15:30 15:45 break

John Baez, Computations and the periodic table By now there is an extensive network of interlocking analogies between physics, topology, logic and computer science, which can be seen most easily by comparing the roles that symmetric monoidal closed categories play in each subject. However, symmetric monoidal categories are just the n = 1, k = 3 entry of a hypothesized ”periodic table” of k-tuply monoidal n-categories. This raises the question of how these analogies extend. We present some thoughts on this question, focussing on how symmetric monoidal closed 2-categories might let us understand the lambda calculus more deeply. Silvia Biasotti (joint work Daniela Giorgi, Michela Spagnuolo and Bianca Falcidieno), Reeb graphs for shape analysis and synthesis Reeb graphs are compact shape descriptions that convey topological information by analysing the evolution of the level sets of a function defined on the shape. Their definition dates back to 1946, and finds its root in Morse theory. Reeb graphs have been proposed as shape descriptors to solve different problems arising in Computer Graphics, and nowadays they play a fundamental role in the field of computational topology for shape analysis. This talk provides an overview of the mathematical properties of Reeb graphs and reconstructs their history in the Computer Graphics context. We will also discuss the enrichment of the Reeb graph with geometrical information about the shape, which give an abstraction of the main shape features. Finally, directions of future research will be discussed. Andrew Blumberg (joint work with Gunnar Carlsson and Michael A. Mandell), Persistent homology and detection of disease clusters with arbitrary shapes Detection of disease clusters is a problem of central importance in public health. The problem is to find clusters of disease cases which are “unusually dense”, given the background population and some model of the baseline disease incidence. Such clusters can indicate developing disease outbreaks and hence locations for public health intervention. The standard method is to apply circular scan statistics, due to Kulldorf and collaborators; this method applies a likelihood ratio test to circular subregions of the region under investigation. Variants of the circular scan statistic methodology are currently in use by public health departments in the United States. Recently, Wieland, et al. introduced a new method based on Euclidean minimum spanning trees for disease cluster detection. This method corrects a central defect of the circular scan statistic: the circular scan statistic is optimized for detecting circular clusters. The minimum spanning tree method has dramatically improved perception of noncircular disease clustering data. However, relating the natural test statistic in this method to the underlying statistical model of the disease distribution is difficult, and in particular certain standard statistical variants are hard to handle (e.g. covariates). Also, a key step in the algorithm is extremely computationally demanding. We introduce a new method for disease cluster detection based on persistent homology. Our algorithm is based on the observation that the “potential clusters” Wieland, et al. derive from minimum spanning trees are in fact precisely persistent components. 8 ALGEBRAIC TOPOLOGICAL METHODS IN COMPUTER SCIENCE (ATMCS) III Thus, we proceed by computing persistent components and then computing a likelihood ratio for the associated simplicial complex. This geometric interpretation of the clustering algorithm permits extremely useful refinements, notably the incorporation of “density” parameters in terms of the presence of higher-dimensional simplices. This allows the algorithm to be tuned via a geometrically meaningful integer parameter (a dimension constraint) to bias towards dense, circular clusters. Experimental results show that our new method achieves superior perception to both the circular scan statistic and the Euclidean minimum spanning tree algorithm. Peter Bubenik, Extremal models of concurrent systems, the fundamental bipartite graph, and directed van Kampen theorems Analyzing concurrent programs, in which multiple processes use shared resources, is notoriously difficult. One approach is to model the state space of such a program using a directed space, and then study its topology. The fundamental category of this space describes the execution space, but it is typically uncountable. Approaches to reducing this category include using components (Fajstrup, Goubault, Haucourt, Raussen) and using injective and projective models (Grandis). I use Grandis’ future retracts and past retracts to construct extremal models of the fundamental category. Minimal extremal models are typically finite. In addition, the fundamental bipartite graph, which captures the homotopically distinct execution paths, is an invariant of the extremal model. Furthermore, extremal models can be constructed in a piece-by-piece manner using a van Kampen-type theorem. Gunnar Carlsson, Topology and data I will survey various ways of using topological methods and points of view to study data sets from various scientific areas. In particular, I will discuss persistent homology and its generalizations, as well as topological methods for mapping out data sets. Andrea Cerri, Applications of size theory in shape comparison Size Theory was proposed in the early 90’s as a geometrical/topological approach to the problem of shape comparison. The main idea is to translate the task of comparing two objects in a database (e.g. images, 3D models or sounds) into the one of comparing two suitable topological spaces M, N (non-empty, compact and locally connected Hausdorff spaces), endowed with two continuous functions φ : M → R, ψ : N → R that are chosen according to the applications. These functions are called measuring functions and can be seen as descriptors of the features considered relevant for the comparison. The pairs (M, φ), (N , ψ) are said to be size pairs and provide a representation of the considered shapes: In Size Theory, such pairs can be compared by size functions, whose role is to capture qualitative aspects of a shape and represent them in a quantitative way. The idea is to study the pairs (M〈φ ≤ x〉,M〈φ ≤ y〉), whereM〈φ ≤ t〉 is defined by settingM〈φ ≤ t〉 = {P ∈M : φ(P ) ≤ t} for t ∈ R: The size function `(M,φ) : {(x, y) ∈ R : x < y} → N is then the function that takes each point (x, y) of the domain into the number of connected components of M〈φ ≤ y〉 containing at least one point of M〈φ ≤ x〉 [1]. By means of Size Theory, we can then model a shape by a size pair, and describe it by considering the associated size function: As a consequence, the comparison of two shapes can be translated into the simpler task of comparing two functions from the half–plane {(x, y) ∈ R : x < y} ALGEBRAIC TOPOLOGICAL METHODS IN COMPUTER SCIENCE (ATMCS) III 9 to the natural numbers. However, a common scenario in applications is to deal withmultidimensional information: Indeed, a shape can be more thoroughly characterizedby means of a set of real functions, each investigating specific features of the shapeunder study. This problem can be faced by observing that size functions are modulardescriptors: In order to study different properties of a shape, we only need to changethe measuring function. Since its introduction, Size Theory has been studied andapplied in quite a lot of applications: An example is given by [2], where the authorspropose an automatic retrieval system for trademark images based on size functions,to support human labor in guaranteeing copyright policy. Other examples on the useof Size Theory in applications can be found in several fields, ranging from leukocyteclassification in medical context [3] to image retrieval in the World Wide Web [4]: Thiswork proposes to be an overview on some meaningful experimental results, in orderto show the capability and the flexibility of this theoretical framework in dealing withconcrete applications. References[1] P. Frosini, and C. Landi, Size Theory as a Topological Tool for Computer Vision, Pattern Recogn. Image Anal. 9(4) (1999), 596–603.[2] A. Cerri, M. Ferri, and D. Giorgi, Retrieval of trademark images by means of sizefunctions, Graphical Models 68 (2006), 451–471.[3] M. Ferri, S. Lombardini and C. Pallotti, Leukocyte classification by size functions,In Proc. of the 2 IEEE Workshop on Applications of Computer Vision, IEEE Com-puter Society Press, Los Alamitos, CA (1994), 223–229.[4] A. Cerri, M. Ferri, P. Frosini, and D. Giorgi, Keypics: free–hand drawn iconickeywords, International Journal of Shape Modelling 13(2) (2007), in press. Roberto De Leo, Implementation of a library to study the asymptoticsof plane sections of periodic surfaces The problem of asymptotics of plane sections of periodic surfaces was first intro-duced in Fifties by solid state physicists.In the quantum theory of metals electron’s quasi-momenta are periodic and boundto lie on a constant-energy periodic surface (Fermi Surface, FS) which characterizesthe metal’s electronic properties; the introduction of a magnetic field obliges the orbitsof quasi-momenta to lie on planes perpendicular to the magnetic field and therefore allof their possible orbits are exactly the intersections between those planes and the FS.A physical quantity called magetoresistance depend qualitatively on the asymptoticsof those orbits and so that their topology is somehow experimentally measureable.The rich topological structure of the problem was discovered only in the last twentyyears by S.P. Novikov and his pupils as a concrete case of his extension of the Morsetheory to closed 1-forms and ultimately led to the conclusion that in the most inter-esting cases the asymptotics of FS’s plane sections are described by a fractal in thespace of directions.In this talk we present our implementation of the algorithms needed to find numer-ically the asymptotics of plane sections once a periodic surface and a (magnetic field)direction are given; the main tasks are the following: 1. producing the full intersectionbetween a (integer) plane and a periodic surface; 2. extracting the critical sections,i.e. those produced by planes tangent to the surface; 3. evaluating the homologicalclass of the critical sections in the 3-torus and on the surface. Finally we shall shortlypresent the most interesting results obtained numerically so far. 10 ALGEBRAIC TOPOLOGICAL METHODS IN COMPUTER SCIENCE (ATMCS) III Herbert Edelsbrunner, Topics in persistent homology Uli Fahrenberg, Simulation hemi-metrics for timed systems, withrelations to ditopology During recent years, interest has emerged in quantitative, or robust, notions of(bi)similarity for probabilistic and timed systems. One motivation for this are someissues pertaining to modeling and implementability; real-world implementations aresubject to imprecision, hence model-checking should be robust with respect to suchimprecision.For real-time systems, several notions of metrics on timed words can be found inthe literature, and also a few notions of bisimulation metrics and simulation hemi-metrics. In this paper, we introduce some new ones, to complete the picture andarrive at a somewhat systematic treatment. We are especially interested in caseswhere one can introduce simulation metrics on timed transition systems and metrics ontimed languages such that the semantics mapping from one to the other is a distance-preserving continuous function.Metrics on timed languages are generally uncomputable, and for bisimulation met-rics, computability is open in most cases, though with one notable example of a com-putable bisimulation metric for timed automata, as shown by Henzinger, Majumdar,and Prabhu in 2005.When turning from bisimulation to simulation, the right notion to use is a hemi-metric, inducing a structure of directed space. One interesting feature here is thatthe topology encodes the (metric) distance, whereas the directed structure encodesthe simulation, hence there is a certain separation of information. Again, we exposedistance-preserving dimaps from timed transition systems to timed languages.The work presented here is in progress; we are especially interested in working outdecidability issues. The relations to directed topology are, we hope, interesting, andcould open up for new application areas of directed topology. Lisbeth Fajstrup, Directed spaces generated by directed cubes Directed topology is studied in many different settings: Locally partially orderedspaces, d-spaces, cubical sets,... The category d− Space, of topological spaces witha selected subset of the set of paths denoted the directed paths or dipaths has niceproperties for directed homotopy, e.g. a Van Kampen theorem proved by M. Grandis.However, it seems to be too big for some purposes. In particular directed coveringtheory, or delooping from the computer science point of view is not well understoodin such a general setting. In “A convenient category for Directed Topology”, F. andJ.Rosicky, we introduce the subcategory d− SpaceB of d-Spaces, which are generatedby directed cubes in the following sense:A directed cube is a product B = I1 × · · · × In, where Ij is the unit interval witheither the discrete order, a ≤ b ⇔ a = b or the interval with the standard order.A directed path is a continuous map γ : I → B, where I has the standard order,respecting the order in each coordinate.B is the full subcategory of d− Space of such cubes.A d-Space X is in d− SpaceB if U ⊂ X open if and only if φ−1(U) is open in Bfor all d-maps φ : B → X where B is in B. Moreover, the dipaths in X should be theimage of dipaths under φ : B → X or concatenations of such.The category d− SpaceB contains geometric realization of cubical complexes, whichare needed in applications as models for Higher Dimensional Automatafor instance. ALGEBRAIC TOPOLOGICAL METHODS IN COMPUTER SCIENCE (ATMCS) III 11 And it avoids some of the pathological examples of d-Spaces, so one should expect thisto be a convenient category. The analogy with compactly generated spaces is obvious,and a study of directed mapping spaces in this setting should be fruitful. Some of theproperties already discovered are:• d− SpaceB is locally presentable.• The inclusion I : d− SpaceB → d− Space has a right adjoint, the Boxificationfunctor, which takes a d-Space X to the d-Space B(X) with topology given asabove by the d-maps of cubes into X (B(X) has more opens than X) and dipathsthe same as in X. For d-map f , B(f) = f as maps of sets.• A dicovering, p : Y → X in d-Space is a d-map with unique lift of dipaths and di-homotopies of dipaths with fixed initial point. A pointed space X ∈ d− SpaceB,x ∈ X is wellpointed if all points in X are reachable by a dipath initiating inx. For a wellpointed space X,x, there is a universal dicovering π : X̃ → Xwith X̃ ∈ d− SpaceB wellpointed, and in fact π : I(X̃)→ I(X) is universal fordicoverings of I(X) in pd-Space, the category of pointed d-Spaces. Michael Farber, Topology of motion planning algorithms I will discuss problems of algebraic topology motivated by the task of designingmotion planning algorithms in robotics. A motion planning algorithm is a rule associ-ating with any pair (A,B) of states of the system a continuous motion of the systemstarting at the state A and ending at the state B. The complexity of the problem ofconstructing a motion planning algorithm is measured by a numerical invariant TC(X)where X denotes the configuration space of the system. The notation TC stands fortopological complexity. TC(X) depends only on the homotopy type of the configura-tion space X of the system and can be estimated and computed using various tools ofmodern algebraic topology including cohomology algebras and cohomology operations.In the talk I will present results about computing the topological complexity TC(X)for several practically important problems: (1) Simultaneous control of many objects;(2) Collision free control of multiple objects moving in R or in R; (3) Collision freecontrol of multiple objects moving along a fixed graph; (4) Motion planning algorithmsfor collision free control of multiple objects in the presence of several moving obstacles. Robin Forman, Topology and the game of twenty questions In the usual game of 20 questions, one player tries to determine a hidden objectby asking a series of ”yes or no” questions. A wide variety of binary searches havethis general form. In applications, one is usually limited to a predetermined set ofquestions, and one is not required to determine the hidden object precisely, but ratheronly up to some equivalence. This is this game we will examine.We will assume that one can complete the task if one is permitted to ask all of theallowable questions. The question we will investigate is Is it possible to do better?That is, can one complete the task without asking all of the allowable questions? Ifnot, the problem is called /evasive/.Our approach is to restate the problem in a more topological form. We will thendefine a new homology theory, generalizing simplicial homology, that captures thedifficulty of solving this problem. The link between the homology theory and theoriginal search problem is provided by a generalization of ”Discrete Morse Theory.”This work is an extension and refinement of a line of work beginning with the paperof Kahn, Saks and Sturtevant “A topological approach to evasiveness.” 12 ALGEBRAIC TOPOLOGICAL METHODS IN COMPUTER SCIENCE (ATMCS) III Jennifer Gamble (joint work with Dr. Giseon Heo) Persistenthomology methods in longitudinal shape analysis In the field of statistical shape analysis, groups of objects are compared in orderto determine whether their mean shapes differ in a statistically significant way. Inthis context, ”shape” refers to geometric information that is invariant to translations,rotations and scaling. A number of statistical and mathematical methods have beendeveloped to facilitate this ”comparison” between groups. Common areas of appli-cation are medicine and biology, with the objects of interest represented in the formof landmark configurations. A usual method is to transform these configurations intosingle points in some ”shape space”, and then perform statistical analyses on the cor-responding tangent space approximations (using a pooled mean shape as the pole).In this setting, performing longitudinal analysis to compare multiple groups over timecan be difficult, as the number of variables is often much larger than the number ofsubjects, with complicated covariance structure. In this poster we will explore the use of algebraic topological methods, particularly persistent homology, to analyze shapechanges in groups over time. An orthodontic dataset involving three treatment groupswill be used as an example. Using the tangent space coordinates we will use thesoftware PLEX to analyze the relationship between our treatment groups over time.Specifically, we will contrast the relationships between the groups (with each subjectsconfiguration represented by a point/vertex in the tangent space) at multiple timepoints, by analyzing the differences in their persistent homology features. In caseswhere there are large deformations between the different shapes, tangent space ap-proximations can be inappropriate. We will also discuss alternative distance measuresdirectly in the shape space, and methods of dimensionality reduction. Philippe Gaucher, Cubes, homotopy, and process algebra In directed algebraic topology, the concurrent execution of n actions is abstractedby a full n-cube. Each coordinate corresponds to one of the n actions. This n-cube maybe viewed as a representable presheaf of the category of precubical sets, as a topological n-cube equipped with some continuous paths modelling the possible execution pathsup to homotopy, and as a commuting n-cube, i.e usually the small category associatedwith the poset of vertices of the n-cube. In fact, we have to remove the identity mapsfor various reasons, e.g., because the full n-cube does not contain any loop. In this talk,all these points of view are related to one another by considering Milner’s calculus ofcommunicating systems (CCS). All operators of this process algebra are given a higherdimensional interpretation. The restriction to dimension 1 corresponds to the usualstructural operational semantics. Daniela Giorgi, Computation and application issues inmultidimensional shape description In Computer Graphics and Vision, computational-topology offers a theoreticalframework for the formalization and solution of problems related to shape analysis,description and comparison. Methods that make use of the properties of real functionsdefined on the shape [1] are often well suited to describe objects that are non-rigidlyrelated to each other. The role of these functions, that we may call measuring func-tions, is to measure quantitative geometric properties of the shape, while taking intoaccount its topology. ALGEBRAIC TOPOLOGICAL METHODS IN COMPUTER SCIENCE (ATMCS) III 13 Recent advances in Size Theory have shown that it is possible to derive a conciseand informative geometrical-topological shape descriptor also in the case of multi-variate measuring functions, that is functions taking values in R [2]. The idea ofusing k-dimensional measuring functions arises from the observation that the shape ofan object can be better characterized by means of a set of functions, each investigatingspecific features of the shape under study. For instance, scientific simulations of realphenomena typically require the analysis of a huge and composite amount of data. Atthe same time, there are properties that are naturally k-dimensional: a first exampleis colour, which lives in the 3-dimensional RGB space. The possibility of working fromthe beginning with k-dimensional functions, instead of merging the information of kseparate functions a posteriori, allows one to produce a single descriptor containing theinformation of the k functions at the same time. In other words, k different functionsconcur to produce a single descriptor.In this talk, we will deal with the main issues related to the application in a dis-crete setting of the concepts introduced in Multidimensional Size Theory, in particularmultidimensional size functions and multidimensional matching distances. A compu-tational scheme coherent with the mathematical model will be given, highlighting thatthe technique proposed is able to deal with different model representations, such assimplicial complexes and digital spaces. Experimental results will be provided so as toillustrate the feasibility of the approach, in terms of storage space, computation time and efficacy of description. Finally, different families of measuring functions will beanalyzed, in the light of their capability to capture salient shape features.