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Scientific and Engineering Applications of Algebraic Topology
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The demands of modern science and engineering have placed us in a position where it is vital to develop methods for qualitative analysis and recognition problems in contemporary contexts, including data [finite metric spaces as samples from experiments, surveys, or sensors], networks [internet traffic, gene regulation, coordinated robotics, communications], and dynamics [systems equipped with only finite resolution or which are stochastic]. The IMA will select up to eight postdoctoral fellows to participate in the program.
Andrew Blumberg | University of Texas, Austin |
Gunnar Carlsson | Stanford University |
Fan Chung Graham | University of California, San Diego |
Robert Ghrist | University of Pennsylvania |
Susan Holmes | Stanford University |
Michael Mandell | Indiana University |
Konstantin Mischaikow | Department of Mathematics, Rutgers University |
Events
April 28 - May 02, 2014 | Topology and Geometry of Networks and Discrete Metric Spaces | Annual Program Workshop |
March 03 - 07, 2014 | Topological Systems: Communication, Sensing, and Actuation | Annual Program Workshop |
February 20 - May 08, 2014 | Geometric and Topological Methods in Variational Calculus | Annual Program Study Group |
February 10 - 14, 2014 | Algebraic Topology in Dynamics, Differential Equations, and Experimental Data | Annual Program Workshop |
December 09 - 13, 2013 | Topological Structures in Computational Biology | Annual Program Workshop |
October 28 - November 01, 2013 | Modern Applications of Homology and Cohomology | Annual Program Workshop |
October 07 - 11, 2013 | Topological Data Analysis | Annual Program Workshop |
October 02 - 04, 2013 | Introduction to Statistics and Probability for Topologists | Tutorial |
Topology was invented as a tool for achieving, inter alia:
- Qualitative analysis: Set-theoretic topology identifies properties [e.g., compactness, connectedness] useful in extensions of analysis beyond finite-dimensional Euclidean space [e.g., manifolds, functional analysis, calculus of variations]. Such properties are said to be topological if they are robust to continuous deformation. The power of these extensions has been demonstrated repeatedly over the last 150 years.
- Geometric pattern recognition: Poincaré found it useful to formalize the notion of loops and holes in a space and their higher dimensional analogues as a way to codify the qualitative properties of spaces. Algebraic topology was subsequently constructed as a rigorous formalization. What was arrived at is a collection of generalizations of the notion of connectivity to higher connectivity information, which are encoded by algebraic objects.
The demands of modern science and engineering have placed us in a position where it is vital to develop methods for qualitative analysis and recognition problems in contemporary contexts, including data [finite metric spaces as samples from experiments, surveys, or sensors], networks [internet traffic, gene regulation, coordinated robotics, communications], and dynamics [systems equipped with only finite resolution or which are stochastic]. Examples include:
- Data of various kinds is being collected at an enormous rate, and in many different forms. Often the data is equipped with a notion of distance that reflects certain notions of similarity, but which may be far from Euclidean [think genomic sequence analysis]. It is also frequently the case that the metrics are not defined by any precise theory, but are chosen in a relatively ad hoc way to reflect the investigator's intuitive notions of similarity. For this reason, it is important to make computations that are reasonably robust to changes in the metric, since one expects that the interesting scientific properties should not change if the metric is changed via deformations.
- In the area of sensor networks, one studies families of sensors with relatively weak computational ability and wishes to study coverage questions. The sensors will likely not even have their own positions available, but rather only information about what the neighboring sensors are. It is very desirable to solve the coverage question, and therefore develop a methodology that solves them given only the adjacency information referred to above. The adjacency information produces an undirected graph, and it is from this information one must develop methods for resolving the coverage question.
- Many problems in biology, from protein folding to gene regulatory networks, can be usefully formulated as qualitative questions about dynamical systems. The large systems in question are given in terms of various different kinds of metrics and are often best formulated with a stochastic component. Methods of understanding the qualitative features using such fuzzy inputs are vital to properly interfacing with biology.
Many efforts to address these problems have been under development over the last decade. There has been a great deal of work in various kinds of persistent homology [a methodology for inferring topological invariants of a geometric object from finite samples with error from the object], the homological properties of sensor networks and their implications for coverage and other questions, and the extension of algebraic topological tools for qualitative analysis of dynamical systems [Conley indices, for example] to tools in the finite approximation and stochastic settings. We believe that the importance of the problems addressed by these methods are of such fundamental importance that a program that will bring together the various groups [topologists, computational geometers, networks experts, statisticians, biologists, and other application domain specialists] who are critical to the further development and implementation of the methods is warranted. We believe that the subject is now at a point where such a gathering would allow decisive progress in a number of different directions.
Vittorio [Victor] Addona | Macalester College | Department of Mathematics, Statistics and Computer Science | 06/08/14-07/18/14 |
Folashade Agusto | Austin Peay State University | Department of Mathematics | 05/25/14-06/14/14 |
Leyda Almodovar | The University of Iowa | Department of Mathematics | 10/01/13-11/02/13 |
Sergio Ardanza-Trevijano | Universidad de Navarra | Department of Physics and Applied Mathematics | 02/08/14-03/08/14 |
Douglas Arnold | University of Minnesota, Twin Cities | School of Mathematics | 09/03/13-06/30/14 |
F. Javier Arsuaga | San Francisco State University | Department of Mathematics | 09/01/13-06/30/14 |
Marzieh Bayeh | University of Regina | Department of Mathematics and Statistics | 01/30/14-03/08/14 |
Greg Bell | University of North Carolina, Greensboro | Department of Mathematics and Statistics | 09/03/13-12/15/13 |
Heather Berlin | Washington University | 06/07/14-07/19/14 | |
Roxane Besse | École Polytechnique Fédérale de Lausanne [EPFL] | Department of Mathematics | 02/17/14-08/08/14 |
Andrew Beveridge | Macalester College | Department of Mathematics, Statistics and Computer Science | 06/09/14-07/18/14 |
Andrew Blumberg | The University of Texas at Austin | Department of Mathematics | 09/03/13-12/15/13 |
Magnus Botnan | Norwegian University of Science and Technology [NTNU] | Department of Mathematical Sciences | 01/20/14-06/15/14 |
Jared Brown | St. Olaf College | Department of Mathematics, Statistics and Computer Science | 06/08/14-07/18/14 |
Vladislav Bukshtynov | Stanford University | Energy Resources Engineering | 05/25/14-06/14/14 |
Gunnar Carlsson | Stanford University | Department of Mathematics | 09/09/13-12/15/13 |
Corrie Jacobien Carstens | RMIT University | School of Mathematical and Geospatial Sciences | 04/01/14-05/02/14 |
Frederick Cohen | University of Rochester | Department of Mathematics | 09/11/13-05/16/14 |
Armindo Costa | University of Warwick | Mathematics Institute | 10/01/13-11/02/13 |
Carina Curto | University of Nebraska | Department of Mathematics | 03/01/14-05/16/14 |
Isabel Darcy | The University of Iowa | Department of Mathematics | 10/01/13-11/01/13 |
Sarah Day | College of William and Mary | Department of Mathematics | 01/10/14-05/30/14 |
Vin de Silva | Pomona College | Department of Mathematics | 09/08/13-05/24/14 |
Yogeshwaran Dhandapani | Technion-Israel Institute of Technology | Department of Electrical Engineering | 09/30/13-11/26/13 |
Lisbeth Fajstrup | Aalborg University | Department of Mathematical Sciences | 10/01/13-11/12/13 |
Michael Farber | University of Warwick | Mathematics Institute | 10/01/13-11/01/13 |
Erica Flapan | Pomona College | Department of Mathematics | 09/01/13-01/01/14 |
Chad Giusti | University of Nebraska | Department of Mathematics | 03/01/14-05/15/14 |
Rebecca Harper | Willamette University | 06/08/14-07/18/14 | |
Brent Hecht | University of Minnesota, Twin Cities | Department of Computer Science and Engineering | 06/08/14-07/18/14 |
Thomas Hoft | University of St. Thomas | Department of Mathematics | 06/08/14-07/18/14 |
Annette Honken | The University of Iowa | Department of Mathematics | 10/01/13-11/02/13 |
Samuel Horlbeck Olsen | Macalester College | 06/08/14-07/18/14 | |
Vladimir Itskov | University of Nebraska | Department of Mathematics | 03/01/14-05/16/14 |
Tomasz Kaczynski | University of Sherbrooke | Département de Mathématiques | 02/01/14-05/03/14 |
Matthew Kahle | The Ohio State University | Department of Mathematics | 01/30/14-05/03/14 |
Sara Kalisnik | Stanford University | Department of Mathematics | 09/28/13-12/06/13 |
Isabelle Kemajou-Brown | University of Minnesota, Twin Cities | School of Mathematics | 05/26/14-06/15/14 |
Mitchell Kinney | Iowa State University | 06/08/14-07/19/14 | |
Irina Kogan | North Carolina State University | Department of Mathematics | 09/09/13-05/30/14 |
Michael LaMar | College of William and Mary | Department of Biology | 01/10/14-05/30/14 |
Tyler Lawson | University of Minnesota, Twin Cities | School of Mathematics | 09/03/13-06/30/14 |
Longfei Li | University of Delaware | Department of Mathematical Sciences | 02/09/14-03/14/14 |
Jun Liu | Southern Illinois University | Department of Mathematics and Statistics | 05/25/14-06/14/14 |
Anders Lundman | Royal Institute of Technology [KTH] | Department of Mathematics | 10/01/13-10/31/13 |
Huy Mai | Brandeis University | 06/08/14-07/19/14 | |
Michael Mandell | Indiana University | Department of Mathematics | 09/03/13-12/15/13 |
Bryan Martin | Macalester College | 06/08/14-07/19/14 | |
Benjamin Mathers | Macalester College | 06/08/14-07/18/14 | |
Christie Mauretour | Florida Gulf Coast University | 06/07/14-07/18/14 | |
Richard McGehee | University of Minnesota, Twin Cities | Geometry Center | 09/03/13-12/31/13 |
Facundo Mémoli | The Ohio State University | Department of Mathematics | 09/04/13-12/14/13 |
Guy-vanie Miakonkana | University of Minnesota, Twin Cities | Institute for Mathematics and Its Applications | 04/06/14-05/09/14 |
Anthea Monod | Technion-Israel Institute of Technology | Department of Electrical Engineering | 09/30/13-11/27/13 |
Vassilios Morellas | University of Minnesota, Twin Cities | Department of Computer Science and Engineering | 06/08/14-07/18/14 |
Crista Moreno | San Francisco State University | Department of Mathematics | 03/01/14-03/31/14 |
Marian Mrozek | Jagiellonian University | Computational Mathematics | 02/08/14-03/06/14 |
Aniceto Murillo | University of Málaga | Departamento de Algebra Geometria y Topologia | 03/04/14-04/30/14 |
Junalyn Navarra-Madsen | Texas Woman's University | Department of Mathematics and Computer Science | 01/28/14-05/28/14 |
Sebastian Öberg | Royal Institute of Technology [KTH] | Department of Mathematics | 09/01/13-11/30/13 |
Christopher Palmer | University of Edinburgh | School of Mathematics | 09/25/13-11/25/13 |
Jose Perea | Duke University | Department of Mathematics | 02/10/14-05/10/14 |
Giovanni Petri | ISI Foundation | Department of Mathematical Sciences | 10/01/13-11/02/13 04/25/14-05/16/14 |
Anya Phillips | Macalester College | 06/08/14-07/18/14 | |
Peter Polacik | University of Minnesota, Twin Cities | School of Mathematics | 09/03/13-06/30/14 |
Martin Raussen | Aalborg University | Department of Mathematical Sciences | 10/01/13-12/15/13 |
Antonio Rieser | Technion-Israel Institute of Technology | Department of Mathematics | 10/01/13-11/01/13 |
Francisco R. Ruiz del Portal | Universidad Complutense de Madrid | Departamento de Geometría y Topología | 03/16/14-04/30/14 |
Fadil Santosa | University of Minnesota, Twin Cities | School of Mathematics | 06/09/14-07/18/14 |
Julian Sass | University of Maryland Baltimore County | 06/08/14-07/19/14 | |
Radmila Sazdanovic | North Carolina State University | Department of Mathematics | 02/02/14-05/23/14 |
Lindsey Schadler | Shippensburg University | 06/07/14-07/18/14 | |
Martina Scolamiero | Royal Institute of Technology [KTH] | Department of Mathematics | 10/01/13-11/02/13 |
Shilad Sen | Macalester College | Department of Mathematics | 06/08/14-07/18/14 |
Ziyue [Zoe] Shuai | Bryn Mawr College | 06/08/14-07/19/14 | |
Gard Spreemann | Norwegian University of Science and Technology [NTNU] | Department of Mathematical Sciences | 01/29/14-07/17/14 |
De Witt Sumners | Florida State University | Department of Mathematics | 10/01/13-11/01/13 |
Carlos Tolmasky | University of Minnesota, Twin Cities | Institute for Mathematics and its Applications | 01/13/14-06/30/16 |
Mariel Vazquez | San Francisco State University | Department of Mathematics | 09/01/13-06/30/14 |
Mikael Vejdemo-Johansson | Royal Institute of Technology [KTH] | Computer Vision and Active Perception Lab | 09/25/13-05/25/14 |
Laura Vonessen | University of Arizona | 06/08/14-07/19/14 | |
Alexander A. [Sasha] Voronov | University of Minnesota, Twin Cities | School of Mathematics | 09/03/13-06/30/14 |
Craig Westerland | University of Minnesota, Twin Cities | School of Mathematics | 09/03/13-06/30/14 |
Julian Wolfson | University of Minnesota, Twin Cities | Division of Biostatistics | 06/08/14-07/18/14 |
Yuan Yao | Peking University | School of Mathematical Sciences | 09/28/13-11/03/13 |
Lori Ziegelmeier | Macalester College | Department of Mathematics | 09/17/13-07/18/14 |