Dalian Winter School on Computational Topology

Date: December 5-20, 2016

Place: Dalian University of Technology (DUT), Dalian, China

Host: School of Mathematical Sciences, Dalian University of Technology

Co-Host: School of Mathematics, Liaoning Normal University

Purpose: The school program will provide with some courses on basic aspects of the computational and applied topology, and some frontier talks on modern results and applications of the theory, to the participants who are interested in the areas of topology and its applications.

Invited Lecturers and Courses:

(1) Oleg Musin (University of Texas Rio Grande Valley, USA)

Course: Introduction to computations in discrete geometry (16 hours)

(2) Frank Lutz (Technical University of Berlin, Germany)

Course: Algorithmic problems and triangulations of 3-manifolds (10 hours)

(3) Miro Kramar (AIMR, Tohoku University, Japan)

Course: Introduction to topological data analysis (16 hours)

(4) Anton Nikitenko (Institute of Science and Technology, Austria)

Course: Introduction to computational topology (10 hours)

(5) Alexander Tselikov (Senior data scientist, VimpelCom Ltd., Russia)

Course: Big data: from theory to practice (6 hours)

(6) Matthias Dehmer (UMIT, Austria)

Course: Big data and data science (4 hours)

Program and Organizing Committee:

Fengchun Lei, Dalian University of Technology, China; fclei@
Oleg Musin, University of Texas Rio Grande Valley, USA; oleg.musin@utb.edu
Andrey Vesnin, Sobolev Institute of Mathematics, Russia; vesnin@math.nsc.ru
Jie Wu, National University of Singapore, Singapore; matwuj@mus.sdu.sg

Homepage: http://math.

Local Organizers:

Fengling Li, dutlfl@163.com

Huadong Yu, yuhuadong@

Youfa Han, hanyoufa@sina.com

Sponsors: The school will be supported in part by Dalian University of Technology and some grants from NSFC.

Contact to: Ms. Fengling Li, dutlfl@163.com. Please send the pre-registration form before November 10, 2016. Limited accommodation support is applicable before the deadline to submit the pre-registration form.

Program Timetable

Course introduction: 

(1) Introduction to computations in discrete geometry (18 hours, given by Oleg Musin):

1)       Lecture 1: The kissing number problem and Delsarte's method. Friday, Dec 9, 10:00 - 11:40.

2)       Lecture 2: Introduction to Computational Geometry. The point location problem. Monday, Dec 12, 10:00 - 11:40.

3)       Lecture 3: Voronoi diagrams, Delaunay triangulations and their applications. Tuesday, Dec 13, 8:00 - 9:40.

4)       Lecture 4: Optimal properties of the Delaunay triangulation. Tuesday, Dec 13, 15:30 - 17:10.

5)       Lecture 5: Sperner's lemma and envy-free fair divisions. Wednesday, Dec 14, 8:00 - 9:40.

6)       Lecture 6: Extensions of the Sperner - KKM lemma and their applications. Wednesday, Dec 14, 15:30 - 17:10.

7)       Lecture 7: Sphere packings and Tammes' problem. Thursday, Dec 15, 10:00 - 11:40.

8)       Lecture 8: Two -- distance sets. Thursday, Dec 15, 15:30 - 17:10.

9)       Lecture 9: Geometrical and topological methods in Image Processing. Discrete analog of the Maxwell - Morse theory. Friday, Dec 16, 10:00 - 11:40.

(2) Algorithmic problems and triangulations of 3-manifolds (10 hours, given by Frank Lutz):

    Computational problems in topology typically do not behave as we would expect them to do. For example, standard packages for homology or persistent homology computations extensively use (NP-hard) discrete Morse theory as a (fast) preprocessing step to avoid (slow, polynomial time) Smith Normal Form computations. Even worse, the recognition problem for manifolds becomes unsolvable beyond dimension three --- yet, in practice, it is rare to ever come across a non-recognizable manifold.

    In this course, we will put the spotlight on

    1) fast heuristic procedures to solve NP-hard or unsolvable problems in topology;

    2) random discrete Morse theory to analyze simplicial complexes;

    3) 3-manifold and other examples from the library of complicated triangulations.

    We will further see that we may encounter an event horizon for standard topological computations. Simple spaces like higher-dimensional simplices or higher barycentric subdivisions of spaces suddenly turn into intractable spaces. This puts a limit to topological data analysis.

(3) Introduction to topological data analysis (16 hours, given by Miro Kramar):

    Nonlinear dynamical systems play an important role in modeling of various processes in fields ranging from physics, chemistry and biology to many other natural and social sciences.  Despite the important role of the nonlinear models and intense efforts of many researchers,  the global dynamics of a large number of these models is still far form being properly understood. Our understanding of the dynamics becomes even more tentative if the governing equations are not known and the study of the system is based on the data collected by the experimentalists.

    In the first part of this course we will start by introducing problems that can be tackled by using topological data analysis. Then we will explain traditional tools of topological data analysis. We will start by introducing different types of complexes (Czech, Rips, etc.) used for data representation. We will show that under certain conditions,  a sampled manifold can be properly reconstructed using the Czech complex. Then we move to the concept of persistent homology. It will allow us to analyze the shape of the data represented by a complex. We will also use persistent homology to quantitatively describe patterns generated by scalar functions. Moreover, we will define a notion of distance between persistence diagrams which quantitatively encodes the differences between the patterns. This will be crucial for our analysis of the dynamics of pattern evolution.

    In the second part of this course we will introduce topological methods for analyzing the global attractors of the nonlinear dynamical systems. We will explain how to use  Morse decompositions to partition the dynamics into recurrent and gradient like parts. To properly understand the structure of the invariant sets contained inside of the recurrent part we will employ the Conley index theory. We will start by presenting this theory for the dynamical systems given by differential equations. We  will focus on the applications and the computational aspects of the theory. Then we will consider its extension to  the analysis of the dynamics for the systems observed by experimentalists. This will be demonstrated on the particular problems that were studied by the lecturer over the last years. At the end, we will present some open problems in the field.

(4) Introduction to computational topology (10 hours, given by Anton Nikitenko):

    The course is aimed at acquainting the listeners with some of the basic problems and objects of computational topology. We will start with simpler low-dimensional questions concerning graphs, point configurations in the plane and two-dimensional manifolds, introducing Voronoi and Delaunay tesselations. We proceed further to shape reconstruction problems in higher dimensions and start by getting to know  the important structures used for these purposes: Čech, Vietoris-Rips and  Delaunay (also known as Alpha-) complexes. We discuss the relevant  generalizations of the classical Morse theory to simplicial complexes, due to R. Forman (1998) and U. Bauer and H. Edelsbrunner (2015) and apply them to study the aforemnetioned complexes. Finally we proceed to computational algebraic topology and study homology of simplicial complexes, coming ultimately to the concept of persistent homology, which shows the evolution of the homology of topological space, and its releavce for identifying features of a geometric shape. A part of the course will be devoted to algorithms.

Reference book: H Edelsbrunner,J Harer, "Computational Topology: An Introduction"

(5) Big data: from theory to practice (6 hours, given by Alexander Tselikov):

    In these lectures we will concentrate on mathematical and algorithmical aspects of data analysis from classical machine learning techniques invented a decades ago to modern approaches which can be applied to solving practical problems. High performance hardware allows us to use vast knowledge base of algorithms not only for task of regression and classification but to task of processing image, video and natural language processing. More than that accumulated data and open source development strategy give researchers opportunities to improve algorithms and create new approaches. We will also cover some aspects of competitive data analysis area which can be used as an intermediate stage between theory and practice to a quick hypothesis testing.

(6) Big data and data science (4 hours, given by Matthias Dehmer)

Lecture 1: Brief overview on computational network analysis and applications

Lecture 2: Aspects of graph entropy measures

Lecture 3: Interconnections between Big data , data science, and network analysis

Lecture 4: Information-theoretic graph measures

Title and Abstract of Frontier Talks: 

Speaker: Miro Kramar (AIMR, Tohoku University, Japan)

Title: Analysis of time scales in data

Abstract: In this talk we will introduce the methods of topological data analysis. Namely, the persistence diagrams which are a relatively new topological tool for describing and quantifying complicated patterns in a simple but meaningful way.  We will demonstrate this technique on patterns appearing in dense granular media. This procedure allows us to transform experimental or numerical data, from experiment or simulation, into a point cloud in the space of persistence diagrams. There are a variety of metrics that can be imposed on the space of persistence diagrams. By choosing different metrics one can interrogate the pattern locally or globally, which provides deeper  insight into the dynamics of the process of pattern formation. We will use these metrics to identify the important time scales at which behavior of the system changes. We will also discuss a physical interpretation of these time scales.

Speaker: Frank Lutz (Technical University of Berlin, Germany)

Title: Roundness of grains in cellular microstructures

Abstract: Polycrystalline materials, such as metals, are composed of crystal grains of varying size and shape. Typically, the occurring grain cells have the combinatorial types of 3-dimensional simple polytopes, and together they tile 3-dimensional space.

We will see that some of the occurring grain types are substantially more frequent than others - where the frequent types turn out to be “combinatorially round”. Here, the classification of grain types gives us, as an application of combinatorial low-dimensional topology, a new starting point for a topological microstructure analysis of materials

Speaker: Anton Nikitenko (Institute of Science and Technology, Austria)

Title: Discrete Morse Theory of Poisson-Delaunay Mosaics

Abstract: A classical object of interest in stochastic geometry is a Poisson-Delaunay mosaic, which is a Delaunay triangulation of a Poisson distributed point cloud. Using generalized discrete Morse theory, we study expected sizes of Poisson-Delaunay complexes with radius bound in R^n and obtain precise values for dimensions up to 4. We also find out that there is, perhaps surprisingly, almost no difference if one takes an n-sphere instead of R^n.

Speaker: Andrey Vesnin (Sobolev Institute of Mathematics, Russia) 

Talk 1

Title: Computation of hyperbolic volumes for polyhedra and 3-manifolds.

Abstract: We will survey basic facts on volumes of hyperbolic polyhedra and 3-manifolds. We will present volume computations with the Lobachevsky function for some interesting cases. Examples of equal volume compact and non-compact hyperbolic 3-manifolds will be discussed.

Talk 2

Title: Computations with hyperbolic structures: right-angled case.

Abstract: We will discuss hyperbolic 3-manifolds which can be decomposed into right-angled polyhedra. We will discuss their volumes, arithmeticity and cohomological rigidity.

Speaker: Jie Wu (National University of Singapore, Singapore)

Title: Braids and Robotics.

Abstract: In this introductory talk, we will discuss topological robotics, braid groups, configuration spaces and their connections to homotopy theory.

Speaker: Alexander Tselikov (Senior data scientist, VimpelCom Ltd., Russia)

Title: BigData: Beyond the hype

Abstract: Can value of big data be significant for whole organization? What limitations of production environment cannot be beaten? We will look into practical use cases of applying big data models in big organization with more than 200 mln clients all around the world.

Speaker: Shiquan Ren (National University of Singapore, Singapore)

Title: The Embedded Homology of Hypergraphs and Applications

Abstract: Hypergraphs are mathematical models for many problems in data sciences. In recent decades, the topological properties of hypergraphs have been studied and various kinds of (co)homologies have been constructed. In this paper, generalizing the usual homology of simplicial complexes, we define the embedded homology of hypergraphs as well as the persistent embedded homology of sequences of hypergraphs. As a generalization of the Mayer-Vietoris sequence for the homology of simplicial complexes, we give a Mayer-Vietoris sequence for the embedded homology of hypergraphs. Moreover, as applications of the embedded homology, we study acyclic hypergraphs and construct some indices for the data analysis of hyper-networks.

Hotel:

1. DUT International Conference Center (DUTICC) (neighbor to the South Gate of DUT),  http://hotel.

2. Yangguangju Hotel (close to DUTICC)

Transportation: Taking a taxi is the easiest way from Dalian International Airport, or Dalian Train Station, or Dalian North Train Station, to the hotel, and the taxi fare is about 30, 35, or 45 Chinese Yuan, respectively.

School of Mathematical Sciences

Dalian University of Technology

Updated by December 01, 2016