ICMS » Workshop

Workshop

Applied and computational topology: ATMCS 5

Jul 02, 2012 - Jul 06, 2012

ICMS, 15 South College Street Edinburgh

Organisers

Name	Institution
Carlsson, Gunnar	Stanford University
Chazal, Frederic	INRIA Saclay - Île-de-France
Farber, Michael	University of Warwick
Ghrist, Robert	University of Pennsylvania
Jardine, Rick	University of Western Ontario
Mischaikow, Konstantin	Rutgers, State University of New Jersey
Vejdemo-Johansson, Mikael	University of St Andrews

ATMCS 5 is the 5th conference in a conference series on Algebraic Topology, its role in Computer Science, and its applications. The acronym changed interpretation for ATMCS 4 in 2010: before then, it was called Algebraic Topological Methods in Computer Science, and since then it has been Algebraic Topology: Methods, Computation and Science.

Previous conferences in the series are:

2001 ATMCS 2001, Stanford, CA, USA
2004 ATMCS 2004, London, Ontario, Canada
2008 ATMCS 2008, Paris, France
2010 ATMCS 2010, Münster, Germany

Furthermore there have been workshops organized under the heading Geometric and Topological Methods in Concurrency (and Distributed Systems) in Aalborg, Denmark (1999 and 2001), Penn State, PA, USA (2000), Toulouse, France (2002), Marseille, France (2003), Amsterdam, the Netherlands (2004), San Francisco, CA, USA (2005), Bonn, Germany (2006) and Aalborg, Denmark (2010).

Applied and computational topology refers to the adaptation of topological ideas and techniques to study problems in science and engineering. The further development of topological techniques for use in applications and the creation of new areas of application in the subject are amongst the goals of this workshop. The workshop will bring together leading researchers in this emerging discipline as well as providing an opportunity for young mathematicians to get involved in it.

In parallel with the academic programme there will be an exhibition of mathematical art, showcasing work by Radmila Sazdanovic, Mikael Vejdemo-Johansson and Madeleine Shepherd & Julia Collins. We are looking for, visual artists, craft workers, designers and film makers who create work with significant mathematical component to complement their work. The Call for Submissions of Mathematical Art is open until 31 May 2012. Download the pdf containing the full details here.

Arrangements

Venue
The workshop will be held at 15 South College Street, Edinburgh. You may also find this map useful for the workshop.

ALL INVITED SPEAKERS SHOULD HAVE RECEIVED AN INVITATION EMAIL FROM ICMS AND SHOULD REGISTER USING THE UNIQUE LINK CONTAINED IN THAT MESSAGE.

Invited speakers:
If your invitation included the offer of accommodation, then ICMS will arrange this and details will be in your letter of invitation. ICMS has reserved rooms for all delegates in University of Edinburgh en suite accommodation. All accommodation is no more than a twenty minute walk from ICMS. Please indicate on your booking form if you wish to take up the offer of these rooms or make your own arrangements.

Meals and Refreshments
Morning and afternoon refreshments will be provided on each day of the workshop. A buffet lunch will be provided at ICMS in the Chapterhouse on the ground floor of 15 South College Street on Monday and Tuesday. A workshop dinner will be held on Wednesday. The workshop grant will cover the cost of this catering.

Registration Fee and Financial Matters
A registration fee of £50 is payable by all invited speakers. This can be paid by completing the details on this credit/debit card payment form. As email is not secure, please print the form and return it by fax using the fax number at the top of the form. ICMS will take your payment and a receipt will be given to you at Registration. Alternatively, we can accept sterling cheques, made payable to Heriot-Watt University but sent to Audrey Brown, ICMS, 15 South College Street, Edinburgh, EH8 9AA. If you have any difficulty in paying in advance, please email Audrey Brown.

INFORMATION FOR ALL OTHER PARTICIPANTS

Registration and Financial Matters
***PLEASE NOTE THAT THIS WORKSHOP IS NOW CLOSED FOR APPLICATIONS***

The registration fee for other participants is 100 GBP. The registration fee covers the cost of lunch on Monday and Tuesday, a wine reception on Monday evening, coffee and tea each day and a workshop dinner on Wednesday evening.

The registration fees can be paid by completing the details on this credit/debit card payment form. As email is not secure, please print the form and return it by fax using the fax number at the top of the form. ICMS will take your payment and a receipt will be given to you at Registration. Alternatively, we can accept sterling cheques, made payable to Heriot-Watt University but sent to Audrey Brown, ICMS, 15 South College Street, Edinburgh, EH8 9AA. If you have any difficulty in paying in advance, please email Audrey Brown: audrey.brown@icms.org.uk.

Accommodation
ICMS cannot reserve accommodation for you. Suggestions for accommodation close to ICMS can be found here.

Visas
ICMS asks attendees to check if they require a visa for entry to the United Kingdom. This government website gives more information. If you do require a visa ICMS will provide you with a letter in support of your application.

Travel
Information about travel to the UK and Edinburgh is available here. Lothian buses charge 1.40 GBP for a single journey and 3.50 GBP for a day ticket. Please note that the exact fare is required and no change is given.

A taxi directly from the airport will cost approximately 18.00 to 20.00 GBP to the city centre for a one-way journey.

From the airport, you can take the airport bus Airlink to the city centre. It will cost you 3.50 GBP for a single ticket, or 6.00 GBP return. More details on the airport bus Airlink is available here. The bus will take you to Waverley Bridge (next to Waverley Railway Station).

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue at ICMS and George Square. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge map. The second railway station is called Haymarket and is at the West End of the city centre.

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies that may occur on your trip.

Lecture theatre facilities
The lecture theatre is fitted with data projection, visualiser (the new generation of over-head), smart-board and blackboard. More information about the facilities are available from Audrey Brown.

Tourist attractions and events
If you are arriving in the days prior to the workshop or staying after the event, you may be interested in some of these Scottish tourist events or attractions.

Programme

Please note that this Programme may be subject to change.

Monday 2 July

09.00 - 09.50	Registration and coffee
09.50 - 10.00	Introductions and welcome
10.00 - 11.00	Frederic Chazal (INRIA Saclay - Île-de-France) Detection and approximation of linear structures in metric spaces
11.00 - 11.30	Coffee/Tea in the Chapterhouse, Level 1
11.30 - 12.30	Stephen Smale (City University of Hong Kong) Topology and immunology
12.30 - 14.00	Lunch provided in the Chapterhouse, Level 1

	Parallel Sessions - Newhaven Lecture Theatre, Level 4
14.00 - 14.30	Matthew Arnold (University of Bristol) Statistical aspects of persistent homology
14.35 - 15.05	Giseon Heo (University of Alberta) Topological analysis of variance with applications in landmark data set
15.10 - 15.40	Yasuaki Hiraoka (Kyushu University) Applications of persistent diagrams to protein compressibility and phylogenetic trees
15.45 - 16.15	Coffee/Tea in the Chapterhouse, Level 1
16.15 - 16.45	Daniel Müllner (Stanford University) Consistent scale selection for exploratory visualization and analysis of data sets
16.50 - 17.20	Monica Nicolau (Stanford University) Unraveling the biology of disease through data transformations and topological data analysis

	Parallel Sessions - Cramond Room, Level 2
14.00 - 14.30	Marek Krcal (Charles University) Computing all maps into a sphere
14.35 - 15.05	Vidit Nanda (Rutgers University) Discrete Morse theory for persistence
15.10 - 15.40	Facundo Mémoli (University of Adelaide) Metric geometry and persistent topology
15.45 - 16.15	Coffee/Tea in the Chapterhouse, Level 1
16.15 - 16.45	Ulrich Bauer (Institute of Science and Technology Austria) Topological simplification problems
16.50 - 17.20	Michal Adamaszek (University of Warwick) Homology and complexity

17.30 - 19.00	Wine reception

Tuesday 3 July

09.00 - 10.00	Nati Linial (Hebrew University of Jerusalem) Random simplicial complexes
10.00 - 11.00	Matthew Kahle (Ohio State University) The topological signature of randomness
11.00 - 11.30	Coffee/Tea in the Chapterhouse, Level 1
11.30 - 12.30	Daniel Cohen (Louisiana State University) Nonaspherical random 2-complexes
12.30	Free afternoon

Wednesday 4 July

09.00 - 10.00	Jean-Daniel Boissonnat (INRIA, Sophia Antipolis) Delaunay-type structures for manifolds
10.00 - 11.00	Sayan Mukherjee (Duke University) Probability measures on the space of persistence diagrams
11.00 - 11.30	Coffee/Tea in the Chapterhouse, Level 1
11.30 - 12.30	Andrzej Szymczak (Colorado School of Mines) Piecewise constant vector field topology
12.30 - 14.30	Lunch provided in the Chapterhouse, Level 1
14.35 - 15.05	Leonidas Guibas (Stanford University) Understanding shapes through mappings
15.10 - 15.40	Coffee/Tea in the Chapterhouse, Level 1

	Parallel Sessions - Newhaven Lecture Theatre, Level 4
15.45 - 16.15	Jesus Gonzalez (Center for Research and Advanced Studies) Immersion dimension and topological complexity of projective product spaces
16.15 - 16.45	Yuli Rudyak (University of Florida) On higher topological complexity and configuration spaces
16.50 - 17.20	Dirk Schuetz (Durham University) Homology of moduli spaces of linkages in high-dimensional euclidean space

	Parallel Sessions - Cramond Room, Level 2
15.45 - 16.15	Wojciech Chacholski (Kungliga Tekniska högskolan) Minimal resolutions of graded modules and multi bar codes
16.15 - 16.45	Claudia Landi (Università di Modena e Reggio Emilia) Uniqueness of models in persistent homology
16.50 - 17.20	Tomasz Kaczynski (Université de Sherbrooke) Suspension of a measuring function
19.00	Workshop Dinner at Playfair Library, Old College, South Bridge, Edinburgh

Thursday 5 July

09.00 - 10.00	Rien van de Weijgaert (Groningen University) The geometry and topology of the cosmic web
10.00 - 11.00	Shmuel Weinberger (University of Chicago) Disordered solids and the dynamics of bounded geometry
11.00 - 11.30	Coffee/Tea in the Chapterhouse, Level 1
11.30 - 12.30	Sanjeevi Krishnan (University of Pennsylvania) Poincaré duality and linear optimization
12.30 - 14.30	Lunch provided in the Chapterhouse, Level 1
14.35 - 15.05	Eric Goubault (CEA LIST, Saclay) Recent advances in directed algebraic topology, with applications to concurrent and distributed systems
15.10 - 15.40	Coffee/Tea in the Chapterhouse, Level 1

	Parallel Sessions - Newhaven Lecture Theatre, Level 4
15.45 - 16.15	Dai Tamaki (Shinshu University) A Salvetti-type model for configuration spaces
16.15 - 16.45	Lisbeth Fajstrup (Aalborg University) Periodicity and the trace space algorithm
16.50 - 17.20	Felix Akinboade (Federal College of Education (Technical),Gombe) Locality and checkability in wait-free computing

	Parallel Sessions - Cramond Room, Level 2
15.45 - 16.15	Hubert Wagner (Jagiellonian University/IST Austria) Persistent homology for image data
16.15 - 16.45	Miroslav Kramar (Rutgers University) The dynamics of granular materials
16.50 - 17.20	William Obeng-Denteh (Kwame Nkrumah University of Science and Technology) Topological dynamics in the contemporary era

Friday 6 July

09.00 - 10.00	Marian Mrozek (Jagiellonian University) Homological persistence of maps
10.00 - 11.00	Primoz Skraba (Jozef Stefan Institute) and Mikael Vejdemo-Johansson (University of St Andrews) TBA
11.00 - 11.30	Coffee/Tea in the Chapterhouse, Level 1
11.30 - 12.30	Vin de Silva (Pomona College) Persistent cohomology and circle-valued functions
12.30 - 14.30	Lunch provided in the Chapterhouse, Level 1
14.35 - 15.05	Herbert Edelsbrunner (Institute of Science and Technology Austria) Adaptive triangulation of a digital image
15.10 - 15.40	Coffee/Tea in the Chapterhouse, Level 1

	Parallel Sessions - Newhaven Lecture Theatre, Level 4
15.45 - 16.15	Pawel Dlotko (Jagiellonian University) Computational (co)homology – applications and recent progress in computations
16.15 - 16.45	Donald Sheehy (INRIA Saclay - Île-de-France) Linear-size approximations to the Vietoris-Rips filtration
16.50 - 17.20	Amit Patel (Rutgers University) Well groups and the homological stability for mappings to manifolds

	Parallel Sessions - Cramond Room, Level 2
15.45 - 16.15	David Balduzzi (Max Planck Institute for Intelligent Systems) An information-theoretic presheaf for distributed learning
16.15 - 16.45	Justin Curry (University of Pennsylvania) Obstruction-theoretic sensing
16.50 - 17.20	SangEon Han (Chonbuk National University) A new type of locally finite topological space and its applications

Presentations:

Presentation Details
Adamaszek, Michal
Homology and complexity
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We look for the boundary between easy (polynomial) and difficult (NP-hard) instances of computational problems for simplicial complexes. We take a combinatorial approach and exhibit a class of clique complexes of a natural class of graphs such that deciding their contractibility is polynomial but deciding triviality of any specified homology group is NP-hard. As a consequence we also prove that computing homology of arbitrary simplicial complexes given as a list of facets is NP-hard. (Joint work with Juraj Stacho).
Akinboade, Felix
Locality and checkability in wait-free computing
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This paper studies notions of locality that are inherent to the specification of a distributed task and independent of the computing environment, in a shared memory wait-free system. A task T =(I,O,D) is checkable if there exists a wait-free distributed algorithm that, given s in I and t in O, determines whether t in D(s), i.e., if t is a valid output for s according to the specification D of T. Determining whether a projection-closed task is wait-free solvable remains undecidable, demonstrating the richness of this class. A locality property called projection-closed is identified, that completely characterizes tasks that are wait-free checkable. A stronger notion of locality considers tasks where the outputs look identically to the inputs at every vertex (input value of a process). A task T =(I,O,D) is said to be locality-preserving} if O is a covering complex of I. This topological property yields obstacles for wait-free solvability different in nature from the classical agreement impossibility results. On the other hand, locality-preserving tasks are projection-closed and therefore always wait-free checkable. A classification of locality-preserving tasks in term of their relative computational power is provided. A correspondence between locality-preserving tasks and subgroups of the edgepath group of an input complex shows the existence of hierarchies of locality-preserving tasks, each one containing at the top the universal task (induced by the universal covering complex), and at the bottom the trivial identity task.
Arnold, Matthew
Statistical aspects of persistent homology
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We are studying statistical aspects of persistent homology, within the framework of "Topological Data Analysis". For points sampled uniformly at random from a topological object in Euclidean space, the persistence barcodes contain some intervals that represent topological "features" and some intervals that represent "noise" which occurs as a result of our uniform sampling strategy. We investigate the null distribution of noisy intervals for uniform samples, by investigating the distribution of the full barcode for a uniform sample from the unit square. This directs us towards algorithmically identifying noisy intervals in a given barcode, so that the remaining intervals are classified as actual features of the underlying space. Related to this is the study of how the noisy intervals change when the sampled points are not uniformly sampled from the object. By using a kernel density estimate of the sample we can correct for the non-uniformity to return barcodes that behave similarly to those we would expect if we could have sampled uniformly. This allows us to identify the noisy intervals using the null distribution described earlier. Some results in both of these areas will be presented.
Balduzzi, David
An information-theoretic presheaf for distributed learning
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Statistical learning theory bounds the expected error of a learning algorithm (on unseen data) based on the number of errors on a finite training set and the complexity of the algorithm, quantified by measures such as VC (Vapnik-Chervonenkis) entropy. Recently, attention has shifted from individual learning algorithms to distributed systems of learners such as deep belief networks, where theoretical guarantees are notably lacking. In my talk, I will formalize learning as performing an abstract measurement. VC entropy turns out to quantify the information generated by the measurement. The focus of the talk is then on the information-theoretic structure of distributed measurements, which is captured by the quale: a family of sections of a suitably defined presheaf. Using the quale I geometrically represent (i) the information content of a measurement; (ii) the extent to which a measurement is context-dependent; and (iii) whether a measurement decomposes into independent submeasurements. Finally, I show that only indecomposable measurements are more informative than the sum of their submeasurements -- which has implications for the VC-entropy of distributed systems and, therefore, bounds on their expected error.
Bauer, Ulrich
Topological simplification problems
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We show that function simplification on surfaces is easy: given a function f on a surface and a tolerance d>0, we construct a function g subject to \|g-f\|<=d such that g has a minimum number of critical points. Our construction relies on a connection between discrete Morse theory and persistent homology. The simplified function g achieves the lower bound dictated by the stability theorem of persistent homology. The solution can be computed in linear time after persistence pairs have been computed. (Joint work with Carsten Lange and Max Wardetzky) On the other hand, we show that isosurface simplification is hard: given a function f on a subset of R^3 and a tolerance d > 0, it is NP-hard to find a function g subject to \|g-f\|<=d such that the t-level set {x: g(x)=t} minimizes the total Betti number. The proof uses a reduction from the homological factorization problem: given a pair of spaces L c K, is there a space L c X c K such that the induced homomorphisms of L c X and X c K are surjective and injective, respectively? This problem is shown to be NP-complete even for simplicial pairs embeddable in R^3. (Joint work with Nina Amenta, Dominique Attali, Olivier Devillers, Marc Glisse, and André Lieutier)
Boissonnat, Jean-Daniel
Delaunay-type structures for manifolds
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To follow
Chacholski, Wojciech
Minimal resolutions of graded modules and multi bar codes
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A polynomial ring over a filed is a PID. This is the main reason why persistent homology can be encoded by a simple bar code. To understand multi persistence one needs to deal with rings of higher projective dimension. However a polynomial ring in several variables is not the right choice. Instead one should look at the representation of the product of the poset of natural numbers. Homological properties of the category of its representations are much easier than for polynomial rings. The aim of this talk is to discuss those properties, in particular illustrate how easy it is to construct minimal resolutions in this category. These resolutions are the multi bar codes. This is a join work with F.Vacarino and M. Scolamiero.
Chazal, Frederic
Detection and approximation of linear structures in metric spaces
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In many real-world applications data come as discrete metric spaces sampled around 1-dimensional linear structures (metric trees or graphs). Building on elementary tools of the theory of delta-hyperbolic spaces introduced by M Gromov in 1987, we provide a framework in which one can measure how much a given data set can be approximated by 1-dimensional metric structures and we provide algorithms to reconstruct such approximating structures. We will also illustrate the performances of our algorithm on various data sets. This is a joint work with Jian Sun (Mathematical Sciences Center, Tsinghua University).
Cohen, Daniel
Nonaspherical random 2-complexes
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Random 2-dimensional subcomplexes of the n-simplex typically either have the homotopy type of a 1-dimensional complex, or have nontrivial second homotopy group.
Curry, Justin
Obstruction-theoretic sensing
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This talk outlines a vision for obstruction-theoretic sensing. Motivation for such a proposal stems from two primary observations: (1) that modern applications increasingly require a systematic way of handling data living over a base space and (2) that the classical machinery of sheaves and vector bundles provides a unified language as well as topological forcing results. Specifically, constructible (co)sheaves are used for their combinatorial structure, effective computability and rich connections to representation theory. The heart of the talk is in describing a new class of pursuit-and-evasion problems, which serves as a testing ground for these ideas.
de Silva, Vin
Persistent cohomology and circle-valued functions
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I will discuss the use of persistent cohomology to discover meaningful circle-valued coordinates on a data set, in the spirit of nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE, and Laplacian Eigenmaps. The construction is closely related to the Abel-Jacobi map in algebraic geometry. There are possible applications to the study and simulation of dynamical systems. My collaborators in this work (some of it completed, some of it in progress) include Morozov, Vejdemo-Johansson, Skraba, Mischaikow.
Dlotko, Pawel
Computational (co)homology – applications and recent progress in computations
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Computational homology and cohomology have recently become important tools in solving many problems in science and engineering. Consequently, new, efficient algorithms are needed to handle the amounts of data coming from real world problems. In this talk some recent developments and algorithms will be presented. I will concentrate on new methods to partially parallelize, distribute and speed up (co)homology and persistent (co)homology computations. Moreover a quick survey of recent applications of presented theory and algorithms will be given.
Edelsbrunner, Herbert
Adaptive triangulation of a digital image
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To enjoy a digital version of the Jordan Curve Theorem, it is common to use the closed topology for the foreground and the open topology for the background of a 2-dimensional binary image. We introduce a single topology that enjoys this theorem for all thresholds decomposing a real-valued image into fore- and background. The topology is easy to construct and it generalizes to n-dimensional images. (Joint work with Olga Symonova.)
Fajstrup, Lisbeth
Periodicity and the trace space algorithm
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The executions of a parallel program is the space of directed paths or traces in a particular d-space. The topological invariants of the trace spaces are invariants of the program. For a non-looped, non-branching program, it is known how to generate a simplicial model of the trace space from a certain index category. For looped processes, the state space is a (directed)torus and the trace space is a disjoint union of trace spaces of deloopings. The index category for the trace space of the deloopings is developed from the once delooped case. When just one process is looped, the index category is generated as words in a regular language. The automaton is constructed.
Gonzalez, Jesus
Immersion dimension and topological complexity of projective product spaces
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It is well known that Farber's topological complexity of real projective spaces agrees with the Euclidean immersion dimension of these manifolds. However this is an isolated phenomenon; even Davis' projective product spaces fail to have such a property. Indeed, for those manifolds, we characterize the immersion dimension and their topological complexity in terms of suitable generalizations of axial maps, from which the (large) differences between both invariants are apparent.
Goubault, Eric
Recent advances in directed algebraic topology, with applications to concurrent and distributed systems
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I will give a survey of some of the recent results in directed algebraic topology, with a view to practical computations of topological invariants (trace spaces, component categories etc.) and their use for proving properties on concurrent and distributed programs.
Guibas, Leonidas
Understanding shapes through mappings
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We discuss a number of ways of understanding shape collections jointly, through the mechanism of (1) finding novel representations and algorithms for maps between the shapes, and (2) investigating the resulting map networks so as to evaluate each map in the context of other maps.
Han, SangEon
A new type of locally finite topological space and its applications
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In the fields of pure and applied topology we have the question: Is there a topological structure which can be used for studying both continuous and digital spaces? As an answer, this talk suggests a new type of locally finite topological space (or space set topology). Since the structure can play an important role in classical, computer, discrete and digital geometry as well as digital topology, the paper proposes a Jordan curve theorem of a simple closed SST-curve. Finally, the paper shows that an SST can be used for studying both continuous and digital spaces so that it plays an important role in both classical and digital topology and further, discrete and computational geometry.
Heo, Giseon
Topological analysis of variance with applications in landmark data set
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It is common to reduce the dimensionality of data before applying classical multivariate analysis techniques in statistics. Persistent homology, a recent development in computational topology, has shown to be useful for analyzing high dimensional (non-linear) data. In this talk, we connect computational topology with the traditional analysis of variance and demonstrate this synergy on a three-dimensional orthodontic landmark data set derived from the maxillary complex. Combining appropriate techniques of both persistent homology and analysis of variance results in a better understanding of the data's non-linear features over and above what could have been achieved by classical means.
Hiraoka, Yasuaki
Applications of persistent diagrams to protein compressibility and phylogenetic trees
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In this talk, we will present our recent research applying persistent diagrams to protein analysis. The first subject is the compressibility of proteins. We will show that we can derive a topological quantity from persistent diagrams of proteins which has a good linear correlation to the experimental data of compressibility. Next subject is phylogenetic trees of proteins. Based on the distance function on persistent diagrams, we can define a tree for a given group of proteins. This tree possesses more information of three dimensional geometric/topological structures of proteins than a conventional phylogenetic trees derived by a one dimensional amino acid sequences. We show some examples which classify hereditary properties. We also discuss our trial to detect hereditarily important regions in a protein structure by persistent diagrams.
Kaczynski, Tomasz
Suspension of a measuring function
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Given a measuring function f on M with values in R, we construct a measuring function g on the based topological suspension of M, also with values in R, and with the property that its rank invariants are those of f with the homology dimension q shifted by one. In particular, the 0-dimensional rank invariant, equivalently, the size function of g is trivial, and its one dimensional rank invariant provides the same information as the size function of f. We use this as a simple model for generating examples aimed at testing various topological invariants of shape for efficiency. In particular, our construction gives some insight into a discussion on what is a better tool in shape comparison: the rank invariant studied for several homology dimensions or the multidimensional size function.
Kahle, Matthew
The topological signature of randomness
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We will survey the emerging field of probabilistic topology. This will include recent work which shows that almost all d-dimensional flag complexes have nontrivial homology (with real coefficients), only in degree [d/2].
Kramar, Miroslav
The dynamics of granular materials
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We present a novel approach based on topological measurements to study force chain structures of particulate systems. We concentrate on systems undergoing compression. Our method distinguishes different types of friction as well as different amounts of polydispersity. These topological measures can be also used to understand the dynamic features of the system and can be correlated to phenomena such as jamming.
Krcal, Marek
Computing all maps into a sphere
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We present an algorithm for computing [X,Y], i.e., all homotopy classes of continuous maps X -> Y, where X,Y are topological spaces given as finite simplicial complexes, Y is (d-1)-connected for some d>1 (for example, Y can be the d-dimensional sphere S^d), and dim X < 2d-1. These conditions on X,Y guarantee that [X,Y] has a natural structure of a finitely generated Abelian group, and the algorithm finds generators and relations for it. We combine several tools and ideas from homotopy theory (such as Postnikov systems, simplicial sets, and obstruction theory) with algorithmic tools from effective algebraic topology (spaces with effective homology). We hope that a further extension of the methods developed here will yield an algorithm for computing, in some cases of interest, the Z_2-index, which is a quantity playing a prominent role in Borsuk--Ulam style applications of topology in combinatorics and geometry, e.g., in topological lower bounds for the chromatic number of a graph. In a certain range of dimensions, deciding the embeddability of a simplicial complex into R^d also amounts to a {Z_2-index} computation. This is the main motivation of our work.
Krishnan, Sanjeevi
Poincaré duality and linear optimization
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In this talk, I will recast and generalize certain dualities in linear optimization, such as max-flow min-cut, as a form of twisted Poincaré Duality for spacetimes and more singular "directed spaces." Flows correspond to the top-dimensional homology, taking local coefficients and values in a sheaves of semigroups, on directed spaces. Cuts correspond to certain distinguished sections of a dualizing sheaf. Thus max-flow min-cut dualities extend to higher dimensional analogues of flows, higher dimensional analogues of directed graphs, and constraints more complicated than upper bounds. I will describe the formal result and some real-world applications.
Landi, Claudia
Uniqueness of models in persistent homology
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We consider the question whether the rank of persistent homology groups of a function f defined on a minifold M uniquely characterizes f, at least up to self-homeorphisms of the manifold. For a function f taking values in IR this is easily seen to be false. However, we can give partially positive answers to this question for vector-valued functions. Our uniqueness problem is clearly strictly related to the decision problem in shape matching, that is, given two patterns, deciding whether there exists a transformation taking one pattern to the other pattern. Rephrased differently, we wish to study to which extent persistent homology can give rise to complete shape invariants.
Linial, Nati
Random simplicial complexes
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For the last several years we have been investigating a model of random simplicial complexes that is the higher-dimensional counterpart of the classical Erdos-Renyi model from graph theory. Several new phenomena have already been discovered, but many more questions remain open. I will try to give an overview of this emerging theory. My talk covers work done jointly with R. Meshulam T Luczak and L Aronshtam.
Mémoli, Facundo
Metric geometry and persistent topology
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We study questions regarding the stability of persistence diagrams arising from pseudo-metric data. One first example is the stability of persistence diagrams of Vietoris-Rips filtrations. These can be proved to be stable in the Gromov- Hausdorff sense. Similar stability results hold for more general filtrations that arise from finite metric measure spaces, in which case the stability is w.r.t. the Gromov-Wasserstein metric. These stability results provide lower bounds for the GH distance, whose computation in general leads to NP-hard problems. These results have applications in data analysis and shape classification.
Mrozek, Marian
Homological persistence of maps
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When a topological space is known only from sampling, persistence provides a useful tool to study its homological properties. In many applications one can sample not only the space, but also a map acting on the space. The understanding of the topological features of the map is often of interest, in particular in time series analysis. We recall the concept of persistence in finite dimensional vector spaces and use it to study the persistence of eigenspaces of maps induced in homology. Our technique is based on the graph approach to computing homology of maps. (This is joint work with Herbert Edelsbrunner and Grzegorz Jablonski)
Mukherjee, Sayan
Probability measures on the space of persistence diagrams
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This paper shows that the space of persistence diagrams has properties that allow for the definition of probability measures which support expectations, variances, percentiles and conditional probabilities. This provides a theoretical basis for a statistical treatment of persistence diagrams, for example computing sample averages and sample variances of persistence diagrams. We first prove that the space of persistence diagrams with the Wasserstein metric is complete and separable. We then prove a simple criterion for compactness in this space. These facts allow us to show the existence of the standard statistical objects needed to extend the theory of topological persistence to a much larger set of applications. We also develop algorithms to compute Frechet means a provide law of large numbers results for these means.
Müllner, Daniel
Consistent scale selection for exploratory visualization and analysis of data sets
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Choosing an appropriate scale is a frequently encountered problem in data analysis, and paradigms in the field support both the choice of strategies to make smart, definite choices and the hierarchical or persistence approach of looking at all scales at once. In the core of the “Mapper” algorithm for visualization and analysis of point cloud data, a scale choice must be made multiple times for overlapping fragments of the data set. We present the concept of a scale graph, where scale choices in neighbouring regions are brought into conjunction to make consistent decisions at local scale, while retaining global flexibility. By selecting an optimal path through the scale graph, we can make more plausible choices, overcome existing weaknesses, validate results more easily and simplify the data analysis process for the user. (Joint work with Gunnar Carlsson, Facundo Mémoli and Gurjeet Singh.)
Nanda, Vidit
Discrete Morse theory for persistence
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Computing persistent homology efficiently has become an integral part of modern topological data analysis. An extension of discrete Morse theory to filtrations produces a much smaller Morse filtration with isomorphic persistent homology groups.
Nicolau, Monica
Unraveling the biology of disease through data transformations and topological data analysis
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The past decade has witnessed developments in the field of biology that have brought about profound changes in understanding the dynamic of disease and of biological systems in general. New technology has given biologists an unprecedented wealth of information, but it has generated data that is hard to analyze mathematically, thereby making its biological interpretation difficult. These challenges have given rise to a myriad novel exciting mathematical problems and have provided an impetus to modify and adapt traditional mathematics tools, as well as develop novel techniques to tackle the data analysis problems raised in biology. I will discuss a general approach to address some of these computational challenges by way of a combination of data transformations and topological methods, to highlight specific biologically driven questions. These methods have been applied in a wide range of settings, in particular for the study of the biology of disease. I will discuss some concrete applications to these methods, including their use to discover a new type of breast cancer and the associated biology that drives the disease, identify disease progression trends, and highlighting the driving mechanisms in acute myeloid leukemia. Much of this is joint work with Gunnar Carlsson.
Obeng-Denteh, William
Topological dynamics in the contemporary era
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Topological dynamics is the study of asymptotic or long term properties of families of maps of topological spaces.This paper will treat us to the concepts of flows,minimal sets, minimal flows,proximality and distality.This is the prerequisite for appreciating examples of continuous actions and groups of homeomorphisms.
Patel, Amit
Well groups and the homological stability for mappings to manifolds
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The stability of homology plays a crucial role in topological data analysis. Given a mapping from a topological space to a manifold, we quantify the stability of the homology of each fiber. The well group is an algebraic construction that plays a fundamental role in this quantification.
Rudyak, Yuli
On higher topological complexity and configuration spaces
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In 2010 I developed the concept of Farber topological complexity, by introducing the so-called higher topological complexity. The symmetrization of this invariant leads to interesting geometry and, in particular, to research some cofiguration spaces. This is a joint work with Basabe, Gonzalez and Tamaki.
Schuetz, Dirk
Homology of moduli spaces of linkages in high-dimensional euclidean space
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We consider closed linkages in d-dimensional euclidean space, up to translations and rotations, and obtain topological information on the resulting moduli spaces and how they depend on the lengths of the links. In particular, we calculate the Poincare polynomial in the case d=5 in terms of the length vector determining the linkage.
Sheehy, Donald
Linear-size approximations to the Vietoris-Rips filtration
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The Vietoris-Rips filtration is a versatile tool in topological data analysis. Unfortunately, it is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an n-point metric space such that the persistence diagram is a good approximation to that of the Vietoris-Rips filtration. The filtration can be constructed in O(n log n) time. The constants depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets. Our approach uses a hierarchical net-tree to sparsify the filtration. We can either sparsify the data by throwing out points at larger scales to give a zigzag filtration, or sparsify the underlying graph by eliminating edges at larger scales to give a standard filtration. Both methods yield the same guarantees.
Smale, Stephen
Topology and immunology
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Topology of spaces of genes will be discussed.
Szymczak, Andrzej
Piecewise constant vector field topology
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The talk will focus on recent work on algorithms for computing Morse decompositions and approximating recurrent dynamics of piecewise constant vector fields, motivated by scientific visualization applications.
Tamaki, Dai
A Salvetti-type model for configuration spaces
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We propose a combinatorial/categorical model for configuration spaces of "good" cell complexes. The model is realized as the classifying space of a topological category by using the notion of cellular and stellar stratified spaces.
van de Weijgaert, Rien
The geometry and topology of the cosmic web
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The Cosmic Web is the fundamental spatial organization of matter on scales of a few up to a hundred Megaparsec, scales at which the Universe still resides in a state of moderate dynamical evolution. Galaxies, intergalactic gas and dark matter exist in a wispy weblike spatial arrangement consisting of dense compact clusters, elongated filaments, and sheetlike walls, amidst large near-empty void regions. While the complex intricate structure of the cosmic web contains a wealth of cosmological information, its quantification has remained a major challenge. In this lecture, we describe recent work towards invoking concepts from computational topology and computational geometry to our understanding of the structure of the Cosmic Web and to new insights into the conditions in the primordial Universe out of which it emerged. Voronoi and Delaunay tessellations form the basis of DTFE, the formalism to infer the density field of an anisotropic and multiscale mass distribution sampled by discrete points. I will describe the new Nexus multiscale morphology filter, based on DTFE, that helps to identify the filaments and walls in the maps of the galaxy distribution. Also, I will shortly describe the Watershed Void Finder, the Morse theory based technique to outline voids in the galaxy distribution. An important aspect of our understanding of the Cosmic Web concerns the connectivity of the various components. This leads us to recent work on the topological analysis of the Megaparsec scale distribution. To this end, we resort to the homology of the weblike structure, and determine the scale-dependent Betti numbers. To infer this from the discrete spatial galaxy distribution (or of particles in computer models of cosmic structure formation) we extract the Betti numbers from alpha shapes. We have studied the alpha complex of the cosmic weblike point patterns, in order to assess the signature of filaments, walls and voids. Of considerable importance within the context of the multiscale nature of the cosmic mass distribution, is the information contained in persistence diagrams. Amongst others, I will describe recent work on persistence properties of Gaussian and non-Gaussian random density fields, which form the initial conditions out of which the cosmic web emerged through the force of gravity.
Wagner, Hubert
Persistent homology for image data
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Persistent homology is a powerful tool and its applications include visualization and image analysis. However, it can be challenging to compute, especially for large 3-dimensional scans, which often contain billions of voxels. In this talk we focus on efficient algorithmic techniques which allow us to handle such datasets even on commodity hardware. These techniques range from efficient data-structures for cubical complexes to data preprocessing based on discrete Morse theory.
Weinberger, Shmuel
Disordered solids and the dynamics of bounded geometry
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I will explain a program of J Bellissard and S Ulgen-Yildirim associating a pre-foliated space to complete manifolds with bounded geometry with (anticipated) applications to quasicrystals and less ordered solids.

Participants

Name	Institution
Adamaszek, Michal	University of Warwick
Adams, Henry	Stanford University
Akinboade, Felix	Federal College of Education (Technical), Gombe
Aleruchi, Victor	University of Port Harcourt
Arnold, Matthew	University of Bristol
Attali, Dominique	CNRS
Bak, Anthony	Stanford University
Balduzzi, David	Max Planck Institute for Intelligent Systems
Bataineh, Khaled	University of Bahrain
Bauer, Ulrich	Institute of Science and Technology Austria
Belchi Guillamon, Francisco	University of Malaga
Boissonnat, Jean-Daniel	INRIA, Sophia Antipolis
Botnan, Magnus Bakke	Norwegian University of Science and Technology
Brodzki, Jacek	University of Southampton
Capitelli, Nicolas	University of Buenos Aires
Carlsson, Gunnar	Stanford University
Chacholski, Wojciech	Kungliga Tekniska högskolan
Chazal, Frederic	INRIA Saclay - Île-de-France
Chernov, Alexey	Durham University
Cohen, Daniel	Louisiana State University
Costa, Armindo	University of Warwick
Curry, Justin	University of Pennsylvania
de Silva, Vin	Pomona College
Dlotko, Pawel	Jagiellonian University
Edelsbrunner, Herbert	Institute of Science and Technology Austria
Fajstrup, Lisbeth	Aalborg University
Farber, Michael	University of Warwick
Ferri, Massimo	University of Bologna
Gamble, Jennifer	North Carolina State University
Ghrist, Robert	University of Pennsylvania
Glisse, Marc	INRIA
Gonzalez, Jesus	Center for Research and Advanced Studies (CINVESTAV)
Goubault, Eric	CEA LIST, Saclay
Greenlees, John	University of Sheffield
Guibas, Leonidas	Stanford University
Gutiérrez, Bárbara	Center for Research and Advanced Studies (CINVESTAV)
Han, SangEon	Chonbuk National University
Henselman, Gregory	University of Pennsylvania
Heo, Giseon	University of Alberta
Hiraoka, Yasuaki	Kyushu University
Jardine, Rick	University of Western Ontario
Kaczynski, Tomasz	Université de Sherbrooke
Kahle, Matthew	Ohio State University
Kalisnik, Sara	Stanford University
Kramar, Miroslav	Rutgers University
Krcal, Marek	Charles University
Krishnan, Sanjeevi	University of Pennsylvania
Kurlin, Vitaliy	Durham University
Landi, Claudia	Università di Modena e Reggio Emilia
Linial, Nati	Hebrew University of Jerusalem
Lipsky, David	University of Pennsylvania
McTague, Carl	University of Southampton
Mémoli, Facundo	University of Adelaide
Mimram, Samuel	CEA, LIST
Mischaikow, Konstantin	Rutgers, State University of New Jersey
Mrozek, Marian	Jagiellonian University
Mukherjee, Sayan	Duke University
Müllner, Daniel	Stanford University
Nanda, Vidit	Rutgers University
Nicolau, Monica	Stanford University
Obeng-Denteh, William	Kwame Nkrumah University of Science and Technology
Oudot, Steve	INRIA Saclay - Île-de-France
Palmer, Chris	University of Edinburgh
Patel, Amit	Rutgers University
Pausinger, Florian	Institute of Science and Technology Austria
Pavesic, Petar	University of Ljubljana
Purohit, Rakeshwar Puroh	MLSU, Udaipur
Ranicki, Andrew	University of Edinburgh
Raussen, Martin	Aalborg University
Rudyak, Yuli	University of Florida
Sazdanovic, Radmila	University of Pennsylvania
Schuetz, Dirk	Durham University
Sheehy, Donald	INRIA Saclay - Île-de-France
Singh, Mahender	Indian Institute of Science Education and Research
Skraba, Primoz	Jozef Stefan Institute
Smale, Stephen	City University of Hong Kong
Szymczak, Andrzej	Colorado School of Mines
Tamaki, Dai	Shinshu University
Teh, Nicholas	University of Oxford
Thatcher, Courtney	Bard College at Simon's Rock
Upadhyay, Ashish Kumar	Indian Institute of Technology Patna
van de Weijgaert, Rien	Groningen University
Vejdemo-Johansson, Mikael	University of St Andrews
Virk, Ziga	University of Ljubljana
Wagner, Hubert	Jagiellonian University/IST Austria
Wedrich, Paul	University of Cambridge
Weinberger, Shmuel	University of Chicago