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A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a discretization of spacetime. Here, by extending our knowledge of growing complex networks to growing simplicial complexes we investigate the nature of the emergent geometry of complex networks and explore whether this geometry is hyperbolic. Specifically we show that an hyperbolic network geometry emerges spontaneously from models of growing simplicial complexes that are purely combinatorial. The statistical and geometrical properties of the growing simplicial complexes strongly depend on their dimensionality and display the major universal properties of real complex networks (scale-free degree distribution, small-world and communities) at the same time. Interestingly, when the network dynamics includes an heterogeneous fitness of the faces, the growing simplicial complex can undergo phase transitions that are reflected by relevant changes in the network geometry.
This talk is based on a joint work with Jerry Buckley. We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.
We compare a few recent skeletonisation methods for unstructured data such as finite sets of points in a metric space. For applications in Computer Vision, such a data cloud may consist of edge pixels detected in a digital image. The first algorithm Mapper has been successfully commercialised by the start-up Ayasdi at Stanford, though its stability properties remained unexplored until recently. The second algorithm requires a scale parameter and outputs an abstract alpha-Reeb graph provably approximating a given data cloud. The third algorithm introduces a Homologically Persistent Skeleton optimally describing the 1-dimensional shape of a cloud across all scales. We present experiments on real images from the Berkeley Segmentation Database. The papers and C++ software are available on author's webpage http://kurlin.org (link is external).
Simplicial complexes are generalized network structures able to encode interactions occurring between two or more nodes. They are emerging as a new tool to describe complex networks with large clustering coefficients and abundant numbers of short loops and have already been used to describe a large variety of complex interacting systems ranging from brain networks, to social and collaboration networks. They are also ideal mathematical objects for the discretisation of geometry as is demonstrated by their wide use in the context of quantum gravity and so they may also open up new scenarios for uncovering the hidden geometry of complex networks.
Here we take a statistical mechanics approach and investigate two ensembles of simplicial complexes based respectively on the placing of hard and soft constraints on local structural properties of the simplicial complexes.
The first ensemble is a generalisation of the network configuration model where networks with a pre-specified generalised degree sequence are sampled uniformly. The second ensemble is an exponential random simplicial complex where we instead constrain the expected generalised degrees.
We derive a relation between the entropy of the two ensembles, showing that they are not in fact asymptotically equivalent. From this relation we are also able to derive an asymptotically exact formula for the number of simplicial complexes that can be constructed with a given sequence of generalised degrees, and find that indeed this number depends heavily on the choice of generalised degrees.
A theorem by J.D.S. Jones from 1987 identifies the cohomology of the free loop space of a simply connected space with the Hochschild homology of the singular cochain algebra of this space. There are very strong relations between the Floer homology of cotangent bundles in symplectic geometry and the homology of free loop spaces of closed manifolds. In the light of these connections, one wants to have a geometric and Morse-theoretic identification of free loop space cohomology and the Hochschild homology of Morse cochain algebras in order to establish relations between Floer homology and Hochschild homology. After describing the underlying Morse-theoretic constructions and especially the Hochschild homology of Morse cochains, I will sketch a purely Morse-theoretic version of Jones' map and discuss its most important properties. If time permits, I will further discuss compatibility results with product structures like the Chas-Sullivan loop product and give explicit Morse-theoretic descriptions of products in Hochschild cohomology in terms of gradient flow trees.
Abstract: If we consider a group presentation with some number of generators and relations of a certain length, what properties do we expect the group to have? The study of "random groups" is concerned with answering such questions. We will discuss how the concept of conformal dimension, an analytically defined invariant of negatively-curved groups introduced by Pansu, can be used to distinguish certain random groups from each other.
Abstract: In this talk, I will discuss a curvature notion in the setting of graphs which, in a more general setting, goes back to Bakry-Emery and is derived from Bochner's formula for the Laplacian on manifolds. I will discuss some basic local and global properties of this curvature notion (behaviour under taking graph products and a finiteness result in the case of strictly positive curvature in the spirit of Bonnet-Myers). I will also present a nice applet to investigate the Bakry-Emery curvature on graphs, which was programmed by David Cushing and George Stagg. This is joint work with David Cushing, Shiping Liu, and Florentin Muench.
Abstract: Topology is one of the oldest and more relevant branches of mathematics, and it has provided an expressive and affordable language which is progressively pervading many areas of mathematics, computer science and physics. Using examples taken from work drug-altered brain functional networks, I will illustrate the type of novel insights that algebraic topological tools are providing in the context of neuroimaging. I will then show how the comparison of homological features of structural and functional brain networks across a large age span highlights the presence of a globally conserved topological skeletons and of a compensation mechanism modulating the localization of functional homological features. Finally, with an eye to altered cognitive control in disease and early ageing, I will introduce preliminary theoretical results on the modelization of multitasking capacities from a statistical mechanical perspective and show that even a small overlap between tasks strongly limits overall parallel capacity to a degree that substantially outpaces gains by increasing network size.
This is a joint talk with the "Complex systems" seminar series.
The topological complexity of a space is a homotopy invariant TC(X) motivated by the motion planning problem from robotics, it was introduced by Michael Farber in 2003. The topological complexity has been computed for diverse classes of examples, yet for many years its value for the Klein bottle was unknown and could only be narrowed down to be one of two possible values. In 2016, D. Cohen and L. Vandembroucq finally computed the topological complexity of the Klein bottle using cohomology methods. In my talk, I will introduce the notion of topological complexity and give an overview over the relevant methods, before sketching the line of argument of Cohen’s and Vandembroucq’s computation.
For a compact Lie group G, we may consider equivariant cohomology theories with values in rational vector spaces. When G is a torus, one may give a complete algebraic model A(G) for them, but this is not the focus of the talk. The aim is to explain how one may recover the space of closed subgroups from the formal properties of the category: the Balmer spectrum is the space of closed subgroups under cotoral inclusion. The two ingredients that get this started are the idempotents in the Burnside ring and the Borel-Hsiang-Quillen Localization Theorem.
Moduli spaces of (sub)manifolds arise as configuration spaces (0-dimensional submanifolds), moduli spaces of Riemann surfaces (2-dimensional submanifolds of R^\infty), or more generally as classifying spaces for diffeomorphism groups of manifolds (d-dimensional submanifolds of R^\infty). I will first explain some recent advances in understanding and computing their cohomology, where for even-dimensional manifolds a great deal is known. I will then explain the sort of techniques which go into proving such results, in the simplest case: configuration spaces.
In this talk I will discuss the usage of persistent homology as a descriptor of physics and dynamics. I will introduce the concept of persistence, discuss various ways it can be computed for variety of inputs. I will present persistence diagrams, and other, more suitable for statistical and machine learning post processing representations of persistence. Finally we I will show a few examples where those tools can be used as descriptor of physical phenomena: in case of phase separation in alloys and analysis of nano-porous materials.
Persistent homology is an algebraic topological tool developed for the analysis of spatial data. It measures changes in topology of a filtration: a growing sequence of spaces indexed by a single real parameter. Persistent homology provides invariants called the barcodes or persistence diagrams that are sets of intervals recording the birth and death parameter values of each homology class in the filtration. Statistical analysis of persistent homology has been difficult because the raw information (the persistence diagrams) are provided as sets of intervals rather than functions. Many research groups are pursuing various approaches to converting persistence diagrams to functional forms that are then amenable to standard statistical techniques. One possibility is based on the persistent rank functions which are analogous to 2D cumulative sums of persistence intervals. A successful application of this technique is the analysis of experimentally imaged configurations of approximately mono-disperse spherical bead packings. The persistence diagrams highlight the regular tetrahedral and regular octahedral configurations of crystalline sphere packing in an unambiguous way and have led to new insights in the grain-scale mechanisms underlying the order-disorder transition in this dissipative, athermal system. Functional principle component analysis of the rank functions also reveals that there is effectively a single axis of variation in the experimental data samples.