish a Quantitative Central Limit Theorem (in Wasserstein distance) for the Euler-Poincare Characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler-Poincare Characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level u is fully degenerate, i.e., the Euler-Poincare Characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. Our results can be written as an asymptotic second-order Gaussian Kinematic Formula for the excursion sets of Gaussian spherical harmonics.

Abstract: Many results in combinatorics, such as the theorem of Szemeredi on arithmetic progressions and the Erdos-Stone theorem in extremal graph theory, can be phrased as statements about families of independent sets in certain uniform hypergraphs. In recent years, an important trend in extremal and probabilistic combinatorics has been to extend such classical results to the so-called `random setting'. This line of research has recently culminated in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. In the talk, we describe a third approach to that also yields natural `counting' counterparts. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then show how to derive many interesting results as fairly straightforward consequences of this abstract theorem.

Based on joint work with Jozsef Balogh and Robert Morris.

Abstract: This talk has three key parts. In the first part, we discuss spanning acycles in simplicial complexes. A spanning acycle can be considered as a higher dimensional analogue of a spanning tree, a fundamental object in graph theory. We prove that many key properties, including the cycle and cut property, of a minimal spanning tree (MST) also hold true for a minimal spanning acycle (MSA). In fact, we show that even MST algorithms such as Kruskal's and Jarnik-Prim-Dijkstra' can be modified naturally for determining MSAs. In the second part, we show that the MSA of a weighted simplicial complex and persistence diagram of the associated simplicial process are closely related. In the third part, we consider a randomly weighted $d-$complex. Unlike in the first two parts, the randomly weighted $d-$complex is an example of a simplicial complex with random weights. For this complex, we show that, under appropriate scaling, the distribution of all the three point sets: nearest neighbour distances, death times in the persistence diagram, and weights in the MSA converge to a Poisson point process.

This is ongoing work with Primoz Skraba and D. Yogeshwaran.

Abstract: Let S be a finite set (the sample space), and f_i: S -> R functions, for i=1,...,k. Given a k-tuple (v_1,...,v_k) in R^k it is natural to ask:

What is the distribution P on S that maximizes the entropy

-SUM P(x) log(P(x))

subject to the constraint that the expectation of f_i be v_i?

In this talk I'll discuss a closed formula for the solution P in terms of a sum over cumulant trees. This is based on a general calculus for solving perturbative optimization problems due to Feynman, which may be of interest in its own right.

The talk will be completely self-contained, requiring only rudimentary knowledge of calculus and probability theory. This is joint work with Tomer Schlank and Ran Tessler.

Abstract: Let $X_n$ be a homogeneous Poisson point process of intensity $n$ in the plane and let $p$ and $q$ be two (deterministic) points in the plane. The point process $X:=X_n\cup\{p,q\}$ generates the so-called Delaunay triangulation DT(X) associated with $X$. This graph is a triangulation of the plane such that there is no points of $X$ in the interiors of the circumdisks of the triangles in $DT(X)$. In this talk, we investigate the length of the smallest path in the Delaunay triangulation starting from $p$ and going to $q$ as the intensity $n$ of the Poisson point process $X_n$ goes to infinity.

This a joint work with Olivier Devillers (INRIA Nancy).

Additionally, participants are invited to attend the following: Monday, April 11, 2016: 3:30 in Amado 232:
Michael Farber, Mathematics, Queen Mary, University of London,
Topology of Large Random Spaces,
at the Mathematics Colloquium

Abstract: The minimum number of bipartite graphs needed to cover a complete n-vertex graph is ceil(log n). In this seminar, we will discuss an extension of this to higher dimensional simplicial complexes.

A simplicial complex X is a set-system which is closed under taking subsets. The members of X are called simplices. The dimension of a simplex s is |s|-1 and the dimension of X is the maximum dimension among its simplices. (Graphs are thus 1-dimensional simplicial complexes.) The n-vertex d-dimensional complete complex K(n,d) is the collection of all subsets of [n] of size at most d+1. One way to view a bipartite graph is as a subgraph of the "coboundary" of a set of vertices, where the coboundary of a set S of (d-1)-dimensional simplices is the collection of all the d-dimensional simplices in K(n,d), each of which contains an odd number of faces from S. (This is equivalent to the homological definition of a coboundary over the field GF(2)).

We will present estimates on the minimum number of coboundaries needed to cover K(n,d).

This is an ongoing joint work with Ilan Newman and Yuri Rabinovich.

Abstract: The study of random topology focuses on describing high-level qualitative properties of random spaces. This field has been rapidly developed in the past decade, and is driven both by deep theoretical questions and state-of-the-art data analysis applications. Topological Data Analysis (TDA) broadly refers to the use of concepts from mathematical topology to analyze data and networks. In the past decade a variety of powerful topological tools has been introduced, and were proven useful for applications in various fields (e.g. shape analysis, signal processing, neuroscience, and genomic research). The theory developed in random topology aims to provide a solid foundation for the statistical analysis of these methods, which to date is at a very preliminary stage. This talk will be divided into three parts. First, I will provide an introduction and motivation to random topology and TDA. Next, we will discuss the theory of random geometric complexes. In particular, we will consider their Betti numbers (counting “cycles” in different dimensions), and phase transitions related to these. Finally, we will present a few examples of combining the theory of random complexes with statistical problems related to TDA. In particular, we are interested in analyzing ``topological noise” that appears in such problems, for the purposes of filtering and hypothesis testing.

Seminar on Probability and Stochastic Processes

Abstract: This work is joint with Peter Sarnak.

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.

Abstract: In 1939, Hermann Weyl derived a now classic formula describing the volumes of tubular sets around manifolds embedded in Euclidean spaces or spheres. In this seminar I will describe a perturbation theory for the Euclidean version of Weyl's tube formula, and approach the problem from two directions.

Initially, we take a general manifold M and perturb it by moving each of its points x a short distance, z(x), along a normal originating from x. We then find an exact expression for the volume of the gap between the original and perturbed manifolds, involving an integral of z over M, obtaining what is eff??ectively an extension of Weyl's original formula to tubes of non-constant radius. This is exploited to obtain a series expansion of this volume in terms of small parameters.

The second direction that we take considers manifolds defined via the level sets of smooth functions. The perturbed manifolds now come from the level sets of perturbations to the initial functions. Again, we derive the a tube formula for the perturbed manifold as well as a series expansion in terms of one small parameter and and integral functionals of the initial function and the perturbation.

This research reported on here is part of the speaker’s MSc research in the Faculty of Electrical Engineering, under the supervision of Robert Adler.

Sunday, June 7, 2015: 14:30 in Amado 814:
Ronen Talmon, Electrical Engineering, Technion,
Common Manifold Learning Using Alternating Diffusion for Multimodal Signal Processing,
at the Nonlinear Analysis and Optimization Seminar

Abstract: Topological and geometric techniques are increasingly becoming an important part of the analysis of high-dimensional, complex data sets. We present current work-in-progress of a new approach to data clustering through approximating the heat flow on a manifold. In particular, given samples from a probability distribution on a submanifold M of Euclidean space, we construct a family of approximations to the heat operator, and then use model-selection techniques in order to pick a 'topologically good' approximation to the number of connected components of M. We then use this to assign each point to one of the components to obtain a clustering of the space. We present several numerical examples, giving experimental support to the conjecture that, for a large number of points, the technique produces a correct clustering with high probability from a uniform distribution on a manifold.

Abstract: In a previous talk, we discussed and motivated various models for random knots. Now we present the results from a new model for random knot diagrams; we assign the crossings to be positive or negative uniformly at random.

Koseleff and Pecker introduced the knot diagram used in the random setting above using a curve parametrized by Chebyshev polynomials, but these can be projected more simply to rational billiard trajectories.

We give a formula for the probability of choosing a knot at random among all knots with bridge index at most 2.

This is joint work with Sunder Ram Krishnan.

Abstract: Roughly speaking, the reach, or critical radius, of a manifold is a measure of its departure from convexity that incorporates both local curvature and global topology. It plays a major role in many aspects of Differential Geometry, and more recently has turned out to be a crucial parameter in assessing the efficiency of algorithms for manifold learning.

We study a sequence of random manifolds, generated by embedding a fixed, compact manifold into Euclidean spheres of increasing dimension via a sequence of Gaussian mappings, and show a.s. convergence of the corresponding critical radii to a constant. Somewhat unexpectedly, the constant turns out to be the same one that arises in studying the exceedance probabilities of Gaussian processes over stratified manifolds.

This is a joint work with Robert Adler, Jonathan Taylor, and Shmuel Weinberger.

Abstract: Composition of fusion algorithms for multiple estimators with unknown cross-correlation is distributive but not associative. We axiomatize information fusion by way of an algebraic structure (known from low dimensional topology) called a quandle, and we illustrate the way in which several popular information fusion schemes fit into this framework. Based on this axiomatizaton, flow of information in an information fusion network can be drawn as a coloured braid or tangle diagram. We suggest two applications: to build fault-tolerant self-optimizing sensor networks, and as an Interactive Proof (IP)- style model for probabilistic computation. The talk is based on joint work with A. Carmi: http://arxiv.org/abs/1409.5505 and http://arxiv.org/abs/1408.2685

Abstract: I will discuss several models of large random simplicial complexes and topological properties which happen with high probability in these models.

Sunday, March 22, 2015: 13:30 in Taub 337:
Alexandre Djerbetian, M.Sc. student, Computer Science, Technion,
Persistent Homology,
at the CGGC Seminar

Monday, March 23, 2015: 3:30 in Sego 1:
Nati Linial, Computer Science, Hebrew University,
Random simplicial complexes: Where combinatorics, geometry and probability meet,
at the Mathematics Colloquium

Abstract: In this talk, I will present recent results about the convergence of persistence diagrams. In the first part of the talk, I will consider the rates of convergence of persistence diagrams defined using the standard distance function and with points sampled from a given distribution. However, the distance fonction to the sampled points is highly non-robust to noise and outliers.To answer this issue, the distance to a measure has been introduced by Chazal et al. (2011). This alternative distance function is a smooth function, it provides useful topological information and it is robust to noise and outliers. In the second part of the talk, I will study the convergence of the distance to the empirical measure and I will apply these results to persistence homology.

Abstract: We will introduce our recent results on the use of persistent homology in the field of complex networks.

In particular we will show how persistence homology can be used to discriminate the variation of the brain functional connectome under the influence of psilocybin extracted from magic mushrooms.

We furthermore give account of our theorem which shows that a persistence module on a finite poset P can be obtained from the filtration of a graph weighted over P.

Abstract: We investigate the limiting behaviour of a subgraph counting process for the study on the formation of random geometric graphs. The subgraph counting process we shall consider counts the number of subgraphs with specific shape, which exist outside an expanding ball as the sample size increases. As an underlying law, we will consider the distributions with regularly varying tail and those with exponentially decaying tail. In either case, the nature of an obtained functional central limit theorem differs depending on how rapidly the ball expands. More specifically, a proper normalization of the central limit theorem and the properties of limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called core - in which the random points are highly densely scattered and form a single giant geometric graph. This is a joint work with Professor Robert J. Adler at Technion.

Abstract: We describe an algorithm for computing a small presentation of the fundamental group of a finite regular CW-space. The algorithm is based on the construction of a discrete vector field on the 3-skeleton of the space. A variant yields the homomorphism of fundamental groups induced by a cellular map of spaces.

We illustrate the computation for knots arising from experimental data on protein backbones. In case of knots the homomorphism is known to be a complete ambient isotopy invariant of the knot. We observe that low-index subgroups of finitely presented groups provide useful invariants of the homomorphism.

Abstract: It is well-known that the G(n,p) model of random graphs undergoes a dramatic change around p=1/n. It is here that the random graph is, almost surely, no longer a forest, and here it first acquires a giant connected component. Several years ago, Linial and Meshulam have introduced the X_d(n,p) model, a probability space of n-vertex d-dimensional simplicial complexes, where X_1(n,p) coincides with G(n,p). Within this model we prove a natural d-dimensional analog of these graph theoretic phenomena. Specifically, we determine the exact threshold for the nonvanishing of the real d-th homology of complexes from X_d(n,p), and show that it is strictly greater than the threshold of d-collapsibility. In addition, we compute the real Betti numbers, i.e. the dimension of the homology groups, of X_d(n,p) for p=c/n. Finally, we establish the emergence of giant shadow at this threshold. (For d=1 a giant shadow and a giant component are equivalent). Unlike the case for graphs, for d > 1 the emergence of the giant shadow is a first order phase transition.

The talk will contain the necessary toplogical backgorund on simplicial complexes, and will focus on the main idea of the proof: the local weak limit of random simplicial complexes and its role in the analysis of phase transitions.

Joint work with Nati Linial.

Abstract: For showing the homology of random simplicial complexes vanishes in a variety of models, the notion of local spectral expansion has proven to be a useful tool. We will introduce this tool and show how it relates to higher dimensional Laplacians and other notions of expansion. We will also highlight some open questions related to various possible extensions of this method.

Abstract: At scales of a few to a few hundred megaparsecs, the Universe has a whispy web like appearance, commonly known as the cosmic web. The seeds of formation of the web are planted in the form of tiny quantum fluctuations in the primordial Universe and the primordial fluctuation field is described as a Gaussian random field to a high accuracy. I will present an investigation into the topological properties of Gaussian random fields through the hierarchical formalism of persistence and homology, and compare them with the more traditional notion of measuring topology through the Euler characteristic. A hierarchical description of the topology of cosmic density fields is interesting because structure formation in the Universe proceeds in a hierarchical fashion.

In addition, I will present a method that draws ideas from Morse theory and persistence to detect and quantify the elements of the cosmic web, in particular the ubiquitous filaments.

Discrete, Computational and Algebraic Topology conference in Copenhagen, including talk by Robert Adler

Abstract: We study random knots and links in R^3 using the Petaluma model, which is based on the petal projections developed by Adams et al. (2012). In this model we obtain a formula for the distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-three knot invariant v3. These are the first precise formulas given for the distributions of invariants in any model for random knots or links. We also use numerical computation to compare these to other random knot and link models, such as those based on grid diagrams.

Joint work with Joel Hass, Nati Linial, and Tahl Nowik.

Abstract: We begin with several random knot models from the literature that will build intuition for knots before going into formal definitions. We project knots onto the plane, together with crossing information, to get knot diagrams, and we present several different structural types of diagrams. These will be used to highlight new, more combinatorial models for random knots.

Abstract: The level sets of probability density functions are of a considerable interest in many areas of statistics. To date, the topological analysis of these sets is mostly limited to their connected components, namely - the clustering problem. In algebraic topology, connected components are merely a small part of a much richer set of topological characteristics known as homology. Briefly, homology is an algebraic framework for studying the topology of sets in terms of components and holes of different dimensions. In this talk we focus on recovering the entire homology structure of level sets from random data.

The main difficulty in recovering the homology is that even a small perturbation in the data can generate a very large error. We discuss these difficulties and present an estimator that is designed to overcome them. We then show that this estimator is consistent, and demonstrate its possible applications for clustering and topological manifold learning. Finally, we show that similar methods can be used in the analysis of nonparametric regression models.

* This talk assumes no prior knowledge in algebraic topology.

Abstract: The study of large scale and complex interconnected systems is of great importance in today's networked world with applications ranging from distributed power generation to deep space exploration. A great challenge for these systems is to understand the interplay between the dynamic properties of the individual systems comprising the networks, the underlying information exchange network, and the interaction protocols governing the collective behavior. In this talk we will explore necessary and sufficient conditions for a network of passive dynamical systems to reach an output agreement, i.e., the trajectories of each system will synchronize. We then show that the steady-state behavior of these systems are in fact solutions to a family of classic network optimization problems, and as a result we draw connections between notions of duality in static optimization to cooperative control. Several examples including vehicle platoons will be used to illustrate these concepts.

Abstract: In this talk, we will present a data-driven method to build the canonical geometry of the signal samples, which is independent of the coordinate system of the signal. This canonical geometry enables us to discover intrinsic latent variables, i.e., hidden variables which are invariant to the instrumental/measurement modalities and to ambient noise.

We exploit the dynamics of the signal to explore the local tangent planes of the low-dimensional manifold of the signal. This information is used to define a Riemannian distance metric, which in turn is used to construct a Laplace operator. Such a construction is equivalent to an inverse problem, which is formulated as a nonlinear differential equation and is solved empirically through the eigenvectors of the Laplace operator.

We will show that applying our method to simulation and real data allows for accurate recovery of latent intrinsic variables. Furthermore, the recovered latent variables have a true physical meaning. In particular, for biomedical signals, which are often difficult to analyze since they tend to be very noisy and highly dependent on the instrumental modality (e.g., the type of sensors used for the signal acquisition), we will show that the intrinsic variables give rise to a better understanding of the human body and brain.

The intrinsic volumes generalize both Euler characteristic and Lebesgue volume, quantifying the size of a set in various ways. A random cubical complex is a union of (possibly high-dimensional) unit cubes selected from a lattice according to some probability model. I will describe a simple model of random cubical complex and derive exact polynomial formulae, dependent on a probability, for the expected value and variance of the intrinsic volumes of the complex. I will also give a central limit theorem and an interleaving theorem about the roots of the expected intrinsic volumes -- that is, the values of the probability parameter at which an expected value is zero. Lastly, I will discuss connections to random fields and applications, especially image recognition and the study of noise in digital images. This work is in collaboration with Michael Werman of The Hebrew University of Jerusalem.

Abstract: We consider the problem of detecting associations between random vectors of any dimension. Few tests of independence exist that are consistent against all dependent alternatives. We propose a powerful test that is applicable in all dimensions and consistent against all alternatives. The test has a simple form, is easy to implement, and has good power. Joint work with Ruth Heller and Yair Heller.

We present a brief overview of Robin Forman's approach Discrete Morse Theory and investigate its relationship both with the classical theory as well with other discretizations. In addition we show a few of its applications to Imaging and Graphics.

Abstract: An important tool in information analysis is dimensionality reduction. There are various approaches for large data simplification by scaling its dimensions down that play a significant role in recognition and classification tasks. The efficiency of dimension reduction tools is measured in terms of memory and computational complexity, which are usually a function of the number of the given data points. Sparse local operators that involve substantially less than quadratic complexity at one end, and faithful multiscale models with quadratic cost at the other end, make the design of dimension reduction procedure a delicate balance between modeling accuracy and efficiency. Here, we combine the benefits of both and propose a low-dimensional multiscale modeling of the data, at a modest computational cost. The idea is to project the classical multidimensional scaling problem into the data spectral domain extracted from its Laplace–Beltrami operator. There, embedding into a small dimensional Euclidean space is accomplished while optimizing for a small number of coefficients. We provide a theoretical support and demonstrate that working in the natural eigenspace of the data, one could reduce the process complexity while maintaining the model fidelity. As examples, we efficiently canonize nonrigid shapes by embedding their intrinsic metric into , a method often used for matching and classifying almost isometric articulated objects. Finally, we demonstrate the method by exposing the style in which handwritten digits appear in a large collection of images. We also visualize clustering of digits by treating images as feature points that we map to a plane.

Abstract: This seminar is concerned with online tethered coverage, in which a mobile robot of size D is attached to a fixed point S by a cable of finite length L. Starting at S, the robot has to cover an unknown planar environment containing obstacles, and return to S with the cable fully retracted. I will first describe an optimal off-line tethered coverage methodology, then introduces the TC algorithm that performs online tethered coverage using position and local obstacle detection sensors. The performance of the TC algorithm is measured by its competitiveness, determined by measuring its total online path length, l, relative to the optimal off-line solution l_opt. I will establish that the TC algorithm has a competitive performance of l <= 2 (L/D) l_opt. I will additionally establish that any online tethered coverage algorithm generates in worst case a~path length of l >= log(L/D) l_opt. Execution example and experiments with a tethered recoiling mechanism illustrate the usefulness of the TC algorithm.

Abstract: A fundamental task in discrete differential geometry is the choice of representation of differential quantities. Different representations will lead to different optimization problems when using DDG in practice, and can considerably influence the scope of feasible applications. We will describe a novel choice of representation, based on functional operators, which although standard in classic differential geometry has only been recently applied to DDG. We will show how operators which take scalar functions to scalar functions can be used to concisely represent smooth maps between shapes and smooth vector fields. We further demonstrate that by using a common operator representation, the intimate connection between maps and vector fields can be leveraged to easily compute vector fields which fulfill intricate global constraints, such as Killing and symmetric vector fields. Finally, we show how the spectral decomposition of the Laplace-deRham operator can be leveraged for combining local and global vector field constraints in a unified framework.

Abstract: We will explain what generating function is and use it to solve some recurrence relationsFollowing Grove and Petersen, we show that a Ricci curvature based triangulation of compact Riemannian manifolds allows their recognition up to homotopy equivalence. We further prove that their method extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension condition $CD(K, N)$. Immediate consequences for imaging are noted. Moreover, we show that the triangulation can be improved to become a thick one and that, in consequence, such manifolds admit measure-sensitive representations of bounded distorsion on $\mathbb{S}^n$, with applications to Information Geometry. Time permitting we shall also explore further applications.

Abstract: Two revolutionary dimension-reduction algorithms were introduced in Science magazine in the year 2000, Isomap (Tenenbaum et al., 2000) and LLE (Roweis and Saul, 2000). These algorithms opened the field of manifold learning, in which non-linear techniques are employed for reduction of high-dimensional data sets that are assumed to sit on a low-dimensional manifold. The goal of manifold learning techniques is to produce a simpler description of the high-dimensional data in order to simplify data analysis and to enable its visualization. Following Isomap and LLE, many different manifold-learning techniques were developed, however, almost no theoretical results on the performance of these algorithms exists. In this talk I will present Isomap and LLE, and discuss some theoretical properties of these algorithms.

Abstract: A random 2-dimensional simplicial complex Y is produced by starting with a complete graph on n vertices, 1, 2, ... , n and adding each 2-simplex 1 \leq i < j \leq n at random, with probability 0 < p < 1, independently of each other. In the talk I will discuss geometric and topological properties of random 2-complexes under different assumptions on the probability parameter p. One of the central questions is whether one can generate randomly aspherical 2-complexes (i.e. such that \pi_2(Y) = 0) and whether random aspherical 2-complexes satisfy the Whitehead Conjecture. This conjecture proposed in 1941 by J.H.C. Whitehead states that any subcomplex of a random 2-complex is also aspherical; it is still open.

One of the main results presented in this talk is the statement that for p < n^{-1/2 - \epsilon} any aspherical subcomplex Y' \subset Y of a random 2-complex Y satisfies the Whitehead conjecture with probability tending to 1 as n --> \infty. We also describe torsion in the fundamental groups of random 2-complexes and their cohomological dimension. Our proofs use Cheeger constants and systoles of simplicial surfaces as well as properties of Gromov hyperbolic groups.

The talk is based on a joint work with Armindo Costa.

Abstract: A common problem in image analysis is to find local significant regions, either for a single image or for the difference between two or more images. In this talk, I will describe how to approach the image inference problem from a multiple testing point of view, and how random field theory comes to the rescue in the estimation of global error rates, such as the family-wise error rate. In particular, I will emphasize the role that topology plays in the solution via the Euler characteristic of the excursion set of a random field. Moreover, I will suggest how the Euler characteristic may enable estimation of topological error rates, turning the signal detection problem into a topological inference one.

Abstract: In this (review) talk I will continue with the description of the interplay between statistical inference and topological data analysis I started on 29/5.

In particular, I will describe (again, but briefly, and in part for those who missed Part I) the concept of persistent homology, this time motivating it from problems from random functions and from cluster analysis.

From there I will go on to discussing the mathematical structure of spaces of barcodes and persistence diagrams (such as stability theorems) and some of the statistical procedures that have grown out of this (such as deciding what the "average" of a collection of barcodes might be.

As in Part I, the main theme will be the challenges that topological data analysis pose for statistics and probability.